Proceedings of International Conference on Physics in ... - KEK

<strong>Proceedings</strong> <strong>of</strong>
<strong>International</strong> <strong>Conference</strong> on
Physics in Intense Fields
PIF2010
24-26 November 2010, KEK, Tsukuba, Japan
Edited by K. Itakura, S. Iso and T. Takahashi
High Energy Accelerator Research Organization
KEK <strong>Proceedings</strong> 2010-13
February 2011
A/H

High Energy Accelerator Research Organization (KEK), 2011
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<strong>Conference</strong> poster
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Group photo taken in front <strong>of</strong> Kenkyu Honkan on 25 th November 2010

Preface
The international conference on Physics in Intense Fields (PIF 2010) was
held from November 24 to 26, 2010, in KEK (High Energy Accelerator Research
Organization), Tsukuba, Japan. The purpose <strong>of</strong> the conference was to discuss
prospects <strong>of</strong> the fundamental physics in strong electromagnetic fields, and the
emphasis was particularly put on its interdisciplinary aspects.
Recent developments <strong>of</strong> the high-intensity lasers open a new window to fundamental
physics as well as applied researches. In particular, the ultra-high
intensity realm <strong>of</strong> quantum electrodynamics (QED) is within reach, and its
non-perturbative nature will be experimentally studied in near future. Investigations
using the high-intensity lasers are intimately tied up with the strongfield
dynamics in other areas <strong>of</strong> physics, such as the quark-gluon plasma <strong>of</strong>
quantum chromodynamics (QCD), astrophysical phenomena in magnetars with
critically strong magnetic fields or dielectric breakdown in strongly correlated
systems. In such circumstances, collaborations and discussions over a wide range
<strong>of</strong> physicists are extremely important and necessary towards understanding <strong>of</strong>
strong-field dynamics.
In the conference, more than a hundred participants gathered from various
countries and from various areas <strong>of</strong> physics, including laser physics, plasma
physics, particle physics, nuclear physics, condensed matter physics, astrophysics
and accelerator physics. In order to share interests among the participants <strong>of</strong>
such wide varieties, several tutorial talks on the basics <strong>of</strong> the intense fields were
given, as well as many contributions on the recent hot topics. Thanks to big
efforts <strong>of</strong> the participants, most <strong>of</strong> them could be included in this proceedings.
On behalf <strong>of</strong> the organizing committee, we would like to thank all the lectures,
speakers and poster presenters for their contributions, participants from
various countries, and the secretaries who worked very hard for the conference.
The conference was financially supported by KEK (directly by the director) and
Sokendai (the Center for the Promotion <strong>of</strong> Integrated Sciences). We especially
acknowledge the director <strong>of</strong> KEK, Atsuto Suzuki, who strongly supported the
conference.
Satoshi Iso KEK, Japan
Tohru Takahashi Hiroshima University, Japan
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Contents
Overview and Tutorial talks
A Recent Development in High Field Science (T. Tajima and G. Mourou) 1
The Heisenberg-Schwinger Effect: Nonperturbative Vacuum Pair Production (G. Dunne) 7
Nonlinear QED
QED in Ultra-Intense Laser Fields (T. Heinzl) 14
Strong-Field Effects in Beam-Beam Interaction in Linear Colliders (K. Yokoya) 19
Second Order QED Processes and Their Radiative Corrections (A. Hartin) 23
Heavy-ion collisions and Quark-Gluon Plasma
Strong Field Dynamics in Heavy Ion Collisions (K. Itakura) 26
Yoctosecond photon pulse generation in heavy ion collisions (A. Ipp) 32
Fields, Instantons, and Currents (K. Fukushima) 36
Critical Behavior <strong>of</strong> Charmonium: QCD Second Order Stark Effect (K. Morita and S.H. Lee) 40
Unruh radia on
On the Unruh Effect (R. Schützhold) 44
Can We Detect ″Unruh Radiation″ in the High Intensity Lasers? (S. Zhang, et al.) 46
Quantum Fields in Accelerated Frames (F. Lenz) 50
Axion-like par cle searches
Shining Light through Walls: en Route towards a New Particle Physics Frontier (A. Lindner) 54
Probing Extremely Light Fields via Resonance Scattering by Focusing Intense Laser (K. Homma) 59
Schwinger mechanism in QGP and condensed ma er
Dynamical View <strong>of</strong> Pair Creation via the Schwinger Mechanism (N. Tanji) 63
Exact Solutions <strong>of</strong> Pair Productions in Strong Electric Field with Finite Width (A. Iwazaki) 67
Strong Field Physics in Condensed Matter (T. Oka) 70
Non-Linear Charge Transport in Plasma under Strong Field (S. Nakamura) 74
Recent developments in Schwinger mechanism
Brilliant Hard γ-Production and e + e – -Creation in Vacuum with Ultra-High Laser Fields:
Testing Theoretical Predictions at ELI-NP (D. Habs, et al.) 78
Numerical Simulation <strong>of</strong> QED Cascades in Intense Laser Fields (N. Elkina and H. Ruhl) 83
Schwinger Limit Attainability with Extreme Light (S.V. Bulanov, et al.) 88
Pair Creation in QED-Strong Pulsed Laser Fields (N. Naumova, et al.) 93
Laser Acceleration up to Black Holes and B-meson Decay (H. Hora, et al.) 97
vii

Recent progress <strong>of</strong> ultra-intense lasers
Present Status <strong>of</strong> Ultra-Intense Lasers and High-Field Physics in the World (H. Takabe) 101
Reaching the Schwinger Limit with X-Rays (C. K. Rhodes, et al.) 107
Magnetars
Nonlinear QED Effects by Strong Magnetic Field in Astrophysics (K. Kohri) 111
Wide-Band X-ray Observations <strong>of</strong> Magnetars (K. Makishima) 116
QCD Origin <strong>of</strong> Strong Magnetic Fields in Compact Stars (T. Tatsumi) 121
New technologies
Recent Progress and Prospects on Laser-Plasma Acceleration <strong>of</strong> Charged Particles (K. Nakajima) 125
Flying Mirror as a Tool to Access Ultra-High Fields (M. Kando, et al.) 130
Poster presenta ons
4-Mirror Laser Stacking Cavity for High Intensity Polarized Photon Generation (T. Akagi, et al.)
Current Status <strong>of</strong> LFEX Laser and Exa-watt Laser Concept at ILE/Osaka (J. Kawanaka, LFEX-Team,
134
EXA-Team, and H. Azechi) 137
X-ray Emission from Magnetars and Its Physical Interpretation (T. Enoto) 141
The Nielsen-Olesen Instabilities in the Glasma (H. Fujii, et al.) 144
First Order Quantum Correction to the Larmor Radiation (G. Nakamura) 147
Fast Vacuum Decay into Particle Pairs in Strong Electric and Magnetic Fields (Y. Hidaka, et al.) 150
Noncanonical Lie Perturbation Analysis for the Relativistic Ponderomotive Force (N. Iwata, et al.) 153
Particle Based Integrated Code EPIC3D for Laser-Matter Interaction (Y. Kishimoto) 156
X-Ray Generation via Laser Compton Scattering by Laser-Accelerated Electron Beam (E.Miura, et al.) 159
Measurement <strong>of</strong> Nanometer Scale Beam Size by the Shintake Monitor (M. Oroku, et al.) 162
Investigating the One-Photon Annihilation Channel in an e – e + Plasma Created from Vacuum in
Strong Laser Fields (A.V. Tarakanov, et al.) 165
Accelerator Test Facility (ATF) and Future Prospect (T. Tauchi) 168
Unruh radiation and Interference effect (Y. Yamamoto et al.) 171
Program <strong>of</strong> PIF2010 174
List <strong>of</strong> participants 178
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energy and Ip the ionization potential. A fraction <strong>of</strong> the
harmonic spectrum is selected to produce pulse durations
down to 100 as [17, 18] pulses, the shortest being at 80 as
[19].
If we want to go even shorter, we need to resort to even
higher intensities and leave the nonlinear bound electron
regime to go into the relativistic regime, which is, for 1 µm
wavelength, greater than 10 18 W/cm 2 . Such intensitesy is
are, today, commonly available using Chirped Pulse Amplification
[20] and Optical Parametric Chirped Pulse Amplification
[21] systems.
In the relativistic regime, electrons oscillating in the
laser field become relativistic and change their “mass” during
their oscillations by a factor proportional to the Lorentz
factor γ, which in turn is also proportional to the normalized
vector potential a0. If a laser pulse can produce this
intensity at a target’s surface, the enormous ponderomotive
laser pressure makes the electron critical surface oscillate
in and out at relativistic velocity. As a consequence, the
light impinging on this oscillating mirror is modulated periodically,
resulting in high harmonics [22, 23]. Relativistic
High Harmonic Generation gives the prospect <strong>of</strong> a much
broader harmonic spectrum, higher efficiency with no cut<strong>of</strong>f
defined by the plasma frequency [22, 24]. This has been
experimentally verified [25] using the long pulse duration
(300 fs) <strong>of</strong> the Vulcan laser and observing the 3200th harmonic
order.
A related scheme was shown based on a few-cycle pulse,
focused on one λ 2 –this is the so called λ 3 -regime [26]–
the relativistic mirror ceases to be planar and deforms due
to the indentation created by the focused gaussian beam.
As it moves, PIC simulation shows, it simultaneously compresses
the sub-cycle pulses and broadcasts them in specific
directions. This technique provides an elegant possibility
to both compress but also isolate individual attosecond
pulses. The predicted pulse duration scales like
T = 600(attosecond)/a0. Here a0 is again the normalized
vector potential, which is about unity at 10 18 W/cm 2
and scales as the square root <strong>of</strong> the intensity. For intensity
<strong>of</strong> the order <strong>of</strong> 10 22 W/cm 2 the compressed pulse could
be <strong>of</strong> the order <strong>of</strong> only a few attoseconds. The same authors
have simulated the generation <strong>of</strong> thin sheets <strong>of</strong> electrons
with γ <strong>of</strong> few tens and with attosecond duration [27].
These electron bunches could provide a way to produce,
by coherent Thomson scattering, efficient beams <strong>of</strong> X-rays
or even γ-rays. A similar concept called ‘relativistic flying
mirror’ has been advocated and demonstrated [28], using
a thin sheet <strong>of</strong> accelerated electrons. Reflection from this
relativistic mirror is highly efficient and instills pulse compression.
COMPRESSION IN THE ULTRA
RELATIVISTIC REGIME
Can we go further in time compression? When one
wishes to go beyond coherent X-rays to gamma rays, the
‘mirror’ that compresses the laser into gamma rays has to
be <strong>of</strong> extremely high density (∼ 10 27 cm −3 ) so that the
laser may be coherently reflected into gamma photons. We
suggest here that this may be achieved by a combination <strong>of</strong>
the relativistically flying mirror just mentioned above with
the implosion <strong>of</strong> this flying mirror so that its density may be
enhanced by ten times in each dimension (thus thousandfold
in its density). We surmise that this may be achieved
by a large energy pulse (∼ MJ) at the ultra-relativistic (even
ions become relativistically moving in the optical fields) intensity
<strong>of</strong> 10 24 W/cm 2 on a partial shell <strong>of</strong> a concave spherical
target. This ultra-relativistic flying mirror [29] with
the imploding shell is conceptually capable <strong>of</strong> coherently
backscattering an injected 10 keV coherent X-ray pulse like
the one mentioned above [26], producing a possibility <strong>of</strong>
coherent gamma rays <strong>of</strong> 100 ys duration.
In relativistic compression, the concept described above
relies only on electrons in a thin flying sheet. Unfortunately,
when we try further compression <strong>of</strong> the pulse
length, the frequency <strong>of</strong> photons enters the domain <strong>of</strong>
gamma rays. Consequently, such a flying or oscillating
electron mirror at the solid density cannot interact
with gamma rays. However, at or near the intensity <strong>of</strong>
10 24 W/cm 2 , ions in the laser field can become relativistic.
This characterizes the onset <strong>of</strong> the ultra-relativistic regime.
Below this regime ions respond to the laser transverse electric
field with harmonic motion. Thus the ion motion is
linear and it cannot directly extract the laser energy. However,
near or above this threshold, ions begin to respond to
the v × B force, which produces the nonlinear thrust force
forward, the ultra-relativistic nonlinearity. Here new opportunities
for pulse shortening arise, based on the ability
to accelerate real matter (both electrons and ions) to approach
the speed <strong>of</strong> light. In turn, it takes significant energy
to move and compress matter into this regime.
We have devised a configuration in which the laser at
the intensity <strong>of</strong> 10 23 − 10 24 W/cm 2 irradiates a thin concave
shell target (30 micron × 30 micron × 100 nm). This
causes a relativistic matter flow (not only electrons but ions
as well). In this regime electrons are held to the same speed
<strong>of</strong> ions. The relativistically moving matter driven by a MJ
laser pulse (red) rapidly converges due to the geometry in
its transverse dimensions. The longitudinal compression
occurs typically in this ultra-relativistic drive (Esirkepov,
2004). Thus we can achieve typically 10×10×10 compression.
This is the ‘ultra-relativistic imploding mirror’. The
relativistic Lorentz factor is estimated to be γi ∼ γe ∼ 30.
Simultaneously, the relativistic mirror (in the usual relativistic
regime, mentioned earlier) driven by a 10 kJ laser
pulse generates coherent X-rays. We let these X-rays collide
into the collapsing shell. The collapsing shell with a
density <strong>of</strong> electrons on the order <strong>of</strong> 10 27 /cm 3 backscatters
X-rays into gamma rays coherently. If the X-ray energy
is 10 4 eV, we obtain coherent gamma rays on the order <strong>of</strong>
4 × 10 7 eV. Such a high energy gamma beam pulse is may
be on the order <strong>of</strong> several times 10 ys to 100 ys. In Fig. 1
we plot all these data points, starting from the world first
laser by Maiman [3] to what we consider in this section.

Figure 1: The Pulse Intensity-Duration Conjecture is shown. An inverse linear dependence exists between the pulse duration
<strong>of</strong> coherent light emission and its intensity <strong>of</strong> the laser driver in the generation volume over 18 orders <strong>of</strong> magnitude.
These entries encompass different underlying physical regimes, whose nonlinearities are arising from molecular, bound
atomic electron, relativistic plasma, and ultra-relativistic, and further eventually from vacuum nature. The blue patches
are from the experiments, while the red from the simulation or theory. This figure is an excerpt from [30].
Remarkably, all these points line up in a single line over
some 18 orders <strong>of</strong> magnitude.[30]
As the pulse is compressed to this extremely short duration
in the foregoing scenario, a modest efficiency could
produce sizable nonlinearities in vacuum, although its nonlinear
coefficient, n2, is 18 orders <strong>of</strong> magnitude smaller
than that <strong>of</strong> a typical optical transparent medium like glass.
If the critical power <strong>of</strong> vacuum is 10 24 W at 1.0 µm, it
will be 6 orders <strong>of</strong> magnitude less for a zeptosecond pulse,
or 10 18 W. Under this condition the vacuum critical power
could be attained with a mere millijoule. It is quite fascinating
to imagine that a filament in vacuum analogous to
those produced in air could be produced with an appropriate
setup (such as counter streaming configuration). As in
air, the filament size would be limited by “vacuum breakdown”
or pair creation, when the intensity would reach
10 29 W/cm 2 corresponding to a filament <strong>of</strong> 10 −5 cm diameter.
Further compression could be obtained with zeptosecond
pulses by self phase modulation in vacuum. If we consider
that the self phase modulation scales like
∆ω
ω
≃ Ln2
c
dI
dt
En
∝ n2
T 2
where ω is the carrier frequency, ∆ω is the self-phase modulation
broadening, n2 the nonlinear index <strong>of</strong> refraction, L
the propagation length and En the pulse energy and T the
input pulse duration. It is clear that in spite <strong>of</strong> a very small
n2 coefficient, extremely short pulse can produce a sizable
nonlinear effect. This confirms what we know already in
the visible range where we had to wait for picosecond- femtosecond
pulses to produce sizable nonlinear effects in the
visible like self-phase modulation or Kerr lens mode locking.
VACUUM NONLINEARITIES
We have learned that: matter exhibits nonlinearities
when exposed to strong enough laser radiation; manifestly
nonlinearities vary depending on the strength <strong>of</strong> the ‘bending’
field (and thus the intensity). The stronger we ‘bend’
the constituent matter, the more rigid the ‘bending’ force
we need to exert; the more rigid the force is, the higher the
(1)

estoring frequency (or the shorter the time scale) is. The
nonlinearities <strong>of</strong> matter may vary, but this response is universal,
ranging over molecular, atomic, plasma electronic
and ionic, and even the stiffest <strong>of</strong> all vacuum, nonlinearities.
Thus, we have traversed nature’s display <strong>of</strong> the universal
behavior <strong>of</strong> the direct correlation between the brevity <strong>of</strong>
pulse being generated and the intensity <strong>of</strong> its driving laser
over the widest intensity range our laboratory has to ever to
<strong>of</strong>fer.
For example, we know that the laser self-focuses above
the critical power with:
χ3 nonlinearity
Pcr = λ2/(2πn0n2) ∼ GW (2)
relativistic plasma nonlinearity
Pcr = mc 5 /e 2 (ω/ωp) 2 ∼ 17(ω/ωp) 2 GW, (3)
vacuum nonlinearity [2]
Pcr = (90/28)cE 2 Sλ 2 /α ∼ 10 15 (λ/λ1µ) 2 GW, (4)
e.g. X-ray <strong>of</strong> 10 keV, Pcr ∼ 10 PW,
where ES is the Schwinger field, above which the vacuum
becomes sufficiently strongly polarized and divulges electron
and positron pairs [31, 32, 33, 34, 35]. If we compare
the critical power <strong>of</strong> self-focusing in a gas with the χ3
nonlinearity with that <strong>of</strong> vacuum, we find that the ratio is
nearly α 6 with no other parameters involved. On the other
hand, the ration <strong>of</strong> the Keldysh power to the Schwinger
power is again α 6 (i.e. EK/ES = α 3 ). We know that the
Keldysh field is the field to create the potential energy <strong>of</strong>
Rydberg energy WB over the Bohr radius aB. On the other
hand, the Schwinger field is to create the potential energy <strong>of</strong>
2mc 2 = α −2 WB over the distance <strong>of</strong> the Compton length
αaB. See Fig. 2.
Figure 2: The exploration <strong>of</strong> vacuum may be learned from
that <strong>of</strong> an atom. While the size <strong>of</strong> the atom is <strong>of</strong> aB (the
Bohr radius) and the potential depth is <strong>of</strong> the Rydberg energy
WB, the size and the potential depth that we wish to
explore <strong>of</strong> vacuum is indicated here in this figure, only by
a factor <strong>of</strong> the fine structure constant α to its some power.
We also know that in atomic physics, there is the Keldysh
parameter [36] γK whereas the equivalent in vacuum is
γV σ = mσωc/eE = 1/a0.(σ = e, or q, electron or quark)
(5)
When the Keldysh parameter is smaller than unity, the
process <strong>of</strong> ionization is non-perturbative;, while greater
than 1, it is multi-photon process [36]. Similarly when
γV σ in vacuum is smaller than 1, the vacuum breakdown
is non-pertubative QED, while greater than 1, it is perturbative
[32, 33, 34]. When we inject an XUV photon
to an atom to ionize it and we apply a sufficiently intense
CEP-locked laser to accelerate the ejected electron,
we can make attosecond resolution streaking by the time<strong>of</strong>-flight
detection <strong>of</strong> the electron [19, 37]. An equivalent
process in vacuum is the above mentioned Nikishov/Ritus
process [32], in which a gamma photon is injected into vacuum
while a sufficiently strong enough laser (or XUV) EM
fields are applied. Nikishov et al.[32, 33, 34] showed that
the Schwinger vacuum breakdown is many orders <strong>of</strong> magnitude
reduced.
If we take advantage <strong>of</strong> this process, it is conceivable
to accomplish electron-positron pair creation from vacuum
with a very strong laser field, but still one that is in a realistically
achievable intensity regime. With the adoption
<strong>of</strong> this process, we see the possibility <strong>of</strong> streaking vacuum
with laser (or XUV coherent photons) with zeptosecond
time resolution. This should open up a possibility to start
the exploration <strong>of</strong> the time-dependent dynamics (in contrast
to the spectroscopy) <strong>of</strong> vacuum in that regime. If we
become more ambitious, perhaps with higher intensity and
shorter time-scale, we might even be able to resolve the
dynamics <strong>of</strong> quarks and gluons out <strong>of</strong> vacuum.
PROBING THE TEXTURE OF QUANTUM
VACUUM WITH ATTOSECOND
METROLOGY IN THE PEV ENERGY
FRONTIER
As discussed above, we can take advantage <strong>of</strong> the process
<strong>of</strong> vacuum breakdown with the assistance <strong>of</strong> a highenergy
photon under strong coherent fields, as investigated
by Nikishov and Ritus some 40 years ago [32]. This is
to use the process that an ultrahigh energy gamma-particle
can assist to break down the vacuum with a substantially
suppressed electric-field threshold compared with the wellknown
Schwinger value. This is the nonlinear QED effect.
The probability <strong>of</strong> the vacuum breakdown is derived as
[
P (E) ∝ exp − 8
3
ES
E
· mc2
ω
where ES the Schwinger field, ω is the gamma energy,
E is the applied electric field in vacuum such as a laser.
With a PeV gamma-ray particle, the exponent factor <strong>of</strong> (6)
is reduced by the ratio <strong>of</strong> MeV to PeV (mc 2 /ω) over
the expression <strong>of</strong> Schwinger’s without the presence <strong>of</strong> a
]
(6)

Figure 3: The attosecond metrology to detect the arrival
time <strong>of</strong> gamma photons within such a time scale is suggested.
It utilizes the property <strong>of</strong> vacuum explored by the
pioneering works by Schwinger, Nikishov et al. This approach
closely parallels with that <strong>of</strong> the atom streaking with
a CEP laser with an XUV photon ionization <strong>of</strong> an electron
from the atom. Here instead <strong>of</strong> an atom, we simply use the
vacuum electron-positron pair.
gamma particle. This means that the vacuum breakdown
field plummets from the value <strong>of</strong> 10 16 V/cm to 10 10 V/cm.
We suggest that by employing time-synchronized somewhat
intense laser field (at 10 10 W/cm 2 ) at the “goal line”
<strong>of</strong> the gamma-photon arrival (see Fig. 3), we cause sudden
breakdown <strong>of</strong> vacuum and its avalanched particles <strong>of</strong> e − e +
as soon as one <strong>of</strong> the high- energy gamma particles arrives.
The PeV gamma particle triggers the vacuum breakdown.
The time scale <strong>of</strong> breakdown is far faster than fs. The
exploitation <strong>of</strong> this phenomenon should allow an ultrafast
signal <strong>of</strong> the PeV gamma-photon arrival. Since the trigger
phenomenon is exponentially sensitive, we could play
a game <strong>of</strong> adjusting the value <strong>of</strong> the laser field to see and
differentiate different types <strong>of</strong> trigger phenomenology and
parameters, depending upon the gamma particle energies.
The creation <strong>of</strong> PeV energy gamma particles is a tremendous
challenge. For example, Pr<strong>of</strong>. Suzuki [38] has challenged
us if this may be accomplished within our life time.
Tajima et al. (2011) has ventured and showed to use the
Laser Wakefield Acceleration (LWFA) based on a MJ class
laser to reach PeV energy electrons. For the probing <strong>of</strong> the
quantum vacuum texture such as Ellis has been spearheading
can be done with a small number <strong>of</strong> electrons generated
gamma particles. Thus we are relieved from the tough luminosity
requirements. However, we need to test a precise
arrival <strong>of</strong> gamma photons at the ’goal line’ [39]. Here we
suggested the exploitation <strong>of</strong> the Schwinger-Nikishov process
to see this time resolution.
CONCLUSIONS
In conclusion, evidences over more than 18 orders <strong>of</strong>
magnitude <strong>of</strong> the Pulse Intensity-Duration Conjecture has
been accumulated experimentally and through simulation.
It shows that the pulse duration goes inversely with the
intensity from the millisecond to the attosecond and zeptosecond
regimes. Most notably it predicts that the short-
est coherent pulse in the zeptosecond-yoctosecond regime
should be produced by the largest laser, like ELI or NIF
and the Megajoule, if they are reconfigurated [40] into femtosecond
pulse systems.
This Conjecture may prove to be an invaluable guide
for future ultra-intense and ultrashort pulse experiments. It
fosters the hope that zeptosecond and perhaps yocto second
pulses could be produced using kJ-MJ systems. It opens up
the possibility <strong>of</strong> taking snap shots <strong>of</strong> nuclear reactions and
<strong>of</strong> peeking into the nuclear interior in the same way that
Zewail [41] examined chemical reactions or Corkum and
Krausz [42] probed atoms. The other exciting prospect is
the possibility <strong>of</strong> studying the nonlinear optical properties
<strong>of</strong> vacuum. This Conjecture in short implies:
1. that the shortest coherent pulse will be produced by
the largest laser, like the ELI or NIF and The Megajoule
pumping a femtosecond Ti: Sapphire or an
OPCPA system.
2. that pulses in the zeptosecond-yoctosecond may be
produced.
3. that even with modest efficiency, extremely high
power rivaling the critical power in vacuum may be
produced. It opens up the regime <strong>of</strong> vacuum nonlinearity.
It ties the three distinct disciplines <strong>of</strong> science, i.e. ultrafast
science, high- field science, and large-energy laser science
together with a single stroke.
We have also discussed an example <strong>of</strong> the PeV energy
frontier using the largest energy laser (e.g. NIF) relaxing
the requirement <strong>of</strong> luminosity. In this example, LWFA enables
us to reach PeV in a manageable, albeit still large
scale, experimental realization [39]. Here again we are utilizing
the strong- field vacuum nonlinearity to help achieve
ultrafast metrology.
ACKNOWLEDGMENTS
We would like to acknowledge the fruitful discussions
with S. Bulanov, E. Goulielmakis, T. Esirkepov, M. Kando,
F. Krausz, A. Suzuki, J. Nees, N. Naumova, E. Moses, S.
Iso, K. Itakura and N. Artemiev. T. Tajima was supported
in part by the Blaise Pascal Foundation and by DFG Cluster
<strong>of</strong> Excellence MAP (Munich Centre for Advanced Photonics).
REFERENCES
[1] http://www.extreme-light-infrastructure.eu
[2] G. A. Mourou, T. Tajima, S. V. Bulanov, Optics in the relativistic
regime. Rev. Mod. Phys. 78, 309-371 (2006).
[3] T. H. Maiman, Stimulated Optical Radiation in Ruby. Nature
187, 493-494 (1960).
[4] R. W. Hellwarth, Advances in Quantum Electronics
(Columbia University Press, New York, 1961).

The Heisenberg-Schwinger effect: Nonperturbative Vacuum Pair Production
Abstract
Gerald V. Dunne
Department <strong>of</strong> Physics, University <strong>of</strong> Connecticut, Storrs, CT 06269, USA
The Heisenberg-Schwinger effect is the non-perturbative
production <strong>of</strong> electron-positron pairs when an external
electric field is applied to the quantum electrodynamical
(QED) vacuum. The inherent instability <strong>of</strong> the vacuum
in an electric field was one <strong>of</strong> the first non-trivial predictions
<strong>of</strong> QED, but the effect is so weak that it has not
yet been directly observed. However, there are exciting
new developments in ultra-high intensity lasers, which may
soon bring us to the verge <strong>of</strong> this extreme ultra-relativistic
regime. This necessitates a fresh look at both experimental
and theoretical aspects <strong>of</strong> the Heisenberg-Schwinger effect.
I describe some new theoretical ideas aimed at making this
elusive effect observable, by careful shaping <strong>of</strong> the laser
pulses.
INTRODUCTION
The experimental observation <strong>of</strong> the Heisenberg-
Schwinger effect, the non-perturbative production <strong>of</strong>
electron-positron pairs from vacuum subjected to an electric
field, would open a new window into the largely unexplored
regime <strong>of</strong> nonperturbative quantum field theory, a
regime in which we can study matter in extreme environments
in a controllable way. This has significant implications
beyond QED, for example in particle physics, plasma
physics and gravitational physics, as is discussed in various
talks at this conference. It is well known that quantum
vacuum fluctuations mean that the QED vacuum behaves
like a polarizable medium that modifies classical behavior,
leading to novel quantum effects [1, 2, 3, 4, 5, 6, 7, 10, 11,
12, 8, 9]. Some <strong>of</strong> these effects, such as the Casimir effect
or the vacuum birefringence effect, are perturbative and
can be well described by perturbative quantum field theory.
The Heisenberg-Schwinger effect is a non-perturbative effect
that cannot be described by any single Feynman diagram;
its essence is a truly non-perturbative process, which
makes it both fascinating and difficult. The process can be
viewed pictorially as in Figure 1: a virtual electron-positron
pair in vacuum is accelerated apart by an external electric
field, becoming a real asymptotic e + e − pair if they gain
the binding energy <strong>of</strong> 2mc 2 from the external field. This
sets the basic scale at which we might expect this process
to become significant: when the work done separating the
pair by a Compton wavelength matches 2mc 2 :
Ec = m2 c 3
e ¯h ≈ 1016 V/cm
Figure 1: Pair production as the separation <strong>of</strong> a virtual vacuum
dipole pair under the influence <strong>of</strong> an external electric
field.
Ic = c
8π E 2 c ≈ 4 × 10 29 W/cm 2
Correspondingly, the probability <strong>of</strong> pair production is exponentially
suppressed by the Heisenberg-Euler factor
[
PHE ∼ exp − π m2 c3 ]
, (2)
e E ¯h
An analogous estimate for atomic ionization [e.g., for H],
again using the approximation <strong>of</strong> a constant electric field,
leads to
P hydrogen [
∼ exp − 2
3
m 2 e 5
E ¯h 4
]
(1)
, (3)
setting the basic scale <strong>of</strong> field strength and intensity near
which we expect to observe nonperturbative ionization effects
in atomic systems:
E ionization
c = m2 e 5
¯h 4
= α3 Ec ≈ 4 × 10 9 V/cm
I ionization
c = α 6 Ic ≈ 6 × 10 16 W/cm 2
Indeed, this is close to the scale <strong>of</strong> atomic ionization experiments,
but in fact intensities three orders <strong>of</strong> magnitude
lower are routinely used. This is because the electric
field in a laser is not constant, and careful shaping <strong>of</strong>
the laser pulses makes ionization experiments possible at
much lower intensities. This simple observation, together
with the fact [see, for example, the talks by G. Mourou
and T. Tajima at this conference] that there are plans at
large laser facilities such as ELI [13], HiPER at Rutherford
Laboratory, the NIF at Livermore, and the XFEL projects
at SLAC and DESY, to approach the 10 25 − 10 26 W/cm 2
intensity regime, motivates us to ask: how critical is the
Schwinger critical field (1)? To answer this question, we
need to review briefly some QFT formalism.
(4)

THE QED EFFECTIVE ACTION
In quantum field theory, the quantum corrections to classical
Maxwell electrodynamics are encoded in the ”effective
action” Γ[A] [14, 15]. For example, the polarization
tensor Πµν = δ2Γ contains the electric permittivity ϵij
δAµδAν
and the magnetic permeability µij <strong>of</strong> the quantum vacuum,
and is obtained by varying the effective action Γ[A] with respect
to the external probe Aµ(x). Γ[A] is defined in terms
<strong>of</strong> the vacuum-vacuum persistence amplitude
[ ]
i
⟨0out | 0in⟩ = exp {Re(Γ) + i Im(Γ)} (5)
¯h
Re(Γ[A]) describes dispersive effects, such as vacuum birefringence,
while Im(Γ[A]) describes absorptive effects,
such as vacuum pair production. The imaginary part encodes
the probability <strong>of</strong> vacuum par production as [14]
Pproduction = 1 − |⟨0out | 0in⟩| 2
[
= 1 − exp − 2
]
Im Γ
¯h
≈ 2
Im Γ (6)
¯h
From a computational perspective, the effective action is
defined as [14, 15]
Γ[A] = ¯h ln det [iD/ − m] = ¯h tr ln [iD/ − m] . (7)
Here, D/ ≡ γ µ Dµ, where the covariant derivative operator,
Dµ = ∂µ − i e
¯hc Aµ, defines the coupling between electrons
and the electromagnetic field Aµ. When the gauge field
Aµ is such that the field strength, Fµν = ∂µAν − ∂νAµ, is
constant, this effective action was computed exactly [and
non-perturbatively] by Heisenberg and Euler [1]. For ex-
ample, for a constant electric field E:
Γ HE [E]
Vol4
= −¯h e2 E 2
8π 2
∫ ∞
0
ds m2
e− eE
s2 s
(
cot(s) − 1 s
+
s 3
(8)
The leading imaginary part comes from the first pole <strong>of</strong> the
cot(s) function:
Im Γ HE
Vol4
∼ ¯h e2 E 2 [ ]
π m2
exp −
8π3 e E
THE EFFECTIVE ACTION IN
INHOMOGENEOUS BACKGROUND
FIELDS
It is essential to understand how this constant field result
(9) is modified for more realistic inhomogeneous fields,
such as those describing ultra-short pulse focussed lasers.
This is a difficult task, as standard perturbative effective
field theory techniques do not apply. The first step in
this direction is motivated by a seminal result <strong>of</strong> Keldysh
[16, 17] for the ionization <strong>of</strong> atoms in a monochromatic
time dependent electric field E(t) = E cos(ωt). This introduces
a new scale to the problem, and Keldysh was able
to compute the ionization probability as a function <strong>of</strong> the
)
(9)
dimensionless adiabaticity parameter γK, that characterized
the fast [γK ≫ 1] and slow [γK ≪ 1] regimes.
[The Keldysh parameter is related to the standard laser
field strength parameter a0 as a0 = 1/γK.] Remarkably,
Keldysh’s WKB result interpolates smoothly between the
non-perturbative tunnel-ionization regime where γK ≪ 1,
and the perturbative multi-photon regime where γK ≫
1. This formalism was generalized to the Heisenberg-
Schwinger effect in QED [18, 19, 20], with an analogous
”adiabaticity parameter”
Ppair prod. ∼
γK ≡ mcω
. (10)
eE
⎧ [
⎨ exp −
⎩
πm2c 3
]
eE¯h , γK ≪ 1
) 2 (11)
2mc /¯hω
, γK ≫ 1
( eE
mω
The γK ≪ 1 regime corresponds to nonperturbative tunneling,
while γK ≫ 1 is the perturbative multiphoton
regime. In the perturbative multi-photon regime, this QED
pair production effect has been observed in a beautiful experiment
(E-144) at SLAC [21], in which a laser pulse collided
with the (highly relativistic) SLAC electron beam,
leading to nonlinear Compton scattering involving 4-5 photons,
producing a high energy gamma photon that decays
into an electron-positron pair. By contrast, it is hoped that
in future laser facilities it will be possible to probe deep
into the nonperturbative regime where γK ≪ 1, to see the
truly nonperturbative Heisenberg-Schwinger effect <strong>of</strong> pair
production directly from vacuum.
The Keldysh approach captures an enormous amount
<strong>of</strong> important physical information. Various methods have
been developed which can be used to compute the pair production
probability when the background electric field depends
on just one coordinate. The problem can be understood
as a one-dimensional scattering problem, based on
Feynman’s interpretation <strong>of</strong> positrons as electrons propagating
backwards in time [22]. Then the probability
can be extracted from the reflection coefficient for
a ”Schrödinger” problem <strong>of</strong> scattering in the time domain.
The reflection probability can be computed exactly
[numerically], as in the quantum kinetic approach
[23, 24, 25, 26], or estimated using semiclassical WKB
arguments [18, 19, 20, 27]. A natural ”inverse question”
arises: can we shape the laser pulses in order to enhance
the pair production effect, or to make it more distinctive?
PULSE SHAPING EFFECTS FOR
TIME-DEPENDENT FIELDS
Continuing the approximation <strong>of</strong> considering a timedependent
electric field, several recent results suggest that
the peak laser intensity at which appreciable vacuum pair
production could be observe is in the 10 25 − 10 26 W/cm 2
intensity range, which is significant since this is the targeted
goal <strong>of</strong> the ELI project, and within range <strong>of</strong> the
HiPER facility. An important set <strong>of</strong> ideas [28, 29] is to
combine multiple pulses, each <strong>of</strong> a lower intensity, and to

take into account the fact that the spatial focussing region is
much larger than the scale <strong>of</strong> the electron Compton wavelength.
Quantitative analyses then suggest a critical intensity
3 or 4 orders <strong>of</strong> magnitude below the Schwinger limit
(1). It has also been shown recently [30] that plane wave
fields <strong>of</strong> finite extent, and therefore having nontrivial shape
dependence, produce interesting and novel effects for vacuum
pair production.
Dynamically Assisted Schwinger Mechanism
Another related but distinct idea is the ”dynamically assisted
Schwinger mechanism” [31], in which a simple superposition
<strong>of</strong> two time-dependent pulses, one strong but
slow, and the other weak but fast, can lead to a significant
enhancement <strong>of</strong> the tunneling process associated with the
Heisenberg-Schwinger effect. This ”dynamically assisted
Schwinger mechanism” was introduced in [31] with fields:
Eslow(t) = E sech 2 (Ωt) ; Efast(t) = ϵ sech 2 (ωt) (12)
with parametric hierarchies: 0 < ϵ ≪ E ≪ Ec, 0 <
Ω ≪ ω ≪ m. Surprisingly, even though both field amplitudes,
E and ϵ, are below the critical field Ec in (1), there is
significant enhancement <strong>of</strong> the pair production rate when
the frequencies follow this hierarchy <strong>of</strong> scales. The nonperturbative
pair production process that we would associate
with the slow strong field interacts with the perturbative
multiphoton pair production process that we would
associate with the fast weaker field to produce a stronger
impact than each process separately. A specific realization
<strong>of</strong> this idea was proposed recently [32], involving a
strong, slow optical laser pulse and a weak, fast X-ray
pulse. Particles in the Dirac sea can perturbatively absorb
a high-frequency photon from the weak, fast field,
thereby lowering the effective tunnel barrier for the nonperturbative
process. This leads to predictions <strong>of</strong> observable
pair-production yields based on current technology.
This dynamically assisted Schwinger mechanism is
closely related to a catalysis mechanism [33], in which one
can view the problem as the propagation <strong>of</strong> photons <strong>of</strong> the
X-ray probe pulse in an intense and effectively constant
electric field provided by the stronger and slower optical
laser pulse. There is a non-zero absorption coefficient for
photon propagation in such a strong field, and from this one
can deduce the rate <strong>of</strong> pair production. Technically, this requires
the computation <strong>of</strong> the imaginary part <strong>of</strong> the photon
polarization tensor in an electric field. As the X-ray frequency
approaches the threshold <strong>of</strong> 2m there is a dramatic
exponential enhancement
{
α
Im(Π) ≈ √ eE exp −
π(π − 2) m2
}
(π − 2) (13)
eE
This exponential enhancement leads to a window <strong>of</strong>
opportunity [33] in the range <strong>of</strong> laser intensity up to
I ≈ 9 × 10 25 W/cm 2 in which this catalyzed Schwinger
mechanism is dramatically enhanced relative to the pure
Heisenberg-Schwinger effect with just the strong optical
laser pulse, without the catalyzing X-ray pulse. This catalysis
mechanism can also be viewed as photon-stimulated
pair-production [34], realizing the more general mechanism
<strong>of</strong> an induced metastable decay process [35].
Interference effects from sub-cycle temporal
pulse structure
Recently it has become clear that the WKB analysis <strong>of</strong>
[18, 19, 20] must be extended to incorporate interference
effects when the temporal pr<strong>of</strong>ile <strong>of</strong> the laser pulse has subcycle
structure. This interference phenomenon is extremely
sensitive to the temporal pr<strong>of</strong>ile <strong>of</strong> the laser pulse; for example,
to the carrier phase <strong>of</strong> the laser pulse, and also to the
presence <strong>of</strong> “chirp” in the pulse [see below]. The technical
reason is that for such fields, the over-the-barrier scattering
problem typically has multiple semiclassical saddle points
[i.e., sets <strong>of</strong> turning points], and the interference between
different saddle points leads directly to oscillatory resonance
behavior in the longitudinal momentum spectrum <strong>of</strong>
the produced particles. Such phenomena are familiar from
strong-field atomic and molecular physics, discussed long
ago in the theory <strong>of</strong> atomic ionization [36, 37, 17], and observed
experimentally in photoionization spectra [38, 39].
In the context <strong>of</strong> QED vacuum pair production, these interferences
effects were first studied numerically in [40],
and given a simple quantitative semiclassical explanation
in [41] in terms <strong>of</strong> the Stokes phenomenon: the interference
produced by the patching together <strong>of</strong> semiclassical approximations
in different regions. Specifically, for an electric
field <strong>of</strong> the form
E(t) = E0 cos(ωt + ϕ) exp
(
− t2
2τ 2
)
(14)
one finds oscillatory behavior <strong>of</strong> the longitudinal electronpositron
momentum spectrum, which becomes pronounced
when ωτ ∼ 4 [i.e., when the number <strong>of</strong> oscillations under
the envelope exceeds 4], and in particular when the
carrier phase ϕ approaches π/2. Most dramatically, when
ϕ = π/2 there are values <strong>of</strong> the electron-positron momentum
at which the probability <strong>of</strong> production vanishes. The
analytic explanation <strong>of</strong> these numerical results lies in the
interference between different semiclassical saddle points.
With just one dominant saddle point [as occurs for a constant
electric field E(t) = E0, or for an electric field with a
simple single-bump structure like E(t) = E0 sech 2 (ωt), or
E(t) = E0/(1 + (ωt) 2 ) 2 ] we have the familiar probability
expression with exponential factor <strong>of</strong> the form e −Sc , where
Sc is the classical (Euclidean) action [27]. With two saddle
points <strong>of</strong> comparable amplitude, the expression generalizes
to [41]
P ≈ e −S(1)
c + e −S(2)
1
c −
± 2 cos(2α) e 2 S(1)
1
c − 2 S(2)
c (15)
where α is an integral connecting different saddle points,
and characterizes the interference. The ± in (15) refers

Figure 2: Turning points in the complex t plane for the ”cosine” [left frame] electric field with carrier phase ϕ = 0,
E(t) = E0 cos(ωt) exp ( −t 2 /(2τ 2 ) ) , and for the ”sine” [right frame] electric field with carrier phase ϕ = π/2, E(t) =
E0 sin(ωt) exp ( −t 2 /(2τ 2 ) ) . Turning points closest to the real axis tend to dominate. In the first case there is one dominant
saddle point, while in the second case there are two, and the interference between them leads to oscillatory behavior <strong>of</strong>
the momentum spectra for electrons and positrons produced by such a laser pulse [41, 42].
to scalar/spinor QED, reflecting the expected opposite sign
dependence <strong>of</strong> interference effects on quantum statistics,
and explaining both qualitatively and quantitatively the numerical
observation in [43]. Including also a chirp parameter
b, by replacing the cosine factor by cos(b t 2 + ωt + ϕ)
one finds a dramatic effect on the form <strong>of</strong> the momentum
spectrum [44]. Besides their theoretical interest, the
acute sensitivity <strong>of</strong> the momentum spectra to the laser
pulse shape may provide distinctive experimental signatures
<strong>of</strong> the Heisenberg-Schwinger effect, and eventually
could provide sub-cycle pulse resolution at extremely short
time scales, using the quantum vacuum fluctuations. Another
suggestion [40] to use these vacuum interference effects
is an all-optical time-domain analogue <strong>of</strong> the doubleslit
experiment, a detailed experimental proposal for which
is in [45].
SPATIAL AND TEMPORAL PULSE
SHAPE EFFECTS
Ideally we would like to be able to compute the imaginary
part <strong>of</strong> the effective action Γ[A] for gauge fields
Aµ(⃗x, t) that represent the full spatial and temporal structure
<strong>of</strong> a realistic laser pulse configuration. This is a nontrivial
question, since temporal inhomogeneities tend to enhance
the rate [because it is easier to tunnel through an
oscillating barrier], while spatial inhomogeneities tend to
suppress the rate [because the field falls <strong>of</strong>f as the particles
accelerate away from the nucleation point]. This
raises an interesting question: how do these competing effects
play out in an ultra-short laser pulse that is tightly
spatially focussed? This is a technically difficult question
to answer, because the conventional WKB and QKE approaches
have not yet been generalized to higher dimensions
in any computationally efficient manner. Nevertheless,
several promising approaches have recently been developed.
Worldline instanton formalism
A natural multi-dimensional semiclassical approach is
provided by Feynman’s worldline formulation <strong>of</strong> the QED
effective action [46, 47]. Feynman formulated a firstquantized
form <strong>of</strong> QED, which amounts to representing
the effective action as a quantum mechanical path integral
over closed loops xµ(τ) in four dimensional spacetime,
with the closed loops being parametrized by the proper
time τ. The propertime parametrization had been developed
earlier by Fock and Nambu [48], and was also used
by Schwinger, in operator form rather than in path integral
form, in his landmark QED computation <strong>of</strong> vacuum
pair production [14]. Feynman’s worldline path integral
formalism has since been extended significantly, primarily
for applications in perturbative quantum field theory [47],
building on analogies and motivation from the Polyakov
formulation <strong>of</strong> string theory. This rebirth has led to many
beautiful advances in our understanding <strong>of</strong> perturbative
scattering amplitudes, but here I describe an application to
non-perturbative processes. For simplicity, consider scalar
QED. The effective action for a scalar charged particle
(charge e, mass m) in a Euclidean classical gauge background
Aµ(x) is the functional:
Γ[A] =
∫ ∞
0
× exp
dT
T e−m2 ∫
T
[
−
∫ T
0
dτ
d 4 x (0)
∫
Dx
x(T )=x(0)=x (0)
(
2 ˙x µ
4 + e Aµ
)]
˙xµ
The main technical difficulty is to compute the quantum
mechanical path integral, a sum over closed trajectories
in four-dimensional Euclidean space. One approach is a
direct Monte Carlo analysis, as has been done for onedimensional
inhomogeneities [49]; this is a powerful approach
since it does not rely on any particular symmetry <strong>of</strong>
the background field, although it is computationally diffi-

cult to extract the exponentially small pair production rate.
A more analytic approach is to make a semiclassical approximation
to the path integral, by solving the classical
equations <strong>of</strong> motion [we set 2e = 1 to simplify notation]:
¨xµ = Fµν(x) ˙xν , (µ, ν = 1 . . . 4) . (16)
In this semiclassical approximation, the path integral
is dominated by a classical loop called a ”worldline instanton”
[a closed-loop solution to the classical Euclidean
equations <strong>of</strong> motion], together with quantum fluctuations
about this loop. The dominant exponential factor in the
imaginary part <strong>of</strong> the effective action is just
exp (−S[xclassical]) (17)
with the action evaluated on the worldline instanton trajectory.
This idea was first applied to the vacuum pair production
problem for a constant electric field in [50], and
later extended to inhomogeneous background field configurations
[51]. The prefactor contributions, which can be
physically significant, are most efficiently computed using
an analogy [52] to the Gutzwiller trace formula <strong>of</strong> nonrelativistic
quantum mechanics, viewing the closed loop as
a closed trajectory in phase space. This worldline instanton
method is very general; the main technical challenge
is finding the closed classical trajectories in a given (Euclidean)
background field.
Wigner function formalism
Given the strong analogies between the problem <strong>of</strong> vacuum
pair production and atomic physics in strong laser
fields, and also to various well-known condensed matter
problems such as Landau-Zener tunneling [53], it is quite
natural to adapt the standard quantum optics approach <strong>of</strong>
the Wigner function [54] to the problem <strong>of</strong> vacuum pair
production. This method has been developed recently in
both the real time and light-cone formalisms [55, 56]. The
essential idea is that the Wigner transform <strong>of</strong> the field twopoint
function contains information about the pair production
process, and the Wigner function satisfies an equation
that generalizes the quantum kinetic equation beyond the
one-dimensional evolution. For vacuum pair production,
consider the vacuum matrix element <strong>of</strong> the equal time density
[including the gauge field parallel transport operator
for gauge invariance]
C(⃗x, ⃗s, t) = ⟨0|e −ie
∫ 1/2
⃗A(⃗x+λ⃗s,t)·⃗s dλ
−1/2
×[Ψ(⃗x + ⃗s
2 , t), ¯ Ψ(⃗x − ⃗s
, t)] |0⟩ (18)
2
and its Wigner transform
W(⃗x, ⃗p, t) = − 1
∫
2
d 3 s e −i⃗p·⃗s C(⃗x, ⃗s, t) (19)
This Wigner function approach maps the problem to phase
space, and leads to coupled equations for the Wigner function,
providing a formalism in which various semiclassical
approximations can be explored. It also opens the possibility
for studying non-equilibrium aspects <strong>of</strong> the pair production
process.
CONCLUSIONS
The observation <strong>of</strong> the Heisenberg-Schwinger effect
presents a fundamental challenge both theoretically and experimentally.
Theoretically we need new non-perturbative
techniques to provide efficient and precise calculations <strong>of</strong>
the expected pair production rate in realistic short-pulse focussed
laser fields. The approach must be sufficiently flexible
and powerful to be able to optimize the pulse shape
to maximize the pair production rate. Other important
theoretical techniques involve intense numerical modeling
<strong>of</strong> solutions and trajectories for particles in laser fields
[57, 58]. An important recent idea concerns the possibility
<strong>of</strong> QED cascade effects, and the possibility <strong>of</strong> an upper
limit on an attainable electric field. This is a fascinating,
difficult and fundamental question that has not yet
been resolved [58, 59, 60, 61]. On the experimental side,
the main challenges are to obtain higher laser intensity, as
close as possible to the critical field limit (1), and to be
able to focus and shape the laser pulse(s) in both the space
and time domain. Recent progress at ELI, HiPER and with
XFEL lasers suggests that we may be very close to entering
this new physical regime <strong>of</strong> ultra-relativistic physics. Beyond
QED, there are fundamental questions to be answered
concerning back-reaction effects, the physics beyond the
Schwinger critical field, and concerning the possible simulation
<strong>of</strong> gravitational effects such as Unruh and Hawking
radiation, using the very large electric field acceleration to
mimic strong gravitational fields. Many <strong>of</strong> these questions
are directly addressed in talks at this conference.
Acknowledgements: I thank the organizers <strong>of</strong> the
PIF2010 conference, especially Satoshi Iso, for organizing
an excellent meeting. I also thank R. Alk<strong>of</strong>er, C.
Dumlu, H. Gies, F. Hebenstreit, G. Mourou, C. Schubert,
R. Schützhold and T. Tajima for collaborations and discussions,
and I acknowledge support from the DOE through
the grant DE-FG02-92ER40716.
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QED IN ULTRA-INTENSE LASER FIELDS ∗
T. Heinzl † , School <strong>of</strong> Computing & Mathematics, University <strong>of</strong> Plymouth, UK
C. Harvey, A. Ilderton and M. Marklund, Department <strong>of</strong> Physics, Ume˚a University, Sweden
Abstract
We present an overview <strong>of</strong> basic QED processes in the
presence <strong>of</strong> an ultra-intense laser background.
INTRODUCTION
The year 2010 has seen the 50th anniversary <strong>of</strong> the laser.
Since its inception it has undergone a very dynamic development
culminating in a multitude <strong>of</strong> everyday applications.
From the physics point <strong>of</strong> view specification parameters
have evolved in many directions, for instance towards
the X-ray regime <strong>of</strong> frequency. For the purpose <strong>of</strong> this conference
and this talk we are particularly interested in ultrahigh
intensities. The historical development <strong>of</strong> these is pictured
in Fig. 1 (adapted from [1]) with a notable breakthrough
in 1985 due to the implementation <strong>of</strong> chirped pulse
amplification (CPA) [2].
Figure 1: Time evolution <strong>of</strong> laser intensity.
The vertical axis on the right-hand side measures intensity
I in terms <strong>of</strong> the dimensionless laser amplitude
a0 = eEλ
mc 2 ∼ I1/2 , (1)
which is the energy gain <strong>of</strong> an electron (charge e, mass m)
across a laser wavelength λ in the r.m.s. field E, in units
<strong>of</strong> its rest energy, mc 2 . Hence, when a0 exceeds unity an
∗ Work supported in part by ERC, Contract No. 204059-QPQV.
† theinzl@plymouth.ac.uk
electron probing the laser field will begin to move relativistically.
It is worth pointing out that ultra-intense lasers produce
the largest electromagnetic fields that are currently available
in the lab. Of course, the downside is that the fields
are pulsed (i.e. “short-lived”) and alternating. An overview
<strong>of</strong> the current magnitudes is given in Table 1.
Table 1: Some typical current magnitudes.
Quantity Magnitude
Power P 10 15 W ≡ 1 PW
Intensity I 10 15 W/cm 2
Electric Field E 10 14 V/m
Magnetic Field B 10 10 G
Planned facilities where these magnitudes will be increased
further include the Vulcan 10 PW project at the
Central Laser Facility <strong>of</strong> Rutherford Lab, UK and the European
Extreme Light Infrastructure where up to 100 PW
are envisaged.
STRONG FIELDS: THEORY
We are interested in elementary processes occurring in
the presence <strong>of</strong> an ultra-intense laser. The appropriate theory
is (a variant <strong>of</strong>) strong-field quantum electrodynamics
(QED) with the laser field being included as an external
background field. The extent to which this theory is under
analytical control depends sensitively on the model chosen
for the laser beam. The simplest model is an infinite,
monochromatic plane wave for which transition amplitudes
can be calculated including an analytic evaluation <strong>of</strong> the
appearing oscillatory integrals [5]. The latter becomes difficult
for pulsed plane waves such that this case presents
more challenging technical difficulties. While pulsed plane
waves have finite extent in time and longitudinal distance
they are still <strong>of</strong> infinite transverse size. Introducing a transverse
pr<strong>of</strong>ile such as for a Gaussian beam certainly represents
a more realistic model but turns out to be difficult to
implement in strong-field QED, the main reason being the
loss <strong>of</strong> too many conservation laws along with translational
invariance. So, for the purposes <strong>of</strong> this talk we will exclusively
be dealing with (infinite or pulsed) plane waves.
From a relativistic field theory point <strong>of</strong> view, which we
have to adopt for a0 > 1, plane electromagnetic waves are
<strong>of</strong> a quite peculiar nature. They are described by a wave
4-vector k that is lightlike or null, i.e. k 2 = 0. The electromagnetic
field strength, F = (E, B), only depends on the
invariant phase, k·x = ωt/c−k·x, where ω is the laser fre-

quency measured in the lab. By Maxwell’s equations, fields
are transverse, k ·F = 0, and, most importantly, inherits the
null properties from k, E 2 − B 2 = E · B = F 3 = 0. This
implies that there is no intrinsic invariant scale characterising
an electromagnetic plane wave field. For this reason
one needs an external probe momentum, p, to build invariants
such that, for instance, a 2 0 ∼ (p · F) 2 [6].
The main effect on such a probe, say an electron, may
actually be understood in classical language. Due to the
Lorentz force the electron will undergo rapid quiver motion
as may be seen by solving for its momentum p(τ) as a function
<strong>of</strong> proper time τ. Averaging over the latter the electron
acquires a quasi-momentum, q ≡ 〈p(τ)〉 = pin + κ(a 2 0) k,
which displays a longitudinal addition to the initial momentum
weighted with an a0 dependent prefactor. Squaring
this expression and using k 2 = 0 one finds that the electron
has become heavier with an effective mass given by
m 2 ∗ = m 2 (1 + a 2 0) . (2)
This fundamental intensity effect has been predicted long
ago [7, 8], but has apparently never been measured. While
the appearance <strong>of</strong> the mass shift may be understood quantum
mechanically it nevertheless has consequences for the
quantum theory. These are best analysed in terms <strong>of</strong> strong
field QED. Its ingredients are the usual particle content
<strong>of</strong> QED namely photons and electrons <strong>of</strong> arbitrary energy
serving e.g. as probes for “quantum diagnostics” when they
are coupled to the external laser field. The basic quantum
effect is then a dressing <strong>of</strong> the electrons via continuous
emission and absorption <strong>of</strong> laser photons. For plane waves
this can be taken into account exactly employing the celebrated
Volkov solution <strong>of</strong> the Dirac equation [9]. In terms
<strong>of</strong> Feynman diagrams the situation is depicted in Fig. 2.
Figure 2: The dressed (Volkov) propagator <strong>of</strong> the electron.
Transition amplitudes are then constructed in terms <strong>of</strong>
Volkov electron lines coupled to probe photons. Laser photons
are no longer explicit but hidden in the dressed propagator
given by the left-hand side Fig. 2. The main issues
to be discussed below will be the dependence <strong>of</strong> these amplitudes
on intensity (a0) and finite pulse duration, all in
a plane wave context. To give an idea <strong>of</strong> the parameter
range involved we present in Fig. 3 a bird’s eye view <strong>of</strong> the
probe energy vs. intensity regimes. Conventional high energy
physics takes place close to the vertical axis, above the
electron pair creation threshold <strong>of</strong> 2mc 2 . In this regime, a
first excursion into high-field physics has been made by the
SLAC experiment E-144 which utilised 30 GeV photons
(obtained via backscattering from the 50 GeV SLAC electron
beam) to produce electron positron pairs in collisions
with a 10 TW (a0 0.4) laser beam [10]. New facilities
such as Vulcan 10 PW or ELI would venture deeply in the
high-intensity region (a0 ≫ 1) and may also come close to
the pair threshold using Compton backscattering from sufficiently
energetic electrons. The latter could in principle
be the result <strong>of</strong> laser wake field acceleration.
Figure 3: Parameter range <strong>of</strong> strong field QED.
A basic example <strong>of</strong> finite pulse duration effects is displayed
in Fig. 4 which shows the mass shift, ∆m 2 ≡
m 2 ∗ − m 2 , in a pulse [11] as a function <strong>of</strong> the number N <strong>of</strong>
cycles per pulse. There is obviously a “switching on” effect
with a sudden increase upon concluding the first cycle after
which the infinite plane wave result (2) is approached. It
should be emphasised that for current ultra-short pulses <strong>of</strong>
a few fs duration the number n is indeed <strong>of</strong> order unity.
Figure 4: The mass shift in a pulsed plane wave.
STRONG FIELDS: EXAMPLES
Nonlinear Compton Scattering (NLC)
The Feynman diagrams for NLC are shown in Fig. 5.
Expanding the Volkov lines according to Fig. 2 we find
that one is essentially summing over all processes <strong>of</strong> the

form e + nγL → e ′ + γ where a photon γ is emitted upon
absorption <strong>of</strong> n laser photons γL by the incoming electron.
Figure 5: Feynman diagrams for NLC.
The process has been analysed in [5, 8] where formulae
for cross sections or emission rates can be found. The main
features <strong>of</strong> NLC may be summarised as follows. There is
no threshold to overcome which implies that there is a classical
limit (nonlinear Thomson scattering) corresponding to
ω ≪ mc2 . In the linear regime (a0 ≪ 1) one finds the
usual Compton upshift for the emitted photon frequency,
ω ′ 4γ2 eω where γe is the electron gamma factor. This is
nowadays being exploited for the generation <strong>of</strong> monoenergetic
gamma rays <strong>of</strong> high peak brillance [12]. The nonlinear
regime (a0 > 1) is characterised by an intensity dependent
cross section, σ(a0), determining the number <strong>of</strong> produced
photons, Nγ ∼ σ(a0)NeNγL . This very fact alters
the Compton upshift the maximum <strong>of</strong> which now becomes
ω ′ n,max 4γ 2 enω/(1 + a 2 0) , n = 1, 2, . . . . (3)
In particular, we note the appearance <strong>of</strong> higher harmonics
(n > 1) and the appearance <strong>of</strong> a 2 0 in the denominator. Thus,
there is a reduction (red-shift) <strong>of</strong> the kinematic Compton
edge which for the first harmonic amounts to
ω ′ max 4γ 2 eω −→ 4γ 2 eω/a 2 0 , (a0 ≫ 1) . (4)
Figure 6: Emission spectrum for NLC.
This redshift w.r.t. linear Compton scattering should be a
clear experimental signal and is highlighted in Fig. 6 (dis-
playing the γ emission spectrum) by a red arrow. The
higher harmonics are visible as side bands to the right <strong>of</strong>
the main (n = 1) spectral peak [13].
Effects due to finite transverse pulse extension are easily
understood qualitatively. The previous situation is typical
for laser beams that are not too strongly focussed such that
the electron beam (radius rb) does not feel the decrease
<strong>of</strong> intensity in transverse direction. Clearly, this requires
rb ≪ w0 with w0 denoting the laser beam waist size (see
Fig. 7, right panel). On the other hand, when the laser beam
is tightly focussed (w0 < rb, Fig. 7, left panel) the electrons
will also probe the boundaries <strong>of</strong> the beam which in
turn will modify the spectra <strong>of</strong> Fig. 6. It is somewhat unfortunate
that the highly nonlinear situation (a0 ≫ 1) corresponds
to a tight focus. This suggests that the experimental
detection <strong>of</strong> the redshift will require a fine tuning compromise
and in particular a very narrow electron beam. For a
detailed study the reader is referred to [14].
Figure 7: Left: Tight laser focus. Right: Wide laser focus.
Laser pair production (PP)
The strong-field QED Feynman diagram for laser induced
PP is obtained from the NLC diagram <strong>of</strong> Fig. 5 via
crossing, i.e. by swapping the outgoing gamma with the
incoming electron which turns into an outgoing positron
(Fig. 8).
Figure 8: Feynman diagram for laser PP obtained from
NLC via crossing.
Expanding the diagram on the right-hand side corresponds
to pair creation stimulated by n laser photons,
γ + nγL → e + e − . Both processes <strong>of</strong> Fig. 8 have been
employed in the experiment SLAC E-144. First, high energy
gammas (30 GeV) have been obtained through nonlinear
Compton upshifting whereupon these were brought
into collision with the laser again. The centre-<strong>of</strong>-mass energy
<strong>of</strong> the colliding photons (for the second harmonic)
was just about enough to produce pairs <strong>of</strong> electrons and

positrons, each <strong>of</strong> effective mass m∗ 1.2 m. A number
<strong>of</strong> about 10 2 positrons has been reported in [15] (see [10]
for a comprehensive overview). In a new theoretical development
this two-step process (NLC + laser PP) has been
reanalysed and led to the prediction <strong>of</strong> pair cascades where
the produced charged particles are continuously being reaccelerated
emitting bremsstrahlung that in turn produces
more pairs [16].
Figure 9: Laser PP rates for a finite wave train (N = 1, 4, 8
from top to bottom).
Pursuing a slightly complementary direction we have recently
investigated the effects <strong>of</strong> finite pulse duration on
laser PP using light-cone field theory [17]. In an infinite
plane wave the triple differential PP rate (w.r.t. longitudinal
and transverse positron momentum, say) is a delta comb
above threshold, with sharp resonance peaks akin to an
ideal interference pattern. One expects this to get washed
out as soon as finite pulse duration becomes noticeable, i.e.
when the number <strong>of</strong> cycle per pulse N = O(1). This is
indeed what one finds as is shown in Fig.s 9 and 10. For a
finite wave train (sharply cut <strong>of</strong>f in k·x) the maxima remain
at the delta comb positions (vertical lines in Fig. 9) while
for a more realistic pulse (smoothly decaying envelope) this
is no longer true (Fig. 10). Notably, in both cases, there is a
signal below the nonlinear (m∗) threshold which seems to
be consistent with Fig. 4 as is the fact that the infinite plane
wave results are recovered when N → ∞. The important
bottom line here is that the spectra are fingerprints <strong>of</strong> the
pulse shapes which might even be turned into a diagnostic
tool.
Figure 10: Laser PP rates for a smooth pulse (N = 1, 4, 8
from top to bottom).
Vacuum birefringence
Our final example addresses an interesting effect due to
external fields below the PP threshold. This possibility
has already been pointed out in the early days <strong>of</strong> QED by
Heisenberg and Euler [18]: “...even in situations where the
[photon] energy is not sufficient for matter production, its
virtual possibility will result in a ‘polarisation <strong>of</strong> the vacuum’
and hence in an alteration <strong>of</strong> Maxwells equations”
(our translation).
In terms <strong>of</strong> strong-field QED Feynman diagrams the
physics involved may be understood via the optical theorem
(Fig. 11). For the case at hand this states that the total
PP rate is given by the imaginary part <strong>of</strong> the polarisation
tensor which, <strong>of</strong> course, is only nonvanishing above threshold.
This appearance <strong>of</strong> an imaginary part corresponds to
an absorptive process (photons ‘decay’). Effects below
threshold, on the other hand, are dispersive, they modify,
in particular, the propagation properties <strong>of</strong> photons probing
the external field. One natural expectation is that the probe
photon polarisation might be affected by the preferred di-

ection(s) associated with the external field. This is indeed
what happens.
Figure 11: The optical theorem for laser PP.
Obviously, to study these effects one needs to calculate
the vacuum polarisation tensor in the presence <strong>of</strong> the relevant
external field (Fig. 11, right-hand side). For slowly
varying fields this has been achieved long ago in Toll’s thesis
[19] based on the Heisenberg-Euler lagrangian [18]. He
has found that there is birefringence <strong>of</strong> the vacuum due to
a phase retardation <strong>of</strong> the probe electric field component
parallel to the background magnetic field component. The
effect is tiny (as are the Heisenberg-Euler corrections to
Maxwell’s equations) and was long thought to be out <strong>of</strong>
reach experimentally [20]. This situation has changed with
the increase in laser intensity as discussed in [21]. Let us
discuss the physics involved in some more detail.
The propagation properties <strong>of</strong> probe photons are determined
by the eigenvalues <strong>of</strong> the polarisation tensor. An
explicit calculation shows that two <strong>of</strong> these are nontrivial
implying birefringence. The eigenvalues translate into two
different indices <strong>of</strong> refraction which, following Toll, may
be written as
n± = 1+ αɛ2 2 2 2
11 ± 3 + O(ɛ ν ) 1 + O(αɛ ) . (5)
45π
There are three small dimensionless parameters involved,
namely (i) the field strength, ɛ ≡ E/ES in units <strong>of</strong> the
Sauter-Schwinger critical field, ES = m 2 c 3 /e 1.3 ×
10 18 V/m, (ii) the probe frequency, ν = ω/mc 2 and the
fine structure constant, α = e 2 /4πc 1/137. The experimental
challenge is to measure the ellipticity acquired
by a linearly polarised probe beam traversing the focus <strong>of</strong>
an ultra-intense beam. As shown in [21] the ellipticity increases
with focus size d, intensity ɛ 2 and probe frequency
ν, the latter suggesting the use <strong>of</strong> probe X-rays. Recent advances
in X-ray polarimetry [22] imply that it may become
possible to observe the effect with intensities <strong>of</strong> 10 23 ...10 24
W/cm 2 which seem in reach for the near future [3, 4].
Conclusion
The examples analysed in this section should have
shown that an experimental programme dedicated to their
study is feasible. NLC is unique in that there are no thresholds
in energy or intensity to be overcome, so its study is
just a question <strong>of</strong> having the resources (electron and laser
beams) in place. Laser PP and vacuum birefringence are
more demanding in intensity but new facilities are under
way. Hence, it seems we are on the verge <strong>of</strong> opening up a
new and exciting interdisciplinary field <strong>of</strong> physics that may
be called laser particle physics.
It is a pleasure to thank the organisers <strong>of</strong> PIF2010 for
the excellent job they did, in particular for creating such an
exciting conference atmosphere.
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Abstract
Strong-Field Effects in Beam-Beam Interaction in Linear Colliders
Various types <strong>of</strong> electron-positron linear colliders are being
studied in the world. In all cases the intense field effects
associated with the beam-beam interaction due to the
tightly focused beams are expected. We summarize the results
<strong>of</strong> the studies.
INTRODUCTION
It was obvious since long ago that the electron-positron
collider after LEP had to be a linear collider rather than
a ring collider. Serious design studies <strong>of</strong> linear colliders
started in mid 1980’s in several laboratories in the world.
To reach the required luminosity the beam must be much
more tightly focused than in ring colliders. It was soon recognized
that the electromagnetic field due to the focused
beam causes various phenomena related to quantum electrodynamics
<strong>of</strong> intense field.
The first obvious fact was that the photon energy <strong>of</strong> the
synchrotron radiation (now called ‘beamstrahlung’) from
the beam field can exceed the energy <strong>of</strong> the primary electrons
if the classical formula <strong>of</strong> synchrotron radiation is
used. The full formula <strong>of</strong> Sokolov and Ternov[2] must be
used.
It was also pointed out the beamstrahlung photon can
create electron-positron pairs in the strong electromagnetic
field. In the linear collider community this is ‘called coherent
pair creation’.
The strong field can cause depolarization <strong>of</strong> electron
(positron) through the process <strong>of</strong> precession in a field and
spin-flip synchrotron radiation.
Moreover, the idea <strong>of</strong> converting an electron-electron
collider into a photon-photon (more <strong>of</strong>ten called gammagamma)
collider was proposed. In this scheme a strong
laser beam is irradiated on the electrons to create highenergy
back-scattered photons. This process also requires
a knowledge <strong>of</strong> high-field quantum electro-dynamics. It
turned out that the nonlinear effects impose strong limitations
on the electron-photon conversion.
All these are unwanted effects for the performance <strong>of</strong> linear
colliders. The theory and simulation had almost been
established by around 1995, although experimental verifications
are very poor still now. In this report we briefly
summarize what we have done in the past.
For numerical examples <strong>of</strong> linear colliders in this paper
we quote three future colliders, namely, ILC (<strong>International</strong>
Linear Collider), CLIC (Compact Linear Collider)
and plasma collider. The technologies for these colliders
∗ kaoru.yokoya@kek.jp
Kaoru Yokoya ∗ , KEK, Japan
are somewhat different but here we only need the beam parameters
at the collision point. Tab.1 shows typical parameters
<strong>of</strong> these colliders (those related to beam-beam interaction
only). Colliders using the plasma acceleration technology
are being plannned as far future projects. Their parameters
are still uncertain. In this table two possible parameter
sets ‘Plasma1’ and ‘Plasma2’ are shown[1].
Table 1: Example Parameters <strong>of</strong> Linear Colliders
ILC ILC CLIC Plasma1 Plasma2
E CM 0.5 1 3 10 10 TeV
N 2 2 0.37 0.1 0.4 ×10 10
σz 300 300 44 1 1 µm
σx 470 550 40 2 2 nm
σy 3.8 2.7 1 2 2 nm
〈Υ〉 0.063 0.109 5.5 2000 9000
δ B 3.9 5 30 25 50 %
nγ 1.71 1.43 2 2.4 2.1
E CM : center-<strong>of</strong>-mass energy, N : number <strong>of</strong> particles per bunch, σz :
rms bunch length, σx : horizontal rms beam size, σy : vertical rms beam
size, 〈Υ〉 : field strength parameter, δ B : energy loss by beamstrahlung,
nγ : number <strong>of</strong> photons per electron.
BEAMSTRAHLUNG
The field strength by the electron (positron) beam at the
collision point is approximately given by
Ne
≈
σz × max(σx, σy)
where the symbols are defined in Tab.1. The field ranges
from ∼ 500 Tesla in ILC, ∼ 10 4 Tesla in CLIC to ∼ 10 6
in plasma colliders (better to use magnetic field because
electric field is cancelled between electron and positron).
This is still low compared with the Schwinger field
(1)
BSch = ESch/c = 4.4 × 10 9 Tesla (2)
However, another Lorentz invariant quantity can be O(1):
Υ ≡ e
m3
(p µ Fµν) 2 = 2
3
¯hω C
E
= λeγ 2
ρ
B
= γ , (3)
BSch where pµ is the electron 4-momentum, m the electron
rest mass, Fµν the electromagnetic tensor, λe the Compton
wavelength, ¯hω C the critical energy <strong>of</strong> radiation, and
E = γmc 2 the electron energy. This parameter varies
along the bunch. Its average can be approximately expressed
by the beam parameters as
〈Υ〉 = 5 Nr
6
2 eγ
. (4)
ασz(σx + σy)

This is listed in Tab.1. The effect is sizable in ILC and is
dominant in colliders above a few TeV.
An example <strong>of</strong> the luminosity spectrum under strong
beamstrahlung is shown in Fig.1. It shows an extreme
case <strong>of</strong> a plasma collider. The peak at low energies comes
from the energy loss by multiple beamstrahlung. The spectrum
for ILC is by far clearly dominated by the high-energy
peak.
Figure 1: An extreme example <strong>of</strong> luminosity spectrum under
beamstrahlung and coherent pair creation. The parameter
Plasma1 is used.
COHERENT PAIR CREATION
When a high-energy photon (beamstrahlung in our case)
travels in an intense electromagnetic field, it can decay into
electron-positron pairs. The one that has the same sign <strong>of</strong>
charge as the oncoming beam is defected by a large angle
due to the Coulomb field and causes serious backgrounds
to the detector. The relevant Lorentz invariant quantity is
χ ≡ e
m3
(k µ Fµν) 2 = ω
m
B
B Sch
where kµ is the 4-momentum <strong>of</strong> the photon and ω its energy.
When Υ is O(1), χ can also be O(1). The beamstrahlung
and coherent pair creation come from the same
diagram seen in different channels as shown in Fig.2.
The spectrum (energy distribution <strong>of</strong> the pair particles)
is given by[3]
dW CP
dE+
= α m
√
3π
2
ω2 ∞
η
K1/3(η ′ )dη ′
E−
+
η = 2
3χ
ω 2
E+E−
+
E+
E+
E−
(5)
K2/3(η)
, E− = ω − E+, (6)
where α is the fine structure constant, E+(E−) the final
positron (electron) energy, Kν the modified Bessel function.
This spectrum (normalized to unity) is plotted in Fig.3
for various values <strong>of</strong> χ.
Figure 2: Beamstrahlung and Coherent Pair Creation. The
double solid line indicates electron in an external field.
Figure 3: Spectrum <strong>of</strong> the coherent pair creation.
There is another process, sometimes called ‘trident cascade’,
<strong>of</strong> creating pairs. The virtual photon associated with
an electron can create pairs under a strong field. This process
has been studied in early 1970’s[4]. When Υ is very
large (e.g., > 1000), the contribution <strong>of</strong> this process may
be even larger than the combination <strong>of</strong> beamstrahlung and
coherent pair creation.
Early studies on beamstrahlung and coherent pair creation
are reviewed in[5].
BEAM-BEAM DEPOLARIZATION
It is relatively easy to obtain polarized beams in linear
colliders than in ring colliders. The most important
source <strong>of</strong> depolarization comes from beam-beam interaction.
There are two mechanisms that causes depolarization,
namely the precession in magnetic field and the spin-flip
synchrotron radiation. Both <strong>of</strong> these processes are wellknown
except the correction <strong>of</strong> the precession formula under
strong field.
The relevant terms in the Thomas-BMT equation is
dS
dt
= e
mγ (γa + 1)B T × S (7)
where S is the spin vector (in the rest frame), B T the transverse
component <strong>of</strong> the magnetic field and a the coefficient
<strong>of</strong> the anomalous magnetic moment.

At high field a is nolonger a constant (a ≈ α/2π) but is
a function <strong>of</strong> Υ [6]:
∞
a(Υ) 2 xdx
=
a(0) Υ 0 (1 + x) 3
∞
x
sin t +
0 Υ
t3
dt (8)
3
= 1+12Υ 2
log 1 1 37
+ log 3−
Υ 2 12 +γ
, (Υ ≪ 1)
E
This is plotted in Fig.4. Since the depolarization is propor-
Figure 4: Anomalous magnetic moment <strong>of</strong> electron as a
function <strong>of</strong> Υ.
tional to a 2 , this correction already gives a sizable (welcome)
effect in ILC at 1TeV, and strongly suppresses the
depolarization by precession in CLIC at 3TeV. The total
depolarization through beam-beam interaction is a fraction
<strong>of</strong> percent in ILC and several percent in CLIC.
The equation (8) has not been experimentally confirmed
in spite the magnetic moments <strong>of</strong> electron and muon at low
field are known extremely accurately. It is quite uncertain
whether it can be measured in the linear collider envioronment
because <strong>of</strong> the complexity <strong>of</strong> beam-beam interaction.
GAMMA-GAMMA COLLIDER
As is shown in Fig.5, An electron-positron collider (in
electron-electron mode) can be converted to a gammagamma
collider by irradiating a laser just before the collision
point (less than 1cm) to get high energy backscattered
photons. The combination <strong>of</strong> longitudinally polarized electron
and circularly polarized laser can create photons with
small energy spread.
§¨©
¡ ¢£¤¥¦ ¡ ¢£¤¥¦
Figure 5: Gamma-Gamma collider scheme
There are three Lorentz invariant quantities made from
the 4-momentum <strong>of</strong> electron pµ and the field tensor Fµν <strong>of</strong>
laser (assume plane wave, wave number kµ): 1
among which there is a relation
Υ = e
m3
(p µ Fµν) 2 (9)
Λ =
2k · p
m2 ξ =
(10)
e
−A µ
Aµ
m
(11)
2Υ = ξΛ. (12)
(Aµ is the vector potential but ξ can be written in gauge
invariant form by using eq(12).)
The 2-dimensional parameter plane <strong>of</strong> Υ, ξ and Λ is
shown in Fig.6. 2 Λ is the parameter <strong>of</strong> the center-<strong>of</strong>-mass
energy <strong>of</strong> the Compton scattering and is a good parameter
in the region ξ ≪ 1, whereas Υ is better in the region
ξ ≫ 1 to describe strong fields.
Figure 6: Parameter plane <strong>of</strong> laser-electron interaction
For gamma-gamma colliders large values <strong>of</strong> Λ (short
laser wavelength) is preferred for obtaining higher energy
photons from the given electron energy. However, if Λ
is too large, the produced photons decay into e + e − pairs
in the same laser field. For this reason, we usually adopt
Λ < 2 + 2 √ 2 = 4.83.
A stronger laser (larger ξ) is preferred for obtaining more
photons but, when ξ is too large, non-linear QED effects
degrades the energy spectrum and polarization <strong>of</strong> produced
photons. Normally we choose ξ 2 < 0.5. Thus, the region
near the center <strong>of</strong> diagram in Fig.6 is chosen.
Fig.7 is an (old) example <strong>of</strong> the spectrum <strong>of</strong> photons for a
gamma-gamma collider. The parameters are : the electron
energy Ee = 250GeV, laser wavelength 1µm (corresponding
to Λ = 4.8), ξ 2 = 0.4. The lower (green) curve shows
1 Laser physicists denote ξ by a and accelerator physicists by K. Λ is
more <strong>of</strong>ten denoted by x.
2 Beamstrahlung is not a radiation in periodic field but is added
in the diagram for comparison by identifying the bunch length as
wavelength/(2π).

the spectrum <strong>of</strong> photons from primary electrons and the upper
(red) curve shows all the photons including those from
repeated Compton scattering. The parameters are chosen
so that the probability <strong>of</strong> Compton scattering for an electron
is about 1 to get high γ-γ luminosity. Hence multiple
scattering is inevitable, producing low energy electrons
which causes background.
Figure 7: An example <strong>of</strong> photon spectrum created by laser-
Compton scattering for a gamma-gamma collider
The simple Compton scattering would produce the maximum
photon energy ω = ΛEe/(1 + Λ)=207GeV. But
due to the large value <strong>of</strong> ξ, this energy is lowered to
ω = ΛEe/(1+Λ+ξ 2 )=194GeV and higher harmonic photons
ω = nΛEe/(1 + nΛ + ξ 2 ) are also produced (n =2
and 3 are visible). In the simulation the formulas for infinite
plane wave laser with the local laser intensity (adiabatic approximation)
is used. This means the value <strong>of</strong> ξ is varying
within the laser beam so that the peaks in Fig.7 are blurred.
Another strong field phenomenon related to gammagamma
colliders is the rotation <strong>of</strong> photon polarization in
a strong laser field. Some gamma-gamma experiments demand
linearly polarized photon. Using linearly polarized
laser, however, would produce blurred photon spectrum.
One possible idea is to produce photons by circularly polarized
laser and to rotate the polarization by another laser
which is linearly polarized.[7]. Circularly polarized laser
rotates the polarization plane <strong>of</strong> linearly polarized photon,
whereas linearly polarized laser interchanges linear and circular
polarization <strong>of</strong> photons. This phenomenon is effective
near the threshould <strong>of</strong> pair creation. Although this idea
does not seem to be practical for gamma-gamma application
but it is theoretically interesting.
SIMULATION TOOLS
In early stages <strong>of</strong> linear collider study a computer code
CAIN[8] was written to treat the beam deformation by
Coulomb field, beamstrahlung, coherent pair creation, and
polarization behavior (trident cascade has not been included
yet). The code GUINEA-PIG[9] has also been written
for the same purpose. The laser-electron interaction has
been included in CAIN in a later stage. The figures 1 and
7 are produced by CAIN. These codes are still evolving
according to the demands but their essential features have
been established long ago.
SUMMARY
The Beam-beam interaction in linear colliders is a place
where we expect various phenomena related to the quantum
electro-dynamics in intense fields. Its study started in
mid 1980’s and is thought to be well established theoretically
by now. Necessary simulation tools have also been
developed. Nonetheless we have to wait for the realization
<strong>of</strong> linear colliders for the experimental verifications.
Up to now these intese field effects are mostly unwanted
phenomena for the performance <strong>of</strong> linear colliders. We
hope we may find useful applications in the future such as
the positron production using the coherent pair creation.
REFERENCES
[1] ICFA-ICUIL Joint Task Force on Laser Acceleration
Meeting, Apr.8-10, 2010 at GSI, Darmstadt, Germany. The
task force report will be published soon (as <strong>of</strong> Jan.2011).
[2] A. A. Sokolov and I. M. Ternov, ‘Radiation from
Relativistic Electrons’, American Institute <strong>of</strong> Physics,
Translation Series, New York, 1986.
[3] V. N. Baier and V. M. Katkov, Sov. Phys. JETP
26(1968)854. W. Y. Tsai and T. Erber,
Phys. Rev. D10(1974)492.
[4] V. I. Ritus, Nucl. Phys. B44(1972)236. V. N. Baier,
V. M. Katkov and V. M. Strakhovenko, Soviet
J. Nucl. Phys. 14(1972)572.
[5] ‘Beam-Beam Phenomena in Linear Colliders’,
K. Yokoya and P. Chen, in Frontiers <strong>of</strong> Particle Beams:
Intensity Limitation. Lecture Notes in Physics 400,
Springer Verlag, (1991) page 414-445.
[6] V. N. Baier, private communication.
[7] G. L. Kotkin and V. G. Serbo,
Phys. Lett. B413(1997)122-129.
[8] CAIN:Conglomérat d’ABEL et d’Interactions
Non-Linéaires. P. Chen, G. Horton-Smith, T. Ohgaki,
A. W. Weidemann and K. Yokoya, Workshop on
Gamma-Gamma Colliders, Berkeley, CA, March 28-31,
1994. SLAC-PUB-6583, July 1994.
Nucl. Instr. Meth. A355(1995)107-110. Contact the present
author for the laset information.
[9] D. Schulte, consult the web page
http://www-sldnt.slac.stanford.edu/nlc/programs/
guinea_pig/Original%20GP%20Home.html
and
http://flc-mdi.lal.in2p3.fr/spip.php?rubrique40

Abstract
Second order QED processes and their radiative corrections
A. Hartin, DESY FLC, Notkestrasse 85, Hamburg, Germany
The effect <strong>of</strong> intense external fields on physics processes
can be taken into account exactly by performing the usual
S-matrix expansion in the bound interaction picture. The
effect so calculated is to predict resonant cross-sections
in second order processes. The Compton scattering in
such a framework is outlined. Resonant infinities in the
tree level process are mitigated by the electron self energy
and the vertex correction. The state <strong>of</strong> the calculation and
inclusion <strong>of</strong> these radiative corrections is discussed.
INTRODUCTION
The Dirac equation for fermions embedded in intense
electromagnetic plane wave fields can be solved exactly.
The wave function solutions include an infinite summation
over interactions with multiple external field photons and
display a quasi-energy level structure. Resonant transitions
between these quasi-energy levels is predicted for quantum
electrodynamical processes with a fermion propagator
and which therefore are at least second order. Transition
probabilities for these higher order processes can be
calculated in the bound interaction picture (BIP) in which
the fermion-external field states interact with free photon
states.
I review here the BIP Compton scattering, a second order
QED process in the Bound Interaction Picture which is
predicted to have multiple resonances in its cross-section.
The BIP Pair Production also displays similar behaviour
being related to the BIP Compton scattering via a crossing
symmetry [1]. The BIP radiative corrections required to
render these resonances finite are the BIP electron self
energy and the BIP vertex correction, the calculation <strong>of</strong>
which is outlined in this paper.
These second order processes and their predicted resonances
could be tested experimentally in the interaction
<strong>of</strong> two lasers - one at least being high intensity - and an
electron beam such as that provided by ATF2 at KEK.
Indeed the cross-section at resonance for second order
processes in the BIP is considerably larger for the 1st order
processes such as one photon pair production and photon
radiation [2].
QED IN THE BOUND INTERACTION
PICTURE
For quantum electrodynamical physics processes that
take place in the presence <strong>of</strong> an external potential A e the
Lagrangian density is written
LQED = − 1
4 F µν Fµν + ¯ ψ(i/∂ + eA + eA e − m)ψ (1)
If the external field is sufficiently strong it is desirable
to consider its effect exactly. This is achieved in the
BIP which considers the interaction <strong>of</strong> the bound fermionexternal
field states with free boson states in the usual time
evolution perturbation theory. The bound fermion-external
field states are obtained by solving the Dirac equation with
an external potential,
e ¯ ψ(i/∂ + A e − m)ψ = 0 (2)
Equation 2 can be solved exactly when the external potential
is a plane wave electromagnetic field. The solutions,
referred to as Volkov solutions [3] are a product <strong>of</strong> the normal
free fermion solution with an extra phase S(x) and a
magnetic moment term Ep(x),
Ψ V p (x) = Ep(x)u(p)
where Ep(x) =
1 − e /Ae
/k
e
2(kp)
iS(x)
(k·x)
and S(x) = −i
0
e(Aep) (kp) − e2Ae2
dφ
2(kp)
For second order processes in the BIP we also need the
fermion propagator in the external field, which turns out
to be the normal propagator sandwiched between Volkov
Ep(x) functions,
(3)
G e (x,x ′
d4p 1
) = Ep(x) Ēp(x
(2π) 4
/p−m+iǫ
′ ) (4)
These new elements can be included in Feynman diagrams
in the usual way (being represented by double

pf f
f
pi
p
x2
x1
k
ki
p
pi
Figure 1: The BIP Compton scattering.
straight lines). In effect, the Volkov Ep(x) functions can
be grouped together around diagram vertices (with incoming
momentum q and outgoing momentum p) in order to
”dress” them,
p
kf
ki
γ e µ(p,q) = Ēp(x)γµEq(x) (5)
COMPTON SCATTERING IN THE BOUND
INTERACTION PICTURE
The Volkov solutions to a fermion <strong>of</strong> momentum p embedded
in an external periodic field A e reveal an energy
level structure depending on the discrete number s <strong>of</strong> external
field photons k that interact with the fermion and given
by the dispersion relation [4],
(p ± sk) 2 = m 2 + |eA e | 2
For Compton scattering <strong>of</strong> a photon ki with such a bound
fermion (figure 1) resonant transitions are possible whenever
the intermediate virtual particle reaches the mass shell.
The kinematics for such transitions are given by the condition
that the propagator denominator is zero,
(6)
(pi + ki − sk) 2 = 2(pi · ki) − 2sk · (pi + ki) = 0 (7)
If the particles are collinear then the resonant condition
simplifies to the condition that the energy <strong>of</strong> the incoming
photon be an integer multiple <strong>of</strong> the external field photon
energy.
RADIATIVE CORRECTIONS
The resonant infinities in the tree level process must be
mitigated by including radiative corrections. The schema
is to calculate the same loop diagrams in the BIP and to
include them to all orders in a geometric series as in the
p
+
k’
p
p
p’
p
k’ p’
+ p + ...
Figure 2: The Corrected Bound fermion Propagator.
case <strong>of</strong> the normal interaction picture (figure 2).
The fermion self energy Σ e p has been calculated for the
case <strong>of</strong> a constant crossed electromagnetic field [5] and
a circularly polarised field [6]. The sum to all orders is
straightforward and the corrected propagator G e p(rc) is
G e p(rc) = G e p + G e pΣ e pG e p + G e pΣ e pG e pΣ e pG e p + ...
k’
d4p 1
= Ep(x)
(2π) 4
/p−Σ e Ēp(x
p− m + iǫ
′ ) (8)
There still remains the question <strong>of</strong> UV divergences in
the BIP self energy. In the literature, a term equivalent
to the normal interaction picture self energy is separated
<strong>of</strong>f and the usual regularization and renormalization
procedures are carried out. However it now appears that
the calculation can be carried out without the appearance
<strong>of</strong> UV divergences 1 .
We need also to include in the BIP Compton scattering,
all terms <strong>of</strong> the same order in the coupling constant in
the S-matrix expansion. One such term required is the
interference term between the BIP vertex correction and
the BIP photon radiation (figure 3)
The BIP vertex correction (figure 4) is a rather complicated
expression including three dressed vertices and infinite
integrations (constant crossed external field) or summations
(periodic external field) over contributions l,r,s
from external field photons at each vertex,
1 This will be the subject <strong>of</strong> a forthcoming paper by the author
2
↔ ×
Figure 3: A S-matrix interference term <strong>of</strong> the same order
as the BIP Compton scattering.
p
p’
~

p f
p
i
Figure 4: The BIP vertex correction.
k f
− ieΓ e µ = 2ie 2
∞
d
−∞
4p (2π) 4 dl dr ds δ4 (pi + lk − pf − kf)
• γ eν (pf,p ′ )
1
/p ′ − m γe µ(p ′ 1
,p)
/p−m γe ν(p,pi) 1
k ′2
where p ′ → p − kf + rk , k ′ → pi − p + sk
Equation 9 has been calculated for special kinematics in
which the radiated photon is parallel to the external field
wave vector. For such a case no UV divergence exists and
the resulting expression contains a term consistent with the
known expression for the anomalous magnetic moment in
a constant crossed field [7]. Work is continuing to extend
the calculation to the general case.
CONCLUSION
Physics with strong external fields such as those present
in intense laser-matter interactions, at the interaction
point <strong>of</strong> modern colliders or in an astrophysical setting, is
experimentally interesting and theoretically challenging.
Calculations which take into account the effect <strong>of</strong> the
external field exactly, are necessary. Such calculations are
performed in the BIP which, in effect, dress the vertices <strong>of</strong>
the corresponding Feynman diagram.
The transition rates for BIP processes containing propagators
indicate the presence <strong>of</strong> resonances which must be
corrected by the inclusion <strong>of</strong> BIP radiative corrections. To
date only the BIP electron self energy (and BIP vacuum
polarization) have been calculated. In order to correct
higher order tree level processes in the BIP, and to prove
that IR divergences cancel assuming a Bloch-Nordiseck
type pro<strong>of</strong>, the BIP vertex correction is also required.
(9)
These calculations and their corrections are underway
and much progress has been made. The ultimate aim is
a realistic calculation <strong>of</strong> BIP Compton scattering crosssection
at resonance and proposals for the experimental
detection <strong>of</strong> such predicted resonances.
REFERENCES
[1] Hartin A 2006 PhD thesis University <strong>of</strong> London
[2] Bamber C et al 1999 Phys Rev D 60(9) 092004
[3] Volkov D M 1935 Z Phys 94 250
[4] Zeldovich Y B 1967 Sov Phys JETP 24 1006
[5] Ritus V I 1972 Ann. Phys. D 69 555-582
[6] Becker W, Mitter H 1976 J Phys A 9(12) 2171
[7] Ritus V I 1970 Sov. Phys. JETP 30(6) 1181

STRONG FIELD DYNAMICS IN HEAVY-ION COLLISIONS
Abstract
In high-energy heavy-ion collisions, there appear two
different kinds <strong>of</strong> strong fields: Strong electromagnetic
fields and strong Yang-Mills (gluon) fields. I explain the
mechanisms how they appear in the collisions, the interesting
interplay between QED and QCD, and finally the
importance <strong>of</strong> the strong field dynamics in understanding
time evolution <strong>of</strong> the full collision events.
INTRODUCTION
The main goal <strong>of</strong> this talk is to convince you that the
high-energy heavy-ion collision (HIC) is the ideal place for
the study <strong>of</strong> strong-field physics. This is primarily because
they provide the “strongest” fields that human being can
create. Let us first overview how strong they are. Table 1
shows the comparison <strong>of</strong> magnetic fields realized in various
situations from weak to strong fields. The strongest steady
magnetic field on earth, B = 4.5 × 10 5 Gauss, is achieved
at National High Magnetic Field Laboratory in Florida [1],
but there are several situations in Nature which create much
stronger magnetic fields. Typical examples include a neutron
star which is a remnant <strong>of</strong> supernova explosion. The
magnetic field <strong>of</strong> a huge star is squeezed to a small region
after explosion to become strong B ∼ 10 12 Gauss.
However, it is still weaker than the so-called “critical”
magnetic field Bc = m 2 e/e = 4 × 10 13 Gauss, beyond
which the nonlinear QED effects must be fully taken into
account. The corresponding electric field Ec = m 2 e/e is
called the “Schwinger field” beyond which the e + e − pair
creation from the vacuum becomes possible [2]. Recently,
it has been recognized that some <strong>of</strong> the neutron stars, called
“magnetars”, would have much stronger magnetic fields.
The surface magnetic fields are estimated as ∼ 10 15 Gauss,
well beyond the critical value [3, 4]. So far, this is the
strongest static magnetic fields in Nature.
Table 1: Comparison <strong>of</strong> magnetic fields (Gauss)
Strength Realized as
0.6 Earth’s magnetic field
100 A typical hand-held magnet
8.3×10 4 Superconducting magnets in LHC
4.5×10 5 Strongest steady magnetic field [1]
∼ 10 12 Surface field <strong>of</strong> neutron stars
(4×10 13 Critical magnetic field <strong>of</strong> electrons)
∼ 10 15 Surface field <strong>of</strong> magnetars
∼ 10 17 Noncentral heavy-ion coll. at RHIC
∼ 10 18 Noncentral heavy-ion coll. at LHC
∗ e-mail: kazunori.itakura@kek.jp
K. Itakura ∗
Theory Center, IPNS, KEK, Japan
Surprisingly, high-energy HIC’s provide much stronger
magnetic fields. They are created in noncentral collisions
and the maximum strength amounts to ∼ 10 17 Gauss at
RHIC in BNL and ∼ 10 18 Gauss at LHC in CERN, far
above the critical value 4 × 10 13 Gauss. Since the origin
<strong>of</strong> magnetic fields is the fast moving nuclei with electric
charges, such strong fields last only for a very short period
(typically only during the passage <strong>of</strong> two colliding nuclei).
However, with the strongest magnetic fields far beyond the
critical value, we expect many interesting phenomena related
to nonlinear QED to occur, as I will explain later. In
particular, it is quite interesting to see how the strong magnetic
field affects the Quark Gluon Plasma (QGP) which
has also a very short life time.
On the other hand, HIC’s generate another strong
fields: “color” electromagnetic fields described by Quantum
Chromodynamics (QCD). The main motivation <strong>of</strong>
HIC’s is to create the QGP by liberating subatomic degrees
<strong>of</strong> freedom, quarks and gluons, from inside <strong>of</strong> nucleons.
QCD is the fundamental theory for the dynamics <strong>of</strong> quarks
and gluons and is the non-Abelian gauge theory with
“color” SU(3) symmetry. Therefore, there exist “color”
electromagnetic fields which are non-Abelain analogs <strong>of</strong>
the ordinary electromagnetic fields. High-energy HIC’s can
create very strong color electromagnetic fields. Since one
cannot directly compare the strength <strong>of</strong> color fields with
that <strong>of</strong> ordinary electromagnetic fields, let us compare them
in unit <strong>of</strong> energy. Table 2 is the comparison <strong>of</strong> the ordinary
and color electromagnetic fields. For the electromagnetic
fields we show √ eB in unit <strong>of</strong> MeV, and for color electromagnetic
fields √ gB in the same energy scale where g
is the coupling strength and B is the color magnetic fields.
Notice that the critical magnetic field in QED is determined
by the electron mass √ eBc = me, but the corresponding
field in QCD will be determined by quark mass mq. Still,
the strength √ gB ∼ 1 GeV at RHIC is well beyond the
critical value mq ∼ a few MeV, suggesting the importance
<strong>of</strong> Schwinger mechanism <strong>of</strong> q¯q pairs. Also we expect that
the dynamics <strong>of</strong> such strong color fields is crucial for the
early time evolution <strong>of</strong> the created matter to the QGP.
Table 2: √ eB for electromagnetic fields (EM) and √ gB
for Yang-Mills fields (YM) in unit <strong>of</strong> energy (MeV)
Strength Realized as
0.5 (= me) Critical mag. field <strong>of</strong> electrons (EM)
∼ 2 − 3 Surface field <strong>of</strong> magnetars (EM)
∼ 10 2 (∼ mπ) Noncentral HIC at RHIC (EM)
∼ 4 × 10 2 Noncentral HIC at LHC (EM)
∼ 10 3 (∼ Qs) Color magnetic fields at RHIC (YM)
(2 − 3)×10 3 Color magnetic fields at LHC (YM)

Figure 1: (Left) Time evolution <strong>of</strong> the matter created in high-energy heavy-ion collisions. (Right) Noncentral collisions.
HIGH-ENERGY HEAVY-ION
COLLISIONS
As already mentioned, the main motivation <strong>of</strong> the HIC’s
is to create the QGP, a local equilibrium matter made <strong>of</strong>
quarks and gluons. In order to throw gigantic kinetic energies
into a small but finite volume, we need to collide
‘large’ objects, namely two heavy ions (bare nuclei without
electrons). Thus, the spatial extent <strong>of</strong> the created state
is, initially, <strong>of</strong> the order <strong>of</strong> that <strong>of</strong> colliding nuclei (e.g., for
Au nuclei, about 10 fm = 10 −14 m). If the energy density
is high enough to reach a very high temperature beyond
the critical value Tc ∼ 170 MeV, one can naively expect
that the QGP will be formed. In reality, however, the created
state with high-energy densities will show quite nontrivial
time evolution: It will quickly change into a hightemperature
QGP through interactions between quarks and
gluons. Then, the QGP will cool down to hadronic matter
as it rapidly expands (see Fig. 1, left). All these processes
take place during a very short time, and the life time <strong>of</strong>
QGP is, at most, <strong>of</strong> the order <strong>of</strong> 10 fm/c = 10 −22 sec.
In order to study the properties <strong>of</strong> QGP, we need to extract
information <strong>of</strong> QGP from the huge number <strong>of</strong> final
hadrons. Since particle production can, in principle, occur
at every stage <strong>of</strong> evolution, it is quite important to understand
the evolution history <strong>of</strong> the created matter in HIC’s.
In particular, the formation <strong>of</strong> QGP is one <strong>of</strong> the most difficult
problems in QCD (or maybe in theoretical physics) because
it requires non-perturbative description <strong>of</strong> non-linear
phenomena in non-equilibrium states. However, many people
believe that the strong field dynamics should be relevant
at least for the early time evolution. In fact, as already mentioned,
there emerge two different kinds <strong>of</strong> strong fields in
HIC’s, and the “color” electromagnetic fields are expected
to play important roles for the transition towards QGP. The
strong color electronic and magnetic fields give rise to the
quark-antiquark pair production (Schwinger mechanism)
and Nielsen-Olesen instability, respectively. We expect
both <strong>of</strong> them will contribute to drive the system towards
thermalization. On the other hand, ordinary magnetic fields
indeed have a significant influence on the created matter,
but would be irrelevant for thermalization, because it does
not appear in the central collision.
STRONG MAGNETIC FIELDS IN
NONCENTRAL HEAVY-ION COLLISIONS
How do they appear?
Note that the atomic nuclei used in HIC’s have large
electric charges 1 Ze and they are moving at very fast speed
close to the speed <strong>of</strong> light. Then, the Coulomb field (electric
field) <strong>of</strong> each nucleus is highly Lorentz contracted to a
strong spike. Time variation <strong>of</strong> the electric fields induces
magnetic fields <strong>of</strong> the same strength. For example, consider
a point charge moving on the z axis at rapidity (velocity)
Y . Then, the magnetic field at the position ⃗x is given by
e ⃗ B(⃗x) = Zα EM
−(⃗x⊥ × ⃗ez) sinh Y
[(⃗x⊥) 2 + (t sinh Y − z cosh Y ) 2 ,
3/2
]
which has a large enhancement factor sinh Y . One can
roughly estimate the magnetic field in noncentral HIC’s
(b ̸= 0) by simply summing the magnetic fields given
above assuming the trajectories <strong>of</strong> two colliding nuclei
shown in Fig. 1, right. Then, one finds the maximum
strength √ eB ∼ 100 MeV in noncentral Au-Au collisions
at RHIC ( √ sNN = 200 GeV) 2 when the impact parameter
b ∼ 10 fm [5]. In fact, such a crude estimate is numerically
confirmed by the UrQMD simulation [6]. As already
commented, this is quite a strong magnetic field, beyond 3
the “critical” value √ eBc = me = 0.5 MeV. Since this
magnetic field arises only from the kinematical configuration
<strong>of</strong> two charges, it becomes strongest when two charges
come closest, and decays rapidly as they recede from each
other. Typically, it lasts only for a few fm/c. However, it is
argued that the life time becomes longer in the presence <strong>of</strong>
the QGP, because the QGP has a large electric conductivity
[7]. If this is indeed the case, life time <strong>of</strong> the strong magnetic
field could be <strong>of</strong> the same order <strong>of</strong> that <strong>of</strong> the QGP,
namely, about 10 fm/c.
1 Z = 79 (Au) at RHIC, 82 (Pb) at LHC and we take e > 0.
2 √ sNN is the center-<strong>of</strong>-mass energy per nucleon-nucleon collision.
The total energy <strong>of</strong> the Au-Au collision is A × √ sNN with A = 197
being the mass number <strong>of</strong> Au.
3 It makes sense to discuss the magnetic field beyond the critical value,
because, unlike the electric field, the vacuum does not break down. However,
perturbative calculation ‘breaks down’ because the insertion <strong>of</strong> n
external magnetic fields in Feynman diagrams contributes as (eB/m 2 e )n
which becomes order 1 for B > Bc = m 2 e/e [3].

Possible effects <strong>of</strong> the strong magnetic fields
Even if the magnetic field lasts only for a short period,
it is extraordinarily strong and should have a large impact
on dynamics. What one would immediately expect
is the nonlinear QED effect. This has been discussed for a
long time in “very peripheral” collisions b > 2RA (RA
is the nuclear radius) [8]. In this case, the impact parameter
b is too large for two nuclei to touch each other,
but multiphoton exchange will occur due to large electric
charges <strong>of</strong> the nuclei that compensate the smallness <strong>of</strong>
α EM = e 2 /4π = 1/137. On the other hand, in the “noncentral”
collisions 0 < b < 2RA, two nuclei indeed touch
each other, and the matter in the reaction region will turn
into a QGP if the collision energy is large enough. Thus,
in this case, we can study properties <strong>of</strong> a QGP in a strong
magnetic field as shown in Fig. 1, right (the orange ellipsoid
represents a QGP). Below we discuss possible observable
effects <strong>of</strong> the strong magnetic field on the QGP.
Gluon synchrotron radiation
First <strong>of</strong> all, notice that quarks/antiquarks have electric
charges (e.g., eu = (2/3)e and ed = −(1/3)e for up
and down quarks), while gluons don’t. Thus, the strong
magnetic field primarily affects quarks and antiquarks and
induces synchrotron radiations [7]. Because <strong>of</strong> the largeness
<strong>of</strong> the QCD coupling, αs ≡ g 2 /4π ≫ αEM, radiations
are predominantly due to gluons, instead <strong>of</strong> photons
as usual. Therefore, this phenomenon is a typical interplay
between QED and QCD. One <strong>of</strong> the observable effects
<strong>of</strong> the gluon synchrotron radiations is the energy loss <strong>of</strong>
quarks and antiquarks. Usually, energy loss <strong>of</strong> fast moving
quarks/antiquarks is assumed to be due to interaction with a
hot medium. So, the synchrotron radiation is considered as
another (less familiar) source <strong>of</strong> the energy loss. However,
this is not just a correction to the ordinary contribution: In
fact, it turned out that it is large enough to be comparable
with the experimentally measured value [7]. For example,
the energy loss per unit length due to synchrotron
radiation is −∆E/ℓ ∼ 0.2 ÷ 0.35 GeV/fm at RHIC and
−∆E/ℓ ∼ 1.5 ÷ 2 GeV/fm at LHC, both <strong>of</strong> which are for
up quarks with transverse momentum p⊥ = 10 ÷ 20 GeV.
Another possible observable effects is the angular distribution
specific to the synchrotron radiation, which is also
discussed in Ref. [7].
Photon decay and photon splitting
Other nonlinear QED effects include photon decay into
fermion pairs and photon splitting. Note that several different
kinds <strong>of</strong> photons are produced in HIC’s. It is quite
important to identify the origin <strong>of</strong> photons to understand
the dynamics <strong>of</strong> HIC’s: Direct photons are produced by
hard collisions at the moment <strong>of</strong> HIC’s and thus carry
the earliest-time information before the formation <strong>of</strong> QGP.
Thermal photons are produced from the QGP. Finally, decay
photons are produced by the decay <strong>of</strong>, say, π 0 at the last
stage <strong>of</strong> HIC’s. So far, all the analyses in relation to pho-
tons assume that the produced photons do not interact with
the QGP and simply carry the information <strong>of</strong> the matter
when they are emitted. However, in a strong magnetic field,
this assumption is no longer true. In order to correctly extract
the information <strong>of</strong> the evolving matter, we have to take
into account the effects <strong>of</strong> strong magnetic fields which will
be present until the end <strong>of</strong> the QGP.
In the vacuum, a real photon cannot decay into either a
lepton pair or two photons. However, in a strong magnetic
field, both become possible because charged fermions are
‘dressed’ by the magnetic field. In the vacuum, it is known
that a fermion loop with odd number <strong>of</strong> photons attached
is zero (Furry’s theorem). In the magnetic field, however,
it is not zero because the fermion is ‘dressed’ and includes
many external lines <strong>of</strong> the magnetic field. Hence these are
both characteristic phenomena <strong>of</strong> the nonlinear QED.
The cross section <strong>of</strong> photon decay into a fermion pair
is easily obtained by changing kinematical variables in the
results <strong>of</strong> the synchrotron radiation, but in the present case,
the e + e − pair production (instead <strong>of</strong> a q¯q pair) is the dominant
process. Decay rate <strong>of</strong> a real photon in HIC’s including
several channels (γ → ℓ + ℓ − or q¯q) was computed in
Ref. [9]. It was also pointed out that such effects would
generate an asymmetry <strong>of</strong> photon emission with respect
to the reaction plane (defined by two colliding nuclei, see
Fig. 1, right). If a photon is produced in the direction close
to the vertical axis <strong>of</strong> the reaction plane, the decay rate is
large. Thus, we will see less photons in the vertical directions,
which generates asymmetry to be measured as elliptic
flow. The calculation was done in a constant homogeneous
magnetic field, but the asymmetry will be enhanced
if one includes the inhomogeneity <strong>of</strong> the magnetic field as
shown in Fig. 1, right. On the other hand, so far, there
is no calculation about the photon splitting in HIC’s. One
naively expects that this can be measured as anomalous enhancement
<strong>of</strong> the photon spectrum in the region <strong>of</strong> smaller
energies, because the energy <strong>of</strong> the parent photon is shared
by two s<strong>of</strong>ter photons.
Chiral magnetic effects
The last example is the chiral magnetic effect [5, 10].
This is not the nonlinear QED effect, but shows an interesting
interplay between QED and QCD. First <strong>of</strong> all, let us
recall that quarks/antiquarks have handedness. For example,
the spin <strong>of</strong> a right-handed quark is parallel to its momentum,
while that <strong>of</strong> a left-handed quark is anti-parallel.
The chirality N5 <strong>of</strong> a matter is defined by the difference
between the numbers <strong>of</strong> right- and left-handed particles:
(
) (
)
N5 ≡ N(qR) + N(¯qR) − N(qL) + N(¯qL) ,
where N(qR) represents the number <strong>of</strong> right-handed
quarks, and similarly for the others. If a strong magnetic
field is present, spins <strong>of</strong> quarks point to the direction <strong>of</strong> the
magnetic field, irrespectively <strong>of</strong> their handedness. When
the numbers <strong>of</strong> left- and right-handed quarks are the same,
no net current is generated in that direction because the

momenta <strong>of</strong> left and right quarks are in the opposite directions.
However, if an excitation <strong>of</strong> the Yang-Mills field
with a topological number change occurs, the handedness
<strong>of</strong> quarks, and consequently the chirality <strong>of</strong> the matter will
change ∆N5 = N5(t = ∞) − N5(t = −∞) = −2Q with
Q being the topological number change. This means that a
net charge current will flow in the direction <strong>of</strong> the magnetic
field. In HIC’s, as I will discuss in the next section, there
appears a nontrivial Yang-Mills configuration with nonzero
topological charge. Therefore, if the strong magnetic field
and the topological number changing Yang-Mills field are
both present, then an electric charge current will flow perpendicular
to the reaction plane, generating asymmetry <strong>of</strong>
charge distribution. The strong magnetic field works to effectively
align the spin. More details are discussed in the
talk by K. Fukushima [10].
STRONG COLOR-ELECTROMAGNETIC
FIELDS: CGC AND GLASMA
Now let us turn to the color electromagnetic field in HIC
that originates from a huge number <strong>of</strong> gluons inherent in
the nuclei before collision. Below, I explain first how such
a state with many gluons (called ‘Color Glass Condensate’,
CGC) appears at high energies, then how the information
<strong>of</strong> CGC is transferred to the strong color fields created in
HIC (called ‘glasma’), and finally how the glasma evolves
towards the QGP.
From CGC to glasma: How do they appear?
What is CGC?
Consider one nucleus that is moving very fast in the z
direction. A nucleus is made <strong>of</strong> nucleons, and they are further
made <strong>of</strong> three quarks. Such a ‘valence’ picture works
well only at low energies or when we use ‘low resolution’
probes. On the other hand, if one increases the scattering
energy, one will instead see a state with a huge number <strong>of</strong>
gluons. These gluons are emitted either directly by the valence
quarks or successively by (already emitted) gluons
(see Fig. 2). Such a highly dense gluonic state is called the
CGC (see Ref. [11] for a review), and is indeed observed
experimentally in the electron deep inelastic scattering <strong>of</strong>f
a proton. It is physically important to distinguish the roles
played by large and small x constituents (x is the fraction <strong>of</strong>
momentum carried by a constituent). Large x constituents
(valence quarks and gluons having large momentum fraction)
are distributed on a Lorentz-contracted nucleus and
their motion is very slow compared to the collision time
scale. Thus we treat them altogether as a static color source
on a transverse disk ρ a (x⊥). We also assume that ρ a can
be taken as random reflecting the unpredictable configuration
<strong>of</strong> colored constituents at the moment <strong>of</strong> collision. On
the other hand, with increasing energy (corresponding to
going to smaller x), gluons with smaller x are successively
created. Then, small x constituents (mostly gluons) are collectively
described as a coherent radiation field A a µ created
Figure 2: Emergence <strong>of</strong> CGC at high energies. With increasing
energies (from left to right), multi-gluon production
occurs, which eventually leads to high-density saturated
gluon matter, CGC.
by the color source ρ a . Namely, we treat
(DνF νµ ) a = J µ , J µ = δ µ+ δ(x − )ρ a (x⊥) , (1)
where J µ is the current induced by a fast-moving color
charge distribution ρ a (x⊥) (its trajectory is taken as x − =
(z −vt)/ √ 2 = 0 when v ≃ c) and F a µν = ∂µA a ν −∂νA a µ −
gf abc A b µA b ν is the field strength. The coherent gluon field
A a µ is a non-Abelian analog <strong>of</strong> the Weizsäcker-Williams
field (or the boosted Coulomb field) <strong>of</strong> a moving electric
charge. Hence, the colliding nuclei at high energies are
necessarily accompanied by strong gluon fields.
Most <strong>of</strong> gluons in CGC have relatively large transverse
momentum called the saturation momentum, Qs ≫ ΛQCD
which corresponds to (the inverse <strong>of</strong>) a typical transverse
‘size’ <strong>of</strong> gluons when they fill up the transverse nuclear
disk (see Fig. 2). One can compute in QCD the energy (or
x) and atomic mass number A dependences <strong>of</strong> Qs as
Q 2 s(x, A) ∝ A 1/3 (1/x) λ , λ ≃ 0.3 , (2)
which is indeed consistent with the x and A dependences <strong>of</strong>
experimentally determined Qs. One expects Q 2 s ∼ 1 GeV
at RHIC ( √ sNN = 200 GeV, Au), while it increases by a
factor 3 at LHC ( √ sNN = 5.5 TeV, Pb). Note also that
1/Qs corresponds to the correlation length on the trans-
verse disk with the correlation <strong>of</strong> two gluons roughly given
by ∼ e−Q2s r2 ⊥. This means that the color field is regarded
as correlated/ordered within this correlation length.
One can roughly estimate the field strength <strong>of</strong> the CGC.
When the CGC is realized, nonlinearity in the Yang-Mills
theory is crucial. For example, the field strength F a µν contains
both ∂A and gAA terms, and these two become <strong>of</strong>
the same magnitude ∂A ∼ gAA in the CGC. Since the
typical momentum scale in CGC is given by Qs, one finds
A ∼ Qs/g that yields very strong color electric (E) and
magnetic (B) fields: E, B ∼ Q2 s/g.

Figure 3: Flux tube structure <strong>of</strong> the glasma
What is glasma?
It is now obvious that the high-energy HIC should be described
as a collision <strong>of</strong> two CGC’s. In the center-<strong>of</strong>-mass
frame (see Fig. 1, left), the current in eq. (1) is replaced by
J µ = δ µ+ δ(x − )ρ1(x⊥) + δ µ− δ(x + )ρ2(x⊥) with ρ1 (ρ2)
being a color source <strong>of</strong> the right (left) moving nucleus 1 (2).
After the collision at t = z = 0, two nuclei receed from
each other still at very high speed, but there appears a highenergy-density
matter in between these two. This matter
is predominantly made <strong>of</strong> gluons reflecting the CGC state
before collision and we expect it to evolve towards QGP.
This transitional state is now called the ‘glasma’ (= glass
+ plasma) [12]. We describe the very early stage <strong>of</strong> time
evolution <strong>of</strong> the glasma by solving source free Yang-Mills
equations in the forward light cone, with the initial condition
specified by the CGC fields <strong>of</strong> each nucleus. In short,
the glasma is a weakly-coupled 4 strong Yang-Mills field
that exhibits quite nontrivial time evolution towards QGP
due to nonlinear interactions.
Flux tube structure in glasma
Let us discuss a characteristic structure <strong>of</strong> the glasma.
Before the collision, each CGC has purely transverse 5 E i
and B i that are orthogonal to each other E · B = E⊥ · B⊥ =
0. However, just after the collision, the field strength suddenly
becomes purely longitudinal [12]. Indeed, the zcomponents
at t = 0 + are given by
E z | τ=0 + = −ig[α i 1, α i 2], B z | τ=0 + = igϵij[α i 1, α j
2 ] , (3)
with α1,2 being the CGC fields in matrix representation
αi = α a i T a , while all the transverse components are vanishing.
Notice that the gluon field just after the collision is
completely determined by the CGC fields. In other words,
CGC provides the initial conditions <strong>of</strong> HIC’s. This also
implies that the glasma field is initially very strong and the
longitudinal color electromagnetic fields E z and B z are <strong>of</strong>
the order <strong>of</strong> Q 2 s/g, the same order as those <strong>of</strong> the CGC
fields. Some comments are in order:
4 Recall that most <strong>of</strong> the gluons in CGC have momenta around Qs,
and Qs increases with increasing energy. Thus, one can treat the CGC in
weak-coupling technique αs(Qs) ≪ 1. This is true for the glasma, too.
5 Only the x and y-components are nonzero, i.e., E i = (E x , E y , 0).
Figure 4: Dynamics <strong>of</strong> color flux tubes.
• In the high-energy limit, there is no z-dependence 6 in
the longitudinal fields E z and B z , reflecting the fact
that the CGC is contracted to an infinitly thin disk.
• The longitudinal fields are correlated on the transverse
plane within the length scale 1/Qs, reflecting the random
configuration <strong>of</strong> the color source on the disk.
• Since both E z and B z are in general nonzero, the product
is nonvanishing E · B ̸= 0, suggesting the (local)
emergence <strong>of</strong> nonzero topological charge.
The first two imply that the glasma has a flux tube structure
as shown in Fig. 3. Unlike the color flux tube connecting
a quark and an antiquark, the glasma flux tube can have a
magnetic field in it. In fact, even a purely magnetic flux
tube is possible if one takes appropriate color structure in
eq. (3). Of course, a purely electric flux tube is also possible.
The third comment is related to the chiral magnetic effects
discussed in the previous section. The glasma allows
local fluctuations <strong>of</strong> the topological charge, which could be
measured with the help <strong>of</strong> strong magnetic fields.
As far as the dynamics <strong>of</strong> the fields at the classical level
is concerned, both the electric and magnetic flux tubes
show the same behavior because the electric-magnetic duality
holds in the forward light-cone where there is no color
charge [13] (see also [14]). The flux tube expands rapidly
towards transverse directions just like the fields in the ordinary
electrodynamics. This is because the glasma is a
perturbative object without the effects <strong>of</strong> confinement. On
the other hand, behaviors <strong>of</strong> fluctuations around the classical
configuration are drastically different in the electric and
magnetic flux tubes, as we will see below.
Towards QGP
If the initial glasma field has no z dependence as ideally
realized in the high-energy limit (see the first comment
above), it never acquires nontrivial pz dependences
after all. This means that the glasma cannot reach thermal
equilibrium (even isotropization) which does not have
any preference on spatial directions. This is a serious problem
in the CGC-glasma description <strong>of</strong> the heavy-ion collision.
This problem has not been fully resolved, but recently
6 Precisely, this should be the η dependence where η is the coordinate
rapidity, but physically it is the same as the z dependence.

many interesting phenomena were found in the dynamics
<strong>of</strong> the glasma, which we expect relevant for thermalization.
In particular, it turned out that the behaviors <strong>of</strong> fluctuations
in the presence <strong>of</strong> electric or magnetic flux tubes are very
important. Note that there always appear quantum fluctuations
even though the glasma field is constant in the zdirection
at the classical level.
Magnetic flux tube: Nielsen-Olesen instability
Consider small fluctuations in a single magnetic flux
tube [13, 14, 15]. One can simplify the situation by considering
fluctuations in a constant magnetic field in an expanding
coordinate system. This problem has been discussed
in the Cartesian coordinates long time ago, and it is
known that the constant color magnetic field undergoes the
Nielsen-Olesen instability[16]. This is indeed the case even
in an expanding coordinate system (see Fig. 4). Fluctuations
a(x⊥, z) with nontrivial z-dependence can grow exponentially
due to nonlinear interactions in the Yang-Mills
theory:
√
gBz τ
a(x⊥, z) ∝ e , (4)
where τ is the propertime and B z is the strong color magnetic
field in a flux tube B z ∼ Q 2 s/g. Since Qs is large
at high energy, the time scale for the fluctuation to grow
is very rapid 1/ √ gB z ∼ 1/Qs. Obviously, this kind
<strong>of</strong> fluctuation must contribute to drive the system towards
isotropization. Note that this picture is consistent with the
result <strong>of</strong> numerical simulations [17]. The Nielsen-Olesen
instability has been reexamined in a box (in the Cartesian
coordinates) with the initial condition similar to the
CGC [18], where a secondary instability was found to occur
as a consequence <strong>of</strong> the enhanced fluctuations.
Electric flux tube: Schwinger mechanism
A completely different phenomenon occurs in an electric
flux tube. Recall that gluons are massless, and that
light quarks have small masses compared to the strength
<strong>of</strong> the color electric field in the glasma √ gE z ∼ Qs.
Then, productions <strong>of</strong> gluons or q¯q pairs are possible in the
presence <strong>of</strong> a glasma field, which is the QCD version <strong>of</strong>
the Schwinger mechanism in QED (see Fig. 4). In fact,
Schwinger mechanism in QCD has been discussed as one
<strong>of</strong> the plausible mechanisms for thermalization in HIC’s.
But recently, this problem acquires renewed interests in
view <strong>of</strong> the glasma. As discussed in [19, 20], this effects
could contribute to thermalization <strong>of</strong> the glasma since the
created charged particles will be accelerated in the color
electric field to obtain nontrivial pz dependence. Once
pairs are created, the external color electric field will be
screened. It is possible to include such kind <strong>of</strong> ‘backreaction’
into the calculation to see the time evolution <strong>of</strong> the
whole system made <strong>of</strong> particles and the background field
[19]. It is also interesting to evaluate the Schwinger mechanism
in the presence <strong>of</strong> magnetic field. In this case, the
lowest Landau level <strong>of</strong> quarks become ‘massless’ and we
expect large enhancement <strong>of</strong> quark pair productions [21].
SUMMARY
In this talk, I discussed the importance and interesting aspects
<strong>of</strong> the strong field dynamics in HIC’s. There are two
different strong fields, i.e., strong electromagnetic fields
and strong Yang-Mills fields, and we can enjoy rich phenomena
caused by them, including the interplay between
these two. I hope many people get interested in this subject,
and participate in the reaseach.
The topics discussed in my talk are already broad, but in
fact there are still many problems left unsolved in relation
to the physics <strong>of</strong> HIC’s. One <strong>of</strong> the important problems
which I did not touch in this talk is the effects <strong>of</strong> ordinary
magnetic fields on the phase transition dynamics in
QCD. It is discussed that the chiral symmetry breaking is
affected by the external magnetic field. Also important is
the application <strong>of</strong> recent development on the QED cascade
discussed in [22] to QCD. This would be one <strong>of</strong> the ideal
subjects for the collaboration between different fields.
REFERENCES
[1] Follow the link “45 tesla” in the “World Records” page:
http://www.magnet.fsu.edu/mediacenter/factsheets/records.html
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[4] K. Makishima, in these proceedings.
[5] D. E. Kharzeev, L. D. McLerran and H. J. Warringa, Nucl.
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[9] K. Tuchin, arXiv:1008.1604 [nucl-th].
[10] K. Fukushima, in these proceedings.
[11] For a recent review, see F. Gelis, E. Iancu, J. Jalilian-
Marian and R. Venugopalan, “The Color Glass Condensate,”
arXiv:1002.0333 [hep-ph].
[12] T. Lappi and L. McLerran, Nucl. Phys. A772 (2006) 200.
[13] H. Fujii and K. Itakura, Nucl. Phys. A809 (2008) 88.
[14] H. Fujii, in these proceedings.
[15] A. Iwazaki, Prog. Theor. Phys. 121 (2009) 809
[arXiv:0803.0188 [hep-ph]].
[16] N. K. Nielsen and P. Olesen, Nucl. Phys. B144 (1978) 376,
S. J. Chang and N. Weiss, Phys. Rev. D20 (1979) 869.
[17] P. Romatschke and R. Venugopalan, Phys. Rev. Lett. 96
(2006) 062302; Phys. Rev. D 74 (2006) 045011.
[18] H. Fujii, K. Itakura and A. Iwazaki, Nucl. Phys. A 828
(2009) 178 [arXiv:0903.2930 [hep-ph]].
[19] N. Tanji, in these proceedings.
[20] A. Iwazaki, in these proceedings.
[21] Y. Hidaka, in these proceedings.
[22] H. Ruhl and N. Elkina, in these proceedings.

Abstract
Yoctosecond photon pulse generation in heavy ion collisions
Heavy ion collisions at RHIC and at the LHC can create
the quark-gluon plasma, a state <strong>of</strong> matter at very high
temperatures. Among a plethora <strong>of</strong> particles that are produced
in these collisions, also light is emitted throughout
the evolution <strong>of</strong> the plasma.
In this talk, the properties <strong>of</strong> this light are discussed and
related to recent efforts towards shorter and more energetic
photon pulses in laser physics. In particular, the time evolution
<strong>of</strong> high-energy photons is studied. These photons
originate from Compton scattering <strong>of</strong> gluons and quarkantiquark
annihilation in the plasma. Due to the internal
dynamics <strong>of</strong> the plasma, double pulses at the yoctosecond
time scale can be generated under certain conditions. Such
double pulses may be utilized for novel pump-probe experiments
at nuclear time scales.
INTRODUCTION
The year 2010 marks the fiftieth anniversary <strong>of</strong> the invention
<strong>of</strong> the laser. Since its invention, not only did the
intensity steadily increase, but also the pulse duration became
shorter and shorter. In fact, pulse duration and intensity
<strong>of</strong> lasers (or derived coherent radiation bursts) turn
out to be correlated over a large range <strong>of</strong> energies and time
scales [1]. This observation provides a good motivation to
present at a conference on Physics in Intense Fields (PIF
2010) a study <strong>of</strong> the shortest possible light flashes that can
be produced in experiments, which is heavy ion collisions
that produce the quark-gluon plasma (QGP) [2].
A few selected milestones <strong>of</strong> the development <strong>of</strong> short
laser pulses are depicted in Fig. 1. Each <strong>of</strong> the time scales
has enabled to access new systems: At the picosecond
to femtosecond timescale, femtochemistry allows for the
time-resolved study <strong>of</strong> chemical reactions [3]. For pumpprobe
spectroscopy, it is essential to have two short pulses
in close succession: The first laser pulse triggers a chemical
reaction, which may involve an excited state and various
short-lived transitions, while the second pulse takes a
snapshot <strong>of</strong> the intermediate state. By varying the interval
between the two pulses, the time-evolution <strong>of</strong> a chemical
reaction can be studied.
At shorter timescales, attosecond science is the window
to capturing electron motion in molecules and atoms [4].
High-order harmonics <strong>of</strong> femtosecond laser radiation have
been shown to be sources <strong>of</strong> trains <strong>of</strong> attosecond extremeultraviolet
pulses [5] that can be used to produce single
attosecond s<strong>of</strong>t X-ray pulses [6]. For example, such single
attosecond X-ray bursts [7] have applications in molec-
∗ ipp@hep.itp.tuwien.ac.at
A. Ipp ∗ , Vienna University <strong>of</strong> Technology, Austria
Pulse duration
ns
ps
fs
as
zs
ys
Nd:glass
CW Dye
CPM
Compression
HHG
Ti:sapphire
QGP
1960 1970 1980 1990 2000 2010 2020
Figure 1: History <strong>of</strong> laser pulse duration. Marked are
selected milestones <strong>of</strong> technologies that allowed to decrease
the laser pulse duration, like Neodymium glass laser
(Nd:glass), Continuous Wave Dye laser (CW Dye), Colliding
Pulse-Mode locked dye laser (CPM), Titanium sapphire
laser (Ti:sapphire), or High-Harmonic Generation
(HHG). For comparison, also the lifetime <strong>of</strong> the Quark
Gluon Plasma (QGP) is indicated as it is produced nowadays
in heavy ion colliders like RHIC or LHC.
ular imaging [8], quantum control [9], or Raman spectroscopy
[10]. By introducing a controlled delay between
two such peaks, the dynamics <strong>of</strong> electron systems could be
studied using pump-probe techniques [6]. Such techniques
also allow for the direct time resolution <strong>of</strong> many-body dynamics,
like the observation <strong>of</strong> the dressing process <strong>of</strong>
charged particles [11]. In the zeptosecond regime, nuclear
processes may become accessible [12]. It has been suggested
that zeptosecond pulses could be created via nonlinear
Thomson backscattering [13, 14], or by employing relativistic
laser-plasma interactions [15, 16], A possible detection
scheme for the characterization <strong>of</strong> short γ-ray pulses
<strong>of</strong> MeV to GeV energy photons down to the zeptosecond
scale has been proposed in Ref. [17].
At even shorter timescales, double pulses <strong>of</strong> yoctosecond
duration <strong>of</strong> GeV photon energy could be created from the
quark-gluon plasma in non-central heavy ion collisions [2].
It turns out that the emission envelope can depend strongly
on the internal dynamics <strong>of</strong> the QGP. Under certain conditions,
a double peak structure in the emission envelope
may be observed. This could be the first source for pumpprobe
experiments at the yoctosecond timescale. The delay
between the peaks is directly related to the isotropization
time, and the relative height between the peaks can be
shaped by varying photon energy and emission angle. Such
pulses could be utilized, for example, to resolve dynamics
on the nuclear timescale such as that <strong>of</strong> baryon resonances
[18]. Conversely, a time-resolved study <strong>of</strong> the emit-

ted photons could provide a window to the internal QGP
dynamics throughout its expansion.
QGP PHOTON PRODUCTION
The Large Hadron Collider (LHC) at CERN has begun
its heavy-ion program in November 2010 [19], just a
few weeks before the PIF 2010 conference. With center<strong>of</strong>-mass
energies <strong>of</strong> 2.76 TeV per nucleon, the collisions
<strong>of</strong> lead ions produce hotter and denser plasmas than previously
achievable at the Relativistic Heavy Ion Collider
(RHIC), where gold ions were used. The temperatures
reached in these collisions are so high that the constituents
<strong>of</strong> atomic nuclei, the neutrons and protons, are split into
their constituents, the quarks and gluons. The interest in
the QGP stems not least from the fact that it is believed to
have filled the entire universe during the first few microseconds
after the Big Bang.
In heavy-ion collisions, the QGP is produced up to the
size <strong>of</strong> a nucleus (∼ 15 fm) for a duration <strong>of</strong> a few tens <strong>of</strong>
yoctoseconds (1 ys = 10 −24 s). The plasma is produced
initially in a very anisotropic state, and reaches a hydrodynamic
evolution through internal interactions only after
some isotropization timeτiso (see Fig. 2). Among the many
particles that are produced, also photons <strong>of</strong> a few GeV energy
are emitted [20, 21].
It was one <strong>of</strong> the early surprises that the observed particle
spectra turned out to agree well with ideal hydrodynamical
model predictions [22]. This led to the initial assumption
that isotropization times may be as low as τiso ≈ 1 ys.
However, it has been pointed out in the mean-time that viscous
hydrodynamic models are still consistent with RHIC
data if isotropization times as large as τiso ≈ 7 ys are assumed
[23], even if the expansion before isotropization
is assumed to be collisionless (“free streaming”). This
should be compared to the lifetime <strong>of</strong> the QGP, which could
amount to 15 ys at RHIC, and which could be as large as
25 ys at LHC.
Figure 2(a-c) shows a schematic view <strong>of</strong> the collision <strong>of</strong>
two heavy ions. The ions are illustrated as relativistically
contracted pancakes. In general, two ions will not collide
head-on-head, but will be displaced by an impact parameter
b. Direct photons are emitted from the expanding QGP
throughout its lifetime [24]. The energy spectrum <strong>of</strong> the
emitted photons extends to the GeV range, and the upper
limit for the temporal duration <strong>of</strong> the GeV photon pulse is
determined by the expansion dynamics <strong>of</strong> the QGP, which
leads to yoctosecond pulses. At the initial stage <strong>of</strong> the collision,
even before the plasma is created, prompt photons are
emitted from nucleon-nucleon collisions in all directions,
see Figs. 2(a) and (d). For an intermediate time after the
collision shown in Fig.2(b), a momentum anisotropy occurs
due to the longitudinal expansion <strong>of</strong> the plasma: Those
particles that originally had momentum components in forward
or backward direction along the beam axis leave the
central region quickly, so that mainly particles with transverse
momenta remain in the plasma. High-energy pho-
Figure 2: Early stages <strong>of</strong> a high-energy collision, involving
pre-equilibrium (first two columns) and equilibrated
QGP phases (last column). Parts (a)-(c) show three snapshots
in time in position space. Shown are the two relativistically
contracted colliding ions that create the quarkgluon
plasma in the overlap region. Curly arrows denote
photon emission and semicircles the detectors. Parts
(d)-(f) are corresponding pictorial representations <strong>of</strong> the
plasma in momentum space. In an intermediate stage <strong>of</strong>
the pre-equilibrium phase, the momentum distribution is
anisotropic, resulting in a change in the angular photon
emission pattern that can give rise to double-peaked photon
pulses [2].
tons that are created in the plasma through Compton scattering
or quark-antiquark annihilation carry preferentially
the momentum <strong>of</strong> the original participants <strong>of</strong> the collision.
Because <strong>of</strong> the momentum anisotropy <strong>of</strong> the quarks and
gluons within the plasma, also the emitted photons are preferentially
emitted perpendicular to the beam axis z, as indicated
in Fig. 2(e). Finally, in Fig. 2(c), the system had
time to isotropize due to collisions within the plasma. The
photons will be emitted again in all directions, as shown
in Fig. 2(f), particularly also into the direction in which
the photon emission was suppressed during the anisotropic
stage.
A photon detector placed towards the beam axis would
therefore measure a time-dependent photon flux. Ideally, in
the intermediate stage in Fig. 2(e), the photon emission in
the direction <strong>of</strong> the detector is suppressed highly enough, so
that the photons emitted in the stages Fig. 2(d) and Fig. 2(f)
are distinct enough in time to form two separate photon
pulses. In order to quantitatively describe the pulse envelope
<strong>of</strong> the emitted photons, detailed calculations are necessary.
Such a calculation has been performed in Ref. [2],
which was based on a one-dimensional expansion model
by Bjorken [25]. This model assumes boost-invariant evolution
<strong>of</strong> the quark-gluon plasma in a central region <strong>of</strong> the
collision. The detector is placed away from the beam axis
by an angle θ within the reaction plane. Direct photons in
the GeV energy range are emitted from the pre-equilibrium
and equilibrated phases <strong>of</strong> the QGP. The leading contribution
to the photon production rateR originates from quarkgluon
Compton scattering and quark-antiquark annihilation.
In principle, higher order s<strong>of</strong>t scattering processes like

emsstrahlung or inelastic pair annihilation would have to
be included as well, but their contributions become dominant
only at lower energies, and are less important at the
higher energies considered [26].
For anisotropic momentum distributions, the photon production
rate R has to be calculated numerically [21]. It
depends on the temperature T <strong>of</strong> the medium, the photon
energy E and momentum k, the fine structure constantα,
and the corresponding quantity for the strong force
αs (with ¯h = c = kB = 1). The rate further depends
on the anisotropy, which is described by a parameter
ξ = p2
2
T / 2 pL − 1 that relates the mean longitudinal
and transverse momenta pL and pT [26, 27]. To integrate
this rate over time, a time evolution model for the preequilibrium
and equilibrated QGP has been used [27]. This
model specifies the time evolution for the energy density
E = E(τ), for the hard scale phard = phard(τ) (which corresponds
to T in the isotropic case), and for the anisotropy
parameter ξ = ξ(τ) as a function <strong>of</strong> the proper time τ.
Qualitatively, the model follows the evolution as outlined
in Fig. 2. For early times, a free streaming phase lets the
anisotropy grow. At late times, the system converges to
an ideal hydrodynamic phase with vanishing anisotropy.
These two phases are linked by a smooth transition which
is controlled by additional model parameters. Thermalization
and isotropization happen concurrently in this model,
τtherm = τiso. The model is thus able to cover both, the
pre-equilibrium phase and the equilibrated QGP phase <strong>of</strong>
the expanding plasma.
dNdtdΩdas 1 GeV 1
100 a p2 GeV
80
b0 fm
60 ΘΠ2
40
20
b
p3 GeV
b0 fm
ΘΠ2
20 0 20 20 0 20
100
80
60
40
20
0
100
80
60
40
20
c Τiso p2 GeV
b9.2 fm
ΘΠ4
4
e p2 GeV
b12.2 fm
ΘΠ8
Τiso 4
d
f
5 p3 GeV
b9.2 fm
Τiso ΘΠ4
20
Τ
p3 GeV
iso
b12.2 fm
ΘΠ8
4
5 0 5 10 155
Τys
0 5 10 15
Figure 3: Photon emission rate as a function <strong>of</strong> detector
time τ. Solid blue lines show a large isotropization time
τiso = 6.7 ys while dashed red lines correspond to a short
isotropization time τiso = 0.3 ys. Parts (a) and (b) display
emission at midrapidity (θ = π/2) for a central collision
with impact parameter b = 0. Parts (c)-(f) show
double-peaked photon pulses obtained for b = 9.2 fm or
12.2 fm, and the vertical dotted line indicates the position
<strong>of</strong> the largerτiso = 6.7 ys. In parts (c) and (d), the detector
direction isθ = π/4, in (e) and (f), it isθ = π/8 [2].
The numerical parameters suitable for calculating LHC
parameters are given as follows: The initial temperature
is assumed to be T0 = 845 MeV with a formation time
τ0 = 0.3 ys. The critical temperature, where the QGP
ceases to exist, is taken asTC = 160 MeV. The isotropization
time is varied in the range τiso = τ0 to τiso = 6.7 ys,
assuming free-streaming at early times. Both possibilities
are not yet ruled out by RHIC data. In order to ensure
fixed final multiplicity, the initial conditions are adjusted as
a function <strong>of</strong> τiso such that the same entropy is generated
as forτiso = τ0.
For central collisions with emission angle orthogonal to
the beam axis (θ = π/2), a typical time evolution <strong>of</strong> the
photon emission rate is depicted in Figs. 3(a) and 3(b). At 2
GeV energy, the photon production from the QGP at midrapidity
is 3 to 4 times as large as the production from the
initial collisions. It is roughly 6 times as large as the production
from the hadron gas [24]. At 3 GeV energy, QGP
photon production is even 50 times larger than the production
from the hadron gas. In the Figs. 3, the origin <strong>of</strong> the
abscissa is the time when a photon emitted from the center
<strong>of</strong> the collision arrives at the detector. Photons arriving
earlier originate from a part <strong>of</strong> the QGP that is closer to the
detector. The pulse shape is mainly determined by the geometry
<strong>of</strong> the lead ion with radius7.1 fm. Any structure on
the yoctosecond timescale is blurred simply by the time for
light to traverse the QGP.
This limit can be overcome in the following ways: By
considering non-central collisions with impact parameter
b, the physical extent <strong>of</strong> the QGP is reduced. Also, an optimization
<strong>of</strong> the detection angle can minimize the traveling
time through the plasma. In forward direction, the initial
shape <strong>of</strong> the QGP is Lorentz-contracted, and light leaves
this initial region quickly. This is partially spoiled due to
the QGP expansion in the same direction. Thus intermediate
emission angles are most promising for which the QGP
appears partly Lorentz contracted but does not expand towards
the detector.
Figures 3(c)-3(f) show the photon emission in the directions
θ = π/4 and θ = π/8. For large impact parameters
b = 9.2 fm and b = 12.2 fm, a striking double-peak structure
appears. The minimum between the two peaks corresponds
roughly to maximum anisotropy within the plasma.
This follows from the fact that the photon emission rate
is suppressed for larger values <strong>of</strong> ξ and smaller values <strong>of</strong>
θ [26]. The distance between the two peaks is approximately
governed by the isotropization time τiso, indicated
by the dotted line in Figs. 3(c)-3(f). The first peak corresponds
to photons emitted from the blue-shifted approaching
part <strong>of</strong> the QGP, while the second peak corresponds to
a slightly red-shifted and time-dilated receding tail <strong>of</strong> the
plasma. For a short isotropization time τiso = τ0 (dashed
lines) the separation into two peaks does not occur. Therefore,
this effect depends delicately on the QGP dynamics.
There are a couple <strong>of</strong> caveats to this model calculation:
Apart from the photons originating from the QGP, in an
actual experiment there is a background <strong>of</strong> photons from

various sources [28]. These include photons produced by a
jet passing through the QGP [20], and could dominate the
effect that is expected from the QGP alone. Since these
photons are produced on a similar yoctosecond timescale,
they would modify the pulse shape on this timescale. Photons
produced from the initial collisions <strong>of</strong> the two ions
can be <strong>of</strong> comparable size in intensity, but they would
only enhance the first peak <strong>of</strong> the double peaks depicted
in Fig. 3(c)-3(f). Other background photons are produced
in the decay <strong>of</strong> pions at later stages <strong>of</strong> the collision. Since
these are produced at much later time scales, they would
not modify a time structure on the yoctosecond timescale.
It would be necessary to take these various photon sources
into account in order to obtain a quantitative prediction <strong>of</strong>
the expected pulse structure.
What kind <strong>of</strong> properties should the detector have in order
to resolve the double pulses? The time-integrated highenergetic
photons are already routinely detected in experiments
[29]. In the GeV energy range, the photon production
rate is <strong>of</strong> the order <strong>of</strong> a few photons per collision [28].
Note that a single GeV photon pulse <strong>of</strong> 10 ys duration corresponds
to a pulse energy <strong>of</strong> only about 100 pJ, but to a
power <strong>of</strong> 10 TW. The photon yield could be enhanced by
considering lower energy photons, but this would also increase
the number <strong>of</strong> unwanted background photons. Alternatively,
the photon yield could be enhanced by increasing
the collision energy. This may have the additional benefit
<strong>of</strong> increasing the relative importance <strong>of</strong> the contribution <strong>of</strong>
thermal photons compared to other kinds <strong>of</strong> photons [28].
The emission envelope is influenced by the geometry, emission
angle, and internal dynamics like the isotropization
time <strong>of</strong> the expanding QGP. The double-peak structure described
may emerge in non-central collisions at an emission
angle close to forward direction, assuming that the
isotropization time is large. In order to detect such short
pulses, new detection schemes would be required. Existing
tools and ideas from attosecond metrology, like pumpprobe
experiments or spectroscopy techniques, may turn
out to be appropriate candidates to be scaled to zepto- or
yoctosecond duration [4, 17]. Also, an experimental determination
<strong>of</strong> the photon emission envelope would serve as
an additional probe <strong>of</strong> the internal dynamics <strong>of</strong> the QGP,
for example by measuring its isotropization time.
CONCLUSIONS
The steady progress in laser physics over the last decades
towards higher intensities and shorter pulse lengths provides
strong motivation to study systems that produce extremely
short flashes <strong>of</strong> light. The QGP is an ideal candidate
because it exhibits non-trivial dynamics on the yoctosecond
timescale, during which GeV photons are emitted.
Under certain conditions, a double-peak structure may
be produced. This could eventually lead to novel pumpprobe
experiments at the GeV energy scale. Alternatively,
measuring the temporal shape <strong>of</strong> the photon emission envelope,
dynamic properties <strong>of</strong> the QGP, like its isotropization
time, could be probed experimentally.
ACKNOWLEDGMENT
I would like to thank my collaborators J. Evers,
K. Z. Hatsagortsyan, and C. H. Keitel for guidance and
fruitful discussions that led to the work that has been presented
[2, 17]. I would further like to thank the organizers
<strong>of</strong> the PIF 2010 for their kind hospitality.
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Abstract
Fields, Instantons, and Currents
Kenji Fukushima
Department <strong>of</strong> Physics, Keio University
3-14-1 Hiyoshi, Kohoku-ku, Yokohama-shi, Kanagawa 223-8522, Japan
A review on the chiral magnetic effect is given with special
emphasis put on the pseudo one-dimensional property
<strong>of</strong> the system under strong magnetic fields. In such a (1+1)dimensional
description as a result <strong>of</strong> the dimensional reduction,
electric fields can be identified as the topological
charge associated with instanton-like gauge configurations.
The chiral magnetic current is, in this picture, nothing but
a current according to familiar Ohm’s law. The currentcurrent
susceptibility is found to be a product <strong>of</strong> the boson
mass in the Schwinger model and the Landau level density.
CHIRAL MAGNETIC EFFECT
It is well-known that special gauge configurations with
non-zero winding number play an important role in the understanding
<strong>of</strong> the vacuum structure in the strong interactions.
The spontaneous breaking <strong>of</strong> chiral symmetry is attributed
to the QCD instanton, which is the origin <strong>of</strong> dynamical
mass generation. The confinement nature is also
explained in terms <strong>of</strong> magnetic monopole condensation in
a special class <strong>of</strong> the gauge choice.
There is no doubt about the existence <strong>of</strong> topological configurations
in QCD physics, but it is a highly non-trivial
question how to “see” such topological effects in real experiments.
The chiral magnetic effect is one <strong>of</strong> the promising
candidates [1, 2, 3]. Let us imagine the following situation;
the QCD vacuum accommodates one instanton that
has a topological charge QW and a (QED) magnetic field
B that is as strong as the QCD energy scale ΛQCD is applied
on the instanton.
Then, the axial anomaly relation (for the single-flavor
case),
implies that
∂µj µ
5
g2
= −
8π2 ∫
d 3 x trFµν F µν , (1)
∆N5 = N5(t = ∞) − N5(t = −∞) = −2QW . (2)
This means that, if we start with the chirally neutral situation
(i.e. N5(t = −∞) = 0), a finite amount <strong>of</strong> chirality is
generated by the topological charge. In the chiral limit in
which Dirac fermions are all massless, the momentum and
the spin are parallel to each other if the chirality is righthanded,
while they are anti-parallel if the chirality is lefthanded.
Therefore, the spin is aligned by the strong B effect,
which makes the momentum also aligned along the B
direction, leading to a non-vanishing value <strong>of</strong> the total momentum
if ∆N5 ̸= 0. In other words, since Dirac fermions
are charged, an electric or baryonic current is produced for
B ̸= 0 and ∆N5 ̸= 0. Such an effect can be expressed
simply as [1]
B
J V = −2QW , (3)
|B|
in the strong B limit, where J V represents the vector current
⟨ ¯ ψγ µ ψ⟩ integrated over the volume.
For the general strength <strong>of</strong> B it is more appropriate
to work in the grand canonical ensemble using the chiral
chemical potential µ5 instead <strong>of</strong> N5. One can actually
prove that [2, 4]
jV = eµ5
B (4)
2π2 holds for any B and µ5. For free Dirac particles under
B ̸= 0 and µ5 ̸= 0, it is possible to confirm that Eq. (4)
is reduced to Eq. (3) in the strong B limit using the relation
(2).
What is detectable in experiments should not be the current
jV itself because the QCD vacuum has both instantons
and anti-instantons and they always fluctuate. In other
words, the parity (P) and the charge-parity (CP) symmetries
are broken only locally at the instanton or antiinstanton,
but those symmetries are restored on average
over all fluctuations in space and time. Thus, the chiral
magnetic current jV is also a local object and its (ensemble
or spatial) average is vanishing. This is the reason why the
chiral magnetic effect is sometimes referred to as the “local
parity violation” in the context <strong>of</strong> the relativistic heavy-ion
collisions.
In this sense the most relevant quantity to the experimental
data [5] is the fluctuation <strong>of</strong> the chiral magnetic current,
i.e. the electric-current susceptibility χj [6]. One-loop
computation results in,
χj = e2 |eB|
, (5)
2π2 which does not depend on µ5 and comes from the Landau
zero-mode alone, interestingly. The derivation from
explicit calculations is lengthy but there is an argument to
take a short-cut to the above expression [6]. For this purpose
let us consider the electric current generation rate,
d(eJV )
dt = V e2 |eB|E
2π2 , (6)
which originates from the correspondence between the chirality
generation and the particle production when fields are
strong enough [7]. The same quantity can be expressed in
the framework <strong>of</strong> the linear response theory as
∫
d(eJV )
= −
dt
d 3 x d 4 x ′ ⟨ d(ejV )(x)
jV (x
dt
′ ⟩
) eA(x ′ ∫
),
= d 3 x d 4 x ′ e 2 ⟨jV (x)jV (x ′ )⟩ E, (7)

where A(x) denotes a vector potential component parallel
to B and thus J V . From the first line to the second line
above, we used E = ˙ A. By equating Eqs. (6) and (7), we
can find chij given by Eq. (5) immediately.
What is addressed in this article is that we can quickly
derive these expressions related to the chiral magnetic effect
once we take the strong B limit and the associated
dimensional reduction.
DIMENSIONAL REDUCTION
Under a strong magnetic field, in general, the transverse
motion <strong>of</strong> charged particles is equivalent to the one in the
harmonic oscillator. The energy level is then discrete due
to the Landau quantization. Spin-1/2 particles have the
Landau zero-mode which would dominate in the dynamics
at energies below the scale ∼ √ |eB|. Such a strong B
enables us to use the so-called lowest Landau-level (LLL)
approximation and to drop the transverse motion at all. In
this limit, thus, we can reduce the (3+1)-dimensional theory
into a form <strong>of</strong> the (1+1)-dimensional one multiplied by
the Landau level density.
In Minkowskian space-time we use the metric g 00 =
−g 11 = 1, g 01 = g 10 = 0, and the 2 × 2 γ-matrices
which satisfy {γ µ , γ ν } = 2g µν . Chirality is characterized
by γ 5 = γ 0 γ 1 = diag(1, −1) in the chiral representation
<strong>of</strong> the γ-matrices. Therefore the upper (lower) element
<strong>of</strong> two-component spinor ψ = (ψR, ψL) t represents the
right-handed (left-handed) particle. In (1+1) dimensions
the particle–anti-particle difference is correlated with the
chirality. That is, in momentum space, the right-handed
component corresponds to a right-moving (p > 0) particle
and a left-moving (p < 0) anti-particle. One can understand
the left-handed component in the same way, i.e. a
left-moving (p < 0) particle and a right-moving (p > 0)
anti-particle.
In (1+1) dimensions the following relation among the
γ-matrices plays an interesting role for the topological currents;
γ µ γ 5 = −ϵ µν γν, (8)
where ϵ 01 = −ϵ 10 = −ϵ01 = ϵ10 = 1, which relates the
vector and the axial-vector currents. As usual, we can write
the vector and the axial-vector currents as
j µ
V = ¯ ψγ µ ψ, j µ
5 = ¯ ψγ µ γ 5 ψ. (9)
Using the relation (8), we have a relation, j µ
5 = −ϵµν jν,
that is explicitly written as [8]
j 1 V = j 0 5, j 1 5 = j 0 . (10)
TOPOLOGICAL CURRENTS
The relation between the vector and axial-vector currents
is very useful because, as we will see in this section, it captures
the essential feature <strong>of</strong> the topologically induced currents
in (3+1) dimensions.
Let us consider the anomaly relation in (1+1) dimensions.
It is well-known that the axial anomaly leads to
∂µj µ
5
= e
2π ϵµν Fµν = e
π E = −2qW , (11)
where the electric field is E = F 10 in our convention. Note
that there is no magnetic field but only the electric field E in
(1+1) dimensions. We here defined the (1+1)-dimensional
topological charge density as qW = −(e/2π)E in accord
to the convention. By integrating Eq. (11) over space-time
and assuming that the current falls sufficiently fast at spatial
infinity, we can recover Eq. (2) easily. We can also prove
that the topological charge, QW = ∫ d 2 x qW (x), takes an
integer number so that the boundary condition in the xdirection
can be maintained.
We also note that we can express Eq. (11) in the form <strong>of</strong>
∂µj µ e
5 (x) = −
π (∂0A 1 − ∂ 1 A 0 ) = −2∂µK µ (x) (12)
with the (1+1)-dimensional Chern-Simons current density
defined by K µ = −(e/2π)ϵ µνAν. From this identification,
the Chern-Simons number in this system is deduced as
∫
ν(t) =
dx K 0 (t, x) = e
2π
∫
dx A 1 (t, x). (13)
Combining these expressions with the relation j1 V = j0 5
(where N5 is the volume integral <strong>of</strong> j0 5), we can immediately
write the vector current integrated over space;
J 1 ∫
V (t) = N5(t) = −2 dt dx qW (t, x), (14)
assuming that N5 was zero at the initial time (i.e. N5(t =
−∞) = 0). This simple relation leads to the current at late
time as given by
J 1 V = −2QW . (15)
This is a result expected when the spin is fully polarized in
the (3+1)-dimensional chiral magnetic effect under a strong
magnetic field (see Eq. (3)). Note that, in (1+1) dimensions
the spin is always fully polarized because there is only one
spatial direction and thus the moving direction (either p >
0 or p < 0) and the chirality <strong>of</strong> particles have one-to-one
correspondence. Here Eq. (14) is nothing but Ohm’s law
because the (1+1)-dimensional topological charge density
is proportional to the electric field as in Eq. (11).
If the spatial component <strong>of</strong> the Chern-Simons current
falls sufficiently fast, the topological charge is written as
QW = ν(t = ∞) − ν(t = −∞). Therefore, (the spatial
average <strong>of</strong>) A1 is the Chern-Simons number and the boundary
condition <strong>of</strong> A1 in the t-direction gives the topological
winding number. Supposed that ν(t = −∞) = N5(t =
−∞) = 0, the topologically induced current is
J 1 V (t) = − e
π
∫
dx A 1 (t, x). (16)
If we identify −eA 0 as the chemical potential µ (regarding
the sign, remember the covariant derivative p 0 − eA 0 and

the dispersion relation p0 = Ep − µ for particles). Equation
(8) implies that eA1γ1 = eA1γ0γ 5 and thus −eA1 can
be identified as the axial (or chiral) chemical potential µ5.
Therefore, we can conclude;
J 1 V = 1
π
∫
dx µ5, (17)
which correctly recovers the (3+1)-dimensional chiral
magnetic current (4) once we multiply this by the Landau
level density, eB/(2π). That is,
jV = µ5
π
−→ jV = |eB|
2π
(in (1+1) dimensions)
· µ5
π
(in (3+1) dimensions), (18)
which coincides with Eq. (4).
Here, it is clear that the longitudinal gauge field A 1 ,
which is the Chern-Simons number in (1+1) dimensions,
plays the role <strong>of</strong> the chiral chemical potential µ5 in (3+1)
dimensions. We note, however, that there is an important
difference; usually µ5 is introduced by hand as a constant,
but in (1+1) dimensions A 1 must have t-dependence to allow
for nonzero QW . We can think <strong>of</strong> a concrete “instanton”
configuration in (1+1) dimensions simply as
A 1 (t, x) = 2πQW t
= −Et, (19)
eL T
where we limit ourselves to the spatially homogeneous case
and denote the spatial and temporal extents as L and T ,
respectively, and then we have
j 1 V (t) = J 1 V (t) eE
= t. (20)
L π
From this, again, if multiplied by the Landau-level degeneracy
we can correctly recover the current generation rate
given by Eq. (6), i.e.
d(ejV )
dt = e2E (in (1+1) dimensions)
π
−→ d(ejV ) |eB|
=
dt 2π · e2E (in (3+1) dimensions), (21)
π
which coincides with Eq. (6).
In the same way we can get a finite axial-vector current
at finite quark chemical potential µ. To see the anomalous
nature the important fact is that the relation between the
density and the chemical potential is given by the quantum
anomaly in (1+1) dimensions, i.e.
n = − eA0
, (22)
π
which results from the anomaly. One can derive this expression
directly from n = ⟨ψ † (x)ψ(x)⟩ by inserting
the gauge field as lim y 0 →x 0 ψ † (y) exp[−ie ∫ dtA 0 ]ψ(x).
From this we can immediately reach,
J 1 ∫
5 = dx n = 1
∫
dx µ, (23)
π
which represents the chiral separation effect. This is again
the anomaly relation exactly same as that in (3+1) dimensions
once multiplied by the Landau level density eB/2π.
SCHWINGER MODEL
So far the arguments and the resulting expressions are
quite general. From now on we shall go into the dynamical
properties calculating microscopic quantities in a solvable
(1+1)-dimensional model, i.e. the massless Schwinger
model. The easiest way to accomplish a calculation in the
Schwinger model is to use mapping onto a free bosonic
theory. In our case, however, the bosonization rule is a
bit more complicated than usual because we deal with not
only fermionic fields (such as the chiral condensate) but
also gauge fields (such as the electric field). So, the Lagrangian
density <strong>of</strong> the corresponding theory should be
L = 1
2 (∂µ θ)(∂µθ) − mγ(∂ µ θ)(∂µϕ) − 1
2 (∂µ ϕ)∂ 2 (∂µϕ)
(24)
with the boson mass,
m 2 γ = e2
. (25)
π
If the ϕ-field is integrated out, we get a theory only in terms
<strong>of</strong> the θ-field that is free and has a mass by mγ. Such a
scalar theory is usually used with the bosonization rule;
j µ
V = ¯ ψγ µ ψ = 1
√ π ϵ µν ∂νθ, (26)
j µ
5 = ¯ ψγ µ γ 5 ψ = − 1
√ π ∂ µ θ, (27)
¯ψψ = −c mγ : cos(2 √ πθ) : (28)
with the normal ordering : :. Now we remark that ϕ in
the Lagrangian density (24) comes from the gauge field,
A µ = −ϵ µν∂νϕ (where ϕ includes an instanton-like configuration
∼ 1
2Et2 which does not satisfy the periodic
boundary condition in the t-direction). Then the electric
field takes a form E = ∂2ϕ. Once we integrate the θ-field
out from the theory, after the Gaussian integration in the
functional formalism, Eq. (27) is replaced by
j µ
5
= − 1
√ π ∂ µ θ → − mγ
√π ∂ µ ϕ = − e
π ∂µ ϕ. (29)
The anomaly relation is then derived as
∂µj µ
5
= − e
π ∂2 ϕ = − e
π E = −2qW , (30)
which is fully consistent with the anomaly relation (11).
In the same manner we can express the vector current in
terms <strong>of</strong> ϕ, and then we find,
j µ
V
e
=
π ϵµν µν ∂ν
∂νϕ = 2ϵ
∂2 qW . (31)
It is easy to confirm that this result is fully consistent with
the previous relation. That is, after the spatial integration
for the µ = 1 component (or ϕ and qW ), the spatial derivative
∂1 drops and the right-hand side simplifies as −2/∂0,
that is just a t-integration. Therefore the right-hand side finally
becomes −2QW together with the spatial integration,
and hence we obtain J 1 V = −2QW .

The above equation gives a microscopic structure <strong>of</strong> the
current in more general cases with spatial modulation. In
momentum space we can re-express this as follows;
j 1 V (ω, k) = −2iω
ω 2 − k 2 qW (ω, k). (32)
This is an interesting relation. If ω → 0 is taken first, we
see that j1 V (0, k) is vanishing. To get the chiral magnetic
current or a finite chiral magnetic conductivity, it is necessary
to take the zero-momentum limit in the order <strong>of</strong> k → 0
first and then ω → 0. This observation is in fact consistent
with the result <strong>of</strong> the one-loop calculation <strong>of</strong> the chiral
magnetic conductivity [9].
We point out that the structure <strong>of</strong> Eq. (32) naturally appears
from the transverse projection. That is, after the oneloop
integration with the gauge potential source, in momentum
space one can find,
j µ
(
V (ω, k) = −
g µν − qµ q ν
q 2
) e
π Aν(ω, k) (33)
with q = (ω, k) and q 2 = ω − k 2 , from which one can
easily find that
j 1 V (ω, k) = − ω2
ω2 − k2 e
π A1 (ω, k). (34)
Because qW = (e/π)∂ 0 A 1 , one can substitute A 1 =
i(2π/e)qW /ω for A 1 above, from which we can immediately
confirm that the above expression is equivalent to
Eq. (32).
From the equivalence to the bosonized theory it is very
easy to read the electric current-current fluctuation too. To
this end we integrate the ϕ-field first, and then what we
have is a free massive scalar theory in terms <strong>of</strong> θ alone.
Then we trivially get,
χj(x − y) = e 2 ⟨j 1 (x)j 1 (y)⟩ = m 2 γ ∂ x 0 ∂ y
0 ⟨θ(x)θ(y)⟩,
(35)
or in momentum space we can express this as
m 2 γ ω 2
χj(ω, k) =
ω2 − k2 − m2 . (36)
γ + iϵ
At a first glance this expression looks different from
Eq. (5). This is because the above expression (36) is a result
after resummation <strong>of</strong> the bubble-type diagrams, while
Eq. (5) is the one-loop result. Roughly speaking m 2 γ appears
in the denominator <strong>of</strong> Eq. (36) as a result <strong>of</strong> infinite
insertion <strong>of</strong> the polarization diagram. This indicates that
we can extract the one-loop result from the leading-order
Taylor expansion <strong>of</strong> Eq. (36) in terms <strong>of</strong> m 2 γ. That actually
leads to
χ one-loop
j (ω, k) = m2γ ω2 ω2 − k2 → m 2 γ = e2
π
(k → 0). (37)
Therefore,
χ one-loop
j
= e2
π
−→ χ one-loop
j
(in (1+1) dimensions)
= |eB|
2π
· e2
π
(in (3+1) dimensions), (38)
which again coincides with the previous result (5).
SUMMARY
The chiral magnetic effect is <strong>of</strong> theoretical and experimental
interest in the context <strong>of</strong> the relativistic heavy-ion
collisions where the strong fields and the topological excitations
(instantons and sphalerons) exist, which leads to
the (electric) current. Such an effect could be observed as
charge asymmetry in experiments.
There are a number <strong>of</strong> works that aim to clarify the
microscopic properties <strong>of</strong> QCD matter related to the chiral
magnetic effect. The explicit computation is usually
lengthy and cumbersome due to the presence <strong>of</strong> the external
magnetic field. We here discussed the dimensional reduction
and demonstrated that many <strong>of</strong> known results can
be reproduced without tedious calculations. As a model
study we picked up the Schwinger model, which is useful
but drops the (3+1)-dimensional gauge dynamics. It is
highly demanded to construct a full effective description
valid for genuine (3+1) dimensional gauge dynamics in the
strong magnetic field limit.
The author thanks D.E. Kharzeev and H.J. Warringa for
fruitful discussions and also he is grateful to G.V. Dunne
for giving useful comments during the workshop.
REFERENCES
[1] D. E. Kharzeev, L. D. McLerran and H. J. Warringa, Nucl.
Phys. A 803, 227 (2008).
[2] K. Fukushima, D. E. Kharzeev and H. J. Warringa, Phys. Rev.
D 78, 074033 (2008).
[3] D. E. Kharzeev, Annals Phys. 325, 205 (2010);
arXiv:1010.0943 [hep-ph].
[4] M. A. Metlitski and A. R. Zhitnitsky, Phys. Rev. D 72,
045011 (2005).
[5] B. I. Abelev et al. [STAR Collaboration], Phys. Rev. Lett.
103, 251601 (2009). N. N. Ajitanand, R. A. Lacey, A. Taranenko
and J. M. Alexander, arXiv:1009.5624 [nucl-ex].
[6] K. Fukushima, D. E. Kharzeev and H. J. Warringa, Nucl.
Phys. A 836, 311 (2010).
[7] K. Fukushima, D. E. Kharzeev and H. J. Warringa, Phys. Rev.
Lett. 104, 212001 (2010).
[8] G. Basar, G. V. Dunne, D. E. Kharzeev, Phys. Rev. Lett. 104,
232301 (2010).
[9] D. E. Kharzeev and H. J. Warringa, Phys. Rev. D 80, 034028
(2009).

CRITICAL BEHAVIOR OF CHARMONIUM: QCD SECOND ORDER
STARK EFFECT ∗
Abstract
Kenji Morita, GSI, Helmholzzentrum für Schwerionenforschung, Darmstadt, Germany †
Su Houng Lee ‡ , Institute <strong>of</strong> Physics and Applied Physics, Yonsei University, Seoul, Korea
We study a mass shift <strong>of</strong> charmonia in hot QCD medium
near and below the critical temperature. We introduce
a formula for the mass shift by the QCD second order
Stark effect based on the operator production expansion.
Then we discuss the behavior <strong>of</strong> the electric field square
on the basis <strong>of</strong> results from lattice QCD. We demonstrate
the mass shift <strong>of</strong> J/ψ serves as a good indicator <strong>of</strong> the
confinement-deconfinement transition. We propose a resonance
gas model for the condensate to describe the medium
with nonzero baryonic chemical potential. We discuss the
mass shift at hadronization temperature and chemical potential
and possible implication for heavy ion experiments.
INTRODUCTION
Understanding the confinement phenomenon in QCD is
<strong>of</strong> fundamental subject in modern physics. Despite the difficulty
<strong>of</strong> solving QCD owing to the strongly coupled nature,
recent development in high-performance computing
enables us to calculate bulk property <strong>of</strong> the hot QCD matter
at T ∼ 200 MeV ∼ 10 12 K by Monte-Carlo simulation
<strong>of</strong> QCD on the lattice. The result at physical quark masses
reveals that the matter at high temperature consists <strong>of</strong> deconfined
quarks and gluons, called quark-gluon plasma,
and that it undergoes a transition into a hadronic gas at
T ∼ 170 MeV. Currently relativistic heavy ion collisions
provide unique opportunity to produce the matter on earth.
However, the matter is far from ideal situation because the
created matter has only short lifetime ∼ 10 −22 sec and we
can detect only finally produced particles such as pions and
nucleons. Therefore it is important to find an observable
which carries information on the transition from QGP to
hadron gas that would happen in the collision process. In
this work, we focus on charmonium and how it reacts with
the change <strong>of</strong> the matter property. Indeed such an idea was
proposed many years ago [1, 2] based on facts that the charmonium
spectrum can be well explained by confinement
force [3] and that the force could be modified by decrease
<strong>of</strong> the string tension and Debye screening in medium. We
have been elaborating a method on the basis <strong>of</strong> the operator
product expansion (OPE) and in-medium change <strong>of</strong>
the gluon condensates which gives leading contribution to
the OPE. The aim <strong>of</strong> this talk is to give an explanation
<strong>of</strong> the role <strong>of</strong> the temperature dependent gluon conden-
∗ Work supported by FIAS and Korea Ministry <strong>of</strong> Education
† k.morita@gsi.de
‡ suhoug@phya.yonsei.ac.kr
sates as an effective order parameter <strong>of</strong> the confinementdeconfinement
transition in QCD and to discuss its consequence
on the in-medium modification <strong>of</strong> charmonium.
QCD SECOND ORDER STARK EFFECT
Interaction <strong>of</strong> a heavy quarkonium with gluons based on
the OPE was formulated by Peskin [4, 5]. The primary
point is to introduce the separation scale k, which is set
to the binding energy k ∼ ϵ <strong>of</strong> the quarkonium. Since
the size <strong>of</strong> the Coulombic bound state with heavy quark
mass m is a0 = 4/(Ncαsm) and the binding energy is
ϵ = 1/(ma 2 0), the separation scale for sufficiently heavy
quark can be large enough for perturbative treatment. Then
the matrix element is calculated through the OPE in which
the short distance process is implemented into the Wilson
coefficients and the s<strong>of</strong>t one is accounted for gauge invariant
local operators. While the formulation can be applied to
scattering cross section <strong>of</strong> the quarkonium-hadron interactions
[5, 6], it reduces to the formula for a mass shift <strong>of</strong> the
quarkonium at rest by change <strong>of</strong> the electric field squared
[7]. The formula with a normalization <strong>of</strong> the wave function
∫ d 3 k
(2π) 3 |ψ(k)| 2 = 1 reads
∆m ¯ QQ = − 1
18
= − 7π2
18
∫∞
dk 2
∂ψ(k)
∂k
0
a 2 0
ϵ
2
k
k2 ⟨
αs
/m + ϵ π ∆E2⟩
med
(1)
⟨
αs
π ∆E2⟩ , (2)
med
where the second line holds for the Coulombic wave function.
As we will see below, the electric condensate increases
as temperature rises up. This implies a downward
mass shift <strong>of</strong> 1S quarkonium irrespective to the detail <strong>of</strong> the
wave function. In the formula with Coulomb wave function,
one sees that a dipole nature <strong>of</strong> the interaction explicitly
appears as a 2 0 scaling. This implies the mass shift does
not depend on the detail <strong>of</strong> the wave function but on the size
<strong>of</strong> the quarkonium. We fix the parameters as mc = 1704
MeV and a0 = 0.271 fm by fitting with J/ψ mass, 3097
MeV and the size <strong>of</strong> the wave function in the Cornell potential
model, ⟨r 2 ⟩ 1/2 = 0.47 fm, implying αs = 0.57.
In the next section, we discuss the behavior <strong>of</strong> the electric
condensate in the pure gluonic system.
PURE GLUONIC CASE
We start with the pure gluonic system as a vivid example
for the relation between the electric condensate E 2 and

confinement-deconfinement transition. It is convenient to
introduce the following quantities [8];
⟨
β(g)
M0(T ) =
2g Ga µνG aµν
⟩
, (3)
T
(
uαuβ − 1
4 gαβ
) ⟨
M2(T ) = −ST G a αµG aµ
⟩
β . (4)
T
These denote the decomposition <strong>of</strong> the gluonic operator
into the trace anomaly part and traceless and symmetric
one, i.e., the scalar gluon condensate and the twist-2 operator
which stand for the leading contribution to the OPE.
Note that the twist-2 operator must be taken into account
for consistency in the case <strong>of</strong> medium in which Lorentz
invariance is absent. The M0 and M2 in the pure gauge
theory are related to the energy density ε(T ) and pressure
p(T ) which were obtained in lattice calculations as
M0 = ε − 3p and M2 = ϵ + p. Then taking the one-loop
expression <strong>of</strong> the beta function leads to the electric condensate
⟨
αs
π ∆E2⟩ =
T
2
11 − 2
3 Nf
M0(T ) + 3
4
α eff
s
π M2(T ). (5)
The equation <strong>of</strong> state is adopted from Ref. [9]. As for α eff
s ,
we use the effective coupling constant αqq(T ) which was
measured by heavy quark free energy [10]. Putting Nf =
0 into Eq. (5), we plot the temperature dependence <strong>of</strong> the
electric condensate near Tc = 264 MeV in Fig. 1. One
sees a rapid rise in the vicinity <strong>of</strong> the transition temperature
which reflects the first order phase transition <strong>of</strong> pure SU(3)
theory. Indeed this behavior could be related with that <strong>of</strong>
the space-time Wilson loop which shows change from the
area law to the perimeter law across Tc [11] through OPE
for the Wilson loop calculated by Shifman [12].
Putting Eq. (5) into Eq. (2), we obtained the mass shift
<strong>of</strong> J/ψ shown in Fig. 2. The downward mass shift occurs
abruptly in the vicinity <strong>of</strong> Tc, signaling the phase transition.
It reaches 40 MeV at Tc and 100 MeV at 1.05Tc.
We note, however, that applicability <strong>of</strong> this method for the
charmonium is questionable beyond this temperature [8].
RESONANCE GAS MODEL
Now we turn to the realistic case including light quarks.
It is known that QCD with physical quark mass exhibits
a crossover transition such that the change <strong>of</strong> the thermodynamic
quantity becomes smoother in the vicinity <strong>of</strong> Tc.
From an experimental point <strong>of</strong> view, the system produced
in heavy ion collisions will be QGP, which undergoes a
transition into a hadronic gas. The temperature and chemical
potential at the hadronization can be extracted from statistical
model analysis <strong>of</strong> particle ratios [13]. We consider
the electric condensate at these points which are just below
Tc.
There are two difficulties to extract the condensate from
the lattice data as done in the case <strong>of</strong> the pure gauge theory.
One is the so-called sign problem. Namely, lattice QCD
(α s /π)E 2 [GeV 4 ]
0.004
0.002
0
-0.002
-0.004
(αs /π)E
0.8 0.9 1
T/Tc 1.1 1.2
2
Figure 1: Electric condensates <strong>of</strong> pure gluonic case near
Tc. The value at low temperature limit is obtained from
that <strong>of</strong> the gluon condensate in vacuum, ⟨(αs/π)G 2 ⟩ =
(0.35GeV) 4 .
∆m [MeV]
0
-50
-100
-150
-200
-250
J/ψ
0.9 0.95 1 1.05
T/Tc 1.1 1.15 1.2
Figure 2: Mass shift <strong>of</strong> J/ψ from the second order Stark
effect with the electric condensate shown in Fig. 1.
cannot perform a simulation at finite chemical potential.
The other is that one has to separate gluonic contribution to
the thermodynamic quantities from those including quarks.
Therefore, we use a resonance gas model based on the linear
density approximation which has been used to estimate
the gluon condensate in the nuclear matter. We define M0
and M2 for a resonance gas, introduced in previous section,
as
M had
0 (T, µ) = ∑
ρi(T, µ)m
i=hadrons
0 i (6)
M had
2 (T, µ) = ∑
ρi(T, µ)miA
i=hadrons
i G. (7)
One sees this expression is linear in the number density <strong>of</strong>
i−th hadrons, ρi. If we put ρi as normal nuclear density
ρ0 = 0.17 fm −3 and pick up only nucleon contribution
in the summation, this formula reduces to the gluon condensates
in the nuclear matter [14]. In Eq. (6), the chiral
limit is taken for the mass <strong>of</strong> hadrons m 0 i to isolate
the gluonic contribution to the trace anomaly. The second
moment <strong>of</strong> the gluon distribution function A i G plays
a similar role that picks up the gluonic part <strong>of</strong> the twist-2
term. Here we take the all resonances given in the Particle
Data Group [15] into account. Masses in the chiral
limit, however, are not known for most hadrons. We use

different masses m 0 i ̸= mi only for the Nambu-Goldstone
bosons and ground state octet and decouplet baryons. We
put m0 π = m0 K = 0 and m0N = 750 MeV [16]; these are
the most important inputs as the contributions to the thermodynamic
quantities are dominated by these hadrons. For
the vector and axial vector mesons, we assume m0 ρ = mρ
and m0 a1 = ma1. We also assume m0 ∆ = m∆. Furthermore,
taking the flavor SU(3) limit, we also put m0 f0 = m0σ, m0 ϕ = m0ω = m0 K∗ = m0ρ. m0 Λ = m0Ξ = m0Σ = m0N ,
m0 Σ∗ = m0Ξ ∗ = m0Ω = m0∆ . In general AiG can differ
for hadrons. We, however, put Ai G (8m2c) = 0.9 for all the
can be shown to deviate little from
hadrons because Aπ G
this value at such a high energy scale.
At vanishing chemical potential, lattice calculations <strong>of</strong>
the equation <strong>of</strong> state with physical quark mass are available
[17, 18, 19]. The results have been compared with
the resonance gas. However, there is still uncertainty in the
lattice results due to the discretization [20]. To take this
uncertainty into account, we introduce an excluded volume
<strong>of</strong> hadrons v0 as a fitting parameter in the model by following
a prescription <strong>of</strong> Ref. [21]. While vanishing excluded
volume, v0 = 0, fits the lattice equation <strong>of</strong> state by
the Wuppertal-Budapest collaboration well [19], v0 = 1.19
fm3 reproduces the scalar gluon condensates by HotQCD
collaboration [18]. Figure 3 shows the comparison <strong>of</strong> the
lattice data and the resonance gas model. From this result,
we may regard our v0 = 0 and v0 = 1.19 fm3 results as
maximum and minimum M0, respectively. After fixing parameters,
we can extend the model to include finite chemical
potential.
Table 1: Temperatures and chemical potentials at chemical
freeze-out in heavy ion collisions at various energies. Data
are taken from Ref. [13].
√ sNN [GeV] T [MeV] µB[MeV]
8.76 156 403
12.3 154 298
17.3 160 240
130 165.5 38
200 160.5 20
Here we consider several sets <strong>of</strong> temperature and chemical
potential which are estimated by the statistical model
[13] and summarized in Table 1. The electric condensates
are obtained by putting M0(T, µB) and M2(T, µB)
[Eqs. (6) and (7)] into Eq. (5) with Nf = 3. We depict
some examples <strong>of</strong> the electric condensate in Fig. 4 for illustrating
the effect <strong>of</strong> the chemical potential. One sees at
µB = 403 MeV, the change <strong>of</strong> the electric condensate is
much larger than the one at µB = 20 MeV. Hence one expects
a larger mass shift at lower collision energies.
Figure 5 shows the mass shift obtained from the electric
condensate <strong>of</strong> the resonance gas and the second order
Stark effect. The band indicates the possible range <strong>of</strong> the
mass shift incorporating the uncertainty <strong>of</strong> the lattice data
M 0 [GeV 4 ]
(ε-3p)/T 4
6
5
4
3
2
1
0
0.004
0.003
0.002
0.001
0
HotQCD N τ =8, p4
HotQCD N τ =8, asqtad
WB, Continuum limit
Model, v 0 =1.19 fm 3
0.57 fm 3
0 fm 3
HotQCD N τ =8, p4
HotQCD N τ =8, asqtad
WB full e-3p (not M 0 )
Model, v 0 =1.19 fm 3
0.57 fm 3
0 fm 3
140 145 150 155 160 165 170 175 180
T [MeV]
Figure 3: Upper: interaction measure (ε − 3p)/T 4 . Lower
: its gluonic part M0. The points are taken from Refs. [18]
for the “HotQCD” data and [19] for the “WB” (Wuppertal-
Budapest) data while each line shows the result corresponding
to various v0.
through varying v0. Since the hadronization temperature is
not monotonic against colliding energy, the mass shift does
not exhibit so in spite <strong>of</strong> the decreasing chemical potential.
Nevertheless, one sees the largest mass shift at the lowest
colliding energy and the magnitude is 10–20 MeV at the
higher ones.
IMPLICATION FOR EXPERIMENTS
Finally we discuss possible implications <strong>of</strong> the results
for heavy ion experiments. Unfortunately, the complexity
<strong>of</strong> the collision process prevents us from directly detecting
the mass shift through the shift <strong>of</strong> the peak in the dilepton
spectrum, even if the detector resolution is fine enough to
cover the magnitude <strong>of</strong> the shift. In this case, a dynamical
model is indispensable to describe the space-time evolution
<strong>of</strong> the created matter and one needs to estimate the number
<strong>of</strong> charmonium which decay inside the medium [2, 22].
Here we consider the possibility <strong>of</strong> the statistical
hadronization <strong>of</strong> charmonium [23, 24] and influence <strong>of</strong> the
mass shift on the charmonium production. In fact, a measurement
in the Pb+Pb collisions at CERN-SPS seems to
support this scenario, because ψ ′ ratio to J/ψ can be well
reproduced by the statistical production while that in the
elementary collision such as p + p cannot be done so [25].
We do not have to take into account charm conservation if
we restrict ourselves to the charmonium-charmonium ratio.
To compare the experimental data, we need to include
J/ψ from decay <strong>of</strong> higher resonances. Since the
second order Stark effect applies to such resonances, pro-

(α s /π)∆E 2 [GeV 4 ]
0.003
0.002
0.001
v 0 =0 fm 3 , µ B =403 MeV
v 0 =1.19 fm 3 , µ B =403 MeV
v 0 =0 fm 3 , µ B =20 MeV
v 0 =1.19 fm 3 , µ B =20 MeV
0
130 135 140 145 150 155 160 165 170
T [MeV]
Figure 4: Electric condensates as functions <strong>of</strong> temperature.
Each line stands for different chemical potential and the
excluded volume.
0
∆m [MeV]
-10
-20
-30
-40
J/ψ
v 0 =1.19fm 3
v 0 =0fm 3
Stark
10 1/2 100
sNN [GeV]
Figure 5: Mass shift <strong>of</strong> J/ψ at hadronization temperature
and chemical potentials for various colliding energies.
duction <strong>of</strong> those resonances is also influenced by the mass
shift. Based on the dipole nature (2), we assume the mass
shift scales with the size <strong>of</strong> the resonance. Then we have
∆mχc ≃ −24 ∼ −49 MeV and ∆mψ ′ ≃ −40 ∼ −82
MeV, respectively. In fact, this crude estimate for χc is
close to more precise one obtained from QCD sum rules
[26]. Assuming the same branching ratio in medium as in
vacuum, we calculate the number ratio <strong>of</strong> ψ ′ to J/ψ including
decay contribution to J/ψ from mass-shifted χc
and ψ ′ . The result is shown in Fig. 6 together with the experimental
data. We plot the result as a function <strong>of</strong> ψ ′ mass
shift which is not known well. One sees an enhancement
<strong>of</strong> the ratio for large ψ ′ mass shift. While larger mass shift
<strong>of</strong> ψ ′ than 100 MeV does not seem consistent with the experimental
data, our rough estimation is still inside the experimental
band. If such an enhancement is confirmed, it
will prove the mass shift <strong>of</strong> the charmonia as a precursor <strong>of</strong>
the confinement-deconfinement transition. Details including
analyses with QCD sum rules have been presented in
Ref. [26].
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Rev. Lett. 57 (1986) 2123.
σ ψ′ /σ J/ψ
0.1
0.08
0.06
0.04
0.02
Exp. data
Statistical production w/ mass shift
No mass shift
0
-200 -150 -100 -50 0
ψ′ mass shift [MeV]
Figure 6: Ratio <strong>of</strong> ψ ′ to J/ψ as a function <strong>of</strong> ψ ′ mass shift.
The horizontal band indicates the experimental data measured
in Pb+Pb collisions at √ sNN = 17 GeV.
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ON THE UNRUH EFFECT ∗
Ralf Schützhold † , Fakultät für Physik, Universität Duisburg-Essen, D-47048 Duisburg, Germany
Abstract
After a brief introduction into the Unruh effect and its
generalization to non-uniform (here circular) acceleration,
we discuss prospects for measuring signatures <strong>of</strong> this effect
in strong lasers.
INTRODUCTION
The Unruh effect [1] describes the striking discovery
that an accelerated observer/detector experiences the
Minkowski vacuum as a thermal bath – implying that
the particle concept depends on the inertial state <strong>of</strong> the
observer/detector. After important preparatory works by
Fulling [2] and Davies [3], Unruh [1] realized this phenomenon
while trying to understand how black holes can
evaporate by emitting Hawking radiation [4]. Around the
same time, the mathematical foundation <strong>of</strong> this striking fact
was established by Bisognano & Wichmann [5] by showing
the relation between the Rindler Hamiltonian and thermality
– but apparently without immediately realizing the
broad physical significance.
However, so far this prediction <strong>of</strong> quantum field theory
has eluded a direct experimental verification, see also [6].
There are some observations regarding the imperfect polarizability
<strong>of</strong> electrons in storage rings which are related to
the Sokolov-Ternov effect [7] and can be interpreted as an
indirect verification <strong>of</strong> the Unruh effect generalized to the
case <strong>of</strong> circular acceleration, see, e.g., [8, 9].
In the following, we briefly discuss a recent proposal
[10, 11] for directly observing signatures <strong>of</strong> the Unruh
effect in the form <strong>of</strong> photon pairs created by electrons
which are accelerated in an ultra-strong laser field, see also
[12, 13]. We start with a short introduction into the Unruh
effect for uniform acceleration and discuss the generalization
to circular acceleration, which is relevant for electrons
in storage rings and the Sokolov-Ternov effect.
DETECTOR PLUS FIELD
Let us consider the following action for the field ϕ and
the detector with excitation energy E, coupled with the
coupling strength g (we use = c = 1)
A = Adetector + Afield
∫ [
E
= dτ
2 σz
]
+ gσxϕ (x[τ])
+ 1
∫
2
d 4 x [(∂µϕ)(∂ µ ϕ)] , (1)
∗ Work supported by DFG under grant SCHU 1557/1.
† ralf.schuetzhold@uni-due.de
where σz and σx are the Pauli matrices and τ is the proper
time along the detector trajectory. In the interaction picture,
the transition Hamiltonian reads (with 2σ± = σx ± iσy)
]
ˆϕ (x[τ]) , (2)
ˆHint(τ) = g(τ) [ e iEτ σ+ + e −iEτ σ−
where g(τ) is smooth switching (on and <strong>of</strong>f) function. Initially
(where the interaction is switched <strong>of</strong>f) both, detector
and field, are in their ground state
|Ψin⟩ = |Ψ(τ ↓ −∞)⟩
= |Ψdetector⟩ ⊗ |Ψfield⟩ = |↓⟩ ⊗ |0⟩ . (3)
Assuming small coupling g, we may derive the final state
via perturbation theory
|Ψout⟩ = |Ψ(τ ↑ +∞)⟩
∫
= |Ψin⟩ − i
dτ ˆ Hint(τ) |Ψin⟩ + O(g 2 ) .(4)
This yields the excitation probability <strong>of</strong> the detector
P↑ = ⟨Ψout| ↑⟩ ⟨↑ |Ψout⟩
=
∫ ∫
dτ dτ ′ g(τ) g(τ ′ ) e iE(τ−τ ′ )
×
× ⟨0| ˆ ϕ (x[τ]) ˆ ϕ (x[τ ′ ]) |0⟩ . (5)
For simplicity, we consider the Wightmann function for a
massless scalar field in 3+1 dimensions
⟨0| ˆ ϕ (x) ˆ ϕ (x ′ ) |0⟩ = − 1
(2π) 2
1
(t − t ′ ) 2 − (r − r ′ , (6)
) 2
where the pole structure (at the light-cone) is understood in
such a way that the Fourier transform <strong>of</strong> the Wightmann
function only contains non-negative energies. Roughly
speaking, (t − t ′ ) 2 is replaced by (t − t ′ − iε) 2 with ε ↓ 0.
UNIFORM ACCELERATION
In the case <strong>of</strong> (eternal) uniform acceleration a, the detector
trajectory in terms <strong>of</strong> the proper time τ reads
t[τ] = 1
a
sinh(aτ) , x[τ] = 1
a
cosh(aτ) , y = z = 0 . (7)
Evaluating the two-point function along this trajectory
⟨0| ˆ ϕ (x[τ]) ˆ ϕ (x[τ ′ ]) |0⟩ = − 1
2(2π) 2
a2 cosh(a[τ − τ ′ , (8)
]) − 1
we find a stationary expression which is periodic along the
imaginary τ− = τ − τ ′ -axis and possesses double poles at
this axis
a[τ − τ ′ ] ∈ 2πi N . (9)

In the stationary case, it is useful to change the integration
variables in Eq. (5) from τ and τ ′ to τ− = τ − τ ′ and
τ+ = (τ + τ ′ )/2. For positive E, we may deform the τ−integration
into the upper complex half plane ℑ(τ−) > 0
and close the integration contour at infinity. As a result, the
τ−-integral is just the sum over all poles with ℑ(τ−) > 0.
Due to periodicity, all residuals are equal and so their sum
yields a geometric series
∞∑
{
P↑ ∝ exp − 2πnE
}
1
=
. (10)
a exp{2πE/a} − 1
n=1
Thus the excitation probability <strong>of</strong> the detector is given by a
thermal spectrum with the Unruh temperature (kB = 1)
TUnruh = a
. (11)
2π
CIRCULAR ACCELERATION
For comparison, let us consider the case <strong>of</strong> circular motion
where the magnitude <strong>of</strong> the acceleration remains constant
but its direction changes all the time. In this case, the
detector trajectory is given by
x[τ] = R sin(γωτ) , y[τ] = R cos(γωτ) , z = 0 ,
t[τ] = γτ =
τ
√ . (12)
1 − R2ω2 The two-point function along this trajectory
− 1
(2π) 2
⟨0| ˆ ϕ (x[τ]) ˆ ϕ (x[τ ′ ]) |0⟩ =
1
γ 2 (τ − τ ′ ) 2 − 4R 2 sin 2 (γω[τ − τ ′ ]/2)
(13)
is again stationary but no longer periodic. Now the poles
lie at γ(τ − τ ′ ) = ±2R sin(γω[τ − τ ′ ]/2). Introducing
the abbreviation φ = γω[τ − τ ′ ]/2, we have φ = β sin φ
where β = ±Rω.
Ultra-relativistic Case
The above transcendental equation for the location <strong>of</strong>
the poles simplifies if the detector velocity approaches the
speed <strong>of</strong> light. Using a Taylor expansion (assuming β > 0)
(
φ = β sin φ = β φ − φ3
6 + O(φ5 )
) , (14)
we find apart from the trivial pole at φ = 0
φ 2 β − 1 3
= 6 ≈ −
β γ2 ❀ τ − τ ′ √
12
= ±
γ2 i .
ω
(15)
This yields the excitation probability, see also [9]
{ √
12 E
P↑ ∝ exp −
γ2 }
. (16)
ω
Even though the spectrum is not exactly thermal, we may
identify an effective temperature (for large E) via
Teff ≈ γ2 ω
√ 12 ≈ a
√ 12 , (17)
where we have used that a ≈ γ 2 ω for β ↑ 1.
SIGNATURES
Instead <strong>of</strong> the excitation probability <strong>of</strong> the detector in
Eq. (5), we may also study the excitation probability <strong>of</strong> the
quantum field itself via the particle number operator ˆ Nk
Pk = ⟨Ψout| ˆ Nk |Ψout⟩
∫
∝
dτ g(τ) e iEτ+ikt[τ]+ik·r[τ]
2
. (18)
Comparing the structure <strong>of</strong> the two expressions for P↑ and
Pk, we find that for each excitation <strong>of</strong> the detector, exactly
one particle has been created [14]. Now, if the detector
decays again after some time, another particle is created.
So in total, the state <strong>of</strong> the detector is the same again – but
a pair <strong>of</strong> particles has been created.
Taking the limit <strong>of</strong> the time between excitation and decay
<strong>of</strong> the detector going to zero, we effectively have a scattering
event. In this limit, we may replace the detector by an
electron, which can scatter photons via Thomson (Compton)
scattering. Thus an accelerated electron would create
pairs <strong>of</strong> photons – which can be interpreted as a signature
<strong>of</strong> the Unruh effect [10, 11], see also [12, 13, 15]. As we
have seen before, this effect is not restricted to uniform acceleration,
but also occurs in the more general case.
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Zh. Eksp. Teor. Fiz. 43, 407 (1986).

Abstract
Can we detect ”Unruh radiation” in the high intensity lasers? ∗
Satoshi Iso † , Yasuhiro Yamamoto ‡ and Sen Zhang § , KEK, Tsukuba, Japan
An accelerated particle sees the Minkowski vacuum as
thermally excited, which is called the Unruh effect. Due
to an interaction with the thermal bath, the particle moves
stochastically like the Brownian motion in a heat bath. It
has been discussed that the accelerated charged particle
may emit extra radiation (the Unruh radiation [2]) besides
the Larmor radiation, and experiments are under planning
to detect such radiation by using ultrahigh intensity lasers
[3, 4]. There are, however, counterarguments that the radiation
is canceled by an interference effect between the
vacuum fluctuation and the radiation from the fluctuating
motion. In this and another reports [5], we review our recent
analysis on the issue <strong>of</strong> the Unruh radiation. In this
report, we particularly consider the thermalization <strong>of</strong> an
accelerated particle in the scalar QED, and derive the relaxation
time <strong>of</strong> the thermalization. The interference effect
is discussed separately in [5].
UNRUH EFFECT AND UNRUH
RADIATION
Quantum field theories in the space-time with horizons
exhibit interesting thermodynamic behavior. The most
prominent phenomenon is the Hawking radiation and the
fundamental laws <strong>of</strong> thermodynamics hold in the black
hole background. A similar phenomenon occurs for a
uniformly accelerated observer in the ordinary Minkowski
vacuum [6]. This is called the Unruh effect. If a particle
is uniformly accelerated in the Minkowski space with an
acceleration a, there is a causal horizon (the Rindler horizon)
and no information can be transmitted from the other
side <strong>of</strong> the horizon. Because <strong>of</strong> the existence <strong>of</strong> the Rindler
horizon, the accelerated observer sees the Minkowski vac-
uum as thermally excited with the Unruh temperature
TU = a
2πckB
= 4 × 10 −23
(
a
1 cm/s 2
)
[K]. (1)
Since the Unruh temperature is very small for ordinary
acceleration, it was very difficult to detect the Unruh effect.
But with the ultra-high intensity lasers, the Unruh effect
can be experimentally accessible. In the electro-magnetic
field <strong>of</strong> a laser with intensity I, the Unruh temperature is
given by
TU = 8 × 10 −11
√
I
2 [K]. (2)
1 W/cm
∗ The report is based on a talk by S.Zhang and [1].
† satoshi.iso@kek.jp
‡ yamayasu@post.kek.jp
§ zhangsen@post.kek.jp
The Extreme Light Infrastructure project [4] is planning to
construct Peta Watt lasers with an intensity as high as 5 ×
10 26 [W/cm 2 ]. The expected Unruh temperature becomes
more than 10 3 K. So it is time to ask ourselves how we can
experimentally observe such high Unruh temperature <strong>of</strong> an
accelerated electron in the laser field.
Chen and Tajima proposed that one may be able to detect
the Unruh effect by observing quantum radiation [2] from
the electron. It is called the Unruh radiation. Since a uniformly
accelerated electron feels the vacuum (with quantum
virtual pair creations <strong>of</strong> particles and anti-particles) as
thermally excited with the Unruh temperature, the motion
<strong>of</strong> the electron fluctuates and is expected to become thermalized(Fig.
1). This fluctuating motion <strong>of</strong> the electron
Figure 1: Stochastic trajectories <strong>of</strong> a uniformly accelerated
electron affected by quantum field fluctuations.
changes the acceleration <strong>of</strong> the electron and may produce
additional radiation (the Unruh radiation) to the ordinary
Larmor radiation. The rough estimation [2] suggested that
the strength <strong>of</strong> the Unruh radiation is much smaller than the
classical one by 10 −5 , but the angular dependence becomes
quite different. Especially in the direction along the acceleration
there is a blind spot for the Larmor radiation while
the Unruh radiation is expected to be radiated more spherically.
Hence they proposed to detect the Unruh radiation
by setting a photon detector in this direction.
The above argument seems intuitively correct, but there
are two problems that should be clarified. The first problem
is the thermalization time <strong>of</strong> the fluctuation. The electromagnetic
field <strong>of</strong> laser are not constant but oscillating. One
may approximate the electron’s motion around the turning
points by a uniform acceleration. This approximation is
valid only when the period <strong>of</strong> the laser is large enough compared
to the relaxation time (or thermalization time) <strong>of</strong> the
particle’s fluctuation. Using a stochastic approach, we ob-

tained the relaxation time <strong>of</strong> the fluctuation and showed that
the relaxation time is longer than the period <strong>of</strong> the laser. In
such a case, we must fully analyze the transient dynamics
<strong>of</strong> the fluctuation to calculate the radiation in the laser field.
The second problem is the interference effect. Since the
Unruh radiation originates in the interaction with the particle
with the quantum fluctuations <strong>of</strong> the vacuum, we cannot
neglect the interference <strong>of</strong> the Unruh radiation and the vacuum
quantum fluctuations. In a simpler model, it has been
known that the Unruh radiation is completely canceled by
the interference effect. The cancellation was shown for
the Unruh detector in both 1+1 and 3+1 dimensions[7, 8].
There was no calculation <strong>of</strong> the interference effect in the
case <strong>of</strong> the uniformly accelerated charged particle since the
calculation needs some technicalities. In the paper [1] we
calculated the interference effect for the charged particle
in the scalar QED and found that some <strong>of</strong> the Unruh radiation
is canceled by the interference effect, but the cancellation
occurs only partially. So we still have a possibility
to detect additional radiation from the uniformly accelerated
charged particle, but the complete understanding
needs more detailed analysis.
In the rest <strong>of</strong> this report we first briefly review the
stochastic model <strong>of</strong> a uniformly accelerated charged particle
and then show how the thermalization <strong>of</strong> the fluctuation
occurs by solving the stochastic equation. Finally we
briefly sketch the calculation <strong>of</strong> the radiation, particularly
put emphasis on the interference effect. More details <strong>of</strong> the
calculation <strong>of</strong> the interference effect and the Unruh radiation
are reviewed in another report <strong>of</strong> the same authors in
the proceedings [5].
THERMALIZATION
We consider the scalar QED. The model is analyzed
in [9] and here we briefly review the settings and the derivation
<strong>of</strong> the stochastic Abraham-Lorentz-Dirac (ALD) equation.
The system composes <strong>of</strong> a relativistic particle z µ (τ)
and the scalar field ϕ(x). The action is given by
∫
S[z, ϕ, h] = − m
∫
+
dτ √ ˙z µ ∫
˙zµ +
d 4 x 1 2
(∂µϕ)
2
d 4 x j(x; z)ϕ(x). (3)
The scalar current j(x; z) is defined as
∫
j(x; z) = e dτ √ ˙z µ ˙zµ δ 4 (x − z(τ)), (4)
We choose the parametrization τ to satisfy ˙z 2 = 1.
The Stochastic Equation
The equation <strong>of</strong> motion <strong>of</strong> the particle is given by
m¨z µ = F µ ∫
−
d 4 x
δj(x; z)
ϕ(x) (5)
δzµ(τ)
where we have added the external force F µ so as to accelerate
the particle uniformly; F µ = ma( ˙z 1 , ˙z 0 , 0, 0). Then
a classical solution <strong>of</strong> the particle (in the absence <strong>of</strong> the
coupling to ϕ) is given by
z µ
0
1 1
= ( sinh aτ, cosh aτ, 0, 0). (6)
a a
The equation <strong>of</strong> motion <strong>of</strong> the radiation field ∂ 2 ϕ = j is
solved by using the retarded Green function GR as
ϕ(x) = ϕh(x) + ϕI(x),
∫
ϕI(x) =
d 4 x ′ GR(x, x ′ )j(x ′ ; z) (7)
where ϕh is the homogeneous solution <strong>of</strong> the equation <strong>of</strong>
motion and represents the vacuum fluctuation. It is responsible
for the particle’s fluctuating motion under a uniform
acceleration. Inserting the solution (7) into (5), we have the
following stochastic equation for the particle
m¨z µ (τ) =F µ (z(τ)) − e⃗ω µ
(
∫
× ϕh(z(τ)) + e
dτ ′ GR(z(τ), z(τ ′ ))
(8)
)
,
where ⃗ωµ = ˙z ν ˙z [ν∂ µ]−¨zµ, which comes from the deviation
<strong>of</strong> the current
∫
(9)
d 4 δj(x; z)
x
δz µ (τ) f(x) = e⃗ωµf(x)| x=z(τ). (10)
The homogeneous part ϕh(z(τ)) <strong>of</strong> the scalar field describes
the Gaussian fluctuation <strong>of</strong> the vacuum, hence, the
first term in the parenthesis <strong>of</strong> (8) can be interpreted as random
noise to the particle’s motion
⟨ϕh(x)ϕh(x ′ )⟩ = − 1
4π2 1
(t − t ′ − iϵ) 2 . (11)
− r2 It is essentially quantum mechanical, but if it is evaluated
on a world line <strong>of</strong> a uniformly accelerated particle
x = z(τ), x ′ = z(τ ′ ), it behaves as the ordinary finite
temperature noise. The second term in the parenthesis <strong>of</strong>
(8) is a functional <strong>of</strong> the total history <strong>of</strong> the particle’s motion
z(τ ′ ) for τ ′ ≤ τ, but it can be reduced to the so
called radiation damping term <strong>of</strong> a charged particle coupled
with radiation field. It is generally nonlocal, but since the
Green function damps rapidly as a function <strong>of</strong> the distance
r, the term is approximated by local derivative terms. After
the mass renormalization, we get the following generalized
Langevin equation for the charged particle,
m ˙v µ − F µ − e2
12π (vµ ˙v 2 + ¨v µ ) = −e⃗ω µ ϕh(z) (12)
where v µ = ˙z µ . This equation is an analog <strong>of</strong> the ALD
equation for a charged particle interacting with the electromagnetic
field. The dissipation term is induced by the
effect <strong>of</strong> the backreaction <strong>of</strong> the particle’s radiation to the
particle’s motion. Note that, if the noise term is absent, the
classical solution (6) with a constant acceleration is still a
solution to the equation (12).

Equipartition Theorem
The stochastic equation (12) is nonlinear and difficult to
solve. Here we consider small fluctuations around the classical
trajectory induced by the vacuum fluctuation ϕh. Especially
we consider fluctuations in the transverse directions.
First we expand the particle’s motion around the
classical trajectory z µ
0 as
z µ (τ) = z µ
0 (τ) + δzµ (τ). (13)
The particle is accelerated along the x direction. In the following
we consider small fluctuation in transverse directions.
By expanding the stochastic equation (12), we can
obtain a linearized stochastic equation for the transverse
velocity fluctuation δv i ≡ δ ˙z i as,
mδ ˙v i = e∂iϕh + e2
12π (δ¨vi − a 2 δv i ). (14)
Performing the Fourier transformation with respect to the
trajectory’s parameter τ
δv i ∫
dω
(τ) =
2π δ˜vi (ω)e −iωτ , (15)
∫
dω
∂iϕh(τ) =
2π ∂iφ(ω)e −iωτ , (16)
the stochastic equation can be solved as
where
δ˜v i (ω) = eh(ω)∂iφ(ω), (17)
h(ω) =
1
−imω + e2 (ω 2 +a 2 )
12π
. (18)
The vacuum 2-point function along the classical trajectory
can be evaluated from (11) as
⟨∂iϕh(x)∂jϕh(x ′ )⟩| x=z(τ),x ′ =z(τ ′ )
= 1
2π 2
= a4
32π 2
δij
((t − t ′ − iϵ) 2 − r 2 ) 2
δij
sinh 4 ( a(τ−τ ′ . (19)
−iϵ)
2 )
It has originated from the quantum fluctuations <strong>of</strong> the vacuum,
but it can be interpreted as finite temperature noise
if it is evaluated on the accelerated particle’s trajectory [6].
The Fourier transformation <strong>of</strong> the symmetrized two point
function is evaluated as
where
⟨∂iϕ(x)∂jϕ(x ′ )⟩S = ⟨{∂iϕ(x), ∂jϕ(x ′ )}⟩/2
= 2πδ(ω + ω ′ )δijIS(ω), (20)
IS(ω) = 1
12π coth
(
πω
)
(ω
a
3 + ωa 2 ), (21)
which is an even function <strong>of</strong> ω. The correlator IS(ω)
should be regularized at the UV, which is large ω or short
proper time difference, where quantum field theoretic effects
<strong>of</strong> electron become important. Full QED treatment is
necessary there.
For small ω, it is expanded as
IS(ω) = a
12π 2 (a2 + O(ω 2 )). (22)
The expansion corresponds to the derivative expansion
⟨∂iϕh(x)∂jϕh(x ′ )⟩S = a3
12π 2 δijδ(τ − τ ′ ) + · · · . (23)
With this expansion, the expectation value <strong>of</strong> the square <strong>of</strong>
the transverse velocity fluctuation can be evaluated as
⟨δv i (τ)δv j (τ ′ )⟩S
= e 2
∫ ′ dωdω
(2π) 2 ⟨∂iφ(ω)∂jφ(ω ′ )⟩S h(ω)h(ω ′ )e −i(ωτ+ω′ τ ′ )
∼ e 2 ∫
dω
δij
24π3 a3e−iω(τ−τ ′ )
) . (24)
(ω2 + a2 ) 2
(mω) 2 + ( e 2
12π
Here we consider the acceleration <strong>of</strong> the electron to be at
the order 0.1 eV, which is much smaller than the electron
mass 0.5 MeV. With the assumption m ≫ a, one can evaluate
the integral and get the following result,
m
2 ⟨δvi (τ)δv j (τ)⟩ = 1 a
2
2πc δij
( 1 + O(a 2 /m 2 ) ) . (25)
Here we have recovered c and . This gives the equipartition
relation for the transverse momentum fluctuations in
the Unruh temperature TU = a/2πc.
Relaxation Time
The thermalization process <strong>of</strong> the stochastic equation
(14) can be also discussed. For simplicity, we approximate
the stochastic equation by dropping the second derivative
term. Then it is solved as
δv i (τ) =e −Ω−τ δv i (0)
+ e
m
∫ τ
where Ω− is given by
0
dτ ′ ∂iϕ(z(τ ′ ))e −Ω−(τ−τ ′ ) , (26)
Ω− = a2 e 2
12πm
(27)
The relaxation time is τR = 1/Ω−. The velocity square
can be also calculated as
⟨δv i (τ)δv j (τ)⟩ =e −2Ω−τ δv i (0)δv j (0)
+ aδij
2πm (1 − e−2Ω−τ ). (28)
For τ → ∞, it approaches the thermalized average (25).
The relaxation time in the proper time can be estimated,
for the parameter a = 0.1 eV and m = 0.5 MeV,
τR = 12πm
a 2 e 2 ∼ 10−5 sec. (29)

Let’s compare this relaxation time with the laser frequency.
The planned wavelength <strong>of</strong> the laser at ELI is around
10 3 nm and the oscillation period <strong>of</strong> the laser field is very
short; 3 × 10 −15 seconds. So the relaxation time is much
longer and the charged particle cannot become thermalized
during each oscillation. Hence the assumption <strong>of</strong> the uniform
acceleration in the laser field is not good. Even in
such a situation, if the electron is accelerated in the laser
field for a long time, an electron may feel an averaged temperature.
The position <strong>of</strong> the particle in the transverse directions
also fluctuates like the ordinary Brownian motion in a
heat bath. The mean square <strong>of</strong> the transverse coordinate
R 2 (τ) = ∑
i=y,z ⟨(zi (τ) − z i (0)) 2 ⟩ is calculated as
R 2 (τ) = 2D
(
τ − 3 − 4e−Ω−τ + e −2Ω−τ
2Ω−
)
. (30)
with the diffusion constant D = 2TU/(Ω−m) =
12/ae 2 ∼ 8 × 10 4 m 2 /s. In the Ballistic region where
τ < τR, the mean square becomes R 2 (τ) = 2TUτ 2 /m
while in the diffusive region (τ > τR), it is proportional to
the proper time as R 2 (τ) = 2Dτ. As the ordinary Brownian
motion, the mean square <strong>of</strong> the particle’s transverse
position grows linearly with time. If it becomes possible to
accelerate the particle for a sufficiently long period, it may
be possible to detect such a Brownian motion in future laser
experiments.
RADIATION AND INTERFERENCE
Now we are ready to calculate the radiation emanated
from the uniformly accelerated charged particle. An important
point is the interference effect between the quantum
fluctuations <strong>of</strong> the vacuum ϕh and the radiation induced by
the fluctuating motion in the transverse directions ϕI. First
let’s consider the two point function
⟨ϕ(x)ϕ(x ′ )⟩ − ⟨ϕh(x)ϕh(x ′ )⟩ (31)
= ⟨ϕI(x)ϕh(x ′ )⟩ + ⟨ϕh(x)ϕI(x ′ )⟩ + ⟨ϕI(x)ϕI(x ′ )⟩.
The Unruh radiation estimated in [2] corresponds to calculating
the 2-point correlation function <strong>of</strong> the inhomogeneous
terms ⟨ϕIϕI⟩. (The same term also contains the
Larmor radiation.) However, this is not the end <strong>of</strong> the
story. As it has been discussed in [7], the interference terms
⟨ϕIϕh⟩ + ⟨ϕhϕI⟩ may possibly cancel the Unruh radiation
in ⟨ϕIϕI⟩ after the thermalization occurs. This is shown
for an internal detector, but it is not obvious whether the
same cancellation occurs for the case <strong>of</strong> a charged particle
we are considering.
The energy-momentum tensor <strong>of</strong> the radiation field can
be obtained from the 2-point function
⟨Tµν⟩ = ⟨: ∂µϕ∂νϕ − 1
2 gµν∂ α ϕ∂αϕ :⟩S. (32)
It is written as a sum <strong>of</strong> the classical part and the fluctuating
part Tµν = Tcl,µν + Tfluc,µν. The classical part corresponds
to the Larmor radiation while the fluctuating part
contains both <strong>of</strong> the Unruh radiation and the interference
terms.
In [1] we calculated the 2-point function including the
interference term, and obtained the energy-momentum tensor.
The result we have obtained is summarized in [5] in
this proceedings. Some terms are partially canceled but not
all. Hence, it seems that the uniformly accelerated charged
particle emits additional radiation besides the Larmor radiation.
The remaining terms after the partial cancellation
are proportional to a 3 and suppressed compared to the Larmor
radiation. It has a different angular distribution, but the
additional radiation also vanishes in the forward direction.
SUMMARY
We have systematically studied the thermalization <strong>of</strong> a
uniformly accelerated charged particle in the scalar QED
using the stochastic method, and calculated the radiation
by the particle. Two main messages <strong>of</strong> are
1. ”Long relaxation time compared to the laser period”
2. ”Importance <strong>of</strong> the interference”
The fluctuation <strong>of</strong> the particle doesn’t become thermalized
during the period <strong>of</strong> the laser, and we need to study transient
dynamics to obtain the radiation from an electron accelerated
in the oscillating laser field. The issue <strong>of</strong> the interference
terms are more subtle. Since the particle fluctuation
originates in the quantum vacuum fluctuation <strong>of</strong> the
radiation field, it can be by no means neglected. Our result
shows that the interference term partially cancels the Unruh
radiation, but some <strong>of</strong> them survives. The remaining
Unruh radiation is smaller compared to the Larmor radiation
by a factor a (acceleration) and has a different angular
distribution. In this sense, it is qualitatively consistent with
the proposal [2]. But as we briefly review in [5], the additional
radiation also vanishes in the forward direction. and
it seems difficult to detect such additional radiation experimentally
so far as the transverse fluctuation is concerned.
Please beware that the longitudinal fluctuations which we
have not calculated yet (because <strong>of</strong> technical difficulties related
to a choice <strong>of</strong> gauge) may change the situation.
REFERENCES
[1] S. Iso, Y. Yamamoto and S. Zhang, arXiv:1011.4191 [hep-th].
[2] P. Chen and T. Tajima, Phys. Rev. Lett. 83 (1999) 256.
[3] P.G. Thirolf, et.al. Eur. Phys. J. D 55, 379-389 (2009).
[4] http://www.extreme-light-infrastructure.eu/
[5] S. Iso, Y. Yamamoto and S. Zhang, in the same proceedings,
“Unruh radiation and Interference effect”
[6] W. G. Unruh, Phys. Rev. D 14, 870 (1976).
[7] D. J. Raine, D. W. Sciama and P. G. Grove, Proc. R. Soc.
Lond. A (1991) 435, 205-215
[8] A. Raval, B. L. Hu, J. Anglin, Phys. Rev. D 53 (1996) 7003.
[9] P. R. Johnson and B. L. Hu, arXiv:quant-ph/0012137. Phys.
Rev. D 65 (2002) 065015 [arXiv:quant-ph/0101001].

Abstract
Quantum fields and static interactions in accelerated frames
Frieder Lenz
Institute for Theoretical Physics III, University <strong>of</strong> Erlangen-Nürnberg
91058 Erlangen, Germany
Properties <strong>of</strong> quantum fields in Rindler space or equivalently
in accelerated frames are explored. Consequences <strong>of</strong>
the inertial forces for the kinematics and the dynamics <strong>of</strong>
scalar particles and photons are discussed and results <strong>of</strong> investigations
<strong>of</strong> interaction energies generated by scalar and
vector particle exchange in Rindler space are presented.
INTRODUCTION
Quantum fields observed in accelerated and in inertial
frames are related to each other by a coordinate transformation.
Nevertheless their properties differ significantly
which is due to the existence <strong>of</strong> a horizon in accelerated
frames (Rindler space). In the following I will present the
results <strong>of</strong> studies [1, 2] <strong>of</strong> both kinematical and dynamical
properties <strong>of</strong> quantum fields. The kinematics <strong>of</strong> noninteracting
quantum fields in Rindler space [3, 4, 5] and
their relation to fields in Minkowski space together with the
interpretation in terms <strong>of</strong> finite temperature quantum fields
[6] have been the subject <strong>of</strong> many investigations [7]. Here
the focus will be on the peculiar properties <strong>of</strong> the spectrum
<strong>of</strong> the particles in Rindler space and their signatures in the
Unruh radiation <strong>of</strong> photons in comparison to blackbody radiation.
The impact <strong>of</strong> the unusual kinematics in accelerated
frames on the dynamics will be demonstrated in the
discussion <strong>of</strong> static interactions.
KINEMATICS
A uniformly accelerated observer (acceleration a) at fixed
coordinates transverse to the acceleration x⊥ moves along
a hyperbola [8]
x 2 − t 2 = 1
a 2 , x⊥ = 0 . (1)
Quantum fields as seen by accelerated observers are most
conveniently described in terms <strong>of</strong> the coordinates obtained
by transformation to the rest frame t, x, x⊥ → τ, ξ, x⊥
t(τ, ξ) = 1
a eaξ sinh aτ , x(τ, ξ) = 1
a eaξ cosh aτ . (2)
By construction, ξ = 0 corresponds to the hyperbolic motion
(1). More generally, a particle at rest in the observers
system at ξ = ξ0 =const. corresponds to the uniformly
accelerated motion in Minkowski space with acceleration
a exp{−aξ0}. Trajectories <strong>of</strong> uniformly accelerated particles
for different values <strong>of</strong> ξ0 are shown in Fig. 1 together
with the lines τ =const. .
II
t
10
5
+
III x I
10 5 5 10
5
10
IV
→
ξ = −τ = −∞
→
←
ξ = const.
← τ = const.
ξ = τ = −∞
Figure 1: Kinematics <strong>of</strong> uniform acceleration
The coordinate transformation (2) is not one-to-one. The
coordinates −∞ < τ, ξ < ∞ cover only one quarter <strong>of</strong> the
Minkowski space, the right ”Rindler wedge”R+ (region I)
R± = x µ |t| ≤ ±x . (3)
Upon reversion <strong>of</strong> the sign <strong>of</strong> x in Eq. (2) the left Rindler
wedge R− (region III) is covered by a corresponding
parametrization. As illustrated in the Figure, the boundaries
x = |t| corresponding to ξ = ±τ = −∞ form an
event horizon. The light cones shown in the Figure indicate
that in region I signals can be transmitted to region II
but not received from it. Signals received from region IV
appear to have originated from the horizon ξ = τ = −∞.
The space-time defined by the coordinate transformation
(2) is called Rindler space and its metric is given by
ds 2 = gµν(ξ)dx µ dx ν = e 2aξ (dτ 2 − dξ 2 ) − dx 2 ⊥ . (4)
The Rindler metric derives its importance from the fact that
essentially any static metric which possesses a horizon can
be approximated near the horizon by the Rindler metric.
This is the case for instance for the Schwarzschild metric.
Acceleration and Schwarzschild radius, or black hole mass,
are related by
a = 1 1
= . (5)
2R 4GM
QUANTUM FIELDS IN RINDLER SPACES
Quantization <strong>of</strong> scalar fields
Quantization <strong>of</strong> <strong>of</strong> a non-interacting scalar field in Rindler
space with the action

S = 1
2
dτ dξ d d−1
x⊥ (∂τ φ) 2 − (∂ξφ) 2
−(m 2 φ 2 + (∂⊥φ) 2 ) e 2aξ , (6)
is standard. The expansion <strong>of</strong> the fields in terms <strong>of</strong> the
normal modes (proportional to the Mc Donald functions)
<strong>of</strong> the associated wave equation is given by
dω
φ(τ, ξ, x⊥) ∼ √ d
2ω d−1 k⊥(a(ω, k⊥)e −iωτ+ik⊥x⊥
m⊥ e aξ
, m 2 ⊥ = (m 2 + k 2 ⊥)/a 2 , (7)
+h.c.) Ki ω
a
and the normalization is chosen such that the commutator
<strong>of</strong> creation and annihilation operators a (†) (ω, k⊥) is standard.
The repulsive exponential barrier in uniformly accelerated
frames is <strong>of</strong> similar origin as the centrifugal barrier
in rotating frames. It prevents unlimited propagation <strong>of</strong> the
wave in positive ξ direction. This repulsion accounts for the
fact that a particle moving with arbitrary constant speed in
Minkowski space is seen by the accelerated observer to approach
ξ = −∞ and the speed <strong>of</strong> light for large times τ. In
the accelerated frame, the transverse velocity <strong>of</strong> the particle
vanishes exponentially for large times (∼ exp{−2aτ})
as a result <strong>of</strong> the forever increasing time dilation induced
by the acceleration. Since m 2 ⊥
appears in the action (6) as
a coupling constant <strong>of</strong> the exponential “potential” , the energy
eigenvalues <strong>of</strong> the normal modes do not depend on the
transverse momentum and the mass, though the eigenfunctions
do. This is reminiscent <strong>of</strong> the degeneracy <strong>of</strong> the Landau
levels <strong>of</strong> a particle moving in a constant magnetic field.
The high degeneracy <strong>of</strong> the eigenstates <strong>of</strong> the Hamiltonian,
including the ground state, is due to the inertial force and
has far reaching consequences. It indicates the presence <strong>of</strong>
a symmetry <strong>of</strong> the Rindler space Hamiltonian. In [1] the
invariance <strong>of</strong> the Hamiltonian under scale transformations
valid even in the presence <strong>of</strong> a mass term has been identified
as the source <strong>of</strong> the degeneracy.
The Unruh heat bath
Starting point for establishing the relation between observables
in inertial and accelerated frames is the identity <strong>of</strong> the
(scalar) fields (φ) and ( ˜ φ) in the two frames
φ(τ, ξ, x⊥) = ˜
φ(t, x) . (8)
t,x=t,x(τ,ξ)
Projection <strong>of</strong> this equation onto the Rindler space normal
modes (6) yields the following relation (Bogoliubov transformation)
between the creation and annihilation operators
in the two frames
a(Ω, k⊥) =
a sinh π Ω
a
1
∞
−∞
dk Ω i √ e a
4πωk
βk
· e πΩ
πΩ
2a −
ã(k, k⊥) + e 2a ã † (k, −k⊥)
. (9)
Observations in the accelerated frame are performed in the
Minkowski vacuum |0M 〉 rather than in the Rindler space
vacuum. A fundamental quantity is the number <strong>of</strong> particles
measured in the accelerated frame which, with the help <strong>of</strong>
(9), is found to be
〈0M |a † (Ω, k⊥)a(Ω ′ , k ′ ⊥)|0M 〉
1
=
Ω
e2π a − 1 δ(Ω − Ω′ )δ(k⊥ − k ′ ⊥) . (10)
In the accelerated frame, a thermal distribution <strong>of</strong> (Rindler)
particles is observed with the temperature determined by
the acceleration
T = a
. (11)
2π
For a black hole (cf. Eq. (5)) this temperature agrees with
the black hole temperature
TBH =
1
8πGM .
Although the above derivation makes use <strong>of</strong> the properties
<strong>of</strong> non-interacting fields, relations between observables in
accelerated frames and at finite temperature can be derived
for interacting fields (cf. [9] for a derivation within the path
integral approach). Here we consider the two point function
<strong>of</strong> a self-interacting scalar field and use its invariance
under Lorentz transformations. The relation (8) between
the fields in Rindler and Minkowski space implies that for
arbitrary points in the right Rindler wedge (cf. Fig. 1) the
values <strong>of</strong> the 2-point functions in Minkowski space and in
Rindler space are equal. The two point functions depend
only on (x − x ′ ) 2 . We express the distance in terms <strong>of</strong> the
Rindler space coordinates
(x − x ′ ) 2 = 2ea(ξ+ξ′ )
a2 ′
cosh a(τ − τ ) − cosh η , (12)
with
cosh η = 1 +
and obtain
aξ aξ e − e ′2 2 + a x⊥ − x ′ 2 ⊥
2e a(ξ+ξ′ )
, (13)
D (x − x ′ ) 2 = D(τ − τ ′ , ξ, ξ ′ , x⊥ − x ′ ⊥)
= i〈0M
T φ(τ, ξ, x⊥)φ(τ ′ , ξ ′ , x ′ ⊥) 0M 〉
a(ξ+ξ 2e
= D
′ )
a2 ′
cosh a(τ − τ ) − cosh η
. (14)
After a Wick rotation to imaginary Rindler time,
τ → τE = −iτ ,
the two point function is periodic with period
β = 2π
a ,
and therefore is a finite temperature 2-point function with
T = 1/β, in agreement with the result (11) for noninteracting
fields.

The relation between acceleration and temperature is subtle.
The Rindler space propagator defined with respect
to the Minkowski ground state coincides with the Rindler
space finite temperature propagator with the value <strong>of</strong> the
temperature determined by the acceleration a (cf. Eq. (11)).
This identity makes also manifest that a change in the acceleration
a does not correspond to a change in temperature
only. The acceleration appears as temperature in the
Boltzmann factor and as a parameter <strong>of</strong> the Hamiltonian. It
controls the range <strong>of</strong> the exponential barrier (cf. Eq. (6)).
Accelerated versus finite temperature photons
Canonical quantization <strong>of</strong> the electromagnetic field in
Rindler space is straightforward though technically involved
[1]. If carried out in Weyl gauge, with the longitudinal
fields determined by the Gauß law, only physical, i.e.,
transverse degrees <strong>of</strong> freedom appear. The relation (10) can
be shown to apply separately for the two polarizations. Although
Eq. (10) is <strong>of</strong> the same structure as the corresponding
expression for the number <strong>of</strong> Minkowski space photons
at finite temperature, the different dispersion law <strong>of</strong> Rindler
photons
∂ ω
∂ k⊥
= 0 gives rise to significant changes in Un-
ruh as compared to blackbody radiation. I illustrate this by
a discussion <strong>of</strong> the expectation value <strong>of</strong> the energy density
<strong>of</strong> the Unruh radiation (g(ξ) = det[gµν(ξ)]) ,
1
|g(ξ)| 〈0M | : HE(ξ, x⊥) + HB(ξ, x⊥) : |0M 〉
= 1
e−4aξ
π2 ωdω ω2 + a2 e ω
11π2
=
Ta − 1 15 T 4 ξ . (15)
As in finite temperature field theory, divergences in H are
avoided by normal ordering with respect to the (Rindler)
ground state. The integrand <strong>of</strong> the frequency integral in
Eq. (15) is shown in Fig. 2 and compared with the corresponding
energy density <strong>of</strong> the blackbody radiation. This
Figure demonstrates the qualitative difference in the spectrum
<strong>of</strong> photons at finite temperature in Minkowski and
Rindler space. The degeneracy with respect to the transverse
momenta gives rise to a non-vanishing energy density
at vanishing frequency.
The quantity Tξ in (15) denotes the Tolman temperature,
Tξ = a
2π e−aξ . (16)
With the local acceleration (cf. the Kinematics section) also
the local temperature Tξ varies with the coordinate ξ. It sat-
√
isfies Tolman’s law [10], Tξ g00 = const., which is valid
in any static space-time. The Tolman temperature is up to
a factor <strong>of</strong> 2π given by the inverse <strong>of</strong> the proper distance to
the horizon
dH(ξ) = 1
a eaξ , (17)
i.e. the Tolman temperature diverges when approaching the
horizon. The spatial variation <strong>of</strong> temperature is necessary
for thermal equilibrium. It compensates the red or blue
shifts <strong>of</strong> photons moving away from or towards the horizon
respectively.
0.15
0.10
0.05
Unruh
Blackbody
0.5 1.0 1.5 2.0
Figure 2: Energy density <strong>of</strong> blackbody radiation at T =
a/2π and <strong>of</strong> the Unruh radiation at acceleration a as a function
<strong>of</strong> the frequency ω in units <strong>of</strong> a.
STATIC INTERACTIONS
ω/a
The propagator <strong>of</strong> a massless non-interacting scalar field in
Rindler space is given by (cf. Eq. (14))
D(x, x ′ ) =
1
4iπ 2 (x − x ′ ) 2 − iδ = D(τ, ξ, ξ′ , x⊥ − x ′ ⊥)
= a2 e −a(ξ+ξ′ )
8iπ 2
Integration over the Rindler time τ,
˜D(ξ, ξ ′ , x⊥ − x ′ ⊥) =
=
∞
−∞
1
. (18)
cosh aτ − cosh η − iδ
a
4πe a(ξ+ξ′ )
dτD(τ, ξ, ξ ′ , x⊥ − x ′ ⊥)
1
sinh η
1 + iη
π
, (19)
yields the static interaction between two scalar sources
Vsc = −κ 2 e a(ξ+ξ′ )
D˜ ′
ξ, ξ , x⊥ − x ′ =
⊥
− aκ2
1 +
4π sinh η
iη
,
π
(20)
with the coupling constant κ. The exponential factors arise
as a consequence <strong>of</strong> the volume factor |g(ξ)| and <strong>of</strong> the
transformation <strong>of</strong> proper time <strong>of</strong> the accelerated sources to
Rindler time.
The static interaction energy is a complex quantity. The
static propagator satisfies the Poisson equation for a pointlike
source in Rindler space. Since the source is real, the
imaginary part <strong>of</strong> the propagator satisfies the corresponding
(homogeneous) Laplace equation and can be written
therefore as a linear superposition <strong>of</strong> zero modes [2]
Im ˜ D(ξ, ξ ′ , x⊥ − x ′ a η
⊥) =
4π2ea(ξ+ξ′ =
) sinh η 1
4π3a
· d 2 ′
ik⊥(x⊥−x
k⊥e ⊥ )
k⊥
K0
a eaξ
k⊥
K0
a eaξ′ . (21)
Thus the imaginary part is due to on-shell propagation <strong>of</strong>
zero-energy massless particles. Unlike in Minkowski space

Abstract
Shining Light through Walls: en Route towards a new
Particle Physics Frontier
Axel Lindner for the ALPS Collaboration, DESY, Hamburg, Germany
“Light-shining-through-a-wall” experiments search for
Weakly Interacting Slim Particles (WISPs). Long standing
quest in particle physics and cosmology may find their
solution in the discovery <strong>of</strong> this new species <strong>of</strong> particles.
In the recent years experiments have achieved unprecedented
sensitivities. The experience gained provides a firm
foundation for future enterprises into unexplored parameter
spaces.
INTRODUCTION TO AXIONS AND WISPS
Since about four decades the so called Standard Model
<strong>of</strong> elementary particle physics faces a triumphant success.
There is not a single earth-bound experiment which really
questions the model. In contrast, precision tests have
probed its constituents and forces to the per mill level or
better. However, evidence is mounting that the known
constituents <strong>of</strong> matter and forces do not fully describe the
world around us. Such arguments arise from astrophysical
and cosmological observations as well as from theoretical
considerations. There are strong convictions among
scientists that new experiments at the high energy frontier
at LHC will provide insight into physics beyond the Standard
Model. Although theoretically well motivated, focusing
the search for new physics onto highest available energies
neglects evidences pointing at the opposite energy
scale. Extensions <strong>of</strong> the Standard Model may manifest
themselves also at meV energy scales, nine orders <strong>of</strong> magnitude
below the mass <strong>of</strong> the electron.
Generally, new very light and very weakly interacting
particles denoted as WISPs (Weakly Interacting Slim Particles)
occur naturally in string-theory-motivated extensions
<strong>of</strong> the Standard Model. There could be bosons and
fermions, charged and uncharged particles. WISPs may interact
with ordinary matter via the exchange <strong>of</strong> very heavy
particles related to very high energy scales and thus give
insight into physics at highest energy scales. The reader is
referred to [1, 2, 3] and references therein for a more detailed
view.
One prime example for a WISP is the QCD axion [4, 5,
6] invented to explain the vanishing electric dipole moment
<strong>of</strong> the neutron, which is a signature <strong>of</strong> CP conservation <strong>of</strong>
the strong interaction. From astrophysical observations its
mass should be below about 1 eV. For the QCD axion such
a low mass implies very weak interactions with the other
constituents <strong>of</strong> the Standard Model [7]. It is striking that a
QCD axion with a mass around 1 µeV is a perfect candidate
for cold dark matter in the Universe [8, 9]. A discovery
<strong>of</strong> the axion could solve long lasting questions <strong>of</strong> particle
physics and cosmology simultaneously. It is worthwhile
to note that also dark energy might be attributed to new
physics at the meV scale [10].
Besides theoretical considerations the existence <strong>of</strong> very
light axion-like particles is suggested by different astrophysical
observations. For example, the cooling <strong>of</strong> white
dwarfs can be modeled significantly better if an additional
energy loss due to axion-like particles is taken into account
[11]. The surprisingly high transparency <strong>of</strong> the Universe
to TeV photons from AGNs at cosmological distances
may be explained by back and forth oscillations <strong>of</strong> photons
into axion-like particles [12]. The heating <strong>of</strong> the solar
corona is not understood, but may be attributed to an energy
flow mediated by axion-like particles [13].
SEARCHING FOR WISPS
WISPs and especially ALPs are searched for in astrophysics
phenomena and laboratory experiments. At present
for most <strong>of</strong> the WISPs the most stringent limits on their existence
originate from astrophysics considerations. The existence
<strong>of</strong> WISPs would for example open up new energy
loss channels for hot environments in stars and thus shorten
lifetimes or cooling cycles [14]. Limits on axions are also
derived from cosmology [15]. Direct searches for axionlike
particles produced in the sun [16] or as constituents <strong>of</strong>
galactic dark matter [17] have greatly progressed in recent
years and reached impressive sensitivities.
However, interpretations <strong>of</strong> astrophysics data are always
hampered by the uncontrolled production mechanism <strong>of</strong>
WISPs. Effective theories have been presented, where the
production <strong>of</strong> some WISP species is suppressed in hot environments
[18, 19]. If such scenarios are true, astrophysics
experiments might fail to detect WISPs while laboratory
experiments could open up this new physics window. Literally,
astrophysics deals with “astronomical” or “cosmological”
distances on the one hand and microscopic distances
in hot dense plasmas on the other hand. Intermediate distances
are only probed in the laboratory.
The are numerous experimental efforts to probe for
WISPs in the laboratory. Typically they are searched for
by looking for new effects in gravitational or QED environments.
The latter ones comprise atomic physics like Lamb
shift, positronium decay, Casimir forces or photon-photon
interaction. Only photon-photon processes are addressed
in the following.
Photon-WISPs interactions
WISPs may interact with photons in different manners as
shown in Fig. 1(see [1]). These interaction may give rise to

=
γ WISP
; ;
ALP HP(mγ ′ > 0)
...
MCP
HP(mγ ′ = 0)
Figure 1: A collection <strong>of</strong> some Feynman diagrams responsible for the mixing term between photons and different hypothetical
“weakly interacting slim particles” (WISPs). Photon oscillations into axion-like particles (ALPs) and massless
hidden photons (HPs) via mini-charged particles (MCP) require the presence <strong>of</strong> a background electromagnetic field, denoted
by crossed circles.
subtle polarization phenomena [20, 21], but also to a very
spectacular “light-shining-through-a-wall” effect (Fig. 2).
γ γ
WISP
Figure 2: Schematic overview <strong>of</strong> a “light shining through a
wall experiment”. The gray blob indicates the mixing term
between photons and the WISP.
In the first part <strong>of</strong> such an experiment WISPs are produced
from intense laser light, either by interaction with
a strong magnetic field or by kinetic mixing. This first
part is separated by a light-tight wall from the second part.
Only WISPs can traverse the wall due to their very low
cross-sections. Behind the wall they could convert back
into photons with exactly the same properties as the light
generating the WISPs. This gives the impression <strong>of</strong> “lightshining-through-a-wall”
(LSW).
The conversion <strong>of</strong> the incident photons to axions or
axion-like particles ϕ in the presence <strong>of</strong> a magnetic field
is governed by the Primak<strong>of</strong>f effect [22]. Behind the absorber,
some <strong>of</strong> these ALPs will reconvert via the inverse-
Primak<strong>of</strong>f process into photons with the initial properties.
In a symmetric LSW setup the probability <strong>of</strong> the Primak<strong>of</strong>f
transition Pγ→ϕ is the same as for the inverse-Primak<strong>of</strong>f
Pϕ→γ. Therefore the LSW probability is just the square <strong>of</strong>
Pγ→ϕ = g 2 B 2 E 2 /(4m 4 ϕ ) · sin2 (m 2 ϕ
L/(4E)) with B the
magnetic field strength, L the length <strong>of</strong> the conversion region
and E the photon energy. Mass (mϕ) and two photon
coupling (g) <strong>of</strong> the ALPs are assumed to be uncorrelated.
With βϕ denoting the velocity <strong>of</strong> the ALP and q = pγ − pϕ
one achieves:
Pγ→ϕ→γ = 1
16β2 (gBL)
ϕ
4
( sin(qL/2)
qL/2
For qL

Figure 3: Schematic view <strong>of</strong> the ALPS LSW experiment. PD denotes various photo detectors, CM the coupling mirror
and EM the end mirror <strong>of</strong> the resonant cavity. See the text and [23] for a description.
tion, cf. Fig. 3. ALPS is the first experiment which successfully
exploits a large-scale optical resonator for WISP
searches. The main parts and their basic functionality are
explained in [1] and reference therein. During the measurement
period in the year 2009 a continuously circulating
power inside the ALPS production region <strong>of</strong> around 1.2 kW
at 532 nm was achieved [23].
ALPS also exploits successfully a new method to cover
the gaps in the sensitivity for higher masses, where the ALP
wave runs out <strong>of</strong> phase w.r.t. the phase <strong>of</strong> the laser beam,
cf. Fig. 4. Introducing Ar gas at a pressure <strong>of</strong> 0.18 mbar
changes the photon momentum and tunes therefore the refraction
index. In the ALPS setup the γ−ALP relative
phase velocity increases thereby to have an extra half oscillation
length. Even if the sensitivity is lowered compared
to vacuum conditions this helps to cover the higher mass
gaps.
ALPS RESULT
ALPS took around 50 data sets (1 h frames) under different
experimental conditions: with magnet on or <strong>of</strong>f, laser
polarization parallel or perpendicular to the magnetic field
and different gas pressures. Details on the methodology
and analysis are described in [1, 23]. From the non observation
<strong>of</strong> any WISP signal a 95 % confidence level on
the conversion probabilty was obtained, ranging between
Pγ→ϕ→γ = 1 − 10 × 10 −25 for the different experimental
setups. Fig. 4 shows the ALPS results for pseudoscalar and
scalar axion-like particles together with the results obtained
from BMV [25], BFRT [26], GammeV [27], LIPSS [28]
and OSQAR [29]. The gaps at higher masses are covered
by the ALPS gas runs as described above. ALPS provide
now the most stringent laboratory bounds on ALPs in the
sub-eV mass range.
Also for hidden photon and minicharged particle
searches ALPS has achieved the highest sensitivity in the
laboratory, cf. Fig. 5. The ALPS LSW results on hidden
photon search fills the gap between lab searches for deviations
from Coulomb’s law and astrophysical bounds.
Remarkable, with the achieved sensitivity ALPS almost
completely rules out the hint <strong>of</strong> WMAP and large-scalestructure
probes with non-standard radiation density contribution
due to hidden photons, cf. [23] and references
therein.
OUTLOOK
The present generation <strong>of</strong> laboratory experiments
searching for WISPs has not found any evidence for those
elusive particles. It’s worth mentioning that this probes
for new physics (if ALPs exist) at the 100 TeV scale 1 already.
The physics case for WISPs is still strengthening
due to ongoing theoretical studies and puzzling observations
in astrophysics as mentioned above. The QCD axion
itself remains a “holy grail” for particle physics. To solve
the CP problem in QCD and understand Dark Matter with
one stroke is very alluring. Perhaps even Dark Energy will
find its explanation in the WISP sector via the detection
<strong>of</strong> “chameleons” [31]. Due to these prospects attempts are
ongoing to largely improve all components <strong>of</strong> LSW experiments.
Usually LSW experiments shine laser light through long
and tight magnet bores. With the help <strong>of</strong> resonant optical
cavities effective laser light powers around 100 kW might
be possible in future. The ALPS collaboration is presently
preparing such a setup.
ALPS has used one spare HERA dipole magnet to
achieve the results mentioned above. At DESY we study
now the possibility <strong>of</strong> an experiment with up to 20+20
HERA dipoles.
Most <strong>of</strong> the present-day LSW experiments use commercial
CCD cameras to search for reconverted photons from
WISPs behind the wall. In the future the detection sensitivity
might be enhanced considerably by using transition
edge sensors (TES) [32, 33]. Here a sensor is cooled down
to about 100 mK and operated in the transition region be-
1 The axion-to-photon coupling g is given by g = α · gγ/πfα, where
fα denotes the new energy scale and gγ a factor derived from theory
expected to vary by about an order <strong>of</strong> magnitude for the QCD axion.

Figure 4: Exclusion limit (95% C.L.) for pseudoscalar
(top) and scalar (bottom) axion-like particles obtained by
the ALPS experiment from vacumm and gas runs together
with the results from various other LSW experiments [23],
see the text for details. Dashed and dotted lines show the
bounds derived from the PVLAS measurement on ALP induced
dichroism and birefringence [30].
tween a superconducting and normal conducting state. Due
to the very low heat capacity <strong>of</strong> such a state the energy deposit
<strong>of</strong> a single photon results in a significant temperature
rise and is well measurable. TES detectors allow for essentially
background-free counting <strong>of</strong> individual photons,
register their arrival times and allow to estimate their energies.
A new technology will be exploited to boost the sensitivity
<strong>of</strong> LSW experiments even further: the challenge is
to realize a second resonant optical cavity in the part <strong>of</strong> an
experiment behind the wall. The idea <strong>of</strong> such a resonantly
enhanced axion photon regeneration was put forward first
in 1993 by F. Hoogeveen and T. Ziegenhagen [34] and independently
rediscovered in 2007 by P. Sikivie, D.B. Tanner
and K. van Bibber [35]. The basic idea is to set up an optical
resonator also in the regeneration part <strong>of</strong> a LSW experiment
very similar to the optical resonator in the first part.
The second resonator effectively increases the conversion
probability <strong>of</strong> a WISP into a photon. To understand this one
Figure 5: ALPS exclusion limit (95% C.L.) for hidden photons
(top) and minicharged particles (bottom) together with
the results from various other experiments [23].
has to consider that the freely propagating WISP-related
wave behind the wall <strong>of</strong> the LSW experiment comprises a
very tiny electromagnetic photon component. Due to this
small component the WISP might convert into a real photon.
An optical resonator enhances this small component
in the same way as the wave amplitude for real photons
is increased. Hence the transition probability <strong>of</strong> WISPs to
photons rises with the power amplification factor <strong>of</strong> an optical
resonator in the second part <strong>of</strong> the LSW experiment
behind the wall. Consequently the sensitivity <strong>of</strong> such a setup
for the coupling constant g improves with the square
root <strong>of</strong> this factor. The technical challenge is to lock the
second cavity to exactly the same frequency and the same
mode as the first cavity (used to enhance the effective laser
photon flux) while keeping it dark in the light <strong>of</strong> photons
searched for as WISP footprints. At ALPS we study different
approaches to realize a regeneration cavity.
Combining all the improvements mentioned above would
result in a greatly improved sensitivity allowing LSW experiments
to surpass present day indirect limits from astrophysics
(g = 10 −10 GeV −1 ) and touch the parameter
region for ALPS indicated by astrophysics phenomena

Figure 6: Estimates <strong>of</strong> sensitivities for the search <strong>of</strong>
axion-like particles for different dipole magnet types (taken
from [36])
.
g < 10 −11 GeV −1 ) as shown in Fig. 6. A detailed analysis
<strong>of</strong> future possibilities can be found in [36]. Other exciting
possibilities for future WISP searches have been presented
at this workshop (see K. Homma’s and T. Tomaru’s contributions
to the proceedings).
ACKNOWLEDGEMENTS
I am very grateful to the organizers <strong>of</strong> PIF 2010 for giving
me the opportunity to participate in and to contribute
to this very exciting meeting with a broad physics scope! I
thank my colleagues <strong>of</strong> the ALPS collaboration for stimulating
and fruitful discussions as well as for the fun working
with experts in very different scientific disciplines.
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Probing extremely light fields via resonance scattering
by focusing intense laser ∗
Kensuke Homma
Graduate School <strong>of</strong> Science, Hiroshima Univ., Hiroshima 739-8526, Japan, and
Fakultät für Physik, Ludwig Maximilians Universität München, D-85748 Garching, Germany
Abstract
Recent astronomical observations suggest that there are
unknown constituents <strong>of</strong> the universe such as dark energy
and dark matter. They may be undiscovered extremely light
fields in the vacuum which only weakly couple to matter.
We suggest a method to observe photon-photon scattering
via resonance states <strong>of</strong> these fields by focusing intense laser
fields in laboratory experiments.
INTRODUCTION
Dark energy and dark matter may be attributable to
undiscovered fields <strong>of</strong> small mass below 1eV [1, 2, 3].
These fields are thought to evade our effort to detect them
in laboratory, because they are supposed to weakly couple
to matter. Among them the most challenging puzzle is the
dark energy problem. The observed dark energy density
is too small to explain it by a simple field theoretical view
point [4]. In order to approach to this problem, let us start
by raising following view points.
The first point is that the minimum energy state may be
different depending on the order parameter to define the energy
state. The second point is that the space-time scales
are totally different between particle physics and cosmology.
In particle physics or field theory we introduce local
fields to define the energy state. We then discuss the energy
state <strong>of</strong> the vacuum through the order parameter such
as particle masses represented by the square term <strong>of</strong> the
polynomial <strong>of</strong> the field in the Lagrangian. On the other
hand, in the case <strong>of</strong> cosmology the observable is geometry.
We relate curvature <strong>of</strong> the vacuum with the gravitational
sources which provide the information on the energy density
and the pressure. For instance, it is not surprising at all,
even if the states <strong>of</strong> soundlessness and darkness, which are
different states <strong>of</strong> sensory organs, do not point an identical
state <strong>of</strong> nature. Similarly it might not be a real problem,
even though we cannot immediately accommodate the connection
on the energy state <strong>of</strong> the vacuum between particle
physics and cosmology as represented by the dark energy
problem. Before claiming the problem, we should admit
that we have probed the vacuum only a little with very different
scales. Thus if we could introduce a new kind <strong>of</strong>
observables or order parameters to define the state <strong>of</strong> the
vacuum in different scales, it would lead us to the deeper
understanding on the state <strong>of</strong> the vacuum. Therefore, it is
∗ Work supported by the Grant-in-Aid for Scientific Research
no.21654035
important for experiments to collect information in different
space-time scales as much as possible.
In spite <strong>of</strong> the fact that the vacuum is not a normal
medium, if we could view the vacuum as if it is a medium,
at least, we can obtain some hints on observables which
have been neither applied to particle physics nor cosmology
so far. Of course, we should remember that we cannot
associate notion <strong>of</strong> motion to this ether like view point. Let
us consider laser-matter interactions where we introduce
polarizations as an order parameter with respect to external
electric fields. All laser physicists are familiar with birefringence
<strong>of</strong> crystals and higher harmonic generation from
them as the nonlinear response <strong>of</strong> atoms to external electric
fields. We may try to apply these observables to investigate
the state <strong>of</strong> the vacuum. We now replace the role <strong>of</strong> a
crystal with the vacuum by introducing notion that the vacuum
is also a special kind <strong>of</strong> medium under influence <strong>of</strong>
external fields. In this case the investigation <strong>of</strong> the high intense
laser-laser interaction in the vacuum is a quite natural
approach to probe the medium-like nature. Therefore, we
may foresee the applicability <strong>of</strong> these known observables
developed for laser-matter interaction.
However, the dynamics in the vacuum is different from
that <strong>of</strong> atoms in matter. First <strong>of</strong> all, we need discuss the
scale dependence <strong>of</strong> the dynamics in the case <strong>of</strong> the vacuum.
In this respect photon is a valuable probe to see natures
<strong>of</strong> the vacuum, since we can introduce the different
frequencies by many orders <strong>of</strong> magnitude in laboratory experiments.
In higher energy side, e.g. higgs particle supposed
to act as a friction to give masses to particles becomes
important. In order to produce this, we would need
∼ 100 GeV as the beam energy. This is only possible
by introducing higher beam momentum such as high energy
colliders. What does happen in 1 eV-100 MeV range?
Here the virtual vacuum polarizations by quantum electrodynamics
and possibly quantum chromodynamics become
the source <strong>of</strong> photon-photon interactions for an instant moment.
Below 1eV, the nature <strong>of</strong> the vacuum response is
not well known. In this energy range exchanged field between
photons become very light. Therefore, they can be
candidates <strong>of</strong> dark matter and/or dark energy. We note
all these interactions are characterized by masses <strong>of</strong> exchanged
fields and the coupling strength to photons.
As the most ambitious motivation to search for a lowmass
field, let us introduce a scalar field in the context <strong>of</strong>
dark energy. Scalar-Tensor-Theory with Λ (ST T Λ) [2] is
one <strong>of</strong> many dark energy models in the market [1]. This

theory provides a natural explanation why the observed
dark energy is so small without any fine tuning. This predicts
decaying Λ with t −2 as a function <strong>of</strong> the age <strong>of</strong> the
universe t. The present age <strong>of</strong> the universe is t = 10 60 in
Planckian unit. Thus it naturally explains why Λ is 10 −120
at present in the same unit. The uniqueness <strong>of</strong> the theory
from an experimental point view is that it can give us a
testable prediction with the explicit allowed mass and coupling
strength. As a result <strong>of</strong> the theory we expect a scalar
field with mass <strong>of</strong> neV range by allowing a few orders <strong>of</strong>
latitudes which couples to matter via gravitational strength.
In past experiments probing a deviation from Newtonian
potential form, massive and huge bodies were used as a test
probes [5]. However, when they measure the gravitational
effects in a short distance, they must suffer from the background
physical process like Coulomb force. In contrast, if
we could use photon-photon scattering as a probe for such
a new kind <strong>of</strong> force, the experiment would be free from the
background physical process, since the total photon-photon
elastic cross section in the optical energy is only 10 −42 b at
most [6].
However, the biggest issue appears, because the huge
and massive probes were actually demanded to have a sensitivity
to the gravitational coupling strength. Nevertheless,
if we could overcome this drawback to use photons
as the test probe, the method opens up a new window to
probe weakly coupling and low-mass fields (finite long
range force). As long as the field has a finite mass and coupling
to matter, we can directly produce low-mass fields as
resonance states such as higgs particle production in high
energy colliders. The production cross section is in principle
free from the constraints by the weak coupling, if the
center <strong>of</strong> mass system energy Ecms <strong>of</strong> two colliding photons
is adjusted to the top <strong>of</strong> the resonance point. In the
following sections we discuss how to realize the photonphoton
collision system and the sensitivity to the coupling
as weak as gravitational coupling for the mass range well
below optical frequency 1 eV.
QUASI-PARALLEL SYSTEM OF
PHOTON-PHOTON COLLISIONS
If we are interested in extremely low-mass ranges even
below meV, we have to reduce the CMS energy <strong>of</strong> photonphoton
collisions compared to the incident photon energy if
optical laser fields are assumed. As long as the resonance is
allowed to decay into only two photons, the scattering process
looks like elastic scattering even if a low-mass field
is exchanged via the resonance state in CMS. Thus there is
no frequency shift in the final state in CMS. However, if we
boost this system to the direction perpendicular to the colliding
axis, the frequency shift takes place along that boost
axis. In the forward direction on the boost axis we expect a
frequency up shift to close to the double <strong>of</strong> the incident frequency,
while a zero frequency photon must be emitted to
the backward direction due to the energy-momentum conservation
independent <strong>of</strong> the dynamics <strong>of</strong> the exchanged
field. We refer this boosted system as the quasi-parallel
system (QPS). This may be interpreted as if the second harmonic
photon is generated from the nonlinear vacuum response.
This could be an interesting analogy to second harmonic
generation due to the nonlinear response <strong>of</strong> a crystal
with a laser injection which was pioneered by Franken
et al. [7]. Inversely if we realize the QPS as a laboratory
frame, the corresponding CMS energy can be very much
lowered. The CMS energy with variables in QPS can be
defined as
Ecms ∼ 2ϑω, (1)
where ϑ is defined as a half incident angle between two
incoming photons with ϑ ≪ 1 and ω is the beam energy
in unit <strong>of</strong> ¯h = c = 1. This relation indicates that we have
two experimental handles to adjust Ecms. If we take the
head-on collision geometry, we have to introduce very long
wavelength as the incident photons. However, it is not too
difficult to introduce the very small incident angle. In such
a case Ecms can be lowered by keeping ω constant. We
also know that the cross section <strong>of</strong> photo-photon scattering
in QED process σqed in QPS is quite suppressed due to the
fourth power dependence on the incident angle which is
expressed as σqed ∼ (α 2 /m 4 e) 2 ω 6 ϑ 4 with the fine structure
constant α and electron mass me [8]. Therefore, the low
frequency photons in QPS is the best system to probe such
a low-mass field.
QPS BY FOCUSING WITH SINGLE
GAUSSIAN LASER BEAM
However, it is difficult to introduce two colliding photon
beams which satisfy the small incident angle based on the
simple geometrical optics due to the wavy nature <strong>of</strong> photons
in the diffraction limit <strong>of</strong> photons. Below meV range
we are naturally led to introduce a geometry by focusing a
single laser beam as illustrated in Fig.1. What is important
here is that in the diffraction limit there are uncertainties on
the incident momentum due to the uncertainty principle, in
other words, there are uncertainties on the incident angles
between two photons among the single beam, even though
photons are in the degenerated state at the output <strong>of</strong> the
laser crystal. This should be contrasted to the case <strong>of</strong> high
energy collider where the momentum spread <strong>of</strong> each colliding
particle or the uncertainty based on the de Broglie
length is negligibly small compared to the magnitudes <strong>of</strong>
relevant momentum exchanges they are interested in. This
different initial condition becomes critically important for
the following discussions.
DYNAMICS OF PHOTON-PHOTON
SCATTERING
As the simplest coupling between two photons and a
low-mass field we focus on the quantum anomaly type coupling
g 2 /M which includes square <strong>of</strong> electric charge g to
couple virtual fermion loops to two external photons and a
dimensional coupling 1/M to low-mass neutral fields [9].

Figure 1: Second harmonic generation in Quasi Parallel
System by focusing a single Gaussian laser beam.
If M is Planckian mass scale <strong>of</strong> 10 27 eV, the coupling expresses
gravitational one. We may discuss possibilities to
exchange scalar and pseudoscalar type <strong>of</strong> fields by requiring
combinations <strong>of</strong> photon polarizations in the initial and
final states [10]. The virtue <strong>of</strong> laser experiments is in the
specifications on the all photon spin states both in the initial
and final states in the two body photon-photon interaction.
This allows us to discuss types <strong>of</strong> exchanged fields in general.
HOW TO OVERCOME THE EXTREMELY
NARROW RESONANCE
The exact resonance condition is the requirement <strong>of</strong> m =
Ecms where m is the mass <strong>of</strong> exchanged field. The squire
<strong>of</strong> the scattering amplitude A can be expressed as Breit-
Wigner(BW) resonance function [11]
|A| 2 = (4π) 2
a 2
χ2 , (2)
(ϑ) + a2 where χ and width a are defined as χ(ϑ) ≡ ω 2 −ω 2 r(ϑ) and
a ≡ (ω 2 r/16π)(g 2 m/M) 2 , respectively. The energy ωr
satisfying the resonance condition can be defined as ωr ≡
m 2 /(1 − cos 2ϑr) [10]. If we take Planckian mass as M,
the width a becomes extremely small. This implies that the
resonance width is too small to hit the peak position <strong>of</strong> the
resonance function. How can we overcome this problem?
We take a unique approach to this situation. Let us remind
you <strong>of</strong> higgs hunting by high energy colliders as an
example to hit the top <strong>of</strong> the resonance. In high energy
colliders the spread <strong>of</strong> the beam energy is much smaller
than the width <strong>of</strong> the resonance function which they try to
probe. On the hand, in collisions at the diffraction limit in
QPS, the resonance width looks almost like delta-function
and the uncertainty on the Ecms is much wider than the
resonance width as we discussed above. We may use the
following feature <strong>of</strong> delta-function. Although the width <strong>of</strong>
the delta-function is infinitesimal, as far as it is integrated
over ±∞, the value <strong>of</strong> integral becomes order <strong>of</strong> unity. In
the case <strong>of</strong> BW function, even if we integrate it over ±a,
the value <strong>of</strong> the integral is just a half <strong>of</strong> the value integrated
over ±∞. This implies that as long as we capture the resonance
peak within a finite range beyond ±a, the value
<strong>of</strong> the integral becomes proportional to a but not a 2 . Actually
in the diffraction limit incident angles <strong>of</strong> incoming
photons are in principle uncertain. Thus we must use the
averaged cross section by integrating the square <strong>of</strong> the scattering
amplitude over a possible range on Ecms determined
by uncertainties on incident angles.
Let us reflect this feature in the case <strong>of</strong> single beam focusing
experiment. We can introduce a probability distribution
function on the possible incident angles between randomly
selected photon pairs among the incident single laser
beam. We then define the range <strong>of</strong> integral on the incident
angel as ∆ϑ ∼ d/(2f) with the beam diameter d and the
focal length f. If this range does not contain the resonance
angle ϑr ≡ m/2ω, that is, ϑr > ∆ϑ, the averaged squared
amplitude |A| 2 becomes proportional to a 2 which indicates
the suppression by M −4 . On the other hand, if ϑr < ∆ϑ is
satisfied, we can obtain the proportionality to a, namely, a
sensitivity enhancement by M 2 compared to the case without
resonance. Thus various focusing parameters to adjust
∆ϑ by controlling the beam diameter and focal length can
introduce a sharp cut<strong>of</strong>f on the cross section which eventually
controls sensitive mass ranges <strong>of</strong> this method through
the relation m < 2ω∆ϑ.
SENSITIVITY TO GRAVITATIONAL
COUPLING STRENGTH
Let us now discuss how much intense laser fields are
necessary to have a sensitivity to gravitational coupling
strength based on the concept <strong>of</strong> photon-photon scattering
in QPS. As the most challenging case let us assume
m ∼ 10 −9 eV which can be a candidate <strong>of</strong> dark energy if
it could couple to matter as weak as gravitational coupling
strength 1/MP ∼ 10 −27 eV −1 [2]. The yield <strong>of</strong> harmonic
generation Y integrated over the solid angle which satisfies
the condition that one <strong>of</strong> final state photons has the
frequency close to 2ω, can be expressed as
Y ∼ Kexp(g 2 m/M) 2 sin 2 ϑL, (3)
where Kexp is a total experimental factor depending on
the the duration time <strong>of</strong> the laser pulse, the energy selection
and the focusing parameter resulting in ϑr/∆ϑ relevant
for the normalization <strong>of</strong> the probability on incident
angles, (g 2 m/M) 2 ∼ (αm/M) 2 ∼ 10 −76 is the factor
after averaging on BW resonance function, sin −2 ϑ is the
factor related with the phase volume integral and the photon
flux factor after integrated over the solid angle, and
L is a luminosity-like factor in the collider concept. The
significant difference <strong>of</strong> L from that <strong>of</strong> collider concept is
that photons are annihilated and created from degenerated

states rather than from the vacuum state |0 >. This implies
that annihilation and creation operators cause √ ¯ N in
each vertex <strong>of</strong> the Feynman diagram with the mean number
<strong>of</strong> photons ¯ N in a laser field by assuming ¯ N is large
enough [12]. On the production vertex <strong>of</strong> the low-mass
field, we expect a factor <strong>of</strong> √ 2
N¯ at the amplitude level due
to two photon annihilation in the degenerated number state.
Taking square <strong>of</strong> this factor gives a similar factor to collider
luminosity which includes a factor proportinal to n 2 where
n is the number <strong>of</strong> charged particles per beam bunch.
In order to have one second harmonic photon per laser
focusing, we would need 10 34 optical photons corresponding
to ∼ 10 16 J with Kexp ∼ 10 −10 for the focal length
f ∼ 1000 m, the beam diameter d ∼ 2 m and the pulse
duration τ ∼ 10 fs. Furthermore, we may also induce decays
<strong>of</strong> the produced resonances by adding a degenerated
photon states with different frequency to let the one <strong>of</strong> two
photons from the resonance decay into the prepared degenerated
state, where creation operator causes the additional
factor <strong>of</strong> √ ¯ N at the scattering amplitude level. Therefore,
if we mix two frequencies in advance with equal intensity
¯N, we may expect the increase <strong>of</strong> the luminosity-like factor
L ∼ ¯ N 3 . In this case the necessary photon intensity
may be very much reduced to order <strong>of</strong> 10 22 optical photons
corresponding to several kJ with the same Kexp factor. We
note that the photon frequency to be observed changes from
the second harmonic <strong>of</strong> the incident photons depending on
the frequency <strong>of</strong> the field to induce decays.
SUMMARY
Higher harmonic generation in Quasi-Parallel-System
can be a novel probe to discuss weakly coupling low-mass
fields. Experimental realization <strong>of</strong> QPS as parallel as possible
is a challenge for future experiments. The dominant
enhancement mechanism is originated by containing the
low-mass resonant peak within the uncertainty on the center
<strong>of</strong> mass energy in the diffraction limit <strong>of</strong> the focused
laser beam. A degenerated field to induce decay <strong>of</strong> resonance
is important to probe coupling as weak as gravity
though higher harmonic generation. Given high intensity
laser fields ∼ 2 kJ per pulse available in Extreme Light
Infrastructure [13], we foresee the breakthrough on the existing
sensitivity to coupling <strong>of</strong> low-mass fields to photons
based on this novel idea.
ACKNOWLEDGMENTS
This work is based on intensive discussions with Y. Fujii,
D. Habs and T. Tajima under support by the DFG Cluster
<strong>of</strong> Excellence MAP (Munich-Center for Advanced Photonics).
REFERENCES
[1] Y. F. Cai, E. N. Saridakis, M. R. Setare and J. Q. Xia,
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[2] Y. Fujii and K. Maeda, The Scalar-Tensor Theory <strong>of</strong> Gravitation
(Cambridge Univ. Press, 2003).
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[13] http://www.extreme-light-infrastructure.eu/.

Abstract
Dynamical view <strong>of</strong> pair creation via the Schwinger mechanism ∗
Naoto Tanji †
Institute <strong>of</strong> physics, University <strong>of</strong> Tokyo, Komaba, Tokyo 153-8902, Japan
Particle production via the Schwinger mechanism has
been studied as a mechanism <strong>of</strong> matter formation in the
context <strong>of</strong> heavy-ion collisions. We describe the particle
pair creation in a strong electric field focusing on its realtime
dynamics. Motivated by the Color Glass Condensate
framework, we investigate the effects <strong>of</strong> a magnetic field
which is parallel to the electric field, and show that the
magnetic field enhances quark production. Also the pair
creation in a boost-invariantly expanding electric field is
discussed.
INTRODUCTION
Particle pair creation from vacuum in the presence <strong>of</strong>
a strong electric field, which is known as the Schwinger
mechanism [1], is one <strong>of</strong> the most remarkable consequences
<strong>of</strong> quantum field theory. This particle production
mechanism has attracted considerable theoretical
and experimental interests, because it concerns the nonperturbative
aspects <strong>of</strong> quantum field theory. However, no
direct observation <strong>of</strong> the Schwinger particle production has
ever been obtained because it requires very strong electric
fields above the critical strength Ec = m 2 e/e ∼ 10 16 V/cm,
which is beyond current technological capabilities.
Although sufficient field strength has not yet been realized
in laboratories as a QED electric field, it may be attainable
as a QCD color field because <strong>of</strong> its strong nature.
Color electric fields are expected to be generated in highenergy
particle collision experiments, such as relativistic
heavy-ion collisions. High energy particle collision experiments
may be a promising playground for the strong field
physics.
Figure 1: A schematic <strong>of</strong> the longitudinal evolution <strong>of</strong> relativistic
heavy-ion collisions.
∗ Work supported by the Japan Society for the Promotion <strong>of</strong> Science
for Young Scientists
† tanji@nt1.c.u-tokyo.ac.jp
Relativistic heavy-ion collision experiments have been
done at Relativistic Heavy Ion Collider (RHIC) and are
starting at Large Hadron Collider (LHC) to explore the nature
<strong>of</strong> matter in extreme condition. A space-time picture
<strong>of</strong> relativistic heavy-ion collisions is illustrated in Fig. 1.
The formation <strong>of</strong> quark-gluon plasma (QGP), which consists
<strong>of</strong> quarks and gluons liberated from confinement, is
expected there. In the description <strong>of</strong> QGP, hydrodynamic
simulations supposing the local thermalization <strong>of</strong> the system
have achieved great successes (see e.g. [2]). However,
a full understanding <strong>of</strong> how high-energy matter is created
and how local thermalization is realized before the hydrodynamic
evolution is still lacking.
One <strong>of</strong> the scenarios <strong>of</strong> matter formation in heavy-ion
collisions is based on the flux-tube model [3]. When two
Lorentz-contracted disks <strong>of</strong> nuclei collide and pass through
each other, they exchange color charges (gluons), and after
the collision, many flux tubes are generated between the
two capacitor plates (Fig. 2(a)). These flux tubes decay
into quarks and gluons by the Schwinger pair creation [4].
Recently also the Color Glass Condensate (CGC) model,
which is an effective theory to describe high-energy nuclei
in saturated region (see e.g. [5] for review), has predicted
the formation <strong>of</strong> color electric fields in the longitudinal
beam direction [6]. The field strength predicted by
the CGC is gE ∼ 1 GeV 2 at RHIC energy. This is strong
enough to cause the Schwinger particle production. Therefore
the Schwinger mechanism attracts renewed interest in
the context <strong>of</strong> the CGC [7]. One <strong>of</strong> the remarkable differences
<strong>of</strong> the color flux tube given by the CGC from that in
the original flux-tube model is the existence <strong>of</strong> the longitudinal
color magnetic fields in addition to the electric fields
[6].
In this paper, we investigate the Schwinger mechanism
focusing on the following issues.
(a) flux tubes
(b) the boost-invariant configuration
<strong>of</strong> electric fields
Figure 2: Color electric fields expected in the early stage <strong>of</strong>
heavy-ion collisions.

Real-time description Because the system <strong>of</strong> heavy-ion
collisions is a dynamic one, we study the nonequilibrium
dynamics <strong>of</strong> the pair creation.
Effects <strong>of</strong> magnetic fields The CGC predicts the generation
<strong>of</strong> longitudinal color-magnetic fields as well as
color-electric fields just after a collision <strong>of</strong> heavy-nuclei,
<strong>of</strong> which state is called Glasma. We examine the effects <strong>of</strong>
such magnetic fields on the pair creation.
Expanding geometry <strong>of</strong> the electric fields Electric fields
expected to be generated in the early stage <strong>of</strong> heavy-ion collisions
exist only between two nuclei receding from each
other at a velocity close to the speed <strong>of</strong> light (Fig. 2(b)).
This situation is quite different from that <strong>of</strong> the Schwinger’s
original work, in which spatially uniform fields are treated.
We study how the dynamics <strong>of</strong> particle production is modified
in this field configuration.
REAL-TIME DESCRIPTION OF PAIR
CREATION
Canonical quantization in background fields
To get a real-time description <strong>of</strong> pair creation, we introduce
an instantaneous particle definition by an explicit
quantization procedure under a background electric field
[8]. The time evolution <strong>of</strong> the quantum state <strong>of</strong> charged
particles is described by the variation <strong>of</strong> that particle definition.
As an illustration <strong>of</strong> it, we first study the QED particle
production in a spatially uniform electric field. Extension
to the quark production under a non-Abelian colorelectric
field is possible [9]. We suppose the electric field
E = (0, 0, Ez(t)) is vanishing at t < 0 and is turned
on at time t = 0. The gauge A 0 = 0 is chosen so that
A 1 = A 2 = 0 and Ez(t) = − d
dt A3 (t). The charged spinor
field ψ obeys the Dirac equation:
[iγ µ (∂µ + ieAµ) − m] ψ(t, x) = 0. (1)
The field quantization is accomplished by imposing the
canonical anti-commutation relation {ψ(t, x), π(t, x ′ )} =
iδ(x − x ′ ), where π(t, x) = iψ † (t, x) is canonical conjugate
momentum. The quantized quark field may be expanded
as
ψ(x) = ∑
s=↑,↓
∫
d 3 [
p +ψ in ps(x)a in p,s + −ψ in ps(x)b in†
]
−p,s ,
(2)
where a in p,s [b in p,s] is the annihilation operator <strong>of</strong> a particle
[antiparticle] with momentum p and spin s, and ±ψ in ps(x)
are classical solutions <strong>of</strong> the Dirac equation (1). The superscript
‘in’ distinguishes the initial condition for ±ψ in ps(x): at
t < 0, +ψ in ps(x) [−ψ in ps(x)] is identical to the positive [negative]
energy solution <strong>of</strong> the free Dirac equation. We set the
state to be in-vacuum |0, in⟩, where no particle exists initially
and which is defined by a in p,s|0, in⟩ = b in p,s|0, in⟩ = 0.
At t > 0, ±ψ in ps(x) evolves under the influence <strong>of</strong> the
electric field and becomes superposition <strong>of</strong> a positive and
negative energy (frequency) state. To describe the pair creation
dynamically, we introduce a time-dependent particle
definition by decomposing the field operator ψ(x) into positive
and negative frequency instantaneously:
ψ(t0, x) = ∑
s=↑,↓
∫
d 3 [
p +ψ (t0)
ps (x)ap,s(t0)
+−ψ (t0)
ps (x)b †
−p,s(t0)
]
,
where +ψ (t0)
ps (x) [−ψ (t0)
ps (x)] is a positive [negative] frequency
solution <strong>of</strong> the Dirac equation under the pure gauge
A 3 = A 3 (t = t0). Instantaneous particle picture is defined
by ap,s(t) and bp,s(t). Of course, ap,s(t) and bp,s(t) agree
with a in p,s and b in p,s at t < 0, respectively. The particle definition
at time t and that <strong>of</strong> the in-state are related by the
time-dependent Bogoliubov transformation:
ap,s(t) = αp,s(t)a in p+eA(t),s
+ βp,s(t)b in †
−p−eA(t),s ,
(3)
b †
−p,s(t) = α ∗ in †
p,s(t)b−p−eA(t),s − β∗ p,s(t)a in (4)
p+eA(t),s ,
<strong>of</strong> which coefficients satisfy
|αps(t)| 2 + |βps(t)| 2 = 1. (5)
Because the creation and annihilation operators are
mixed by the Bogoliubov transformation, the vacuum
expectation <strong>of</strong> the number operator may be nonzero:
⟨0, in|a † 2 V
ps(t)aps(t)|0, in⟩ = |βps(t)| (2π) 3 , where V is the
volume <strong>of</strong> the system. This means that particle creation
happens. We define a particle pair distribution function as
fps(t) = ⟨0, in|a † ps(t)aps(t)|0, in⟩ (2π)3
V
= ⟨0, in|b †
−ps(t)b−ps(t)|0, in⟩ (2π)3
V
= |βps(t)| 2 .
Because <strong>of</strong> the charge and the momentum conservation,
antiparticles have always opposite momentum to particles.
The distribution function can not exceed unity: fp,s(t) =
|βp,s| 2 = 1 − |αp,s| 2 ≤ 1, because <strong>of</strong> the constraint (5).
This is a manifestation <strong>of</strong> Pauli’s exclusion principle.
Pair creation in constant electric fields
In Fig. 3, the time evolution <strong>of</strong> the momentum distribution
function (6) under a constant electric field Ez(t) =
E is plotted. Hereafter, all figures are shown in the
dimension-less unit scaled by √ eE [or √ eE0 in a nonsteady
field case, where E0 is an initial field strength].
After the switch-on <strong>of</strong> the electric field, particles are created
with approximately vanishing longitudinal momenta.
Their occupation numbers, in other words, the ( heights <strong>of</strong>
the distributions can be approximated by exp − πm2
)
⊥
eE ,
where m⊥ is the transverse mass defined by m 2 ⊥ ≡ m2 +p 2 ⊥ .
(6)

(a) longitudinal momentum distribution with fixed
transverse momentum p⊥ = 0
(b) transverse momentum distribution with fixed
longitudinal momentum √ eEpz = 1
Figure 3: Time evolution <strong>of</strong> the momentum distribution <strong>of</strong> created particles under the constant electric field. m2
2eE = 0.1.
These features are consistent with the semi-classical tunneling
description <strong>of</strong> pair creation [10]. However, the longitudinal
momenta that particles bring when they are created
are not exactly zero but broadened around zero due
to quantum fluctuation. After created, particles are accelerated
by the electric field according to the classical equation
<strong>of</strong> motion: pz = eEt + const. In contrast, there is no
acceleration in the transverse directions and the transverse
momentum distributions exhibits a Gaussian-like form.
Back reaction
Previously, we have treated the pair creation in the constant
electric field. However, if charged particles are created,
they generate further electromagnetic fields and the
original field should be modified. This effect, called back
reaction, is not negligible under a strong field where pair
creation happens intensively. To take account <strong>of</strong> the back
reaction, we treat an electromagnetic field as a dynamical
variable obeying the Maxwell equations. The uniformity
<strong>of</strong> the system reduces the Maxwell equations into the single
equation
dEz
dt = −d2 A 3
dt 2 = −jz, (7)
Figure 4: Time evolution <strong>of</strong> the longitudinal momentum
distribution. The effect <strong>of</strong> the back reaction is taken into
account. m2 = 0.01 and e = 1.
2eE0
where jz is the charge current generated by created particles
and antiparticles.
We have numerically solved the coupled equations <strong>of</strong> (1)
and (7). The results are shown in Figs. 4 and 5. The longitudinal
momentum distribution (Fig. 4) oscillates in momentum
space, and also the current and the electric field
(Fig. 5) show oscillations. These oscillations can be understood
as usual plasma oscillation.
Other than the plasma oscillation, which is a classical
dynamics, also quantum effects such as the Pauli blocking
and interference between matter fields play remarkable
roles in the time evolution. Because <strong>of</strong> the interference, the
momentum distribution shows fine oscillations at later time
(Fig. 4).
EFFECTS OF MAGNETIC FIELDS
Since the formation <strong>of</strong> longitudinal color-magnetic fields
as well as color-electric fields is predicted by the framework
<strong>of</strong> the CGC [6], we study the effects <strong>of</strong> a longitudinal
magnetic field on the pair creation. As illustrated in Fig. 6,
under the longitudinal magnetic field, transverse momentum
p⊥ is discretized into the Landau levels as
p 2 ⊥ −→ (2n + 1)eB (n = 0, 1, 2, . . . ). (8)
Notice that even the lowest Landau level depends on the
magnetic field strength B for the case <strong>of</strong> scalar particles.
Since the transverse momentum acts as an effective mass
m2 ⊥ = m2 + p2 ⊥ , scalar particles become effectively heavy
under a strong magnetic field, so that their pair production
(a) current density (b) electric field
Figure 5: Time evolution <strong>of</strong> the charge current density and
the electric field. e = 1. (a = m2
2eE0 ).

Figure 6: A schematic <strong>of</strong> the Landau levels and the spinmagnetic
field interaction.
Figure 7: Magnetic field dependence <strong>of</strong> the charge current
density. m2
2gE0
= 0.01 and g = 1.
is strongly suppressed. In contrast, owing to spin-magnetic
field interaction, the lowest level <strong>of</strong> spinor particles is zero
and thereby independent <strong>of</strong> B. Therefore, the production
<strong>of</strong> spinor particles in that level is not at all suppressed. Not
only the creation <strong>of</strong> the lowest level is not suppressed, the
magnetic field enhances field quantities such as the current
and the total particle number. That is because the number <strong>of</strong>
modes degenerating in a unit transverse area is proportional
to B.
In Fig. 7, the magnetic field strength dependence <strong>of</strong> the
charge current density is exhibited. The current density is
indeed enhanced by the magnetic field. Furthermore, the
enhanced current makes the time scale <strong>of</strong> the plasma oscillation
shorter through the back reaction. This mechanism
may have significance in the time evolution <strong>of</strong> the Glasma.
PARTICLE PRODUCTION IN
EXPANDING ELECTRIC FIELDS
So far, we have investigated the pair creation in the spatially
uniform electric fields. However, in a system <strong>of</strong>
heavy-ion collisions, color electric fields are expected to
be generated between two color-charged nuclei receding
from each other at a velocity close to the speed <strong>of</strong> light
(Fig. 2(b)). A characteristic <strong>of</strong> this electric field is the invariance
under the Lorentz-boost in the longitudinal direction.
In the case <strong>of</strong> spatially homogeneous fields, particles
and antiparticles with (approximately) vanishing longitudinal
momentum are created. If this is true also in the boostinvariant
electric field confined in the forward light cone,
these particle-antiparticle pairs clearly break the boost invariance
<strong>of</strong> the background field by introducing one spe-
cific frame, namely their center-<strong>of</strong>-mass frame. How is this
situation modified if an electric field exists only inside the
forward light cone? Actually, under the boost-invariantly
expanding electric field, particles are created not as an
eigenstate <strong>of</strong> the longitudinal momentum, which violates
the boost-symmetry, but as a superposition <strong>of</strong> several momentum
modes preserving the boost-symmetry [11]. The
particles have the scaling velocity distribution vz(t, z) =
z/t from the first instance they are created. This velocity
distribution is the same as the flow velocity <strong>of</strong> the boostinvariantly
expanding fluid <strong>of</strong> QGP, i.e. the Bjorken flow
[12]. Therefore, our result would narrow the gap between
the pre-equilibrium stage <strong>of</strong> heavy-ion collisions and the
state <strong>of</strong> the boost-invariantly expanding QGP.
SUMMARY
In this paper, we have investigated the real-time dynamics
<strong>of</strong> the Schwinger particle production and its phenomenological
applications to matter formation in relativistic
heavy-ion collisions.
Because <strong>of</strong> the back reaction, the electric field varies in
time. Using the initial field strength gE0 ∼ 1 GeV 2 , which
is expected in the framework <strong>of</strong> the CGC, one can estimate
the time scale <strong>of</strong> this field variation to be a few fm/c. This
time scale is the same order as that <strong>of</strong> the pre-equilibrium
stage <strong>of</strong> heavy-ion collisions ∼ 1 fm/c. This result would
verify the importance <strong>of</strong> the Schwinger mechanism at the
initial stage <strong>of</strong> heavy-ion collisions.
We have also discussed the enhancement <strong>of</strong> the quark
production by the longitudinal magnetic field, and the
emergence <strong>of</strong> the boost-invariant velocity distribution from
the expanding electric field. These results would have a
significance to understand the formation process <strong>of</strong> QGP.
REFERENCES
[1] J. S. Schwinger, Phys. Rev. 82 (1951) 664.
[2] R. Hwa, X. Wang, Quark-Gluon Plasma 3, World Scientific,
2003.
[3] F. E. Low, Phys. Rev. D12 (1975) 163; S. Nussinov, Phys.
Rev. Lett. 34 (1975) 1286.
[4] A. Bialas, W. Czyz, Phys. Rev. D31 (1985) 198; K. Kajantie,
T. Matsui, Phys. Lett. B164 (1985) 373; G. Gat<strong>of</strong>f, A. K.
Kerman, T. Matsui, Phys. Rev. D36 (1987) 114.
[5] E. Iancu, R. Venugopalan, arXiv:hep-ph/0303204.
[6] T. Lappi, L. McLerran, Nucl. Phys. A772 (2006) 200.
[7] D. Kharzeev, E. Levin, K. Tuchin, Phys. Rev. C75 (2007)
044903; P. Castorina, D. Kharzeev, H. Satz, Eur. Phys. J.
C52 (2007) 187; K. Fukushima, F. Gelis, T. Lappi, Nucl.
Phys. A831 (2009) 184.
[8] N. Tanji, Ann. Phys. 324 (2009) 1691.
[9] N. Tanji, Ann. Phys. 325 (2010) 2018.
[10] V. Popov, Sov. Phys. JETP 34 (1972) 709; A. Casher,
H. Neuberger, S. Nussinov, Phys. Rev. D20 (1979) 179.
[11] N. Tanji, arXiv:1010.4516 [hep-ph].
[12] J. D. Bjorken, Phys. Rev. D27 (1983) 140.

Exact solutions <strong>of</strong> pair productions in strong electric field with finite width
Abstract
A. Iwazaki, Nishogakusha University, Chiyoda-ku 3-6-16, Tokyo, 102-8336, Japan
We show that chiral anomaly is a very useful tool for discussing
Schwinger mechanism when collinear strong electric
and magnetic fields are present. We can obtain number
densities <strong>of</strong> particles without calculating their wave functions.
By taking into account back reaction <strong>of</strong> the particles
we can explicitly show solutions <strong>of</strong> the electric field even
when it has finite size or when the pair creations occur in
heat bath.
CHIRAL ANOMALY AND SCHWINGER
MECHANISM
The Schwinger mechanism[1] is a non-perturbative phenomena
<strong>of</strong> particle pair productions under strong electric
field. Namely, when the unifrom electric field E is present,
the potential eEx <strong>of</strong> charged particles has no lower bound
and becomes negative infinity as x → −∞. Then, it is
energetically favorable that for example, electron-positron
pairs are spontaneously produced and they partially screen
the electric field. Since such a production is suppressed by
the factor exp(−m 2 /eE) where m(e > 0) is the electron
mass (charge), the phenomena is non-perturbative; we can
not expand it in the power series <strong>of</strong> eE.
In general, in order to discuss such non-perturbative
phenomena we need to obtain wave functions <strong>of</strong> charged
particles under electric fields[2]. When the electric field
has spatially nontrivial configulation or complicated time
dependence, it is quite difficult to obtain the wave functions.
Furthermore, if there is a magnetic field in addition
to the electric field, Dirac equation becomes too complicated.
On the other hand, when the magnetic field is sufficiently
strong for the particles to occupy only the lowest
Landau level, the phenomena are simplified. This is because
only the motions in the longitudinal direction parallel
to the magnetic field are allowed; the motions in the
transvers directions are frozen owing to the magnetic field.
Thus, the phenomena occur in spatially one dimension.
In this report we explain the utility[3] <strong>of</strong> chiral anomaly
for the discussion <strong>of</strong> Schwinger mechanism when such a
strong magnetic field is present. In particular, without solving
the complicated field equations, we can find physically
meaningful quantities associated with the pair production
simply by using the chiral anomaly,
∂t(nR − nL) = e2
4π 2 ⃗ E · ⃗ B, (1)
where nR and nL denote the number density <strong>of</strong> right and
left chiral fermions (hereafter we consider electrons and
positrons). We have assumed [3] that both <strong>of</strong> electric and
magnetic fields are much larger than the mass <strong>of</strong> electrons
and that the particles occupy only the lowest Lnadau level.
The anomaly equation implies that chirality is not conserved
when both electric (E) and magnetic fields (B) are
present. The chirality is equal to the helicity <strong>of</strong> the particle
when particle mass is negligible. Thus, the anomaly implies
that the temporal evolution <strong>of</strong> the difference, nL−nR,
is given by ⃗ E · ⃗ B.
Here we should note that electrons (positrons) under
the strong magnetic field have spins anti-parallel (parallel)
to the magnetic field. On the other hand, when the
pair production <strong>of</strong> electron and positrons arises, electrons
(positrons) are accelerated to the direction anti-parallel
(parallel) to the electric field. Therefore, when the electric
field and magnetic field are parallel to each other, their spin
and momentum are pointed to the same direction. Hence,
all <strong>of</strong> produced particles are right handed; nR ̸= 0 and
nL = 0. Obviously their number densities are equal to nR.
Thus, the production rate <strong>of</strong> the particles is governed by the
anomaly equation. This is the reason why the equation <strong>of</strong>
the chiral anomaly describes the pair production when both
strong electric and magnetic fields are present. We explain
it in several cases.
We should stress that all <strong>of</strong> quatum effects on Schwinger
mechanism are involved in the anomaly equation. Hence,
by simply solving classical equations along with the
anomaly equation we can obtain quantities including quatum
effects.
Time dependent homogeneus electric field
Assuming that there are no particles before t = 0 and
the uniform electric field is switched on at t = 0, the
pair production starts to occur and the number densities <strong>of</strong>
electrons ne and positrons np(= ne) increase with time;
2ne = nR. It is given [4] from the anomaly equation as
ne(t) =
∫ t
0
′ e2
dt
8π2 E(t′ )B (2)
where the dependence <strong>of</strong> E on t is arbitrary. In this calculation
we have not taken into account back reaction <strong>of</strong> produced
particles. We can treat the back reaction by solving
a Maxwell equation ∂tE = −J with the electric current
J = 2ene <strong>of</strong> electrons and positrons. (The velocity <strong>of</strong> the
particles is the light velocity in vacuum so that J is given as
J =charge×velocity×number density.) Then, ne satisfies
the equation,
(
∂ 2 t + e3B 4π2 )
ne = 0. (3)
Solution can be found with the initial conditions ne(t =
0) = 0 and ∂tne(t = 0) = e 2 E(t = 0)B/8π 2 where

E(t = 0) is the electric field switched on at t = 0.
In this way we can easily obtain the number density <strong>of</strong>
produced electrons when homogeneous electric and magnetic
field are sufficiently strong so as to neglect the mass
<strong>of</strong> electrons. This is a simple example[2] <strong>of</strong> the utility <strong>of</strong>
the anomaly equation.
Electric flux tube
It is remarkable that we can explicitly obtain the number
density <strong>of</strong> electrons even if the electric field is tube
like and back reactions <strong>of</strong> the produced particles are important.
The back reactions are reducing the energy <strong>of</strong> the
electric field by the generation <strong>of</strong> charged particles and <strong>of</strong>
azimuthal magnetic field, which is induced by the electric
current <strong>of</strong> the particles.
Next we explain how the number density is obtained by
using the chiral anomaly in such a complicated situation[3].
We assume an axial symmetric electric field E(r, t)
switched on at t = 0, where r denotes cylindrical radial
coordinate. Then, electrons and positrons are produced
and their axial symmetric electric current generates azimuthal
magnetic field Bθ(r, t). We also assume that the
back ground strong magnetic field B is uniform and static.
These quantities are governed by Maxwell equations,
∂tBθ(r, t) = ∂rE(r, t)
∂tE(r, t) = ∂r(rBθ(r, t))
− J(r, t), (4)
r
with the initial conditions Bθ(r, t = 0) = 0 and E(r, t =
0) = E0 exp(−r 2 /R 2 ), where R denotes the width <strong>of</strong> the
electric field and J(r, t) is the electric current carried by
electrons and positrons, given by J = 2ene.
We should make a comment that the form <strong>of</strong> the current
J = 2ene may be obtained by imposing the condition <strong>of</strong>
the energy conservation,
∫
∂t d 3 {
1
x
2 (E2 + B 2 }
θ) + ϵ = 0 (5)
where ϵ denotes the energy
∫
density <strong>of</strong> electrons and
t
positrons; ϵ = nepF = ne 0 dt′ eE(t ′ ). That is, positrons
and electrons are produced with momentum p = 0 and accerelated
by the electric field so that their Fermi momentum
pF is equal to ± ∫ t
0 dt′ eE(t ′ ). Thus, the Fermi distribution
at zero temperature is formed. Obviously, the energy density
<strong>of</strong> each particle is given by 1
2ne ∫ t
0 dt′ eE(t ′ ).
We rewrite the condition in the following,
∫
d 3 x(E∂tE + Bθ∂tBθ + ∂tϵ) (6)
=
∫
d 3 {
x E ( ∂tE − 1
r ∂r(rBθ) ) ∫
}
+ ∂tϵ (7)
= d 3 ∫
x(−JE + ∂tϵ) (8)
= d 3 xE(−J + 2ene) = 0 (9)
where we have used the Maxwell equations and the equa-
tion ∂tnepF = γBE ∫ t
0 dt′ eE = E ∫ t
0 dt′ ∂tne = neE
with γ ≡ e2
8π 2 . Since E(t = 0) can be taken arbitrary, we
find that the current J is given by 2ene.
Using eqs. (1) and (4), we derive the equation for the
electric field E under the effects <strong>of</strong> the back reaction,
∂ 2 t E(r, t) =
(
∂ 2 r + ∂r
r
)
e3
− B E(r, t). (10)
4π2 We see that the effect <strong>of</strong> the back reaction gives rise to an
effective mass term m 2 ≡ e3
4π 2 B <strong>of</strong> the electric field.
By solving eq. (10) and using the chiral anomaly (1), we
explicitly obtain the number density <strong>of</strong> electrons,
ne(r, t)
= e2 E0R 2
16π 2
∫
∞
and the azimuthal magnetic field
Bθ(r, t)
2 E0R
= −
2
0
∫ ∞
0
kdk sin(t√ k 2 + m 2 )
√ k 2 + m 2
J0(kr)e −k2R 2
/4
,
kdk sin(t√k2 + m2 )
√ J1(kr)e
k2 + m2 −k2R 2 /4
,
where J0 and J1 denote Bessel functions. We can see that
these solutions are reduced to the ones in the case <strong>of</strong> the
homogeneous electric field as R → ∞.
In Figs. 1 and 2, we show the spatial and temporal behaviors
<strong>of</strong> these solutions with R = 10 and t in unit <strong>of</strong> R.
nr
0.8
0.6
0.4
0.2
0.0
0 5 10 15
r
20 25 30
Figure 1: number densities ne(r) at t = 0.2 (small dots)
and at t = 0.6 (large dots)
Br
0.4
0.3
0.2
0.1
0.0
0 5 10 15
r
20 25 30
Figure 2: azimuthal magnetic fields Bθ(r) at t = 0.2 (small
dots) and at t = 0.6 (large dots)

Pair production in heat bath
Finally, we give another example for the utility <strong>of</strong> the
chiral anomaly, Schwinger mechanism in heat bath. That
is, the pair production arises in heat bath and the produced
particles are thermalized immediately after their production.
Thus, the distribution <strong>of</strong> the particles is given by a
Fermi distribution with finite temperature.
For simplicity, we assume that the electric field is uniform
and we take into account only back reaction <strong>of</strong> reducing
the electric field energy by pair productions. In
this case we need electric current in the heat bath when
we solve a Maxwell equation ∂tE = −J. As explained
above, we impose the condition <strong>of</strong> the energy conservation,
∫
3 1
∂t d x( 2E2 + ϵ) = ∫ d3x(−EJ + ∂tϵ) = 0 in order to
find J. Here the energy density is given by
∫ ∞
p
ϵ = γ dp
(11)
1 + exp(p − µ(t))β
0
with β = 1/T where T is the temperature, where µ(t) is
the chemical potential which depends on the number density
ne(t) through the formula,
∫ ∞
1
ne = γ dp
. (12)
1 + exp(p − µ(t))β
0
Using the formulae ∂tϵ = ∂nϵ∂tne = ∂nϵγEB in the condition
<strong>of</strong> the energy conservation, we find that J = γB∂nϵ.
Therefore, using the anomaly equation and Maxwell
equation, we obtain the equation <strong>of</strong> ne,
∂ 2 t ne + 2γeB ne exp(neβ/n0)
= 0, (13)
exp(neβ/n0) − 1
with n0 ≡ eB/8π 2 . It is easy to see that the formula in
the heat bath is reduced to the one in vaccum when we
take β → ∞; effective mass becomes m = √ e 3 B/4π 2
as we have shown above; the mass means the frequency <strong>of</strong>
ne(t) ∝ sin(mt).
Although we can not analytically solve the equation, numerical
solutions are available. We have shown the temporal
behaviors <strong>of</strong> the number density (Fig. 3) and the electric
field (Fig. 4) in both vacuum and heat bath. We can see
that the electric field decays more rapidly in the heat bath
than in vacuum. Similarly, we see that the number density
<strong>of</strong> electrons and positrons is smaller in the heat bath
than in vacuum. This is caused by the fact that according
to the Fermi distribution, each electron and positron can
have much larger energies in the heat bath with finite β
than in vacuum with β = ∞ when it is produced. Since a
pair production in the heat bath decreases the energy <strong>of</strong> the
electric field more than in vacuum, the electric field decays
more rapidly in the heat bath than in vacuum. Therefore,
we find that Schwinger muchanism proceeds more effectively
in heat bath than in vacuum.
CONCLUSION
To summarize, we have shown that simply using the chiral
anomaly we can obtain physically interesting quantities
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0.5 1.0 1.5 2.0 2.5
Figure 3: number densities ne(t) with arbitrary scale in
vacuum (dash) and in heat bath (line)
1.0
0.5
0.5
0.5 1.0 1.5 2.0 2.5
Figure 4: electric fields E(t) with arbitrary scale in vacuum
(dash) and in heat bath (line)
<strong>of</strong> Schwinger mechanism without calculating wave functions.
Thus, we can discuss pair productions under electric
flux tube in vacuum or homogeneous electric field in
heat bath, which could not be obtained with the method <strong>of</strong>
the evaluation <strong>of</strong> wave functions. The simplification comes
from the fact that the problem in Schwinger mechanism
is reduced to one dimentional one when strong magnetic
field is present. Such a collinear strong magnetic field and
electric field (B, E ≫ (electron mass or quark mass) 2 )
are produced by high-energy heavy-ion collisions [5]. In
the collisions, corresponding strong fields are color electric
and magnetic fields. Thus, they decay into quarks by
Schwinger mechanism very rapidly (< 1fm/c).
REFERENCES
[1] J. Schwinger, Phys. Rev. 82 (1951) 664.
[2] N. Tanji, Ann. Phys. 324 (2009) 1691; see references therein.
[3] A. Iwazaki, Phys. Rev. C80 (2009) 052202.
[4] S.P. Gavrilov and D.M. Gitman, Phys. Rev. D53 (1996) 7162.
[5] E. Iancu, A. Leonidov and L. McLerran, hep-ph/0202270.

Abstract
Strong field physics in condensed matter ∗
There are deep similarities between non-linear QFT
studied in high-energy and non-equilibrium physics in condensed
matter. Ideas such as the Schwinger mechanism and
the Volkov state are deeply related to non-linear transport
and photovoltaic Hall effect in condensed matter. Here, we
give a review on these relations.
INTRODUCTION
In strong field physics, researchers are interested in the
change <strong>of</strong> the “quantum vacuum” due to strong external
fields. A typical example is the decay <strong>of</strong> the QED vacuum
in strong electric fields due to the Schwinger mechanism
[1]. When a strong enough electric field is applied
to the vacuum, pair creation <strong>of</strong> electrons and positrons
takes place and the insulation breaks down. The threshold
for this phenomena is known as Schwinger’s critical
field and is given by Eth = m 2 e/e = 1.3 × 10 16 V/cm.
Since the critical field is extremely strong, direct experimental
verification is still a challenge. On the other
hand, in the condensed matter community, there is an increasing
interest in non-equilibrium phase transitions and
non-linear transport in strongly correlated electron systems
(Fig. 1)[2, 3, 4, 5, 6, 7]. In the experiments, one also applies
strong electric fields and the original insulating phase is destroyed.
However, the threshold for dielctric breakdown
is orders <strong>of</strong> magnitude smaller than the Schwinger mechanism
in QED since the excitation gap is far smaller. This
makes condensed matter systems to be an idealistic play-
Field strength
E
pair creation
Schwinger limit
~1 eV/a
induced-polarization
V
nonlinear transport
dielectric breakdown
photovoltaic Hall effect
(nonlinear)-optics
~1 eV Photon energyΩ
T. Oka † , University <strong>of</strong> Tokyo, Japan
“Keldysh line”
ξ E ~ Ω
photo-induced
phase transition
Figure 1: Several phenomena in condensed matter physics
in strong electric fields plotted in the (E, Ω)-space.
∗ Work supported by Grant-in-Aid for Scientific Research on Priority
Area “New Frontier <strong>of</strong> Materials Science Opened by Molecular Degrees
<strong>of</strong> Freedom”.
† oka@cms.phys.s.u-tokyo.ac.jp
ground to test and develop theoretical ideas in non-linear
QFT. Non-linear physics has been studied rather independently
in the two fields, high energy and condensed matter,
during the past few decades, and several parallel ideas were
developed. The aim <strong>of</strong> this article is to explain some <strong>of</strong> the
correspondences (Table 1).
PAIR CREATION IN STRONG ELECTRIC
FIELDS
Hesenberg-Euler’s effective action and the nonlinear
extension <strong>of</strong> the Berry’s phase approach to
polarization [4]:
We study lattice electrons in homogenious electric fields.
In the time-dependent gauge, this can be realized by adding
a time dependent phase to the hopping term in the lattice
Hamiltonian. For example, for a one-dimensional model, a
typical Hamiltonian is given by
H(Φ) = −
L∑ ∑
i=1
σ
(e iΦ c †
i+1σ ciσ + e −iΦ c †
iσ ci+1σ)(1)
+U ∑
ni↑ni↓ + ∑
Vini.
i
We impose periodic boundary condition and the phase Φ is
proportional to the magnetic flux through the ring (L number
<strong>of</strong> sites). The time derivative <strong>of</strong> the magnetic flux is related
to the applied electric field through F (t) = eaE(t) =
dΦ(t)/dt, where e is the charge quantum and a the lattice
constant. U represents on-site Coulomb repulsion and Vi
the local potential. The hopping term is set to unity. The
Hubbard model (U > 0, Vi = 0) at half-filling is in the
Mott insulating phase for positive U in one dimension.
Here, we study what happens to the an insulator when we
apply strong electric fields. We denote the eigenstates <strong>of</strong>
the Hamiltonian H(Φ) by |ψn(Φ)⟩, n = 0, 1, . . . and study
the time evolution starting from the groundstate |ψ0(Φ)⟩.
The groundstate-to-groundstate amplitude defined by
Ξ(t) ≡ ⟨ψ0(Φ(t))|e −i
∫ t
0 H(Φ(s))ds |ψ0(0)⟩e i
∫ t
0 E0(Φ(s))ds
is <strong>of</strong> central importance. In the long time limit, an asymptotic
behavior (d is dimension) Ξ(t) ∼ e itLd L is expected
to take place where L is the condensed matter version <strong>of</strong>
the Heisenberg-Euler effective Lagrangian. The imaginary
part describes quantum tunneling where Γ(F )/L d ≡
2Im L(F ) gives the speed <strong>of</strong> the exponential decay <strong>of</strong> the
vacuum (groundstate). This quantity is proportional to the
decay rate <strong>of</strong> the Loschmidt Echo L(t) = |Ξ(t)| 2 . The
i
(2)

Table 1: Related ideas in strong field physics
High Energy Condensed Matter
Schwinger mechanism in QED Landau-Zener tunneling in band insulators
Heisenberg-Euler effective Lagragian Non-adiabatic geometric phase, Loschmidt Echo
Vacuum polarization Extended Berry’s phase theory <strong>of</strong> polarization
Pair creation in interacting systems (e.g. QCD) Many-body Schwinger-Landau-Zener mechanism
in strongly correlated system
Dirac particles in circularly polarized light Photovoltaic Hall effect
Furry picture Floquet picture
real part ReL is written in terms <strong>of</strong> a non-adiabatic phase
called the Aharonov-Anandan phase (which we denote γ)
that the wave function acquires during the time-evolution.
For band insulators (U = 0) in dc-electric fields, the effective
Lagrangian becomes [4]
∫
dk
Re L(F ) = −F
BZ (2π) d
γ(k)
, (3)
2π
∫
dk
Im L(F ) = −F
(2π) d
1
ln [1 − p(k)] , (4)
4π
BZ
where the momemtum integral is over the Brillouin Zone
(BZ). There is an interesting parallel theory developeded
in the condensed matter comunity. This is known as the
Berry’s phase theory <strong>of</strong> polarization[8, 9, 10, 11, 12], where
the ground-state expectation value <strong>of</strong> the twist operator
2π −i e L ˆ X , which shifts the phase <strong>of</strong> electron wave functions
on site j by − 2π
L j, plays a crucial role. It was revealed that
the real part <strong>of</strong> a quantity
w = −i
2π
ln⟨0|e−i L
2π ˆ X
|0⟩ (5)
gives the electric polarization Pel = −Rew [10] while its
imaginary part gives a criterion for metal-insulator transition,
i.e., D = 4πImw is finite in insulators and divergent
in metals [11]. The present effective Lagrangian can be
regarded as a non-adiabatic (finite electric field) extension
<strong>of</strong> w. To give a more accurate argument, we rewrite the
effective Lagrangian in the time-independent gauge
L(F ) ∼ −i¯h
τL ln
(
i −
⟨0|e ¯h τ(H+F ˆ i X)
|0⟩e ¯h τE0
)
(6)
for d = 1. Let us set τ = h/LF and consider the
small F limit. For insulators we can replace H with the
groundstate energy E0 to have L(F ) ∼ wF in the linearresponse
regime. Thus the real part <strong>of</strong> Heisenberg-Euler’s
expression[13] for the non-linear polarization P (F ) =
−∂L(F )/∂F naturally reduces to the Berry’s phase formula
Pel in the small field limit F → 0. Its imaginary part
gives the criterion for photo-induced metal-insulator transition,
originally proposed for the zero field case.
Many-body Schwinger-Landau-Zener mechanism
in stongly correlated insulators:
Next, let us consider dielectric breakdown in a strongly
correlated system. In the one-dimensional Mott insulator
where the groundstate is a state with one electron per site,
the relevant charge excitations are doublons, i.e,. doubly
occupied sites, and holes, i.e., sites with no electron. Pairs
<strong>of</strong> doublons and holes play a similar role as the pair <strong>of</strong> electrons
and positrons in the Schwinger mechanism. Indeed,
it has been shown that dielectric breakdown in Mott insulators
takes place due to pair production <strong>of</strong> charge excitations
through quntum tunneling, which is called the many-
(a)
Im
Φ
Re E
imaginary time path
(b)
35
30
25
20
Fth 15
10
5
Φ(t ) *
n=1
n=0
0 2π/L 4π/L
Φ(t 0)
LZ
Fth ITM
Fth 0
0 5 10 15 20
U
Φ = Φ(t)
Re Φ
Figure 2: (a) Many-body energy levels against the complex
AB flux Φ for a finite, half-filled 1D Hubbard model
(L = 10, N↑ = N↓ = 5, U = 0.5). Only charge excitations
are plotted. Quantum tunneling occurs between the
groundstate (labeled as n = 0) and a low-lying excited state
(n = 1) as the flux Φ(t) = F t increases on the real axis,
while the tunneling is absent for the states plotted as dashed
lines. The wavy lines starting from the singular points (×)
at Φ(t ∗ ) represent the branch cuts for different Riemann
surfaces, along which the solutions n = 0 and n = 1
are connected. In the DDP approach, the tunneling factor
is calculated from the dynamical phase associated with
adiabatic time evolution (DDP path) that encircles a gapclosing
point at Φ(t ∗ ) on the complex Φ plane. (b) Threshold
electric field obtained by the imaginary time method
(solid) and the naive Landau-Zener formula (dashed).

IV-characteristics conductance
Figure 4: (upper) DC current. (lower left) IVcharacteristics.
(lower right) Conductance.
Especially, in a circularly polarized light, a gap opens at
the Dirac point [7]. This has an important physical consequence
since a gap <strong>of</strong> a 2+1 dimensional Dirac electron
is related to parity anomaly and is detectable through transport
measurements, i.e., the Hall effect. In 2+1 dimensions,
the Hall conductivity can be written as a momentum integral
<strong>of</strong> the Berry curvature (∼ Chern density) over the
Brillouin zone. This is known as the TKNN formula[32],
and is know extended to ac-driven transport via the Floquet
picture (∼ Furry picture) [7]
σxy(Aac) = e 2
∫
dk
(2π) d
∑
fα(k) [ ∇k × Aα(k) ]
. (10)
z
α
Here, Aα(k) ≡ −i⟨⟨Φα(k)|∇ k |Φα(k)⟩⟩ is the photoinduced
artificial gauge field. In the Floquet picture, the
Green’s function incorporates the effect <strong>of</strong> photon absorption
and emission (Fig. 3 (a)), and Hall conductivity is
given by the bubble diagram in the non-interacting case,
which is nothing but the parity anomaly diagram. The
photo-induced Berry curvature shown in Fig. 3 (c) acts as
an artificial magnetic field and becomes finite when the circularly
poralized light is introduced.
The current in the presence <strong>of</strong> circularly poralized light
in a graphene ribbon attached to two electrodes is plotted
in Fig. 4. The calculation has been done by combining the
Keldysh green’s function method with the Floquet picture.
The Hall current, which is originally absent, increases as
the strength <strong>of</strong> light becomes stronger. The numerical result
supports our understanding <strong>of</strong> the photovotaic Hall effect
obtain by the extended TKNN formula (eqn.(10)).
We would like to acknowledge Naoto Tsuji, Martin Eckstein
and Philipp Werner for enlighting discussions. It is
a pleasure to thank Gerald Dunne for illuminating discussions
during PIF2010.
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(1993).
[10] R. Resta, Phys. Rev. Lett. 80, 1800 (1998).
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[12] M. Nakamura and J. Voit, Phys. Rev. B 65, 153110 (2002).
[13] W. Heisenberg and H. Euler, Z.Physik 98, 714 (1936).
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V. E. Korepin, The One-Dimensional Hubbard Model (Cambridge,
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Nijs, Phys. Rev. Lett. 49, 405 (1982).

Abstract
Non-linear charge transport in plasma under strong field ∗
S. Nakamura † , Department <strong>of</strong> Physics, Kyoto University, Kyoto 606-8502, JAPAN
We study nonlinear charge transport in a strongly interacting
system <strong>of</strong> charges under the presence <strong>of</strong> external
electric field, by using the AdS/CFT correspondence. We
show that the pair-creation process assisted by the external
field can cause negative differential resistivity.
INTRODUCTION
Recent development <strong>of</strong> high-intensity lasers is opening a
new window for studies on strong-field dynamics. One <strong>of</strong>
the main subjects in the strong-field physics is pair creation
<strong>of</strong> charged particles assisted by the strong external field. In
this talk, we consider a pair-creation process in a strongly
interacting system under the presence <strong>of</strong> an external field
and its relationship to the nonlinear charge transport. We
shall show that the pair-creation process affects the nonlinear
charge transport in such a way that the negative differential
resistivity (NDR) 1 can be realized.
NDR is a nonlinear phenomenon in charge transport
where the electric field (E) decreases with increasing current
density (J), and vice versa (Fig. 1). This has been observed
in various materials and devices[2]. Since electronic
devices that exhibit NDR are useful in electric circuits, the
understanding <strong>of</strong> NDR is important from the viewpoint <strong>of</strong>
industrial applications, as well. However, theoretical study
on NDR has difficulties coming from the following facts:
• NDR is a nonlinear phenomenon where we need to go
beyond the linear response theory.
• The system is far from equilibrium owing to the dissipation
caused by the finite current.
• Nonperturbative analysis is necessary if the NDR is
associated with the transition <strong>of</strong> the vacuum such as
the metal-insulator transition.
We employ the AdS/CFT correspondence [3, 4, 5] to overcome
these difficulties.
The typical J-E characteristics we obtain falls into the
category <strong>of</strong> the S-shaped NDR 2 which is sketched at Fig. 1.
NDR is realized between B and C in Fig. 1. J(E) is a
∗ Talk based on the original work in Ref. [1]. The present work was
supported by MEXT KAKENHI (21105006), Grant-in-Aid for Scientific
Research on Innovative Areas “Elucidation <strong>of</strong> New Hadrons with a Variety
<strong>of</strong> Flavors,” and by the Grant-in-Aid for the Global COE Program
“The Next Generation <strong>of</strong> Physics, Spun from Universality and Emergence”
<strong>of</strong> MEXT <strong>of</strong> Japan.
† nakamura@ruby.scphys.kyoto-u.ac.jp
1 This may also be referred to as negative differential conductivity
(NDC) in some literature.
2 This corresponds to the SNDC in Ref. [2]
J
A
C
Figure 1: Schematic J-E characteristics with NDR.
multivalued function <strong>of</strong> E. Experimentally, the multivalued
behavior is obtained by measuring E as a function <strong>of</strong>
the controlled current density J; note that the J-E curve is
“N-shaped” if the axes are swapped. The function E <strong>of</strong> J is
still a single-valued function, and the NDR is well-defined
if J is controlled. If we control E instead, the NDR branch
is unstable and the hysteresis is observed. In this talk, we
regard J as a control parameter and E is determined as a
result <strong>of</strong> dynamics.
B
ADS/CFT CORRESPONDENCE
The AdS/CFT correspondence is a conjectured equivalence
between strongly-interacting non-abelian quantum
gauge theories and higher-dimensional classical gravitational
theories. This enables us to analyze the nonperturbative
nature <strong>of</strong> the gauge theories in terms <strong>of</strong> general relativity.
The correspondence is at the level <strong>of</strong> the microscopic
theory, and it is potentially possible to describe nonequilibrium
process. What is surprising is that the notion <strong>of</strong> equilibrium
appears in the gravity side in terms <strong>of</strong> black holes.
The coarse graining is automatically performed when we
solve the Einstein’s equation to obtain the black hole solution.
These features <strong>of</strong> the AdS/CFT correspondence
tempts us to apply it to nonlinear nonequilibrium physics
<strong>of</strong> the strongly-interacting gauge theories. Indeed, the nonlinear
conductivity <strong>of</strong> a global charge in some system has
already been computed by using the AdS/CFT correspondence
[6]. 3
The AdS/CFT correspondence has also disadvantages.
One <strong>of</strong> them is that we cannot analyze U(1) gauge theory
like QED. The conventional analysis in the AdS/CFT
correspondence is restricted to SU(Nc) gauge theories at
the large-Nc limit. However, we can still introduce nondynamical
external fields <strong>of</strong> the U(1) gauge theory. The
SU(Nc) gauge theory can also be idealized in such a way
that the interaction among the charges is <strong>of</strong> Coulomb type.
3 See also, for example, Refs. [7, 8].
D
E

What we shall do in the present work is to consider an idealized
gauge theory that has the Coulomb type interaction
among the charges in the presence <strong>of</strong> the U(1) external
electric field.
MICROSCOPIC THEORY
Our idealized gauge theory is the (3+1)-dimensional
SU(Nc) N =4 super-symmetric Yang-Mills (SYM) theory
with Nf flavors <strong>of</strong> fundamental N =2 hypermultiplets. This
is a supersymmetric cousin <strong>of</strong> Quantum Chromodynamics
(QCD), but the inter-quark potential is <strong>of</strong> Coulomb type at
zero temperature with infinite current quark mass due to
the conformal nature <strong>of</strong> N = 4 SYM. The supersymmetry
is broken at finite temperatures. The number <strong>of</strong> colors Nc
is taken to be infinity with the ’t Ho<strong>of</strong>t coupling gYMN 2 c
kept fixed, where gYM is the Yang-Mills coupling constant.
We define λ ≡ 2gYMN 2 c in this article, and take the strongcoupling
limit λ ≫ 1. In our setup, the quarks and the antiquarks
are strongly correlated owing to the large λ. The
quarks carry the global U(1) baryon (U(1)B) charge (or the
quark charge), and we analyze the conductivity associated
with this charge. The above setup is employed to make the
AdS/CFT correspondence applicable.
Ignoring the super-partners, the theory contains SU(Nc)
adjoint gluons and the Nf species <strong>of</strong> SU(Nc) fundamental
quarks (and the anti-fundamental antiquarks). The gluons
and the quarks play the roles <strong>of</strong> photons and the electrons
in QED, respectively. The antiquark may be regarded as a
counterpart <strong>of</strong> positrons or holes, depending on the system
we make an analogy.
The system is in the deconfinement phase <strong>of</strong> gluons
(which means that the degree <strong>of</strong> freedom <strong>of</strong> the gluonic
sector is O(N 2 c )) at nonzero temperatures, but the quark
and antiquark may still form the bound states depending on
the parameters <strong>of</strong> the theory. If they form the bound states,
the system is an insulator since the bound states are neutral.
The system becomes a conductor if the bound states
are unstable and the charge carriers are liberated. This system
shares several features similar to those <strong>of</strong> the excitonic
insulators [9, 10] or sQGP [11]. We find that the system
shows NDR due to the pair-creation process <strong>of</strong> the charge
carriers. Our result suggests a possibility to observe NDR
in some excitonic insulators or in some quark-hadron systems,
as we shall discuss later.
Let us consider how to realize a nonequilibrium steady
state (NESS) with a constant current in the conductor
phase. Since our quarks/antiquarks interact strongly with
the gluons, the kinetic energy <strong>of</strong> the quarks/antiquarks will
be dissipated. Because <strong>of</strong> the dissipation, the system will
be heated up if we maintain a constant current. However,
we can realize a steady state with a constant current by taking
the probe limit Nc ≫ Nf . The degree <strong>of</strong> freedom <strong>of</strong>
the gluonic sector is O(N 2 c ), whereas that <strong>of</strong> the flavor sector
(the quark/antiquark sector) is O(NcNf ); the gluonic
sector has infinitely large degrees <strong>of</strong> freedom in comparison
with the flavor sector at this limit. As a result, the
dissipated energy from the flavor sector is absorbed into an
infinitely large reservoir <strong>of</strong> gluons and the system is wellapproximated
as a NESS for the time period shorter than
O(Nc) [12]. The gluonic sector acts as a “heat bath” for
the flavor sector in this sense. Note that the interaction between
the charge carriers and the “heat bath” is taken into
account in our setup.
GRAVITY DUAL
The gravity dual <strong>of</strong> the foregoing microscopic theory is
the so-called D3-D7 system [13], where the Nf D7-branes
are embedded in the background geometry given by a direct
product <strong>of</strong> a 5-dimensional AdS-Schwarzschild black hole
(AdS-BH) and S 5 . The flavor sector is governed by the
dynamics <strong>of</strong> the D7-branes, whereas the gluonic sector is
described by the AdS-BH. We take the string tension to be
1 for simplicity, namely, 2πl 2 s = 1, where ls is the string
length. The metric <strong>of</strong> the AdS-BH part is given by
ds 2 = − 1
z 2
(1 − z 4 /z 4 H )2
1 + z 4 /z 4 H
dt 2 + 1 + z4 /z 4 H
z 2
d⃗x 2 + dz2
, (1)
z2 where z is the radial coordinate <strong>of</strong> the black hole. The horizon
is located at z = zH and the boundary is at z = 0.
The Hawking temperature that corresponds to the temperature
√ <strong>of</strong> the gluonic sector (heat bath) is given by T =
2/(πzH). ⃗x denotes the 3-dimensional spatial directions.
The S5 metric is dΩ2 5 = dθ2 + sin 2 θdψ2 + cos2 θdΩ2 3,
where 0 ≤ θ ≤ π/2, and dΩd is the volume element <strong>of</strong>
the unit d-dimensional sphere. The radius <strong>of</strong> the S5 has
been taken to be 1, which is equivalent to the choice <strong>of</strong>
λ = (2π) 2 .
The D7-branes are wrapped on an S3 part <strong>of</strong> the S5 . We
choose our space-time coordinates in such a way that the
S3 is located at ψ = 0. Let us choose the worldvolume
coordinates <strong>of</strong> the D7-brane to be the same as the spacetime
coordinates. We also assume that the external U(1)B
electric field E is applied along the x direction. We have
a U(1) gauge field Aµ on the D7-branes, which couples to
the U(1)B current. The relationship between the external
field E and the resulting current J along the x direction
is given by the GKP-Witten prescription [4, 5] as (see also
J
2 N z2 +O(z 4 ), where
we have employed the gauge ∂xAt = 0. N is given by
N = Nf TD7(2π2 ), where TD7 is the D7-brane tension. In
our choice <strong>of</strong> λ = (2π) 2 and 2πl2 s = 1, N = NcNf /(2π) 2 .
We consider the vanishing quark-charge density in most
cases and we set the other components <strong>of</strong> the vector potential
to be zero unless specified.
The D7-brane action with the present setup is explicitly
Ref. [6]) Ax(z, t) = −Et+const.+ 1
written as
∫
SD7 = −N
dtd 3 xdz cos 3 [
θ |gtt|gxxgzz
(
− gzz( ˙ Ax) 2 − |gtt|(A ′ x) 2) ] 1/2
, (2)
where the prime (the dot) denotes the differentiation with
respect to z (t). We have already integrated the S 3 part

under the assumption <strong>of</strong> the symmetry along it. gtt, gxx
and gzz are the induced world-volume metric, and they are
equal to the background metric (1) except for gzz = 1/z 2 +
θ ′ (z) 2 .
NONLINEAR CONDUCTIVITY
It was found [6] that the on-shell D7-brane action becomes
complex unless we choose a specific combination <strong>of</strong>
J and E; the relationship between J and E is determined
by the reality condition <strong>of</strong> the on-shell action, hence, J is
obtained as a nonlinear function <strong>of</strong> E. The on-shell action
is given by [6] ¯ SD7 = −N ∫ dzdtd 3 x √ ¯gzz|gtt| −1√ F1F2
with F1 = |gtt|gxx − E 2 and F2 = |gtt|g 2 xx cos 6 ¯ θ −
gxxJ 2 /N 2 , where ¯gzz is the induced metric given by ¯ θ,
which is the on-shell configuration <strong>of</strong> θ(z). Since both F1
and F2 cross zero somewhere between the boundary and
the horizon, the only way to make ¯ SD7 real is to choose
J and E so that F1 and F2 cross zero at the same point
z = z∗. The hypersurface given by z = z∗ is <strong>of</strong>ten
called the “singular shell”. Then, the reality condition
F1(z∗) = F2(z∗) = 0 gives us the relationship between
J and E in the form <strong>of</strong> J = σ0E [6], where
σ0 = N T (e 2 + 1) 1/4 cos 3 ¯ θ(z∗). (3)
Our task is to solve the equation <strong>of</strong> motion (EOM) for θ
to obtain the explicit representation. ¯ θ(z) can be expanded
as ¯ θ(z) = mqz + O(z 3 ), where mq is the current quark
mass [13], which is a parameter <strong>of</strong> the microscopic theory.
mq is related to the gap in condensed matters.
Let us choose the temperature to be T = √ 2/π so that
zH = 1, e = E/2 and z∗ = √ E 2 /4 + 1 − E/2. 4 We
further fix NcNf = 40. NcNf governs the pair-creation
rate <strong>of</strong> the charge carriers as we shall explain later. We
need to solve the EOM for θ numerically. The boundary
condition we employ is θ(z)/z|z=0 = mq and we request
the absence <strong>of</strong> singularity in the D7-brane configuration.
For earlier studies on the nonlinear conductivity by using
this method, see for example, Refs. [7, 8].
RESULTS
Examples <strong>of</strong> J-mq curves at several values <strong>of</strong> E are
shown in Fig. 2. Of course, mq has a unique value at a
given model and we need to choose some particular value
<strong>of</strong> mq. We find that there are two different possible values
<strong>of</strong> J at given mq and given E in some parameter region,
which indicate the multi-valued nature <strong>of</strong> J(E). Furthermore,
if we increase E along the given mq, the smaller
J decreases while the larger J increases; the smaller-J
branch shows NDR, whereas the larger-J branch has a positive
differential resistivity. Note that the smaller-J branch
4 In this article, we have employed the natural units c = ¯h = kB = 1.
If our scale unit is meV (mili eV), T ∼ 5 K. If we identify the unit
quark charge with the unit charge <strong>of</strong> electrons, the effective fine-structure
constant read from the Coulomb interaction in the inter-quark potential is
∼ 1.
mq
1.34
1.33
1.32
1.31
1.30
1.29
E0.12
E0.20
E0.15
1.28
0.000 0.005 0.010 0.015 0.020 0.025
Figure 2: J-mq curves at E = 0.12, 0.15, and 0.20. mq is
maximum at a nonzero but small value <strong>of</strong> J.
E
0.25
0.20
0.15
0.10
0.05
0.00
0.000 0.005 0.010 0.015 0.020 0.025
Figure 3: J-E curve at mq = 1.315. Ec = 0.11 in this
case. NDR appears in J ≤ 0.0031 and is absent for E ≥
0.19.
is a very narrow window in the full part <strong>of</strong> the J-mq curve.
For example, the J-mq curve at E = 0.2 extends until
J = 0.288, and the width <strong>of</strong> the smaller-J branch along the
J axes is less than 2% <strong>of</strong> the full part. The detailed analysis
shows that the highest value <strong>of</strong> mq approaches around
1.310 at the E → +0 limit, suggesting that Ec = 0 if
mq < 1.310. This is consistent with the fact that the system
is a conductor at sufficiently small mq in comparison
with T (or sufficiently high T in comparison with mq).
An example <strong>of</strong> J-E relation at mq = 1.315 is given in
Fig. 3. 5 The system is an insulator for E < Ec = 0.11. If
E ≥ Ec, the insulation is broken and we observe a current.
NDR is realized in the smaller-J region. We always have
the J = 0 branch on top <strong>of</strong> the vertical axis. Therefore,
our interpretation is that Fig. 3 shows the B-C-D region <strong>of</strong>
Fig. 1 with the axes swapped; our NDR falls within the Sshaped
NDR. There may be a small tunneling current that
almost overlaps with the vertical axis, but we could not detect
it within our numerical precision. We leave the detailed
analysis on the tunneling current in a future work.
It is important to clarify what is the physically essential
process in our NDR. Let us consider the doped cases.
We can also “dope” the system by introducing finite quarkcharge
density [14, 15]. In this case, the system is always a
5 If we choose our scale unit to be meV, the critical electric field Ec
in Fig. 3 is Ec ∼ 5 × 10 −1 V/m, and the current density realized at
E = Ec is J ∼ 1 × 10 −4 mA/mm 2 .
J
J

conductor. The current is given by [6]
√
J = σ2 0 + d2 /(e2 + 1) E, (4)
where d is related to the quark-charge density ρ through
d = ρ/( π
√
2
2 λT ). Owing to the doped charges, any small
E causes a current and we observe Ohm’s law in the small-
J region. If we raise J, we may again observe NDR owing
to the nontrivial behavior <strong>of</strong> σ0. It is indeed the case if d
is small enough not to smear the contribution <strong>of</strong> σ0. In this
case, the curve in Fig. 3 will be “N-shaped”, (S-shaped in
the sense <strong>of</strong> Fig. 1) starting at the origin. The point is that
the d-dependent term in the square root in (4) does not have
any structure to produce NDR. Therefore, the σ0-part in (4)
is crucial for NDR. It is understood that the current due to
the σ0-part is caused by the pair creation <strong>of</strong> the charge carriers.
The reasons are as follows: it contributes the current
with the total system being kept neutral, and it vanishes if
the mass <strong>of</strong> the charge carriers mq is infinite. [6] 6 As a conclusion,
the pair-creation process is essential for our NDR.
DISCUSSION
We can suggest a phenomenological model <strong>of</strong> NDR. The
phenomenological origins <strong>of</strong> NDR are classified into three
types (except for the tunnel effect for some semiconductor
junctions) in Ref. [2]: 1) nonlinearity <strong>of</strong> mobility, 2)
nonlinearity <strong>of</strong> carrier density, and 3) nonlinearity <strong>of</strong> the
electron temperature. 7 We have found that our NDR originates
in the pair creation <strong>of</strong> the charge carriers but not in
the normal current <strong>of</strong> the doped charges. This means that
the above feature 2) is crucial in our NDR. Although further
study is necessary to reach the final conclusion, it is
natural to assume that both the normal current and the paircreated
current contribute 1) and 3) regardless <strong>of</strong> the origin
<strong>of</strong> the charge carriers. If this assumption is right, 1) and 3)
do not seem to be important in our NDR. The behavior <strong>of</strong>
our NDR is in the category <strong>of</strong> the “SNDC” in Ref. [2] and
it may be attributed to the impact ionization explained in
Ref. [2]. The proposal <strong>of</strong> the many-body avalanche model
<strong>of</strong> NDR[17, 18] also matches our picture. It is important
to study further the connection between our results and the
phenomenological models <strong>of</strong> NDR.
We can also see our results from the viewpoint <strong>of</strong> the
quark-hadron physics and that <strong>of</strong> the excitonic insulators.
Let us consider the sQGP state [11] where the quarkantiquark
bound state exists in the deconfinement phase <strong>of</strong>
gluons. Our results suggest that the quarks are liberated
at the critical value <strong>of</strong> the electric field and their current
may show NDR. We may also have a chance to observe a
qualitatively similar NDR in excitonic insulators after the
insulation breaking. It is important to study how general
this NDR is, in quark/meson systems and in the systems <strong>of</strong>
6 We also point out that σ0 is proportional to NcNf . This suggests
that it may be a one-loop contribution <strong>of</strong> the quarks, as in the perturbative
computation <strong>of</strong> the pair-creation rate. Note that the quark loops have been
taken into account to the 1-loop order in the probe approximation.
7 See also Ref. [16].
charge-anticharge bound states, in the presence <strong>of</strong> strong
external fields.
We expect that the present system is a good theoretical
playground for studies on nonlinear charge transport
and nonequilibrium steady states. The AdS/CFT correspondence
can be a new tool for studying nonequilibrium
physics as we have demonstrated here.
The author thanks the organizers <strong>of</strong> PIF2010 conference.
Discussions with the participants <strong>of</strong> various research areas
such as plasma physics, strong-field dynamics and laser
physics are quite fruitful in planning further studies along
the direction <strong>of</strong> the present work.
REFERENCES
[1] S. Nakamura, Prog. Theor. Phys. 124 (2010), 1105.
[2] E. Schöll, Nonlinear Spatio-Temporal Dynamics and Chaos
in Semiconductors (Cambridge University Press, Cambridge
2001).
[3] J. M. Maldacena, Adv. Theor. Math. Phys. 2 (1988), 231
[Int. J. Theor. Phys. 38 (1999), 1113].
[4] S. S. Gubser, I. R. Klebanov and A. M. Polyakov, Phys. Lett.
B 428 (1998), 105.
[5] E. Witten, Adv. Theor. Math. Phys. 2 (1998), 253.
[6] A. Karch and A. O’Bannon, J. High Energy Phys. 0709
(2007), 024.
[7] J. Erdmenger, R. Meyer and J. P. Shock, J. High Energy
Phys. 0712 (2007), 091.
[8] T. Albash, V. G. Filev, C. V. Johnson and A. Kundu, J. High
Energy Phys. 0808 (2008), 092.
[9] N. F. Mott, Philos. Mag. 6 (1961), 287.
[10] See also for a review, S. A. Moskalenko and D. W. Snoke,
Bose-Einstein Condensation <strong>of</strong> Excitons and Biexcitons
(Cambridge Univ. Press, Cambridge 2000).
[11] M. Gyulassy and L. McLerran, Nucl. Phys. A 750 (2005),
30.
[12] A. Karch, A. O’Bannon and E. Thompson, J. High Energy
Phys. 0904 (2009), 021.
[13] A. Karch and E. Katz, J. High Energy Phys. 0206 (2002),
043.
[14] S. Nakamura, Y. Seo, S. J. Sin and K. P. Yogendran, J. Korean
Phys. Soc. 52 (2008), 1734.
[15] S. Kobayashi, D. Mateos, S. Matsuura, R. C. Myers and
R. M. Thomson, J. High Energy Phys. 0702 (2007), 016.
[16] H. C. Law and K. C. Kao, IEEE Trans. Electron Devices 17
(1970), 562.
[17] Y. Tokura, H. Okamoto, T. Takao, T. Tadaoki and G. Saito,
Phys. Rev. B 38 (1988), 2215.
[18] T. Oka, H. Kishida and H. Aoki, talk given at JPS 2010 Annual
Meeting, March 20th (2010).

Brilliant hardγ-production ande + e−-creation in vacuum with ultra-high
laser fields: Testing theoretical predictions at ELI-NP
Dietrich Habs, Peter Thirolf, Nina Elkina and Hartmut Ruhl
Fakultät für Physik, Ludwig-Maximilians-Universität München, D-85748 Garching, Germany
Abstract
We want to measure the hard-γ production, when a
brilliant bunch <strong>of</strong> 600 MeV electrons is injected into
the intense focus <strong>of</strong> two counter-propagating lasers with
10 24 W/cm 2 in vacuum. In a second step these hard-γ photons
can produce e + e − pairs in the same laser field in
vacuum. We describe an experiment planned for ELI-NP,
where we want to test for the first time non-perturbative
high-field QED effects.
INTRODUCTION
One <strong>of</strong> the main goals <strong>of</strong> the project Extreme Light Infrastructure
(ELI) [1] is to study the “boiling <strong>of</strong> the vacuum”
[2], i.e. the electron-positron pair creation in the vacuum
and photon production with ultra-high laser fields. At
the Extreme Light Infrastructure - Nuclear Physics (ELI-
NP) facility [3] we presently plan for laser intensities <strong>of</strong>
1024W/cm2 , equivalent to an electric field strength E =
4.7 · 1015V/m, or normalized vector potentials a = 103 ,
which are still much too small to produce directly pairs
from the vacuum. Here the laser field is characterized by
the Lorentz invariant dimensionless normalized vector potential
a:
a = eEλL
me2πc2 = eAµA µ
mec2 , (1)
whereλL is the laser wavelength andAµ is the laser vector
potential. In comparison to the Schwinger intensity <strong>of</strong>Is =
5·10 29 W/cm 2 , or the Schwinger fieldES:
Es = m2 ec 3
e¯h = 1.3·1018 V/m (2)
the extremely strong suppression by the exponential factor
exp[−π ·(Es/E)] ≈ 10 −1000
prevents the observation <strong>of</strong> pair creation, when focusing
the high-power lasers <strong>of</strong> ELI-NP into vacuum. Thus we
need some kind <strong>of</strong> seeding. The special feature <strong>of</strong> ELI-
NP is, that at the same time a Compton-backscattering γbeam
facility [5] will be installed, where a very brilliant,
intense, classical electron beam is produced in a warm electron
linac with up to 600 MeV and is used to produce γquanta
with maximum energiesEγ = 19 MeV by backscattering
laser light. Thisγ-facility can be operated in coincidence
with the high-intensity laser pulses with a repetition
rate <strong>of</strong> 1/min. Here one first might think <strong>of</strong> “dynamically
(3)
assisted pair creation” [6], injecting into the intense lowfrequency
laser focus at the same time the weak-intensity,
high-frequency γ-quanta, resulting in a new strongly reduced
exponential suppression factor:
exp[−(π −2)·(Es/E)] ≈ 10 −350 . (4)
Though this is a largely reduced hindrance factor, it is still
much too small to generate for the given repetition rates and
γ-intensities any observable effects. Narozhnyi [15, 16]
predicted for an additional injection <strong>of</strong> very high-energy
γ-quanta with energyEγ an exponential hindrance factor:
exp[−(8/3)·(Es/E)·(mec 2 /Eγ)] ≈ 10 −1 . (5)
So, if we would inject γ-quanta with much higher energy
Eγ = 1000 · mec 2 =500 MeV, the exponential suppression
would basically vanish and we could observe pair
creation. However, it would be extremely difficult to produce
such high-energy γ-quanta by Compton backscattering
with high harmonic laser pulses from the 600 MeV
electron beam with sufficient intensity. Is there another
way to inject γ-quanta <strong>of</strong> up to 500 MeV into the highintensity
laser focus at ELI-NP with sufficient intensity?
If one looks at the Compton backscattering <strong>of</strong> laser photons
<strong>of</strong> energy EL and an intensity (characterized by a)
from the classical electron beam with energy Ee = γe ·
mec 2 , one obtains for the γ-energyEγ:
Eγ = n· 4γ2 eEL
1+a 2
with the harmonic number n. While one obtains for the
600 MeV electron beam with γe = 1200 and small values
<strong>of</strong> the normalized laser vector potential a γ-energies <strong>of</strong> a
few MeV, one finds that for increasinga ≈ γe the Doppler
boost for the γ-energy or the factors γ 2 e and a2 cancel out
andEγ ≈ n·4·EL drops to the optical regime. However,
for such large values <strong>of</strong> γe and a, new high-field effects
come into play and Eq. (6) is no longer valid.
If large external electromagnetic fields are present electrons
can gain large energies. In that case any field in their
rest frame can be considered as constant and crossed due
to the transformation properties
E ′
|| = ′
E || , E
⊥ = γ E⊥ +v ×
B
B ′
|| = ′
B || , B
⊥ = γ
(6)
, (7)
B⊥ − 1
c2 v ×
E . (8)

This implies that E 2 − c 2 B 2 ≈ 0 and E · B ≈ 0. Any
constant crossed field can be transformed into a pure magnetic
field in an appropriate reference frame. Radiation
emission by electrons or positrons in a constant magnetic
field is naturally controled by the dimensionless parameter
B⊥ǫ/(mEs), where B⊥ is the magnitude <strong>of</strong> the external
magnetic field normal to the particle momentum, ǫ is
the energy <strong>of</strong> the particle and m is the mass. This can be
understood by observing that the peak <strong>of</strong> the emitted radiation
energy <strong>of</strong> an energetic electron in a constant magnetic
field versus its kinetic energy is approximately given
by (¯hω0/ǫ)(ǫ/m) 3 = B⊥ǫ/(mEs), where ω0 = eB⊥/ǫ.
In the lab frame this parameter becomes the quantum effi-
ciency parametersχ
χ = e¯h
−(F µν pν) 2
m 3 c 4 , (9)
where pν is the electron or positron 4-momentum. The
total transition rate for radiation in strong external fields
scales likem 2 χ 2/3 /ǫ, where the spectrum <strong>of</strong> the emittedγradiation
now extends to ever larger energies the larger the
parameterχis [7, 9]. If this Lorentz invariantχ gets close
to unity, intense γ emission with 10 8 times higher γ energies
compared to Eq. (6) sets in and a significant fraction <strong>of</strong>
the electron energy is emitted over small radiation lengths
with high energy γ-quanta [9, 7]. We will focus the 600
MeV classical electron beam directly into the high-power
laser focus and produce an intense, brilliant high-energyγ
beam directly inside the high intensity laser focus. Thus at
ELI-NP we will pursue a two-fold strategy: (1) we want
to study the high field processes to produce new brilliant,
high-energyγ-beams as a function <strong>of</strong>aandγe and then (2)
the new γ-beam will cause pair creation in the vacuum in
the same intense laser focus.
One could also think <strong>of</strong> producing the high-energy electron
beam by laser acceleration, but considering the fluctuations
in presently produced laser-accelerated electron
beams (e.g. pointing variations), it appears much more reliable
to explore these new high-field forces and the production
<strong>of</strong> high-energyγ-quanta with a very brilliant, very
stable, classical high-energy electron beam.
With these high-energy quanta γh and the laser photons
γL even hadronic QCD processes likeγh+n·γL → π 0 →
γ1+γ2 are energetically allowed, and we may learn something
about the hadronic QCD content <strong>of</strong> the vacuum with
strong laser-dressed processes.
In the following we will first decribe the experimental
setup at ELI-NP to explore these new high-field processes,
and then we show the theoretically predicted rates <strong>of</strong>γ production
ande + e − pair creation.
EXPERIMENTAL SETUP AT ELI-NP FOR
DETECTING HARDγ-PRODUCTION
AND PAIR CREATION
In Fig. 1 we show in a schematic way the experimental
setup. We focus the electron beam with a triplet lens into
Figure 1: Experimental setup, showing the two focusing
mirrors with the high-field focus and the triplet lense which
focusses the electron beam (red) into the laser focus. Behind
the laser focus, a dipole magnet deflects the electron
beam, while theγ beam (green) continues straight on.
Figure 2: Extended view <strong>of</strong> the experimental setup, showing
the sampling measurements on the γ beam, where in
thin foilsγ-quanta are converted toe + e − pairs, which then
are deflected in small magnets and measured in calorimeters.
the laser focus. The electron bunches have a maximum energy
<strong>of</strong> 600 MeV, corresponding toγe ≈ 1200, a maximum
charge <strong>of</strong> 250 pC (≈ 10 9 e/s) with a typical bunch length
<strong>of</strong> 1.5 ps or 450 µm , a diameter <strong>of</strong> 5µm and a normalized
emittance <strong>of</strong> ǫn = 0.2 mm mrad. We use two counterpropagating
laser pulses from two synchronized APOL-
LON lasers [11] with 15 fs FWHM, where we can realize
different Lorentz-invariant intensity parameters F and G
from the field tensorF µ,ν in the focus
µν
FµνF
F = −
2E2 s
G = ǫµνλκF µν F λκ
8E 2 S
= E 2 −c 2 B 2
E 2 S
= c E · B
E 2 S
1 m
(10)
(11)
Especially interesting is a configuration, where the electric
E fields <strong>of</strong> both oppositely circular polarized lasers are
added in the focal plane, while the magnetic B fields cancel

each other. Thus the E field rotates around the laser axis
and its amplitude varies slowly with the envelope <strong>of</strong> the
laser pulse. The two parabolic mirrors focus the two lasers
to a radius <strong>of</strong> about 1 µm, and we have a maximum field
strength characterized by a normalized vector potential a≈
1000. Thus within the high-field volume <strong>of</strong> the laser pulse
about ≈ 10 5 electrons are contained. In the simulations
(discussed below) we obtain, that each electron in the highfield
regime produces about 20 high-energyγ-quanta with
an exponential spectrum, reaching up to about 600 MeV
with the special quantalγ emission processes [7]:
e+n·γL → e′+γh
(12)
These γ-quantaγh exhibit the very small opening angle <strong>of</strong>
the electron beam <strong>of</strong> ≈ 1/(2γ) ≈ 10µrad and a diameter
<strong>of</strong> the laser focus <strong>of</strong> about≈ 2µm.
The process is controlled by the relativistic invariant parameters
and
χe = e¯h
m3 ec4
−(Fµνpν e) 2 = E
·
ES
Ee
mec2 (13)
κγ = e¯h
m3 ec4
− Fµνkν 2 E
γ = ·
ES
Eγ
mec2 (14)
is the
where Fµν is the four-tensor <strong>of</strong> the laser field, pν e
four-momentum <strong>of</strong> the high energy electron and kν γ is the
four-momentum <strong>of</strong> theγ quantum.
The duration <strong>of</strong> the γ-pulse is about 15 fs, like the laser
pulse. In Fig. 1 and Fig. 2 we show how the primary electron
bunch is deflected by a dipole magnet behind the laser
focus and then propagates straight to the electron beam
dump. The high-energy γ-pulse traverses several identical
detector units before it is stopped in the γ-beam dump.
Each detector unit consists <strong>of</strong> a thin foil, which converts
about oneγ-quantum into an electron-positron pair. In this
way we can sample the spectrum <strong>of</strong> the intense, 15 fs γbunch
containing about 106 γ-quanta. The forward-going
e + e− pairs are opened up by a local magnetic field. Thus
the positron hits the calorimeter array on the lift side <strong>of</strong> the
γ-beam, while the electron hits the calorimeter array on the
right side. From the deposited energy, the position <strong>of</strong> the
shower in the calorimeter and the vertex in the thin foil we
can reconstruct the γ-ray spectrum and determine the spot
size <strong>of</strong> the γ beam. By this sampling technique we avoid
a pile-up <strong>of</strong> the γ-pulses in the detectors. When reducing
the field strength <strong>of</strong> the laser pulse or reducing the electron
energy, we obtain from simulations a fast decrease <strong>of</strong> these
high field processes. Also a strong dependence on the laser
pulse duration is predicted. This will allow us for the first
time to test the assumed physical processes <strong>of</strong> this highfield
regime. At the same time we will be able to predict the
properties <strong>of</strong> the brilliant, intense γ beams for other laser
and electron beam parameters. Theseγ-beams have typical
peak brilliances <strong>of</strong>≈ 1024 /[(mm·mrad) 2 ·s·0.1%BW].
/N 0
N e + e
¢
0 5 10 15
t,
20 25 30
¡1
10
9
a =10
8
7
6
5
4
3
2
1
3
a =1.2 £10 3
a =1.5 £10 3
Figure 3: Simulated yield <strong>of</strong> electron positron pairs as a
function <strong>of</strong> time for different values <strong>of</strong> the dimensionless
field amplitudea = 10 3 , 1.2×10 3 , 1.5×10 3 . The energy
<strong>of</strong> the primary electron beam is600MeV .
The polarisation <strong>of</strong> the brilliant γ-beam is determined by
the polarisation <strong>of</strong> the laser beams.
Furthermore, we show in Fig. 2 the broad-acceptance
magnetic spectrometers with which we want to measure
the energies and spatial distribution <strong>of</strong> the electrons and
positrons. We expect for our parameters only a few e + e −
pairs per shot produced in the multiphoton Breit-Wheeler
reaction
γh +n·γL → e + e −
(15)
For our field strength characterized by a≈1000 and electron
energies with γe ≈1200, the pair creation is still
marginal. Exponentially increasing QED cascades [14] are
expected only for higher laser field strengths. Still these
measurements will present a strong test <strong>of</strong> the new highfield
pair creation simulations.
The spectra <strong>of</strong> the electrons may also be used to align the
two laser pulses and the electron bunch properly in space
and time, because the deflection, acceleration or deceleration
<strong>of</strong> the electrons depend sensitively on the combined
fields <strong>of</strong> the two lasers. The electrons also probe the outer
fringe fields <strong>of</strong> the lasers.
Thus we hereby use the high-field laser focus in two
functions at the same time: (1) to generate the high-energy
γ-quanta and (2) to induce their pair decay.
SIMULATIONS
For the final comparison between experiment and simulation
we need the precise parameters <strong>of</strong> the laser focus and
the electron bunch. We not only have to describe the highfield
laser interaction for the production <strong>of</strong> high-energyγquanta
and e + e − pairs properly, but also the dynamics <strong>of</strong>
the electrons in the laser field like acceleration, deceleration
and the motion in the ponderomotive potential. At

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Figure 4: Simulated spectra <strong>of</strong> generated γ-quanta,
positrons and electrons for a focus with a rotating E field
and10 24 W/cm 2 and a 600 MeV electron bunch.
present we performed first order calculations, which give
us the main spectra <strong>of</strong> the high energy γ-quanta, electrons
and positrons to design the spectrometers. In Monte Carlo
computer simulations, solving the transport equations for
the high-field laser interaction using Landau-Lifshitz-like
forces or quantal interactions [7, 10, 9], we obtained our
predictions. We estimate about 2 electron-positron pairs
per laser shot and about Nγ=20 high-energy photons per
shot. Fig. 3 shows the strong exponential increase <strong>of</strong> the
numbers <strong>of</strong> pairs with the normalized vector potentialaand
the duration <strong>of</strong> the laser pulse. It demonstrates that we are
just starting to see positrons and that a small improvement
<strong>of</strong> the parameters will lead us in the QED cascade regime.
In Fig. 4 on sees the exponentially decreasingγ-spectrum,
reaching up to 600 MeV. The high-energy γ-quanta then
lead to the e + e − production. The electron and positron
spectra show, that they are accelerated in the rotating field,
but at the same timeγ-emission results in lower energies.
COMPARISON WITH THE E144 SLAC
EXPERIMENT ON HIGHγ-PRODUCTION
AND PAIR CREATION
The high-field theory will be tested in the ELI-NP experiment,
which is very different from the E144 SLAC experiment
[12, 13]. In both experiments the dominant process
for pair creation is a two-step process, where an electron
first produces a high-energy γ-quantum in the laser
field and in the second step via a Breit-Wheeler reaction
the high energyγ-quantum is converted into ane + e − pair.
In Table 1 we compare the parameters <strong>of</strong> the two experiments.
While the laser intensity in the SLAC experiment
was 1.3 · 10 18 W/cm 2 , we expect to have ≈ 10 24 W/cm 2
and corespondingly the normalized vector amplitude in the
SLAC experiment was a = 0.36, while we will have a
much larger value <strong>of</strong> a = 1000. Our laser field config-
uration can be chosen very flexibly with the two Lorentz
invariantsF andG, while in the SLAC case only one laser
beam with crossed E and B fields and F=0 was used. On
the other hand, in the SLAC experiment much more highenergetic
electrons <strong>of</strong> 46.6 GeV were used, while we will
have only 0.6 GeV. For many considerations again only
the Lorentz-invariant quantities χe and χγ are important.
While in the SLAC experiment the produced high-energy
γ-quanta had about 30 GeV, in the ELI-NP case we will
have only quanta below 600 MeV. Theχvalues <strong>of</strong> the ELI-
NP experiment are larger and thus the creation times for
new particles are shorter. While the number <strong>of</strong> electrons
in the accelerated bunch is similar, the number <strong>of</strong> electrons
reaching the high-field laser interaction in the ELI-NP experiment
will be a factor <strong>of</strong>10 4 smaller.
Table 1: Comparison <strong>of</strong> the parameters <strong>of</strong> the ELI-NP and
E144 SLAC experiments [12]
parameter ELI-NP E144 SLAC
norm. vec. potent. a 1000 0.36
laser intensity ≈ 10 24 W/cm 2 1.3·10 18 W/cm 2
laser width 15 fs 1600 fs
σx ≈ 1µm 25µm
σy ≈ 1µm 40µm
Ee 0.6 GeV 46.6 GeV
Ne 1.5·10 9 e 7·10 9
repetition rate 0.02 Hz 10-20 Hz
χe 1.7 0.3
χγ ≈1 0.2
photons absorbed
in pair cr. npair ≈ 10 9 ≈ 5
CONCLUSIONS AND OUTLOOK
We are preparing for a very intense laser focus in vacuum
at ELI-NP with a large freedom to choose the invariant
field parameters. Furthermore, we require very good
vacum in the surrounding <strong>of</strong> the laser focus using cryopumping.
Thus this setup will represent an ideal laboratory
for probing the real and imaginary part <strong>of</strong> the vacuum
in high laser fields. In addition, we will add a very brilliant,
intense high-energy electron beam and a brilliant polarized
high-energyγ-beam for seeding the high-field laservacuum
interaction. Thus all components <strong>of</strong> QED cascades
[14] can be probed individually in order to test theoretical
predictions. We may even improve the high-energyγ spectrum
from its exponential shape to a spectrum with more intensity
at higher energies, by modulating the electron density
and electron energy <strong>of</strong> the large bunch in the 100 fs
range, before the 15 fs high field interaction occurs.

ACKNOWLEDGMENTS
This work is based on intensive discussions with H. Gies,
R. Schützhold, T. Heinzl, T. Tajima ans Z. Zamfir. It
was supported by the DFG Cluster <strong>of</strong> Excellence MAP
(Munich-Center for Advanced Photonics).
REFERENCES
[1] http://www.extreme-light-infrastructure.eu/.
[2] Physics Today, June 2010, p. 20.
[3] http:/www.eli-np.ro/excecutive committee-meeting-april12-
13.php (2010).
[4] J. Schwinger, Phys. Rev. 82, 664 (1951)
[5] C. Barty et al., Development <strong>of</strong> MEGa-Ray technology at
LLNL: http:/www.eli-np.ro/executive committee-meetingapril-12-13.php
(2010).
[6] R. Schützhold et al., Phys. Rev. Lett. 101, 130404 (2008).
G.V. Dunne et al., Phys. Rev. D 80, 111301(R) (2009).
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(1964), JETP 25, 1135 (1967).
[8] L.D. Landau and E.M. Lifschitz, Klassische Feldtheorie,
Chapt. 76.
[9] N.V. Elkina et al., arXiv:1010.4528v1 21okt (2010).
[10] N. Elkina and H. Ruhl, contribution to the PIF2010 conference.
[11] J.P. Chambaret, The Extreme Light Infrastructure Project
ELI and its Prototype APOLLON/ILE “the associated laser
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[12] D.L. Burke et la., Phys. Rev. Lett. 79, 1626 (1997).
[13] C. Bula et al., Phys. Rev. Lett., 76, 3116 (1996).
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[16] V.N. Baier et al., arXiv:0912.5250v1[hep-ph] (2010).

Abstract
Numerical simulation <strong>of</strong> QED cascades in intense laser fields ∗
Due to the dramatic progress in laser technology [1] a
novel area <strong>of</strong> laser-matter interaction at ultra-high intensity
is arising. The Extreme Light Infrastructure project
(ELI) [2] is aiming at getting access to intensity levels up to
10 26 W/cm 2 . Therefore it is timely to investigate the structure
<strong>of</strong> the QED vacuum and the interaction <strong>of</strong> charged
particles with extreme fields. In particular, cascades <strong>of</strong>
electrons, positrons, and photons may arise limiting the increase
<strong>of</strong> the laser intensity beyond a given threshold due
to the depletion <strong>of</strong> the laser. In addition, it is found that the
super-intense laser field is capable <strong>of</strong> restoring the energy
<strong>of</strong> electrons and positrons and the dynamical quantum efficiency
parameter by efficient acceleration <strong>of</strong> electrons and
positrons in the laser field. This novel restoration effect
may become a dominating feature <strong>of</strong> laser-matter interaction
at ultra-high intensities. In the present paper we report
about the current status our simulation framework based on
semi-classical transport equations for electrons, positrons
and photons.
INTRODUCTION
The purpose <strong>of</strong> the present contribution is to present
the current status <strong>of</strong> our simulation code capable <strong>of</strong> modeling
new physical phenomena at ultra-high laser fields
(I ≥ 10 24 W/cm 2 ) as they will be available in the
near future (ELI). The presence <strong>of</strong> strong electromagnetic
fields reveals a range <strong>of</strong> quantum electrodynamic effects,
which may significantly change the dynamics <strong>of</strong> relativistic
plasma. At laser intensities in excess <strong>of</strong> I ≥ 10 24 W/cm 2
non-classical effects like radiation reaction, vacuum polarization,
and electron-positron pair creation due to cascading
become important and ultimately dominant. Due to the
ubiquitous complexity <strong>of</strong> the interaction at ultra-high fields
simulations will have crucial importance. A concerted research
effort will be required because conventional plasma
modeling techniques, while adequate for classical laserplasma
interaction or very small numbers <strong>of</strong> particles, cannot
access the length and time scales relevant to illumnate
the real structure and dynamics <strong>of</strong> strong field QED.
The paper is organized as follows. First we review and
outline elementary quantum processes used in our code.
They are single photon emission by electrons and positrons
and pair creation by hard photons in strong laser fields
<strong>of</strong> arbitrary configuration. Next we proceed with stating
appropriate kinetic equations, which include elementary
quantum processes. The presence <strong>of</strong> high intensity laser
∗ This work was supported by DFG project RU-633/1-1 and the
Cluster-<strong>of</strong>-Excellence ’Munich Centre for Advance Photonics’ (MAP)
N. Elkina and H. Ruhl
LMU München
fields allows a semiclassical approach for the description
<strong>of</strong> the particle motion in electromagnetic fields. This leads
to the formulation <strong>of</strong> semi-classical transport equations. As
a first step for the incorporation <strong>of</strong> nonlinear QED effects
into the modeling <strong>of</strong> electron-positron-photon plasma we
state a set <strong>of</strong> kinetic equations to describe QED effects. The
kinetic equations can be reformulated as a set <strong>of</strong> moment
equations for quasi-elements. Since phase space can grow
rapidly in ultra-strong fields adaptive management <strong>of</strong> phase
space by adaptive weights for quasi-elements has to be introduced
in order to keep the required resolution in phase
space. Next we present results <strong>of</strong> preliminary simulations
in order to illustrate the power <strong>of</strong> our numerical approach.
Finally a short conclusion is given.
ELEMENTARY QED EFFECTS IN
STRONG LASER FIELDS
The process <strong>of</strong> cascading can be represented by two
coupled processes <strong>of</strong> hard photon emission and pair production
via the absorbtion <strong>of</strong> n ”s<strong>of</strong>t” photons (<strong>of</strong> energy
¯hω ∼ 1 eV) from the laser field
e ± + n¯hωl → γ + e ± , (1)
γ + n¯hωl → e + + e − . (2)
The created charged particles are ultra-relativistic (γ ≫
1) and are exposed to electromagnetic fields <strong>of</strong> ultrarelativistic
(a0 ≫ 1, χ ∼ 1) but still sub-critical (F ≪ ES)
intensities, where F = √ F µν √
Fµν = ⃗E 2 − B⃗ 2 . The en-
ergy <strong>of</strong> hard photons in the simulations are assumed to be in
the range <strong>of</strong> ¯hω > mc 2 . The theoretical approach used in
this work is based on the fact that in the rest frame <strong>of</strong> highly
relativistic charged particles any field can be considered as
constant and crossed. This implies that E 2 − H 2 ≈ 0 and
⃗E · ⃗ H ≈ 0. Constant crossed fields can be derived from
a vector potential <strong>of</strong> the kind A µ = a µ k · x, where a µ is
a constant polarization vector. The functional form for the
vector potential <strong>of</strong> a plane wave is A µ = a µ (k · x), where
a µ would now be a function <strong>of</strong> k·x. The appropriate transition
amplitudes on a tree level are calculated with the help
<strong>of</strong> Volkov solutions for the fermions [4]. However, for constant
crossed fields one better makes use <strong>of</strong> the solution <strong>of</strong>
the Dirac equation for a constant external magnetic field,
since any constant crossed field can be transformed into
a pure magnetic field in an appropriate reference frame.
Radiation emission by electrons or positrons in a constant
magnetic field is naturally controled by the dimensionless
parameter B⊥ϵ/(mEs), where B⊥ is the magnitude <strong>of</strong> the
external magnetic field normal to the particle momentum,

ϵ is the energy <strong>of</strong> the particle, m is the mass <strong>of</strong> the particle,
and Es = m2 /(e¯h) is the Schwinger field. This
can be understood by observing that the peak <strong>of</strong> the emitted
radiation energy <strong>of</strong> an energetic electron in a constant
magnetic field versus its energy is approximately given by
(¯hω0/ϵ) (ϵ/m) 3 = B⊥ϵ/(mEs), where ω0 = eB⊥/ϵ.
Further details can be found in [5]. In the lab frame this
parameter becomes the quantum efficiency parameters χ
χ = e¯h
√
− (F µν pν) 2
m3 , (3)
where pν is the electron or positron 4-momentum. In the
classical radiation realm the emitted photon energies are
much smaller than the charged particle energy. Hence,
χ ≪ 1 holds. In the quantum realm χ ≫ 1 is valid. The
transition rate for radiation emission depends on a second
quantum parameter κ given by
κ = e¯h
√
− (F µν kν) 2
m3 , (4)
where kν is the photon 4-momentum. The angle integrated
transition rate for photon emission is
where
dWγ
dω
(∫ ∞
m2
= −α
ϵ2 dz Ai(z) (5)
x
[
2
+
x + κ √ ] )
x ∂zAi(x) ,
[
κ
x =
χ (χ − κ)
] 2
3
, 0 ≤ κ < χ . (6)
The rate holds for radiation emission by electrons or
positrons. In the simulations we assume that
′
⃗k = ⃗p + ⃗p (7)
holds for the momenta. For the energy balance we find
q + ω = ϵ ′
+ ϵ . (8)
Making use <strong>of</strong> Eqns. (7) and (8) the energy taken from the
external laser field is
q =
√
ϵ 2 + 2ωϵ ′ (1 − v ′ cos θ) − ϵ ,
where θ is the angle between the emitted photon and the
outgoing electron or positron ⃗p ′
and v ′
= |⃗p ′
|/ϵ ′
is the
velocity <strong>of</strong> the electron or positron after the radiation process.
For electrons or positrons retaining large momenta ⃗p ′
along ⃗k after the emission process v ′
cos θ ≈ 1 holds. The
external field has to deliver only little energy, q ≈ 0, in that
case. Given the electron or positron 4-momentum and the
external field context χ can be calculated. In a second step
κ is obtained for permissible photon emission and hence x.
With the help <strong>of</strong> the crossing symmetry the angle integrated
transition rate for e + e−-pair creation is given by
where
dW e + e −
dϵ−
= α m2
ω2 (∫ ∞
y
+
[
κ
y =
χ (κ − χ)
[
2
y − κ √ ]
y
] 2
3
dz Ai(z) (9)
)
∂zAi(y) ,
, 0 ≤ χ < κ . (10)
The following kinematic relations are used when pairs are
created
⃗ k = ⃗p+ + ⃗p− , (11)
q + ω = ϵ+ + ϵ− . (12)
Making use <strong>of</strong> Eqns. (11) and (12) the energy taken from
the external laser field is
√
q = ϵ2 − + 2ωϵ+ (1 − v+ cos θ) − ϵ− ,
where θ is the angle between the photon and the emitted
positron, and v+ = |⃗p+|/ϵ+ is the velocity <strong>of</strong> the positron.
Given a photon 4-vector and the field context κ can be obtained.
Next it is possible to calculate χ for any permissible
electron energy and hence y.
If χ and κ are large enough we find that photons generate
electrons and positrons and the latter again photons.
The chain process leads to exponential growth <strong>of</strong> particles
in the simulation. The solution <strong>of</strong> the underlying transport
equations for uniform circular polarized light is approximately
N e + e −(t) = N e + e −(0) e Γ t , (13)
where the growth rate Γ can be estimated as
Γ = α µ 1
√
m ω c2 4 ,
¯h
µ = E
.
α Es
(14)
Here E is the external field.
Quantum efficiency becomes very large in circular
purely electric fields. Hence we consider two counterpropagating
circular laser beams in the center <strong>of</strong> which
there is a rotating electric field. Figure 1 shows the situation
and the simulation results. Along the purple line electrons
are injected into the laser focus at 600 MeV. Red and
green lines represent secondary electrons and positrons. As
a main result we find that the mean energy per particles is
not decreasing in the course <strong>of</strong> the cascade development.
The contrary could be expected since the number <strong>of</strong> secondary
particles grows exponentially. The finding implies
enhanced energy deposition. In fact, the empirical observation
<strong>of</strong> almost constant mean energy per particle represents
exponentially growing energy deposition in the evolving
plasma.

Figure 1: Cascading in a laser focus at I = 3 · 10 24 W/cm 2
initiated by primary electrons <strong>of</strong> energy ε = 600 MeV injected
into the laser focus. The multiplicity <strong>of</strong> charged particles
is about 2.5, which represents the onset <strong>of</strong> non-linear
QED in the laser-focus.
KINETIC SIMULATION OF THE QED
PLASMA
Relevant processes in ultra-intense laser fields are the
generation and annihilation <strong>of</strong> photons during the interaction
<strong>of</strong> the laser with electrons and positrons and e + e−-pair creation with the help <strong>of</strong> the laser and existing photons. Appropriate
transport equations can be formulated. They are
solved with the help <strong>of</strong> the quasi-elements.
Neglecting higher order effects as the annihilation <strong>of</strong>
pairs and classical radiation reaction a system consisting
<strong>of</strong> electrons, positrons, and radiation can be described by
transport equations <strong>of</strong> the following kind
(
∂t + ⃗v · ∂⃗x ± ⃗ )
F · ∂⃗p f±(⃗x, ⃗p, t) (15)
∫
=
ω>ω0 ∫
−f±(⃗x, ⃗p, t)
∫
+
d 3 k W ⃗ E, ⃗ B
γ ( ⃗k, ⃗p + ⃗k) f±(⃗x, ⃗p + ⃗k, t)
ω>ω0
ω>ω0
where ⃗ ( )
F = |e| ⃗E + ⃗v × B⃗
d 3 k W ⃗ E, ⃗ B
γ ( ⃗k, ⃗p)
d 3 k W ⃗ E, ⃗ B
e + e −( ⃗ k, ⃗p) fγ(⃗x, ⃗ k, t) ,
is the Lorentz force and
(
∂t + ∂ω
∂⃗ )
· ∂⃗x fγ(⃗x,
k ⃗k, t) (16)
∫
= d 3 p W ⃗ E, ⃗ B
γ ( ⃗k, ⃗p) [f+(⃗x, ⃗p, t) + f−(⃗x, ⃗p, t)]
−fγ(⃗x, ⃗ ∫
k, t)
d 3 p W ⃗ E, ⃗ B
e + e −( ⃗ k, ⃗p) ,
which for ω < ω0 have to be coupled to Maxwell’s equa-
tions with the current
∫
⃗j(⃗x, t) = e d 3 p ⃗v [f+(⃗x, ⃗p, t) − f−(⃗x, ⃗p, t)] . (17)
It has to be made sure that radiation stored in ⃗ E, ⃗ B and
fγ does not lead to double counting <strong>of</strong> radiation, hence the
ω > ω0 threshold.
S<strong>of</strong>t photons are described classically by means <strong>of</strong> the
Maxwell equations for the electromagnetic fields
1 ∂
c
⃗ E
∂t = ∇ × ⃗ B − 4π ∑
∫
⃗p±
q±
c mγ f±d⃗p , (18)
− 1 ∂
c
⃗ B
∂t = ∇ × ⃗ E , (19)
∇ · ⃗ E = −4π ∑ ∫
q± f±d⃗p , (20)
∇ · ⃗ B = 0 . (21)
The complete set <strong>of</strong> equations with the discussed restrictions
can be solved for intensities I ≥ 10 24 W/cm 2 , which
are planned for ELI. For illustration a simulation setup is
shown in Figure 1, where the cascade is initiated by an electron
beam <strong>of</strong> energy ε = 600 MeV colliding with a focused
laser field <strong>of</strong> two circularly polarized counter-propagating
Gaussian laser beams. The observed multiplicity is about
2.5.
EVENT GENERATOR
To solve the transport equations we make use <strong>of</strong> an extended
version <strong>of</strong> the Particle-In-Cell (PIC) method [6].
The PIC method samples the phase space with a finite number
<strong>of</strong> quasi-particles and proceeds by integrating kinetic
equations in time by advancing quasi-particles along their
characteristics within phase space.
We trace the motion <strong>of</strong> quasi-electrons and quasipositrons
in momentum and coordinate space, whereas for
hard photons we utilize the ray tracing approximation. Our
numerical algorithm works as follows: At each time step
tn = n ∆t we advance the positions and momenta <strong>of</strong> all
the particles present in the simulation box by solving their
equations <strong>of</strong> motion. With the help <strong>of</strong> the event generator
we check if an electron or a positron is going to emit a photon
between tn < t < tn + ∆t and if any <strong>of</strong> the photons
present is going to produce a pair. With the help <strong>of</strong> ⃗p n and
⃗E n at tn the efficiency parameter χ n is evaluated. Next
the total transition probability Wγ is computed. In order
to remove the infrared divergencies we restrict the integrations
over photons to k0 > ε in the transport equations.
We assume that electrons or positrons present in the simulation
box emit a photon between tn < t < tn + ∆t if
Wγ ∆t < r, where 0 ≤ r < 1 is a uniformly distributed
random number. If the above inequality is fulfilled the energy
εγ = ¯hωγ <strong>of</strong> the emitted photon is obtained as a root

<strong>of</strong> the sampling equation
1
εγ ∫
Wγ
ε0
dWγ(εγ)
dεγ = r
dεγ
′ , (22)
where 0 ≤ r ′ < 1 is a second random number. The direction
<strong>of</strong> propagation <strong>of</strong> the newly emitted photon is assumed
to be parallel to the ⃗p n <strong>of</strong> the emitting electron or positron.
Their momenta after emission are found from the conservation
laws. For pair creation the event generator works
the same way apart from the regularization parameter ε0,
which is not needed in this case.
ADAPTIVE MANAGEMENT METHOD
FOR QUASIPARTICLES
In ultra-intense laser fields the number <strong>of</strong> quasi-elements
Npart cannot be considered constant. The evolution <strong>of</strong> particle
number depends critically on the laser and seed particle
parameters. Seed particles can be electrons, positrons,
and photons. The control parameters are the quantum efficiency
parameters χ and κ. The qualitative threshold for
the onset <strong>of</strong> cascading is χ ∼ κ ∼ 1. At those conditions
the number <strong>of</strong> particles grows exponentially. In
order to handle exponential grows with a limited number
<strong>of</strong> quasi-particles we implement a new techniques <strong>of</strong> refinement/coarsening
<strong>of</strong> phase space represented by quasiparticles.
Our approach to keep phase space adequately resolved
consists <strong>of</strong> two basic algorithms for particle splitting
and merging.
Splitting
Splitting <strong>of</strong> quasi-particles is straightforward. The splitted
quasi-particle’s weight, momentum and energy are the
sum <strong>of</strong> the same quantities <strong>of</strong> the resulting smaller quasiparticles.
Particle splitting is used to increase the number <strong>of</strong>
quasi-particles per computational cell. Quasi-particles are
split into two equal smaller quasi-particles. The momenta
and/or positions <strong>of</strong> the produced quasi-particles should be
slightly different from each other in order to reach better
statistical sampling. This approach is probably not the most
accurate one but conserves exactly all moments <strong>of</strong> the distribution
function.
Merging
The merging algorithm is somewhat more complicated.
Roughly speaking we have to replace the original set <strong>of</strong> N
quasi-particles by a new set <strong>of</strong> M particles, where N > M.
Under this transformation the basic properties <strong>of</strong> the distribution
function have to be kept, i.e. f(N) ≃ f(M). This
can be achieved by making use <strong>of</strong> the constraints defined
by the conservation <strong>of</strong> moments. Let us assume that the
distribution function <strong>of</strong> the electrons, positrons or photons
is represented by a certain number <strong>of</strong> quasi-particles
f(px, py, pz, t) = ∑
wi δ(⃗p − ⃗pi(t)) S(⃗r − ⃗ri(t)) , (23)
i
where wi is the weight <strong>of</strong> the quasi-particle, which can
change in the course <strong>of</strong> the simulation and S is the shape
factor <strong>of</strong> the quasi-particle in coordinate space [6]. Taking
moments <strong>of</strong> this distribution function, one can obtain the
weight or mass, the momentum, the energy, and so on <strong>of</strong>
the plasma. Let us denote the weight <strong>of</strong> N quasi-elements
in a cluster <strong>of</strong> quasi-particles to be merged by W, the momentum
by ⃗ P , and the energy by ϵ. Then the conservation
<strong>of</strong> mass implies
N∑
wi = W = const .
i=1
Momentum conservation is implied by
N∑
wi ⃗ui = ⃗ P = const
i=1
and energy is conserved if
N∑
i=1
wi
√
1 + u2 i = ε = const .
To keep W , ⃗ P , and ε conserved the group <strong>of</strong> N original
quasi-particles has to be merged to at least two remaining
quasi-particles as is shown in Figure 2. The weight
w 1/2 and momentum ⃗p 1/2 <strong>of</strong> the two new quasi-elements
is given by
w1,2 = W
2 , ⃗p1,2 = ⃗ P
2 ± ⃗ Q . (24)
In 2D the direction <strong>of</strong> the vector ⃗ Q can be defined as follows
( )
−Py
⃗Q = |Q| , (25)
Px
where the quantity | ⃗ Q| can be obtained from the energy
conservation condition as
| ⃗ Q| = 1√
ε2 − P 2 − W 2 . (26)
2
Thus, in 2D momentum space the momenta for newly
merged particles can be found exactly. In the 3D case the
perpendicular component ⃗ Q lies in the plane perpendicular
to ⃗ P , i.e. the number <strong>of</strong> possible directions <strong>of</strong> ⃗ Q is infinite.
In order to assign the vector direction we suggest to use the
local anisotropy <strong>of</strong> the phase space represented by the cluster
<strong>of</strong> quasi-particles. Making use <strong>of</strong> the mean variation
<strong>of</strong> the distribution function we can choose the direction <strong>of</strong>
⃗Q. The test results <strong>of</strong> the code show that even for the case
<strong>of</strong> exponential growth <strong>of</strong> the number <strong>of</strong> the real particles
due to cascading, particle resampling based <strong>of</strong> conservation
<strong>of</strong> the local phase space topology can significantly reduce
the number <strong>of</strong> computational particles while retaining
the features <strong>of</strong> the distribution function and keeping mass,
momentum, and energy conserved.

Figure 2: Clustering <strong>of</strong> quasi-particles with adaptive
weights (color-coded) in phase space.
Figure 3: Development <strong>of</strong> the e ± γ cascade in a rotating
electric field. The number <strong>of</strong> created pairs as a function <strong>of</strong>
time obtained from an unrefined simulation (green) [3] and
one with phase space refinement (blue).
The two basic algorithms <strong>of</strong> particle splitting and merging
are used to control the number <strong>of</strong> computational quasiparticles.
The results are compared in Figure 3, where the
number <strong>of</strong> the real particles represented by a finite number
<strong>of</strong> quasi-particles agrees with the results <strong>of</strong> an unrefined
simulation reported in [3].
SUMMARY AND CONCLUSIONS
We have presented the current status <strong>of</strong> a new simulation
framework for full scale simulations <strong>of</strong> QED cascading
in strong laser fields. A novel feature <strong>of</strong> cascades in
an ultra-strong laser field is the generation <strong>of</strong> electrons,
positrons, and energetic (hundreds <strong>of</strong> MeV) photons at intensity
levels far below the Schinger limit. Since the strong
external field is capable <strong>of</strong> accelerating charged particles to
ultra-relativistic energies cascades can be triggered by ini-
tially slow electrons and positrons that are injected into the
strong field region. The appearance <strong>of</strong> cascades raises principal
question about the accessibility <strong>of</strong> higher laser fields
since the appearance <strong>of</strong> charged particles can deplete the
laser field. This question requires further investigation by
taking exact energy conservation and the reverse processes
<strong>of</strong> pair annihilation into account, which can both limit the
growth <strong>of</strong> the number <strong>of</strong> particles. We also plan to extend
our model further by the incorporation <strong>of</strong> the elastic collisions
between particles and Compton scattering, which
may be important for the design <strong>of</strong> new γ-sources.
REFERENCES
[1] Drake, P., High-energy-density physics, Physics Today 63, 28
(2010).
[2] http://www.extreme-light-infrastructure.eu/
[3] Elkina, N. V., Fedotov, A. M., Kostyukov, I. Y., Legkov,
M. V., Narozhny, N. B., Nerush, E. N., Ruhl, H., QED cascades
induced by circularly polarized laser fields, accepted
for publication by Phys. Rev. ST. Accel. Beams, 2011 and
http://arxiv.org/abs/1010.4528.
[4] Nikishov, A. I., Ritus, V. I., Sov. Phys. JETP 25, 1135 (1964).
[5] Lifshitz, E. M., Pitaevskii, L. P., and Berestetskii, V. B.,
Landau-Lifshitz Course <strong>of</strong> Theoretical Physics, Vol. 4: Quantum
Electrodynamics, Reed Educational and Pr<strong>of</strong>essional
Publishing, Oxford, 1982.
[6] Ruhl, H., Introduction to Computational Methods in Many
Body Physics, eds. M. Bonitz and D. Semkat, Rinton Press,
(2001).

Abstract
SCHWINGER LIMIT ATTAINABILITY WITH EXTREME LIGHT ∗
S. V. Bulanov † , T. Zh. Esirkepov, J. K. Koga, KPSI-JAEA, Kizugawa, Kyoto, Japan
S. S. Bulanov, Univ. <strong>of</strong> California, Berkeley, USA
A. Thomas, Univ. <strong>of</strong> Michigan, Ann Arbor, USA
High intensity colliding laser pulses can create abundant
electron-positron pair plasma. This process can prevent
reaching the critical field <strong>of</strong> Quantum Electrodynamics
at which vacuum breakdown and polarization occur.
Considering the pairs are seeded by the Schwinger mechanizm,
it is shown that the effects <strong>of</strong> radiation friction and
the electron-positron avalanche development in vacuum depend
on the electromagnetic wave polarization. For circularly
polarized colliding pulses these effects dominate
not only the particle motion but also the evolution <strong>of</strong> the
pulses. While for linearly polarized pulses, where the electrons
(positrons) oscillate along the electric field, these effects
are not as strong. There is an apparent analogy <strong>of</strong>
these cases with circular and linear electron accelerators
with the corresponding constraining and reduced roles <strong>of</strong>
synchrotron radiation losses.
INTRODUCTION
The lasers nowadays provide one <strong>of</strong> the most powerful
sources <strong>of</strong> electromagnetic (EM) radiation under laboratory
conditions and thus inspire the fast growing area
<strong>of</strong> high field science aimed at the exploration <strong>of</strong> novel
physical processes [1]. Reaching intensity <strong>of</strong> the order <strong>of</strong>
10 23 W/cm 2 and above will bring us to experimentally unexplored
regimes. At such intensities the laser interaction
with matter becomes strongly dissipative, due to efficient
EM energy transformation into high energy gamma rays
[1, 2]. These gamma-photons in the laser field may produce
electron-positron pairs via the Breit-Wheeler process
[3]. Then the pairs accelerated by the laser generate high
energy gamma quanta and so on [4], and thus the conditions
for the avalanche type discharge are produced at the
intensity ≈ 10 25 W/cm 2 . The occurrence <strong>of</strong> such ”showers”
was foreseen by Heisenberg and Euler [5]. In Ref.
[6] a conclusion is made that depletion <strong>of</strong> the laser energy
on the electron-positron-gamma-ray plasma (EPGP) creation
could limit attainable EM wave intensity and could
prevent approaching the critical quantum electrodynamics
(QED) field. This field [5, 7] is also called the Schwinger
field, ES = m 2 ec 3 /e¯h corresponding to the intensity <strong>of</strong> ≈
10 29 W/cm 2 .
The particle-antiparticle pair creation by the Schwinger
field cannot be described within the framework <strong>of</strong> perturbation
theory and sheds light on the nonlinear QED prop-
∗ Work supported by the Ministry <strong>of</strong> Education, Culture, Sports, Science
and Technology (MEXT) <strong>of</strong> Japan, Grant-in-Aid for Scientific Research,
Project No. 20244065.
† bulanov.sergei@jaea.go.jp
erties <strong>of</strong> the vacuum [8]. Understanding the vacuum breakdown
mechanisms is challenging for other nonlinear quantum
field theories [9] and for astrophysics [10]. Reaching
this field limit has been considered as one <strong>of</strong> the most
intriguing scientific problems. Demonstration <strong>of</strong> the processes
associated with the effects <strong>of</strong> nonlinear QED, such
as vacuum polarization and vacuum electron-positron pair
production, will be one <strong>of</strong> the main challenges for extreme
high power laser physics [1, 11].
Below we discuss the attainability <strong>of</strong> the Schwinger field
with high power lasers (see also Ref. [12]). We compare
the role <strong>of</strong> radiation dissipative effects in the motion <strong>of</strong>
electrons (and positrons) produced via the Schwinger effect
and show their dependence on the EM wave polarization.
3D EM FIELD CONFIGURATION
Pair creation is determined by the Poincare invariants
[13] F = (E 2 − B 2 )/2, G = (E · B) and requires the
first invariant F be positive. This condition can be fulfilled
in the vicinity <strong>of</strong> the antinodes <strong>of</strong> colliding EM waves,
or/and in the configuration formed by several focused EM
pulses, [14]. This EM configuration locally can be approximated
by an oscillating TM mode with poloidal electric
and toroidal magnetic fields. The magnetic field in spherical
coordinates R, θ, ϕ is given by
a0 sin(ω0t)
B(R, θ) = eϕ
(8πR) 1/2 Jn+1/2(k0R)L l n(cos θ), (1)
where a0 = eE0/mecω0, k0 = ω0/c, Jν(x) and L l n(x) are
the Bessel function and associated Legendre polnomials.
The electric field is equal to E = ik0(∇ × B). In cylindrical
coordinates r, ϕ, z the z-component <strong>of</strong> the electric
field oscillates in vertical direction, ∼ a0 cos(ω0t), the ϕcomponent
<strong>of</strong> the magnetic field vanishes on the axis being
linearly proportional to the radius, ∼ (a0/8)k0r sin(ω0t),
and the radial component <strong>of</strong> the electric field is relatively
small, ∼ 0.1a0k 2 0rz cos(ω0t). The EM field and first
Poincare invariant F(r, z) are shown in Fig. 1. We see
that the EM field is localized in a region <strong>of</strong> width less than
the laser wavelength, λ0 = 2π/k0. The second invariant is
equal to zero, G = 0.
Probability <strong>of</strong> electron-positron pair creation
Using expression for the probability <strong>of</strong> electron-positron
pair creation [5, 7] and expanding F(r, z) in the vicinity <strong>of</strong>
its maximum we find that the pairs are created in a small
4-volume near the electric field maximum with the charac-

10
k 0z
0
-10
0
k 0r
a) 1
10
0
-10
0
F/a0 2
Figure 1: a) The vector field shows r- and z-components <strong>of</strong>
the poloidal electric field in the r, z plane for the TM mode.
The color density shows the toroidal magnetic field distribution,
Bϕ(r, z). b) The first Poincare invariant F(r, z).
teristic size
1
0.5
πr 2 0z0t0 ≈ 53/2 λ 4 0
16π 5 c
k 0r
( a0
aS
10
k 0z
10
b)
) 2
. (2)
Here, we introduce aS = eES/meω0c = mec 2 /¯hω0. Integrating
over the 4-volume the probability <strong>of</strong> the pair creation
[15] we obtain the number <strong>of</strong> pairs produced per wave
period,
53/2 4π3 a4 (
0 exp − πaS
)
, (3)
a0
i. e. the first pairs can be observed for an one-micron wavelength
laser intensity <strong>of</strong> the order <strong>of</strong> 2×10 27 W/cm 2 , which
corresponds to a0/aS ≈ 0.075, i.e. a characteristic size, r0,
approximately equal to 0.04λ0.
Electron orbit near magnetic null point
In the region, where the magnetic field vanishes, the
electron oscillates along the electric field. For an electron
generated at small but finite radius r0 ≪ λ0 the magnetic
field bends its trajectory outwards. By solving the electron
equations <strong>of</strong> motion linearized about the solution corresponding
to ultrarelativistic electron oscillations in the zdirection,
i.e. a0ω0t ≫ 1, we can find the electron trajectories,
which are described in terms <strong>of</strong> modified Bessel
functions, Iν(x),:
pz(t) = meca0ω0t, (4)
pr(t) = mec a0k0r0ω0t
23/2 (
ω0t
I1
23/2 )
,
r(t) =
(5)
a0r0
(
ω0t
I1
23/2 23/2 )
+
[ (
a0r0ω0t ω0t
I0
16 23/2 ) (
ω0t
+ I2
23/2 )]
. (6)
Here r0 is the initial electron coordinate, which is <strong>of</strong>the
order <strong>of</strong> λ0(5a0/4π 3 aS) 1/2 . The instability growth rate is
approximately equal to half the EM field frequency, ω0/2,
pr ≈
( a0
8
)
k0r0(ω0t) 2 , (7)
i. e. the electron remains in the close vicinity <strong>of</strong> the zeromagnetic
field region leaving it along the z-direction.
EM RADIATION EMISSION
Frequency spectrum
The electron oscillating along the electric field emits
the high frequency EM radiation with the power ≈
(2πre/3λ0)ωemec 2 γ 2 e proportional to the square <strong>of</strong> electron
energy. In order to find the angular distribution and
frequency spectrum <strong>of</strong> the radiation in this case we should
take into account its dependence on the retarded time:
t ′ = t − n · r(t)/c. Here n is the unit vector in the direction
<strong>of</strong> observation and r(t) is the electron coordinate.
Introducing the angle η between vectors n and r(t),
n · r(t) = |r(t)| cos η, (8)
we can find that in the direction <strong>of</strong> electron oscillations,
η = 0, the radiation intensity vanishes. The maxima <strong>of</strong> the
radiated power correspond to the angle ηm, for large γe,
inversely proportional to the particle energy: ηm ≈ 1/2γe.
Fourier components for 4-vector potential <strong>of</strong> the EM
field according to Ref. [13] are
A µ (ω) = e
R
∫+∞
−∞
u µ
c exp
{ [
iω
t − 1
n · r(t)
c
where u µ = p µ /meγe is the four-velocity and
]}
dt, (9)
r(t) = ey(c/ω0)Arcsin [βm sin(ω0t)] , (10)
with βm = a0/ √ 1 + a2 0 is the electron coordinate.
Expanding the phase in expression (9),
{ ( )
}
cos θ
Φ(t) = ω t − Arcsin [βm sin(ω0t)] , (11)
ω0
on small parameters, γ −1
e,m and ω0t, for θ = θm ≈ 1/2γe,m,
we obtain
Φ(t) ≈ ω
[
(1 − βm cos θ) t +
( βm cos θ
6γ 2 e,mω0
)
(ω0t) 3
]
.
(12)
Using the Airy integral, we can find the y-component <strong>of</strong>
the 4-vector potential <strong>of</strong> EM field (9). Taking into account
smallness <strong>of</strong> the angle θ ≈ θm, θ = ψ/2γe,m ≪ 1, and
presenting cos θ in the form cos θ ≈ 1 − ψ2 /2(2γe.m) 2 we
can obtain the radiation power density pL(ω, ψ).
The power emitted by the electron is given by the integral
∫+∞
pL(ω) = pL(ω, ψ)dψ. (13)
−∞

To find it we use the integral calculated in Ref [17] and
obtain
dpL 16πre
=
dω 33/2 mec
λ0
2
( ) 2 ( )
ω 2 ω
FL
, (14)
ω0 3
where
∫
FL(z) = z
z
ω0γ 2 e,m
∞
K 5/3(η) dη − zK 2/3 (z). (15)
Maximum frequency <strong>of</strong> the radiation emitted by oscillating
electron corresponds to ωm ≈ 0.21ω0γ 2 e,m. As we see,
compared to the case <strong>of</strong> circular polarization, the linearly
oscillating electron emission is weaker with the photon frequency
in a factor γe,m smaller.
Radiation friction effects
To take into account the radiation friction we use equation
<strong>of</strong> motion <strong>of</strong> a radiating electron [13]. We can estimate
the regime where the radiation friction can become
relatively large by comparing the energy losses with the
maximal energy gain <strong>of</strong> an electron accelerated by the
electric field, E ˙(+)
≈ ω0 mec2a0, i.e. ω0 mec2a0 =
εradω0mec2 γ2 e , where
εrad = 4πre/3λ0, (16)
with re = e2 /mec2 - classical electron radius. As is apparent,
although an electron moving along the oscillating
electric field loses energy, radiation friction effects may be-
come important only at a0 = 2ε −1
rad
, i.e. at the electric field
E0 = 3m 2 ec 4 /e 3 , which is <strong>of</strong> the order <strong>of</strong> the critical electric
field <strong>of</strong> classical electrodynamics (see also Ref. [15]).
This is 137 times larger than the field ES.
DIMENSIONLESS PARAMETERS IN QED
In QED the charged particle interaction with EM fields is
determined by relativistically and gauge invariant parameters
[18] χe = [(Fµνpν) 2 ] 1/2 /mecES. The parameter, χe,
characterizes the probability <strong>of</strong> the gamma-photon emission
by the electron with Lorentz factor γe. It is <strong>of</strong> the
order <strong>of</strong> the ratio E/ES in the electron rest frame <strong>of</strong> reference.
Another parameter, χγ = [(Fµν¯hkν) 2 ] 1/2 /mecES,
is similar to χe with the photon 4-momentum, ¯hkµ, instead
<strong>of</strong> the electron 4-momentum, pµ. It characterizes the probability
<strong>of</strong> the electron-positron pair creation due to the collision
between the high energy photon and EM field. QED
effects come into play when the energy <strong>of</strong> a photon emitted
by an electron becomes comparable to the electron kinetic
energy, i.e., for ¯hωm = mec 2 γe. In a linearly polarized
oscillating electric field the maximum frequency <strong>of</strong>
emitted photons, ωm, is proportional γ 2 0, and, therefore,
quantum effects should be incorporated into the theoretical
description at the electron energy corresponding to the
gamma-factor γ L Q = mec 2 /0.21 ¯hω0, which is above the
Schwinger limit. We see that in the case <strong>of</strong> electron motion
in a linearly polarized oscillating electric field neither
radiation friction nor quantum recoil effects are important.
THRESHOLD OF AVALANCHE
DISCHARGE
Reaching the threshold <strong>of</strong> an avalanche type discharge
with EPGP generation discussed in Refs. [4, 6] requires
high enough values <strong>of</strong> the parameters χe and χγ defined
above because for χγ ≪ 1 the rate <strong>of</strong> the pair creation is
exponentially small [19],
( 2 m
W (χγ) ≈ α
ec4 ) (
χγ exp − 8
)
. (17)
3χγ
¯h 2 ωγ
In the limit χγ ≫ 1 the pair creation rate is given by
( 2 m
W (χγ) ≈ α
ec4 )
(χγ) 2/3
¯h 2 ωγ
(18)
(for details see Ref. [18]). Here ¯hωγ is the energy <strong>of</strong> the
photon which creates an electron-positron pair.
Since for γe ≥ γQ the photon is emitted by the electron
(positron) in a narrow angle almost parallel to the electron
momentum with the energy <strong>of</strong> the order <strong>of</strong> the electron energy,
the parameters χe and χγ are approximately equal
to each other. The parameter χe can be expressed via the
electric and magnetic field as (see Ref. [18])
χ 2 (
e =
E
γe
ES
+ p × B
mecES
) 2
( ) 2
p · E
−
. (19)
mecES
In order to find the threshold for the avalanche development
we need to estimate the QED parameter χe. The condition
for avalanche development corresponding to the parameter
χe should become <strong>of</strong> the order <strong>of</strong> unity within one
tenth <strong>of</strong> the EM field period (e.g. see Ref. [6]). Due to the
trajectory bending by the magnetic field the electron transverse
momentum changes as p⊥ ≈ (a0/16)k0r0(ω0t) 2 ,
where k0r0 = (2.5a0/πas) 1/2 , Eq. (2). Assuming ω0t
to be equal to 0.1 π, we obtain from Eq. (19) that χe becomes
<strong>of</strong> the order <strong>of</strong> unity, i.e. the avalanche can start, at
a0/aS ≈ 0.105, which corresponds to the laser intensity
4 × 10 27 W/cm 2 . The radiation losses in this limit can be
described as the synchrotron losses <strong>of</strong> an electron with the
energy ≈ mec 2 moving in the magnetic field a0(k0r0)/8.
Using formulae for synchrotron radiation [13], it is easy
to show that they do not become significant until a0 ≈
5 × 10 4 . At that limit the Schwinger mechanism provides
approximately 5 × 10 5 pairs per one-period.
In the case <strong>of</strong> two colliding circularly polarized EM
waves the resulting electric field rotates with frequency ω0
being constant in magnitude. The power emitted by the
electron is ≈ εradω0mec 2 γ 4 e. This is a factor <strong>of</strong> γ 2 e larger
than in the case <strong>of</strong> linear polarization. The properties <strong>of</strong>
radiation emitted by rotating electron are well known from
the theory <strong>of</strong> synchrotron radiation [13, 15] and from Ref.
[16]. In the limit γe ≫ 1 the emitted power is proportional
to the fourth power <strong>of</strong> the electron energy. The radiation
is directed almost along the electron momentum being localized
within the angle inversely proportional to the electron
energy: δη ≈ 1/γe. The frequency spectrum given

y the well known expression [13] has a maximum frequency,
ωm = 0.29ω0γ 3 e , proportional to the cube <strong>of</strong> the
electron energy. This is a factor <strong>of</strong> γe larger than in the
case <strong>of</strong> linear polarization. For the electron rotating in the
circularly polarized colliding EM waves the emitted power
becomes equal to the maximal energy gain at the field amplitude
a0 = arad = ε −1/3
rad . For the laser wavelength
λ0 = 0.8 µm εrad = 2.2 × 10−8 . The normalized amplitude
arad is ≈ 400 corresponding to the laser intensity
Irad = 4.5 × 1023W / cm2 .
We represent the electric field and the electron momentum
in the complex form:
and
E = Ey + iEz = E0 exp ( −iω0t) (20)
p = py + ipz = P exp ( −i(ω0t − φ)) , (21)
where φ is the phase equal to the angle between the electric
field vector and the electron momentum. In the stationary
regime, when the electron rotates with constant energy, the
equations for the electron energy, γe = [1+(P/mec) 2 ] 1/2 ,
and for the phase φ have the form
a 2 0 = ( γ 2 e − 1 ) ( 1 + ε 2 radγ 6) e , (22)
tan φ = − 1
εradγ3 . (23)
e
In the limit <strong>of</strong> weak radiation damping, a0 ≪ ε −1/3
rad ,
the absolute value <strong>of</strong> the electron momentum is proportional
to the electric field magnitude, P = meca0, while
in the regime <strong>of</strong> dominant radiation damping effects, i.e. at
a0 ≫ ε −1/3
rad , it is given by P = mec (a0/εrad) 1/4 . For
the momentum dependence given by this expression the
power radiated by an electron is Pγ,C = ω0mec 2 a0, i.e.
the energy obtained from the driving electromagnetic wave
is completely re-radiated in the form <strong>of</strong> high energy gamma
rays. At a0 ≈ ε −1/3
rad we have for the gamma photon energy
¯hωγ = 0.29¯hω0a3 (
3 2
rad ≈ 0.45¯hω0 mc /e ) . For example,
if λ0 ≈ 0.8 µm and a0 ≈ 400 the circularly polarized laser
pulse <strong>of</strong> intensity Irad = 4.5 × 1023 W/cm2 generates a
burst <strong>of</strong> gamma photons <strong>of</strong> energy about 20 MeV with the
duration determined either by the laser pulse duration or by
the decay time <strong>of</strong> the laser pulse in a plasma.
In Fig. 2a we show a dependence <strong>of</strong> γ and φ on the
EM field amplitude, a, for the dimensionless parameter
εrad = 10−8 , obtained by numerical solution <strong>of</strong> Eqs. (22,
23). Here the horizontal axis is normalized on ε −1/3
rad and
the vertical axis is normalized on (am/εrad) 1/4 .
The parameter χe can be expressed via the electric field,
E, as (see Ref. [18])
χe = |E|
(
m
mecES
2 ec 2 + |P| 2 sin 2 ) 1/2
φ , (24)
where φ is an angle between the electron momentum and
the electric field. As we see from Fig. 2 the angle φ tends
to zero at large electric field.
ϕ
Figure 2: Dependence <strong>of</strong> γ and φ on the EM field amplitude,
a, for the dimensionless parameter εrad = 10−8 . The
horizontal axis is normalized on ε −1/3
rad and the vertical axis
is normalized on (am/εrad) 1/4 .
Since in the case <strong>of</strong> circular polarization ωm is proportional
to the cube <strong>of</strong> electron gamma-factor quantum effects
should be incorporated into the theoretical description
at γe ≈ γ C Q = (mec 2 /0.29 ¯hω0) 1/2 ≈ 1300. For γe = a0
this limit is reached at the intensity <strong>of</strong> ≈ 3.4 ×10 24 W/cm 2 .
The electron motion should be described within the framework
<strong>of</strong> quantum mechanics. These effects change the
radiative loss function (see Ref. [18]). In the quantum
regime, it is necessary to take into account not only radiative
damping effects but also recoil momentum effects,
which change the direction <strong>of</strong> motion <strong>of</strong> the electron because
the outgoing photon carries away the momentum
¯hkm = ¯hωm/c.
In the regime when the radiation friction effects are important,
i.e. when a0 ≫ ε −1/3
rad , the angle φ between the
electron momentum and the electric field is small being
equal to ( εrada3 ) −1/4,
0 i. e. the electron moves almost
in the electric field direction. The electron momentum is
given by P = mec (a0/εrad) 1/4 . This yields an estimation
χe ≈
( a0
γ
a 2 S εrad
b)
a
) 1/2
. (25)
This becomes greater than unity for a0 > εrada2 S ≈
5.5 × 103 , which corresponds to the laser intensity equal
to 6 × 1025W/cm2 . In Ref. [6] an avalanche threshold
intensity several times lower has been found neglecting
the effects <strong>of</strong> the radiation friction force (see also [19]).
However, the radiation friction time is <strong>of</strong> the order <strong>of</strong>
(
trad = 1/ω0 εrada3 ) 1/2,
0 which for a0 ≈ 5.5 × 103 is
approximately one tenth <strong>of</strong> the laser period. Hence the radiation
friction effects do not prevent the EPGP cascade development
for circularly polarized colliding waves. Such a
prolific electron-positron pair and gamma ray creation [4]
should result in the EPGP generation.
While creating and then accelerating the electronpositron
pairs the laser pulse generates an electric current
and EM field. The electric field induced inside the EPGP

cloud with a size <strong>of</strong> the order <strong>of</strong> the laser wavelength, λ0
can be estimated to be
Epol = 2πe(n+ + n−)λ0. (26)
Here n+ ≈ n− are the electron and positron density, respectively.
Coherent scattering <strong>of</strong> the laser pulse away from
the focus region occurs when the polarization electric field
becomes equal to the laser electric field. This yields for the
electron and positron density n+ ≈ n− = E/4πeλ0. The
particle number per λ 3 0 volume is about a0λ0/re. This is a
factor a0 smaller than required for the laser energy depletion.
CONCLUSION
In conclusion, the high enough laser intensity pulse
with arbitrary polarization plus high enough density <strong>of</strong>
seed electrons, e.g. generated in the laser interaction with
solid targets can provide necessary and sufficient conditions
for the avalanche development, [4]. Instead, in vacuum,
when the seed electrons(positrons) are created via
the Schwinger mechanism, we see a fundamental difference
between the circularly and linearly polarized waves.
In the case <strong>of</strong> the circularly polarized EM wave the electron
radiation is strong and the threshold for the avalanche
is low enough for avalanche starting at the laser intensity
well below the Schwinger limit. Since, as noted in Ref.
[4], the electron-positron avalanche parameters are insensitive
to the seed electrons (positrons), the parameters <strong>of</strong> the
Schwinger created pairs become hidden and can hardly be
revealed. Contrary to this, in the linearly polarized EM
wave is more favorable for the realization and reaching
<strong>of</strong> ”pure” Schwinger electron-positron pair creation. An
electron moving along the electric field with velocity and
acceleration parallel to the field emits much fewer photons
with substantially lower energy neither experiencing
the radiation friction nor quantum recoil effects. We see
an analogy <strong>of</strong> these cases with circular and linear electron
accelerators with the corresponding constraining and reduced
roles <strong>of</strong> synchrotron radiation losses. The electronpositron
pair creation in the Breit-Wheeler type process is
also suppressed because the key parameters χe and χγ dependence
on the electron and photon momentum, in the
laser field with the same intensity,is much weaker. The
linear polarization has apparent advantages for approaching
the regimes <strong>of</strong> electron-positron pair creation in vacuum
and nonlinear vacuum polarization important for fundamental
science.
We thank S. G. Bochkarev, V. Yu. Bychenkov, P. Chen,
G. Dunne, N. V. Elkina, E. Esarey, A. M. Fedotov, V. F.
Frolov, D. Habs, T. Henzl, M. Kando, Y. Kato, K. Kondo,
G. Korn, N. B. Narozhny, W. Rozmus, H. Ruhl, and A. I.
Zelnikov for discussions.
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PAIR CREATION IN QED-STRONG PULSED LASER FIELDS ∗
I. V. Sokolov † , SPRL, University <strong>of</strong> Michigan, Ann Arbor, MI 48109, USA
N. M. Naumova ‡ , LOA, ENSTA - Ecole Polytechnique - CNRS, 91761 Palaiseau, France
J. A. Nees, CUOS and FOCUS Center, University <strong>of</strong> Michigan, Ann Arbor, MI 48109, USA
G. A. Mourou, ILE, ENSTA - Ecole Polytechnique - CNRS, 91761 Palaiseau, France
Abstract
Electron-positron pair creation is among the QEDeffects
known to occur in a strong laser pulse interaction
with a counter-propagating electron beam. In this
regime multiple pairs may be generated from a single beam
electron, some <strong>of</strong> the newborn particles being capable <strong>of</strong>
further pair production. Radiation back-reaction prevents
avalanche development and limits pair creation. The system
<strong>of</strong> integro-differential kinetic equations for electrons,
positrons and γ-photons is solved numerically.
INTRODUCTION
The effects <strong>of</strong> quantum electrodynamics (QED) may occur
in a strong laser pulse interaction with a counterpropagating
electron beam. In the well-known experiment [1]
these effects were weak and barely observable. If the laser
pulse intensity is increased up to J ≥ 5 · 10 22 W/cm 2 the
QED effects control the laser-beam interaction and result
in multiple pair production from a single beam electron.
QED-strong laser fields
In QED an electric field, E, should be treated as strong if
it exceeds the Schwinger limit: E ≥ ES = mec 2 /(|e| ¯ λC)
(see [2]). Such field is potentially capable <strong>of</strong> separating a
virtual electron-positron pair providing an energy, which
exceeds the electron rest mass energy, mec 2 , to a charge,
e = −|e|, over an acceleration length as small as the Compton
wavelength, ¯ λC = ¯h/(mec) ≈ 3.9 · 10 −11 cm. Typical
effects in QED strong fields are: electron-positron pair creation
from high-energy photons, high-energy photon emission
from electrons or positrons and the cascade development
(see [3, 4]) resulting from the first two processes.
QED-strong fields may be created in the focus <strong>of</strong> an
ultra-bright laser. Consider QED-effects in a relativistically
strong pulsed field [3]:
|a| ≫ 1, a = eA
, (1)
mec2 with A being the vector potential <strong>of</strong> the wave. In the laboratory
frame <strong>of</strong> reference the electric field is not QEDstrong
for achieved laser intensities, J ∼ 10 22 W/cm 2 [5],
and even for the J ∼ 10 25 W/cm 2 intensity projected [6].
∗ One <strong>of</strong> us (I.S.) is supported by the DOE NNSA under the Predictive
Science Academic Alliances Program by grant DE-FC52-08NA28616.
† igorsok@umich.edu
‡ natalia.naumova@ensta-paristech.fr
Nonetheless, a counterpropagating particle in a 1D wave,
a(ξ), ξ = ωt−(k·x), may experience a QED-strong field,
E0 = |dA/dξ|ω(E − p∥)/c, because the laser frequency,
ω = c/ ¯ λ, is Doppler upshifted in the frame <strong>of</strong> reference
co-moving with the electron. Herewith the electron dimensionless
energy, E, and its momentum are related to mec2 ,
and mec correspondingly, and subscript ∥ herewith denotes
the vector projection on the direction <strong>of</strong> the wave propagation.
The Lorentz-transformed field exceeds the Schwinger
limit, if χ ∼ E0/ES ≥ 1. Numerical values <strong>of</strong> the parameter,
χ, may be expressed in terms <strong>of</strong> the local instantaneous
intensity <strong>of</strong> the laser wave, J:
χ = 3 ¯λC
2 ¯λ (E − p
∥)
da
dξ ≈ (E − p∥) 1.4 · 103 √
J
1023 [W/cm 2 ] .
(2)
This dependence shown in Fig.1 demonstrate that χ parameter
on the order <strong>of</strong> tens can be achieved with KEK
or SLAC electron beams using available or foreseen in the
future ultraintense laser pulses.
Figure 1: Dependence <strong>of</strong> χ on energy <strong>of</strong> electrons counterpropagating
to laser pulses <strong>of</strong> various intensities, according
to Eq.(2).
Radiation back-reaction
The creation <strong>of</strong> pairs in QED-strong fields is a particular
form <strong>of</strong> the radiation losses from charged particles. At high
χ an intermediate stage in the pair creation process is the
emanation <strong>of</strong> a high-energy photon by a charged particle:
e → γ, e. This photon is then absorbed in the strong field,
generating an electron-positron pair: γ → e, p.

A way to quantify the irreversible radiation losses has
been found in [7]. Specifically, in the 1D wave field the
transfer <strong>of</strong> energy, ∆E, from the wave to a particle may be
interpreted as the absorption <strong>of</strong> some number <strong>of</strong> photons.
Accordingly, the momentum from the absorbed photons is
added to the parallel momentum <strong>of</strong> the particle. So, both
energy and parallel momentum are not conserved, however,
their difference is: ∆(E − p∥) = 0. To get the Lorentzinvariant
formulation, introduce the four-vector <strong>of</strong> the particle
momentum, p = (E, p), and the wave four-vector,
k = ( ω
c , k) for the 1D wave field. Their four-dot-product,
(k · p) = ω(E − p∥)/c, is conserved in any particle interaction
with the 1D wave field, including its motion, photon
emission, pair creation etc. The sum <strong>of</strong> this quantity over
all particles in the final state is equal to that for the particles
in the initial state.
The radiation losses, thereby limit the cascading pair
creation. Particularly, emission <strong>of</strong> s<strong>of</strong>ter γ photons even
may be described within the radiation force approximation,
which is traditionally used to account for the radiation
back-reaction (see [8, 9, 10, 11]).
The discussed processes are described by the kinetic
equations for the involved particles (electrons, positrons, γphotons).
For circularly polarized 1D wave <strong>of</strong> constant amplitude,
the system <strong>of</strong> three 1D integro-differential kinetic
equations is reducible to a large system <strong>of</strong> ODEs, which is
solved here numerically.
ELECTRON IN QED-STRONG FIELD
The emission probability in the QED-strong 1D wave
field may be found in Sections 40,90,101 in [12]. However,
to simulate highly dynamical effects in pulsed fields,
one needs a reformulated emission probability, related to
short time intervals (not (−∞, +∞)), which is rederived
in Appendix A in [13] with careful attention to consistent
problem formulation.
Again, the energy, ¯hω ′ , and momentum, ¯hk ′ , <strong>of</strong> the emitted
photon are normalized to mec 2 and mec. The fourdot-product,
(k · p), is the motional invariant for an electron
and it is also conserved in the process <strong>of</strong> emission:
(k · pi) = (k · k ′ ) + (k · pf ). A subscript i, f denotes the
electron in the initial (i) or final (f) state.
In the 1D wave field the emission probability may be
conveniently related to the interval <strong>of</strong> the wave phase, dξ,
which should be taken along the electron trajectory. The
interval <strong>of</strong> time, dt, and that <strong>of</strong> the electron proper time,
dτe, are related to dξ as follows: dτe = dt/E = dξ/[c(k ·
p)]. The phase volume element for the emitted photon is
chosen in the form d2k ′ ⊥d(k·k′ ). The emission probability,
dWfi/(dξd(k · k ′ )), is integrated over d2k ′ ⊥ , therefore, it
is related to the element <strong>of</strong> the phase volume, d(k · k ′ ) (see
detail in Appendix A in [13]):
dWfi
d(k · k ′ )dξ = α (∫ ∞
r K5/3(y)dy + κrK2/3(r) )
√
3πλC(k ¯ · pi) 2
, (3)
κ = (k · k′ )χe
(k · pi) , r = (k · k′ )
χe(k · pf ) , χe = 3
2 (k·pi)
da
dξ ¯ λC.
Here Kν(r) is the MacDonald function and α = e 2 /(c¯h).
Collision integral
In strong fields we introduce χ-parameter not only for
electrons but also for γ-photons and relate the emission
probability to dχγ ∝ d(k · k ′ ):
χγ = 3
2 (k · k′
)
da
dξ ¯
dWfi
λC, = α
da
dχγdξ dξ we→γ,e χe→χγ ,
√ [
∫
(4)
∞ ]
3
χγrK2/3(r) + K5/3(y)dy ,
w e→γ,e
χe→χγ =
2πχ 2 e
(5)
Here r = χγ/[χe(χe − χγ)], χγ ≤ χe. The electron parameter,
χe, is taken for the initial state and its value in the
final state is χe − χγ.
The distribution functions for electrons and photons may
be also integrated over p⊥ and k ′ ⊥ correspondingly. Thus
integrated functions are distributed over (k · p), (k · k ′ ). We
can parameterize locally these distributions via χe ∝ (k·p),
χγ ∝ (k · k ′ ) and introduce the 1D distribution functions,
fe(χe) and fγ(χγ).
The collision integral (see [14]) describes the change
in the particle distributions due to emission and accounts
for the electrons, leaving the given phase volume, dχe,
and those arriving into it within the interval, d ˜ ξ =
α|da/dξ|dξ = 2αcχedτe/(3 ¯ λC):
δfe(χe)
d˜ =
ξ
∫ ∞
χe
fe(χ)w e→γ,e
χ→χ−χedχ−fe(χe) ∫ χe
0
δfγ(χγ)
d˜ =
ξ
∫ ∞
χγ
r
w e→γ,e
χe→χ dχ,
fe(χ)w e→γ,e
χ→χγ dχ. (6)
PHOTON IN QED-STRONG FIELD
The absorption probability for photons in the 1D field is
derived in Appendix B in [13]. An electron-positron pair
(e,p) is generated in the photon absorption with the conservation
law: (k · k ′ ) = (k · pe) + (k · pp).
The phase volume element for the created electron, again
is chosen in the form d2p⊥d(k · p). The absorption probability,
dWfi/(dξd(k · pe)), is integrated over the transversal
momenta components and related to the element <strong>of</strong> the
phase volume <strong>of</strong> electron, d(k · pe), resulting in the following
collision integral:
δ − fe,p(χe,p)
d ˜ ξ
=
∫ ∞
χe,p
fγ(χγ)w γ→e,p
χγ→χe dχγ, (7)
∫ χγ
δ−fγ(χγ) d˜ = −fγ(χγ)
ξ
0
Here r = χγ/[χe(χγ − χe)], χe = χγ − χp ≤ χγ and
√ [
∫ ∞ ]
3
χγrK2/3(r) − K5/3(y)dy .
w γ→e,p
χγ→χe =
2πχ 2 γ
w γ→e,p
χγ→χe dχe. (8)
r
(9)

Figure 2: Distribution functions <strong>of</strong> electrons and positrons, fe,p(χ), and a spectrum <strong>of</strong> emission, χγfγ(χ)/χ0, after the
interaction <strong>of</strong> 8-GeV electrons with one, five and ten cycles <strong>of</strong> a laser pulse <strong>of</strong> intensity J ≈ 2 · 10 23 W/cm 2 (so that
χ ≈ 2E[GeV ] — see Eq.(2)). Here fe − fp is the distribution <strong>of</strong> the beam electrons and ∫ (fe − fp)dχ = 1.
Figure 3: Pair production (upper panel) and energy exchange
between electrons, photons, positrons (lower panel)
as functions <strong>of</strong> phase for the simulation presented in Fig.2.
SOLUTION FOR KINETIC EQUATIONS
As long as the distribution functions are integrated over
the transversal components <strong>of</strong> momentum and expressed in
terms <strong>of</strong> the motional integrals, (k · pe,p), their evolution is
controlled by the collision integrals:
(
δ + + δ − + δ (rf))
fe,p,γ.
∂fe,p,γ( ˜ ξ, (k · pe,p,γ))
∂ ˜ =
ξ
(10)
The derivatives, ∂/∂ ˜ ξ, are taken at constant (k · p). The
term δ (rf) accounts for the radiation losses for the photons
with χγ ≤ ϵ ≪ 1 excluded from the other two terms in-
tentionally (see details in Ref.[7]). Eqs.(10) are easy-tosolve
for the 1D wave field <strong>of</strong> any shape, however, for
circularly polarized wave <strong>of</strong> constant amplitude the solution
is especially simple. In this case (k · p) are different
from χ by a constant factor, and Eqs.(10) may be solved
with derivatives, ∂/∂ ˜ ξ, at constant χ for the functions,
fe,p,γ( ˜ ξ, χe,p,γ).
We solve Eqs.(10) numerically, by discretizing them at a
uniform grid, χj = j∆χ, j = 1, 2, 3..., N, with the choice
<strong>of</strong> ∆χ = 0.1. The ˜ ξ-dependent distribution functions at
this grid obey the system <strong>of</strong> 3N ODEs, which is integrated
numerically.
NUMERICAL EXAMPLES
KEK parameters: χ = 30
At initialization, electrons with fe(χe) = δ(χe − χ0),
χ0 = 30, counterpropagate in the circularly polarized wave
field with |da/dξ| = 220. This choice corresponds to the
8-GeV electron beam and the laser intensity <strong>of</strong> J ≈ 2 ·
1023 W/cm 2 for λ = 0.8µm, to be achieved soon.
In Fig.2 the beam-wave interaction is traced during ξ
2π =
10 cycles <strong>of</strong> the incident laser pulse (≈ 27 fs). The initial
beam electron energy is rapidly converted into γ-photons
with high χγ, which then rapidly produce pairs, the typical
rates <strong>of</strong> the processes being <strong>of</strong> the order <strong>of</strong> the inverse
light period. However, the larger fraction <strong>of</strong> the new particles
is born at χ ≤ 1, with strongly reduced pair production
rate. Slow absorption <strong>of</strong> photons with χγ ∼ 1−2 maintains
pair production even after tens <strong>of</strong> wave periods, as shown in
Fig.3 (upper panel). In Fig.3 the lower panel shows the evolution
<strong>of</strong> the functions ∑
∑
i
χife,i/χ0,
i χifp,i/χ0,
∑
and
i χifγ,i/χ0, which represent the energy portions in the
corresponding sorts <strong>of</strong> particles. These plots demonstrate
that for 10-cycle laser pulse ≈ 90% <strong>of</strong> the initial electron
energy is converted into photons, the rest part is split between
electrons and positrons.
SLAC parameters: χ = 90
Using 46-GeV electron beam and laser pulses <strong>of</strong> intensity
J ≈ 5 · 10 22 W/cm 2 , the value <strong>of</strong> χ = 90 can be

Figure 4: Pair production vs time for 46.6 GeV electrons interacting with laser pulse <strong>of</strong> intensity J ≈ 5 · 10 22 W/cm 2 .
Plots are presented in linear and log − log scale. Dashed line is ∝ ξ 2 .
achieved in their interaction. An evolution <strong>of</strong> pair production
for these parameters is shown in Fig.4 (see other plots
for the same parameters in Ref.[7]). The results for such
high initial value <strong>of</strong> χ demonstrate almost quadratic dependence
<strong>of</strong> pair production. This observation is in agreement
with simple estimations as follows.
The change in the pair production is proportional to the
number <strong>of</strong> rigid photons:
dN e − e +
dξ
∝ Nγ
dWabsorption
.
dξ
In the same time, the change in the number <strong>of</strong> rigid photons
is proportional to initial number <strong>of</strong> electrons:
dNγ
dξ
∝ Ne,0
dWemission
.
dξ
Assuming a constant value for initial number <strong>of</strong> electrons,
Ne,0, we obtain dependences for the number <strong>of</strong> rigid photons
and the number <strong>of</strong> pairs:
what agrees with our results.
Nγ ∝ ξ, N e − e + ∝ ξ 2 ,
CONCLUSION
We see that the laser-beam interaction may be accompanied
by multiple pair production. The initial energy <strong>of</strong> a
beam electron is efficiently spent for creating pairs with
significantly lower energies as well as s<strong>of</strong>ter γ-photons.
This effect may be used for producing a pair plasma. It
could also be employed to deactivation after-use electron
beams, reducing radiation hazard.
The way to solve the kinetic equations is accurate and
it does not employ the Monte-Carlo method. The solution
can be used to benchmark numerical methods designed to
simulate processes in QED-strong laser fields.
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LASER ACCELERATION UP TO BLACK HOLES
AND B-MESON DECAY *
H. Hora, # Deptm. Theor. Phys. UNSW Sydney, Australia
R. Castillo, T. Stait-Gardner, BMSci. U. Western Sydney Campbelltown, Australia
D.H.H. H<strong>of</strong>fmann, Dept. Nucl. Phys. TU Darmstadt, Germany
G.H. Miley, Dept. Nucl. Plasma & Radiolog. Engin. Univ. Illinois USA
P. Lalousis, IESL/FORTH, Heraklion, Greece
Abstract
Studies about laser produced pair production are followed
up from early stages. The pair production by vacuum
polarization was discussed with laser produced
acceleration up to the values at black holes leading to the
discovery <strong>of</strong> a difference between Hawking and Unruh
radiation. It was clarified that production <strong>of</strong> anti-hydrogen
is at least million times more efficient than by present day
accelerator technology. Another application <strong>of</strong> the
ultrahigh laser fields is to focus them into the collision
area <strong>of</strong> the LHC with the possibility to study the details <strong>of</strong>
the B-meson dacay. This <strong>of</strong>fers an access to detect more
details about CP violation and Bs mesons and possible
signs <strong>of</strong> new particles on the horizon. The available lasers
with picosecond pulses are developed to exawatts power
what is interesting also for studying ultra-intense shock
waves in astrophysics and resulting nuclear reactions.
INTRODUCTION AND INITIAL RESULTS
Using the very high intensity laser radiation with the
electric and magnetic fields E and H far above any values
applied before, led to very many new physics phenomena
and last not least to the realization that the opened
nonlinear physics opens a new dimension <strong>of</strong> exploration
where indeed the linear physics needs to be based on
higher accuracy data than needed before [1]. Examples
appeared where results in linear physics were completely
wrong compared to the truth in nonlinear physics in
contrast to earlier happening gradual differences or
approximations only. This all developed not only due to
techniques to produce higher and higher laser intensities,
mostly realized by chirped pulse amplification CPA [2],
but also from realizing to generate relativistic effects.
After first considerations how to produce relativistic
conditions for pair production <strong>of</strong> electrons [3] the
conditions were elaborated for the laser fields producing
quiver motion <strong>of</strong> electrons with energies above mc 2 [4].
The steps to conclude the conditions for producing
anti-protons [5] were parallel to estimations [6] and
experiments where indications for the generation <strong>of</strong> the
very first laser produced positrons were reported [7].
With respect to the quiver motion and drift for proton pair
production, the advantages <strong>of</strong> long wave length laser
pulses were <strong>of</strong> interest [8]. After estimations with anti-
* Dedicated to Pr<strong>of</strong>essor Chiyoe Yamanaka, Osaka
University, to his 88 th year.
# h.hora@unsw.edu.au
1
hydrogen for space research became known, the solution
with lasers were <strong>of</strong> interest because the efficiency was
more than one million times higher with lasers due to the
available much higher particle density than with
accelerator techniques. It was proved that a mission to the
next fix star within a reasonable time <strong>of</strong> 50 years can only
be done with laser produced anti-hydrogen fuel [9].
Thanks to the CPA technique, sub-picosecond laser
pulses <strong>of</strong> 2 PW produced the first considerable number <strong>of</strong>
positrons [10] finally arriving [11] at record intensities
positron beam above any other method.
PAIR PRODUCTION BY VACUUM
POLARIZATION
Pair production in vacuum was from the beginning
considered [3][4][5] where a laser intensity above 10 28
W/cm 2 was needed [12] and specified to the well known
higher value later. The acceleration <strong>of</strong> the electrons by the
electric field <strong>of</strong> the laser was close to the values <strong>of</strong> the
Hawking radiation and the Unruh radiation at the black
holes. This was studied in connection with the black body
radiation which fields are <strong>of</strong> the same order and where the
electrons at thermal equilibrium were not longer
following the Fermi-Dirac statistics [13]. Further studies
clarified that there was a difference between the Hawking
and the Unruh radiation [14] with a relation to the
Casimir effect [15][16]. These results were based on the
theory <strong>of</strong> electron acceleration in vacuum [17] as a
basically nonlinear effect [1]. The essential aspects <strong>of</strong>
these studies are as follows.
The Unruh effect is a phenomenon whereby an
accelerated observer travelling through a true vacuum
state—that is the ground state |0> which will be referred
to here as the Minkowski vacuum—will experience
themselves to be immersed in a thermal blackbody
distribution <strong>of</strong> particles [16]; Before comparing the
thermal radiation experienced by an accelerated observer
to the Hawking radiation <strong>of</strong> a black hole a brief digression
into the physical nature <strong>of</strong> the vacuum is appropriate.
The Minkowski vacuum is a physical vacuum with
pairs <strong>of</strong> virtual particles manifesting for short durations
continuously and, unlike the pre-quantum field theory
vacuum, has observable effects on physical systems (e.g.
the fine structure <strong>of</strong> the atomic hydrogen spectrum and
the Casimir effect). Taking the Casimir effect as an
example, two parallel mirrors placed in a vacuum will
experience an attractive force inversely proportional to
the forth power <strong>of</strong> the distance separating them as a result

<strong>of</strong> the quantum fluctuations <strong>of</strong> the vacuum. Essentially
long wavelength virtual particles cannot manifest between
the conducting mirrors resulting in a decreased energy
density between the mirrors compared with the vacuum
surrounding them where there is no such restriction. The
Casimir effect is symbolic <strong>of</strong> the physical nature <strong>of</strong> the
quantum vacuum.
The quantum field is best decomposed for an
accelerated observer using a different basis than the
standard momentum basis used in quantum field theory;
this basis being related to the standard basis by the
Bogoliubov transformations. These transformations play
an integral part in analyses <strong>of</strong> the Unruh effect. The
particle number operator differs too and does not give
zero when applied to the Minkowski vacuum state (which
is not identical to the Rindler vacuum state). The result is,
as stated above, that an accelerated observer in a pure
vacuum will experience themselves in a heat bath with a
blackbody distribution.
Thus a state without particles to an inertial observer
will be seen to contain particles by an accelerated
observer. The dependence <strong>of</strong> temperature upon
acceleration is, T = 2πckBa/ħ, where c is the velocity <strong>of</strong>
light, and a is the acceleration. If a is interpreted as the
acceleration at the event horizon <strong>of</strong> a black hole then the
same equation describes the temperature <strong>of</strong> the thermal
radiation emitted from a black hole via the process <strong>of</strong>
Hawking radiation. The similarity <strong>of</strong> the equations and
the equivalence principle <strong>of</strong> general relativity hint that the
mechanisms for the radiation may be the same but this is
not the case. Consider the following.
Hawking radiation is sometimes described as resulting
from pair production near the horizon <strong>of</strong> a black hole with
one <strong>of</strong> the virtual particles escaping and becoming real
and the other disappearing into the black hole [15]. All
observers experience Hawking radiation while only
accelerated observers experience the Unruh effect.
Furthermore, an observer on earth is effectively in an
accelerated coordinate system via the equivalence
principle and hence should observe the surrounding
vacuum to have a temperature due to the Unruh effect but
the earth does not emit Hawking radiation and neither do
other gravitational bodies without event horizons. The
Unruh effect results from a different mechanism to that <strong>of</strong>
Hawking radiation; it is local, being experienced only by
accelerated observers [14].
PETAWATT LASER PULSES FOR
B-MESON DIAGNOSTICS
The present day available PW laser pulses <strong>of</strong> subpicosecond
duration and the next higher powers can be
used for important studies <strong>of</strong> the details <strong>of</strong> B-meson
diagnostics because their lifetimes are on the same time
scale. This diagnostics at collider beam interactions with
lasers was studied before [18] for the conditions <strong>of</strong> the
Large Electron Positron (LEP) collider and can now be
extended for the conditions <strong>of</strong> B-mesons, e.g. at the Large
Hadron Collider LHC or similar B-meson factories [19].
2
A prototype <strong>of</strong> this technique was given by the
interaction <strong>of</strong> 10 16 W/cm 2 laser intensities in low density
helium [20]. It was expected from theory that a radial
emission <strong>of</strong> electrons from the focus should convert half
<strong>of</strong> the quiver energy <strong>of</strong> the electrons into energy <strong>of</strong>
translative motion. The measured radially emitted keV
electrons corresponded exactly to the expected theory.
The conservation <strong>of</strong> the momentum <strong>of</strong> the photons leads
to a slightly forward direction parallel to the laser axis.
This was measured in [21] in agreement with the earlier
prediction [22]. In the same way, the charged particles
generated in the focus <strong>of</strong> the collider when being in the
focus <strong>of</strong> the laser beam, will get an upshift <strong>of</strong> energy and
a change <strong>of</strong> direction. The PW laser pulses and even the
better exawatt (EW) laser pulses <strong>of</strong> few fs duration [23]
can then follow up the timing <strong>of</strong> generating or
annihilating process <strong>of</strong> the B-meson generation and the
decay processes. The importance is evident for further
analyzing the different types <strong>of</strong> B-mesons where insights
<strong>of</strong> the CP violation strange bosons Bs will be interesting
on the way <strong>of</strong> a “possible new particles on the horizon”
[24].
The theory is based on electro-dynamic interaction <strong>of</strong>
the laser radiation with the particles as known from
plasma interaction as the nonlinear force given by [25]
fNL = ∇•[EE + HH − 0.5(E 2 + H 2 )1
+ (1+(∂/∂t)/ω)(n 2 −1)EE]/(4π)
− (∂/∂t) E × H/(4πc) (1)
(see Eq. 8.88 <strong>of</strong> Ref. 1991 [25]) where 1 is the unity
tensor, c the vacuum speed <strong>of</strong> light. The value n is the
(complex) refractive index determined by the laser
frequency ω and the electron-ion collision frequency ν <strong>of</strong>
a plasma
n = 1 – (ne/nec)/(1 + iν/ω), (2)
where ne is the electron density, nec is the critical electron
density where the plasma frequency ωp is equal to the
laser frequency ω. The dielectric properties <strong>of</strong> the vacuum
polarization are to be included appropriately for the pair
production in vacuum. The derivation <strong>of</strong> this force with
inclusion <strong>of</strong> the dielectric plasma properties for the nontransient
case since 1969 [25] was based on momentum
conservation. The final complete transient case, Eq. (1), is
known since 1985 based on symmetry where it was
proved later that this and only this is the Lorentz and
gauge invariant description <strong>of</strong> the nonlinear force.
For simplified one-dimensional geometry and
perpendicular laser irradiation, the force (1) can be
reduced to the time averaged value
fNL = − (∂/∂x)(E 2 +H 2 )/(8π)
= − (ωp/ω) 2 (∂/∂x)(Ev 2 /n)/(16π), (3)
where Ev is the amplitude <strong>of</strong> the electric field in vacuum.
The last expression is reminding <strong>of</strong> the formulation <strong>of</strong> the
ponderomotive force in electrostatics and is sometimes
called “radiation pressure acceleration”.
The relativistic limits for the emission <strong>of</strong> the charged
particles from the collider area with a laser focus are
given for laser intensities <strong>of</strong> neodymium glass lasers [19]

(1) charged B-mesons
Irel = 1.2×10 25 W/cm 2 ∆ε = 2.41 keV
(2) protons or antiprotons from the B-decay
Irel = 3.9×10 26 W/cm 2 ∆ε = 424 eV
(3) charged π-mesons from B-mesons decay:
Irel = 2.73×10 23 W/cm 2 ∆ε = 31.5 keV
The size <strong>of</strong> the lasers for PW-fs pulses are comparably
compact such that the diagnostics with an additional laser
focus may not be a too difficult problem. The signals
from the detectors for comparable cases with and without
the laser will then be done by functional analytical
folding <strong>of</strong> the information about the time dependence <strong>of</strong>
creation, decay and annihilation processes <strong>of</strong> the
numerous types <strong>of</strong> charged particles,
Figure 1. Genuine two fluid hydrodynamic computations
[32][33] <strong>of</strong> the ion density in solid DT after irradiation <strong>of</strong>
a laser pulse <strong>of</strong> 10 20 W/cm 2 <strong>of</strong> ps duration at the times 22
ps (dashed) and 225ps after the initiation.
EXAWATT LASER PULSES FOR SHOCK
WAVES AND NUCLEAR REACTIONS
Studies with the advanced PW to EW laser pulses are
important also for exotic conditions <strong>of</strong> shock waves in
astrophysics [26], ultrahigh accelerations and for related
interactions including nuclear mechanisms. The essential
difference to the usual thermal pressure generation
processes in plasmas is the direct conversion <strong>of</strong> laser
energy into particle motion. This can be seen from the
nonlinear forces including the optical response since 1969
[25] expressed in Eq. (1). The then predicted ultrahigh
accelerations were first measured by Sauerbrey by the
Doppler effect at target interaction with above TW-ps
laser pulses. The nonlinear force driven accelerations
were 10 20 cm/s 2 [27] in contrast to comparable
accelerations with thermal-pressures <strong>of</strong> 10 15 cm/s 2 [28].
The high acceleration was in full agreement with the
theory [29] and could then be used to ignite solid state
density fusion fuel deuterium tritium DT [30]. This is a
3
rather simplified scheme <strong>of</strong> igniting hydrogen-boron11
with producing less radioactive radiation per generated
energy than burning coal [31].
In order to show the velocity <strong>of</strong> the reacting fusion
flame – similar to cases in astrophysics – Fig. 1 shows the
computations <strong>of</strong> the ion density in frozen DT at ps laser
irradiation <strong>of</strong> 10 20 W/cm 2 . The reaction front at the
interaction following the ps ignition at later times when
propagating through the solid density DT can be seen
where compressions up to four times the solid state are
generated within the moving short depth shock wave.
This numerical result automatically agrees with the factor
four <strong>of</strong> the Rankine-Hugoniot shock wave theory. The
shock velocity <strong>of</strong> 1550 km/s is in the range known for this
type <strong>of</strong> interaction. For later times the fusion flame shows
more and more a deviation <strong>of</strong> the density pr<strong>of</strong>ile differing
from the simplified shock wave theory. This is evident
from the output <strong>of</strong> the fast velocity <strong>of</strong> the generated alpha
particles when moving into the untouched solid DT by
gradually changing there the conditions <strong>of</strong> densities and
temperatures. However the velocity <strong>of</strong> the entire flame is
remarkably unchanged. More properties are given in the
references, however, the genuine two-fluid computations
arrive at many more details than known from the onefluid
computation [30][31]. It is important to note that
these studies are aimed to apply ps laser pulses in the
range <strong>of</strong> 30 PW up to nearly EW. Generalizing the
preceding computations [30][31], the genuine two-fluid
hydrodynamics [32][33] is used in order to follow up the
details <strong>of</strong> the generated very high electric fields in the
shock fronts and to confirm most <strong>of</strong> the other results
calculated before with the usual one fluid hydrodynamics.
The results are interesting for astrophysical cases and for
shock ignition <strong>of</strong> fusion [34] where in contrast to the
thermal pressure process, the new research now was
generalized to non-thermal nonlinear force direct
conversion <strong>of</strong> laser energy into plasma motion to reach
the ultra-high accelerations.
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Phys. Rev. Lett. 98, 155001 (2007)

Present Status <strong>of</strong> Ultra-intense Lasers and High-Field Physics
in the World
INTRODUCTION
It is the 50 th year anniversary <strong>of</strong> the invention by
Theodor Maiman, who demonstrated laser generation in
1960 with a ruby rod as shown in Fig. 1. At that time,
most <strong>of</strong> scientists believed that the gas is the best for
lasing media, while Maiman took a different way against
all and succeeded to be the first runner in this big race.
Fig. 1 Maiman and his world first laser made <strong>of</strong>
Ruby rod. It was June in 1960.
In the last fifty years, the scale, power, and focused
intensity dramatically increased from 1kW to 10 PW, the
12 order <strong>of</strong> magnitude increase. The laser is now widely
used in fundamental and applied science. In addition, the
laser is also used for many devices in commercial goods.
In the present paper, we focus on the highest intensity
laser at the present time and to be constructed in the near
future and discuss about what kind <strong>of</strong> extreme experiment
we can do. The vacuum breakdown and resultant
highly-relativistic electron-positron pair plasma
production are main topics <strong>of</strong> this paper. We have reached
the intensity <strong>of</strong> 2.2×10 22 W/cm 2 <strong>of</strong> focused laser at Univ.
Michigan [1]. The electric field <strong>of</strong> this intensity is 1.4×
10 12 V/cm. The energy <strong>of</strong> an oscillating electron in this
electric field can be calculated to be 10 2 mc 2 . It is natural
that with the appearance <strong>of</strong> such strong lasers wishes to
carry out a variety <strong>of</strong> experiments related to Non-liner
QED and QCD.
H. Takabe, ILE, Osaka University, Japan
In the present paper, we review the present status <strong>of</strong>
existing and planned ultra-intense laser facilities in the
world at first, and describe the physics scenario <strong>of</strong> the
production <strong>of</strong> electron pair plasma. It is mentioned that
when the laser intensity becomes 100 times the present
record and it becomes above 10 24 W/cm 2 , vacuum
breakdown becomes essential to creation <strong>of</strong> highly
relativistic electron pair plasma. The avalanche effect
enhances the pair creation <strong>of</strong> the vacuum.
M U -PW L LASERS T I IN THE WORLD
A great progress in constructing 10 PW laser systems
in Europe is going by the leadership <strong>of</strong> G. Mourou, the
founder <strong>of</strong> the pulse compression technique with CPA [2].
It is amazing that EU decided to fund three 10 PW laser
facilities in the EU member countries <strong>of</strong> the East Europe,
Czech, Hungary, and Romania. Total budget for
construction is about 800 M Euro. Laser property <strong>of</strong> each
<strong>of</strong> them is almost same but with different main subjects
[3]. Czech’s aims at high-brightness sources <strong>of</strong> x-rays and
particles[4], Hungary’s atto-second XUV/X-ray source
and its application [5], and Romania’s laser-induced
nuclear physics[6]. They are three ELI modules and the
site selection was done on October 1, 2009. They will also
play important role in technology development for
constructing 200PW laser system ELI (Extreme Light
Infrastructure) [7]. It is announced that the technology
and site <strong>of</strong> ELI are to be determined after 2012.
On the other hand, UK will construct 10 PW system
by modifying and expand the present Vulcan laser system
and it will be completed due by 2014-15 [8]. France plans
to construct a single beam 10 PW laser APOLLON as
international collaboration system with main contribution
by three institutions, LULI, LOA, and Institute <strong>of</strong> Optics
and it will be completed due by 2012-13. Vulcan is <strong>of</strong>
300J, 30fs, 1shot/15 min, and APPLLON is <strong>of</strong> 150J, 15 fs,
1shot/min.
In Asia, APRI in Korea has 1PW laser system with 47
J/30fs at 0.1 Hz [9]. In Japan, 1 PW laser is in
Kansai-J A E [10] A and 10 PW laser LFEX with 10 kJ/ 1ps
single shot base [1 1] is under construction in Osaka
University.

In USA, LLNL has a user’s facility, Titan laser which
is 1PW system with 400fs--10ps, up to 530 J and
2shots/hour [12]. At LLE, University <strong>of</strong> Rochester, an
ultra-intense laser system OMEGA-EP is now in
operation. It is <strong>of</strong> 1 P multi-kJ, W , 1ps and intensity
higher than 10 20 W/cm 2 We demonstrated that the Bethe-Heitler process is main
process to produce the pairs, although it is the two step
process in the gold foil. We predicted the positron
spectrum and compared with Cowan’s experimental data.
What we found is that the experimental data looks like
[13]. In addition, the world- shifted by about 10 MeV to the higher energy side as you
biggest laser NIF will have four beams to be multi-P It W . can see in Fig. 2 [18]. The Monte-Carlo calculation
is called NIF-ARC [14]. NIF-ARC is originally motivated
for diagnostic purpose for NIF ignition campaign (NIC),
but it is also open to users in the world only for the
fundamental science.
PAIR PLASMAS
Einstein said “Imagination is more important than
knowledge”. We can spread the world <strong>of</strong> imagination with
the appearance <strong>of</strong> such ultra-intense lasers. The first
experiment with PW laser was done by T. Cowan et al.
[15] and he reported photo-nuclear fission <strong>of</strong> Uranium
and positron production in a gold foil. In order to analyze
his experimental data on positron production [16], we
pursued joint research starting from modeling the Trident
and Bethe-Heitler processes in Fokker-Planck equation to
relativistic electrons as main source for pair creation [17].
Figure 2: (a) The longitudinal momentum
distributions <strong>of</strong> positron (red) and electrons (black) at
back <strong>of</strong> the target. The target rear is at x=0. The solid
line indicates the sheath field normalized by 10 12 V/m.
(b) Positron spectrum calculated by PIC. The blue
dotted line is the electron spectrum used in PIC
calculation.
usually used in HEP field [17] predicted almost the same
as Fokker-Planck case.
In order to see the physics causing the difference, we
thought that since the laser-produced relativistic electron
density is much higher than the case <strong>of</strong> accelerator beams,
a strong electric field formation by charge separation
effect, namely; plasma effect, becomes important to
accelerate the created positron at the rear side <strong>of</strong> the target
foil [18]. We pointed out this is critically different point <strong>of</strong>
laser case compared to the accelerator case. We have
carried out two-dimensional PIC (Particle-in-Cell)
simulation [20] with the initial condition calculated with
the Fokker-Planck equation and successes to reproduce
the experimental positron spectrum as shown in line with
PIC. PIC simulation calculated Maxwell equations con-
sistently with charged particles and the electric field the
potential <strong>of</strong> about 20 MV is found to be produced. It
should be noted that since the electric field is the vertical
direction <strong>of</strong> the target rear surface and the positrons are
accelerated predominantly to this direction, we obtain a
jet like positrons with Lorentz factor about 10-20 the
number <strong>of</strong> which is the same as those <strong>of</strong> AGN-jets
(Cosmological jets) [21]. Recently, H. Chen et al. pub-
lished several papers and they also found the importance
<strong>of</strong> the electric field and jet-like emission <strong>of</strong> positrons [22].
We are now collaborating to demonstrate the charge
Fig. 3 Positron number scaling from 100’s Joule
region to Several KJ laser case.

neutral pair plasma production as shown in Fig. 3 with 10
kJ class PW laser NIF-ARC to be completed in 2013 [23].
VACUUM BRAKDOWN
With more increase <strong>of</strong> laser intensity higher than
10 22 W/cm 2 , we have a possibility to use all laser energy
to create the pair plasma fireball. This physics is based on
an exciting physics <strong>of</strong> the vacuum breakdown with laser
fields. In this case, we use the vacuum as target to
produce the pair plasma. The pair plasma creation itself is
a tiny topic, but the demonstration <strong>of</strong> “Vacuum
Breakdown” is more dramatic target for the research.
The vacuum breakdown started to be one <strong>of</strong> the arguing
topics just after the paper <strong>of</strong> relativistic quantum
mechanics with Dirac equation [24]. N . B o W h . r ,
Heisenberg, and so on predicted the vacuum breakdown
as “Gedanken experiment”. It is now going to be a real
experiment thanks to the progress <strong>of</strong> laser technology.
When the laser intensity exceeds 10 24 W/cm 2 , we
can expect the following three different physics scenario
with increase <strong>of</strong> the laser intensity.
(1) Vacuum breakdown due to the pair avalanche
triggered by the induced pair production
(2) Vacuum breakdown due to the pair avalanche
by spontaneous pair production
(3) Vacuum breakdown without the avalanche near
and over the Schwinger limit
When the concept <strong>of</strong> Dirac sea is proposed, distinguished
physicists predicted that if a strong electric field is
imposed in the vacuum, electron-positron pairs appears
because <strong>of</strong> the tunneling effect <strong>of</strong> quantum particles.
Heisenberg’s uncertain principle requires the existence <strong>of</strong>
quantum noise energy even in a complete vacuum. This
means any particle with a finite energy shows its face in a
very short time in the vacuum because the uncertain
principle requires ∆ε∆t ≥ 1/2ℏ. If we assume ∆ε = mc 2 ,
then we obtain ∆t ~ 10 -21 s. If during this extremely short
time, we can give the energy <strong>of</strong> the order <strong>of</strong> the rest mass
and separate the pair over a substantial distance to avoid
spontaneous annihilation, real pair is created.
The critical value <strong>of</strong> such extremely strong electric
field E can easily obtained with the relation

field triggering such phenomena is calculated with
particle and Monte-Carlo hybrid code [27] and it is
concluded to be about 10 24 W/cm 2 , the value <strong>of</strong> which is
very lower than the Schwinger limit <strong>of</strong> Eq. (4). It should
be noted that prolific pair creation occurs thanks to the
avalanche effect explained later.
In order to know the radiation emission by an
electron in a strong electromagnetic field, we have to
solve the solution for the Lagrangian

small for the electric field weaker than Schwinger limit <strong>of</strong>
Eq. (3).
It is very important to know that for the plane wave
<strong>of</strong> laser field in the vacuum, the two scalars S and P
disappear and no tunneling effect can be expected even
with the laser intensity higher than the Schwinger limit <strong>of</strong>
Eq. (4). So, we are required to deform the laser field from
the plane wave relation to maximize the Lorentz
invariants S and P, especially the scalar P in Eqs. (8) and
(9). In order to optimize with realistic optics, not only the
colliding laser scheme but also strongly tightly focused
laser field is proposed [32]. In addition, multiple colliding
laser pulses method is also proposed and evaluated
quantitatively how less laser energy is enough to produce
one pair [33]. We have proposed another tightly focused
laser with use <strong>of</strong> a radially polarized laser [34].
In the above three schemes, it is concluded that the
tunneling effect starts with the laser intensity more than
10 26 -10 27 W/cm 2 . Although the tunneling effect is not
substantial like Schwinger limit, the seed pairs are pro-
duced in the pure vacuum and the prolific pair production
and the resultant vacuum breakdown might be expected
by the avalanche mechanism. Of course, very precise cal-
culation is necessary with the computational modeling
briefly mentioned previously. In this case, solving the
Maxwell equations self-consistently is very important to
see if the laser field is scattered out without being focused
like the case <strong>of</strong> pure vacuum. This is because the
appearance <strong>of</strong> the pairs means average refractive index
changes as a function <strong>of</strong> time at the focused point.
When the laser intensity reaches the Schwinger limit
Eq. (4), the pair creation by the tunneling effect becomes
dominant and spontaneous creation <strong>of</strong> the pairs occur at
all the place where the value <strong>of</strong> Eq. (7) is large enough.
In this case, the interesting point <strong>of</strong> research is how the
avalanche effect is still important for the pair creation
process. In addition, we also expect the creation <strong>of</strong> low
energy hadron and hadron pairs such as π and π ± .
What kind <strong>of</strong> lepton pairs and hadron mixture or quark-
gluon mixture plasma is produced is a very interesting
subject.
LABORATORY COSMOLOGY AND
MODELING QUARK-GLUON PLASMA
We can expect to do a variety <strong>of</strong> model experiments
for open questions in high-energy astrophysics. The
investigation <strong>of</strong> the plasma properties <strong>of</strong> highly rela-
tivistic electron positron plasmas and jets is essential for
example, AGN jets and the jets related to the gamma-ray
bursts [35]. In the case <strong>of</strong> “quiet” pair production in gold
foil as described relating to Fig. 2, the am-bipolar field
accelerates the positron to produce relativistic pair plasma
in the jet-like structure. This is good to model the
propagation <strong>of</strong> pair plasma jets in the Universe.
In the case <strong>of</strong> “violent” pair creation via the vacuum
breakdown, it is not clear what kind <strong>of</strong> energy distribution
and angular distribution <strong>of</strong> the pair plasma is produced
and how we can control them. In addition the distribution
<strong>of</strong> gamma-ray is also open question. It may not be in the
thermodynamic equilibrium state, while the physical
dynamics in the highly relativistic “pair fireball” is very
interesting matter to study liner and nonlinear collective
phenomena as plasma physics. Whether a collisionless
shock formation may observed after the nonlinear stage is
very hot topics in the GRB physics [36]. For example, this
plasma would be a good test bed to verify and validate the
computational modeling. It is very challenging, while the
improvement <strong>of</strong> such simulation code compared to the
real experiment is very much beneficial also to the
modeling <strong>of</strong> Quark-Gluon plasma (QGP), where the color
force is dominant than the electric force. It is said that
non-Abelian system like QGP would become thermo-
dynamic equilibrium in an extremely short time [37]. In
addition, we may be able to identify if the avalanche
effect plays an important role in the formation <strong>of</strong> QGP in
extremely tiny space and time.
As schematically shown in Fig. 6, the QGP created
through vacuum breakdown by color field is <strong>of</strong> the order
<strong>of</strong> 10 -12 cm (=10fm) and the life time <strong>of</strong> QGP is about
10fm/c = 3×10 -21 s. In the RHIC [38] and LHC [39]
experiment for QGP, therefore, it is difficult to measure
the plasma state and, for example, the equation <strong>of</strong> state <strong>of</strong>
Fig. 6 Laser produced electron pair fireball will be a
model experiment <strong>of</strong> quark-gluon plasma (QGP)
dynamics.

QGP is studied by using the particle energy and angle
distribution determined just after the phase transition <strong>of</strong>
liquid phase <strong>of</strong> QCD to free particle state which is called
freeze-out [40]. In case <strong>of</strong> laser driven vacuum breakdown,
the lepton pair plasma is generated and its spatial
scale is about 10 -4 cm (=1µm) and life time is about 10 -15 s
(1fs) as schematically shown in Fig. 6. Since the plasma
size and lifetime are million times bigger and longer than
t h we can e develop the Q diagnostics G to P directly ,
[1] V Yan<strong>of</strong>sky . et al., Opt. Express16, 2109n (2008)
[2] G.A.Mourou, Phys. Today 51,No.1,22(1998).
[3] http://www.extreme-light-infrastructure.eu/
[4] Czech: http://www.eli-beams.eu/
[5] Hungary:
http://www.eli-beams.eu/news-from-hungary/
[6] Romania: http://eli.ifa-mg.ro/
[7] ELI: http://www.eli-beams.eu/evropsky-projekt/
eli-v-kostce/
[8] http://www.clf.rl.ac.uk/
[9] http://apri.gist.ac.kr/eng/index.php
[10] http://wwwapr.kansai.jaea.go.jp/outline.html
[ 1 1 ] http://www.ile.osaka-u.ac.jp/
[12] https://jlf.llnl.gov/html/facilities/titan/titan.html
[13] http://www.lle.rochester.edu/
[14] https://e-reports-ext.llnl.gov/pdf/349575.pdf
[15] T.E. Cowan et al., Phys. Rev. Lett. 84, 903 (2000).
[16] T.E. Cowan et al., Laser Part. Beams 17, 773
(1999).
[17] K. Nakashima and H. Takabe, Phys. Plasmas 9, 1505
(2002).
measure the properties <strong>of</strong> the pair plasma. For example,
we can measure the turbulent spectrum <strong>of</strong> the highly
nonlinear stage <strong>of</strong> the pair plasma with a bright x-ray
source in the range <strong>of</strong> atto-second pulse technology to be
developed in Hungary [5]. Deep understanding <strong>of</strong> such
plasma and verification and validation <strong>of</strong> the computational
modeling <strong>of</strong> the vacuum breakdown, avalanche
effect, photon-lepton interaction, and a variety <strong>of</strong> physics
<strong>of</strong> highly relativistic pair plasma are sure to be very much
beneficial for understanding the QGP physics. We can say
the vacuum breakdown physics is a model experiment <strong>of</strong>
QGP in relatively small scale laser facility.
CONCLUSION
With the progress <strong>of</strong> intense laser technology, the
physics to be solved in laser-matter interaction has
already become relativistic regime for electrons. In
several years, we will come to the regime where laservacuum
interaction becomes important and the 80-years
standing theory <strong>of</strong> the vacuum breakdown will be really
experimentally studied. We concluded that the electron
pair creation can be expected from 10 24 W/cm 2 with use
<strong>of</strong> seed electrons and thanks to the resultant avalanche
effect. When the laser intensity exceeds more than 10 26
W/cm 2 [18] T. Cowan et al, 2002: H. Takabe, in “Hadron and
Nuclear Physics 09”, Edt. A. Hosaka et al., p.
388-403 (World Scientifics, 2010)
[19] Y. Sentoku, private communication.
[20] For example; L. L. Stephan, Computer Physics
Communications 72, Issues 2-3, November 1992,
Pages 144-148
[21]
[22]
[23]
[24]
A. Mizuta, S. Yamada, and H. Takabe,
Astrophysical J. 606 804 (2004).
H. Chen et al., Phys. Plasmas 16, 122702 (2009).
References there in.
H. Chen, H. Takabe et al, NIF proposal (2010) ,
not published, approved by NIF committee.
P. Dirac, The Principle <strong>of</strong> Quantum Mechanics.
[25]
(Oxford Univ. Press, 1930).
A. R. Bell and J. G. Kirk, Phys. Rev. Lett., 101,
200403 (2008)
[26] Landau & Lifshitz, “Theory <strong>of</strong> Classical Field”,
Chap. 9, S.75.
[27] J. G. Kirk, A. R. Bell, and I. Arka, Plasam Phys.
, we can really break down the pure vacuum with
[28]
[29]
[30]
[31]
[32]
Control Fusion 51 085008 (2009).
H. Takabe, “Relativistic Plasma Physics”, J.
Plasma Fusion Res. 78 427-438 (2002), in
Japanese
N. Elkina and H. Ruhl, in this proceeding
N. B. Narozhny et al., JETP Letters
R. Ruffini et al., Physics Reports 487, 1-140
(2010)
A. M. Fedotov, Laser Physics 19, 214 (2009)
help <strong>of</strong> pairs appearing by quantum tunneling effect. W e [33] S. S. Bulanov et al., Phys. Rev. Lett. 104, 220404
also pointed out that this lepton fireball physics will be (2010)
beneficial to modeling quark-gluon plasmas.
[34] G. Miyaji, PhD Thesis, January, 2004
[35] For example; F. D. Seward and P. A. Charles,
REFERENCES
“Exploring the X-ray Universe” 2 nd edition
(Cambridge, 2010)
[36] T. N. Kato, Astrophysical J. 668 974-979, (2007);
P. Chang, A. Spitkovsky, and J. Arons 674 378,
(2008).
[37] M. Asakawa et al., Phys. Rev. Lett. 96, 252301
(2006)
[38] RHIC: http://www.bnl.gov/rhic/
[39] LHC: http://lhc.web.cern.ch/lhc/
[40] S. A. Bass and A. Dumitru, Phys. Rev. C 61,
064909 (2000)

REACHING THE SCHWINGER LIMIT WITH X-RAYS*
Charles K. Rhodes, John Boguta, Alex B. Borisov, Shahab F. Khan, James W. Longworth, John C.
McCorkindale, Sankar Poopalasingam, Ervin Racz # and Ji Zhao
Laboratory for X-ray Microimaging and Bioinformatics, Department <strong>of</strong> Physics, University <strong>of</strong>
Illinois at Chicago, Chicago, IL 60607-7059, USA
Abstract
The derivation <strong>of</strong> an elementary figure <strong>of</strong> merit shows
that attainment <strong>of</strong> an intensity corresponding to the
Schwinger/Heisenberg Limit (~ 4.6 x 10 29 W/cm 2 ) is
significantly facilitated by the use <strong>of</strong> coherent x-ray
sources in the kiloelectronvolt regime. For the Xe(L)
system at ~ 4.5 keV, a minimum pulse energy <strong>of</strong> ~ 1.5 J
and corresponding peak power P0 ~ 300 PW are estimated.
INTRODUCTION
The history <strong>of</strong> high-intensity nonlinear interactions, that
commenced in 1961 with the observation <strong>of</strong> second
harmonic radiation [1] at 347.2 nm in crystalline quartz,
spans a range <strong>of</strong> ~ 10 18 in experimental intensity and
remains a stable, robust province <strong>of</strong> laser-based research
after a half century. The generation <strong>of</strong> focal intensities in
the 10 20 -10 21 W/cm 2 range is presently a routine
achievement. Over a period <strong>of</strong> ~ 25 years, a path <strong>of</strong>
research was cut through this field <strong>of</strong> nonlinear
phenomena that led to the development <strong>of</strong> a multikilovolt
(~ 4.5 keV) x-ray amplifier <strong>of</strong> exceptional peak brightness
[2-5] that is excited by femtosecond KrF* (248 nm)
pulses and whose experimentally based power scaling
limit for a compact laboratory instrument falls in the
multipetawatt realm [6]. The existence <strong>of</strong> advanced highenergy
KrF* technology [7], that has been independently
developed for fusion applications, could be readily
adapted to extend the coherent x-ray power level into the
200 - 500 PW regime.
* Sponsored by Defense Advanced Research Projects
Agency, Microsystems Technology Office (MTO),
Program: Ultrabeam - The Scaling <strong>of</strong> Coherent
Amplification to the Gamma Ray Region, issued by
DARPA/CMO under Contract No. HR0011-10-C-0105.
"The views and conclusions contained in this document
are those <strong>of</strong> the authors and should not be interpreted as
representing the <strong>of</strong>ficial policies, either expressly or
implied, <strong>of</strong> the Defense Advanced Research Projects
Agency or the U.S. Government."
rhodes@uic.edu
# KFKI Research Institute for Particle and Nuclear
Physics, EURATOM Association, P.O. Box 49,1525,
Budapest, Hungary
DISCUSSION
The use <strong>of</strong> coherent x-rays for the attainment <strong>of</strong>
intensities approaching the Schwinger Limit [8] <strong>of</strong> ~ 4.6 x
10 29 W/cm 2 , a condition initially discussed by Sauter [9]
and Heisenberg and Euler [10], is supported by very
powerful scaling relationships. A summary <strong>of</strong> the key
physical parameters is presented in Fig. (1); the
assessment gives the comparison <strong>of</strong> an infrared source
(ħω ≅ 1eV) with a corresponding source delivering (ħω
≅ 4.5 keV) x-rays. The chief outcome is a figure <strong>of</strong> merit
that measures the propensity to generate a high intensity,
and the x-ray illuminator is favored over the infrared
system by a factor that exceeds 10 21 , a result predicated
on a relationship proportional to ω 6 . Furthermore, it is
known that the overall physical situation governing the
attainment <strong>of</strong> the Schwinger condition is generally
assisted by the existence <strong>of</strong> particularly propitious
geometries <strong>of</strong> irradiation [11,12].
An additional advantage <strong>of</strong> an x-ray wavelength is the
very large elevation <strong>of</strong> the intensity characteristic <strong>of</strong> the
relativistic regime that flows from a fundamental scaling
proportional to ω 2 . Basically, the electric field E
corresponding to the relativistic regime is given by the
condition
eE
mc ω
= 1 (1)
which, for Xe(L) radiation with ħω ≅ 4.5 keV, yields an
intensity <strong>of</strong> ~ 10 25 W/cm 2 . This value is regarded both as
(a) sufficiently high to produce observable consequences
from the dynamics <strong>of</strong> the vacuum [13] and (b) free <strong>of</strong> the
cascade limit for intensities up to the Schwinger/
Heisenberg value, in contrast to the infrared case [14].
The ability to produce stable self-channeled propagation
<strong>of</strong> intense pulses <strong>of</strong> radiation in an underdense
plasma is a well established phenomenon at quantum
energies below ~5 eV [15-18]. An immediate consequence
<strong>of</strong> the availability <strong>of</strong> a high power source <strong>of</strong> coherent
x-rays is the possibility <strong>of</strong> extending the generation
<strong>of</strong> these highly confined stable [16] modes <strong>of</strong> deeply
penetrating propagation to x-ray wavelengths in materials
at solid density. If such channels can be pro-duced with
multi-kilovolt x-rays in high-Z solids, the production <strong>of</strong>
power densities on the order <strong>of</strong> ~ 10 30 W/cm 3 are
projected [19,20].

Fig. (1): Presentation <strong>of</strong> the physical parameters
associated with the development <strong>of</strong> a figure <strong>of</strong> merit for
the propensity to produce a high focal intensity. The
combined consideration <strong>of</strong> the quantum energy (ħω), the
radiative rate, and the characteristic area, produces a
strong scaling relationship favoring short wavelengths
that is proportional to ω 6 . Accordingly, in the comparison
<strong>of</strong> the infrared (ħω ≅ eV) to the x-ray range (ħω ≅ 4.5
keV) associated with the Xe(L) system, the figure <strong>of</strong>
merit increases by a factor greater than 10 21 .
PHOTON STAGING CONCEPT
The key concept for the production <strong>of</strong> ultrahigh
intensities with x-rays is “Photon Staging”. Simply stated,
this is the channeling process outlined in Fig. (2) elevated
in both frequency ω and electron density ne. The
governing scalings are illustrated in Fig. (2); basically, it
is the channeling <strong>of</strong> x-rays in solids, a phenomenon that
raises the critical power Pcr to values in the range <strong>of</strong> ~ 0.1-
1 PW. Since the plasma density is ne ~ 4 – 5 x 10 24 cm -3
in high-Z solids, the corresponding channel diameters are
compressed to ~ 100 Ǻ in materials like Fe, Au, and U.
At the wavelength λx ~ 2.9 Å, for which the critical
electron density is ncr = 1.33 × 10 28 cm −3 , all fully ionized
condensed matter is underdense, including uranium. The
key requirement [15,16] for the production <strong>of</strong> a channel in
the underdense regime is the generation <strong>of</strong> a peak power
Po exceeding the critical power Pcr necessary for the
development <strong>of</strong> the confined propagation with the
relativistic/charge-displacement mechanism.
The stability <strong>of</strong> the propagation is assured by the
existence <strong>of</strong> a robust eigenmode [16]. For the case <strong>of</strong> λx
≅ 2.9 Å and fully ionized uranium, Pcr ≅ 49 TW. With a
pulse length τx ~ 50 as, a value that is well within the
projected performance [5,6] <strong>of</strong> the Xe(L) system, the
critical power corresponding to uranium can be achieved
with a pulse energy as small as Ex ~ 3.0 mJ.
Fig. (2): “Photon Staging”, channel formation in solids
with x-rays. Channel formation becomes possible with
pulse energies <strong>of</strong> ~ 10 – 100 mJ for corresponding x-ray
pulse lengths <strong>of</strong> ~ 10 – 100 as. Enormously enhanced
power compression is the leading outcome.
FINDINGS
The power and intensity scaling properties <strong>of</strong> channels
produced in uranium by Xe(M) and Xe(L) radiation,
respectively at ħω ≅ 1 keV and ħω ≅ 4.5 keV, are
summarized in Figs. (3) and (4). The calculations <strong>of</strong> the
propagation <strong>of</strong> Xe(L) radiation in uranium channels show
that a power <strong>of</strong> ~ 300 PW is sufficient to reach the
Schwinger value <strong>of</strong> ~ 4.6 x 10 29 W/cm 2 , the
corresponding power for Xe(M) is ~ 750 PW. The overall
findings for the peak power in Be and U for Xe(M) and
Xe(L) needed to reach the Schwinger intensity are
presented in Fig. (5). The practicality <strong>of</strong> this achievement
is documented in Table I. The minimum total energy
required to reach the Schwinger level for the 248 nm
driver technology is estimated to be 10 - 15 J, a value that
can certainly be attained with the utilization <strong>of</strong> KrF*
technology at its present level <strong>of</strong> development [3-7].
Although mechanisms <strong>of</strong> radiative loss, such as Compton
scattering, Bremsstrahlung, pair production, and the
excitation <strong>of</strong> nuclear reactions, have not been included in
this elementary analysis, these losses can be significantly
mitigated by the use <strong>of</strong> composite low-Z/high-Z targets.

Fig. (3): Channel production in solid U with a Xe(L)
pulse having an energy <strong>of</strong> 1.5 J and an incident power P0
= 300 PW. The peak intensity produced is ~ 5.6 x 10 29
W/cm 2 , a value indicating that the Schwinger Limit is
attainable.
Fig. (4): Channel production in solid U with a Xe(M)
pulse having an energy <strong>of</strong> ~ 7.5 J and an incident power
P0 = 750 PW. The peak intensity produced is ~ 5.5 x 10 29
W/cm 2 , a value indicating that the Schwinger limit is
attainable.
Fig. (5): Power scaling for Xe(M) and Xe(L) in solid Be
and U associated with the attainment <strong>of</strong> an intensity at the
Schwinger limit. Since the area <strong>of</strong> the channel scales
approximately as the inverse <strong>of</strong> the plasma density [16],
the case <strong>of</strong> Be requires ~ 10-fold higher incident power.
The optimal case, illustrated in Fig. (3), corresponds to
Xe(L) in U with an incident peak power P0 ≅ 300 PW.
Table I: Tabulation <strong>of</strong> practical considerations for the
construction <strong>of</strong> Xe(M) and Xe(L) x-ray systems capable
<strong>of</strong> reaching the Schwinger Limit. The range <strong>of</strong> system
parameters shown corresponds to the intensity range <strong>of</strong>
10 26 - 5 x 10 29 W/cm 2 . *The minimum pulse energy
shown is not corrected for the influence <strong>of</strong> radiative losses.
SYSTEM PARAMETER Xe(M) Xe(L)
Wavelength (eV) 900 4500
Pulse Length (as) 10 5
Minimum Pulse Energy
(J)*
~0.01–5 ~0.01–1.5
Intensity Limit (W/cm 2 ) 10 26 – 5 x 10 29
E-M Cascade Limit No
Coherent Beam Addition
Required
No
Pulse Rate (Hz) ~ 1
Optical Beam
Compression/Focusing
Required
(Gratings/Optics)
Prepulse Contrast Control
Needed
Aperture Scale
(Wavelength)
Basic New Physics in
Source and Interactions
CONCLUSIONS
No
No
~ 30 cm
(248 nm)
Attosecond Excitation <strong>of</strong>
Progagation/ Cluster
Excitation/Dicke
Effect/Channel Focusing
We conclude that an intensity <strong>of</strong> the magnitude <strong>of</strong> the
Schwinger Limit can be attained and that the basic
physical scaling greatly favors the use <strong>of</strong> coherent x-rays.
Since the cosmological constant representing the “dark
energy” is now experimentally established [21] to be ΩΛ
≅ 0.73, the ability to probe directly the nature <strong>of</strong> the
complex entity that we traditionally consider as the
“vacuum” can only be expected to yield pr<strong>of</strong>ound insights
into the basic concept <strong>of</strong> space.
ACKNOWLEDMENT
Two authors, A.B. Borisov and C.K. Rhodes acknowledge
informative conversations with S.V. Bulanov.

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Abstract
Non-linear QED effects by strong magnetic field in astrophysics
Kazunori Kohri
Cosmophysics group, Theory Center, IPNS, KEK, Tsukuba 305-0801, Japan
Department <strong>of</strong> Particle and Nuclear Physics, GUAS, Tsukuba, 305-0801, Japan
In this talk we have reviewed possible non-linear QED
effects in supercritical magnetic fields with B ≫ 3 × 10 13
G which sometimes appear in astrophysics. Under that circumstance,
the Landau levels and the propagator <strong>of</strong> electron
are significantly affected, which can induce some nontrivial
modifications, e.g., in equation <strong>of</strong> state <strong>of</strong> electron,
or in refractive indices <strong>of</strong> photon in the magnetized plasma.
In particular, it should be quite attractive that the so-called
“photon splitting” would occur in that situation, which
could have dominated energy-loss process <strong>of</strong> high-energy
photons only in such an extremely strong magnetic field.
INTRODUCTION
Astrophysical objects which have supercritical magnetic
fields B > Bc with Bc ≡ m 2 e/e ≃ 3 × 10 13 have been
reported. Here me denotes electron mass and e means
electron charge. They are known as S<strong>of</strong>t Gamma-ray Repeaters
(SGRs) and Anomalous X-ray pulsars (AXPs) with
having Period <strong>of</strong> O(1) – O(10) sec and Period Derivative
O(10 −12 )–O(10 −10 ). Nowadays they are collectively
called “magnetars” [1].
It has been known that pulsars have strong magnetic
fields with the order <strong>of</strong> B ∼ 10 12 G and periods with the
order <strong>of</strong> O(1) sec. Therefore we may think that magnetars
could belong to an another special class <strong>of</strong> pulsars or
neutron stars.
In terms <strong>of</strong> the non-linear quantum electrodynamics
(QED), such a strong magnetic field is exclusively attractive.
That is because we cannot make use <strong>of</strong> the standard
perturbation theory, i.e., the perturbative approach to calculate
the scattering amplitudes, the self-energies and so
on. Then usual results obtained in a weak-magnetic field
limit can be completely different from those in the strongmagnetic
field limit.
In the language <strong>of</strong> quantum field theory, the order parameter
<strong>of</strong> the perturbation B/Bc becomes no longer sufficiently
small in a supercritical magnetic field. In a weakmagnetic
field limit, we can expand physical quantities as
a power series with respect to the order parameter. For example,
when we consider the full orders <strong>of</strong> the electron
propagator (which might be called “dressed propagator”)
in a weak magnetic field, external magnetic field lines can
couple to the undressed propagator in the higher order calculations
(see Fig. 1). Then the next-order term is obtained
by multiplying a numerical number <strong>of</strong> the order <strong>of</strong> B/Bc
to the undressed term. 1
1 Readers can refer APPENDIX A for a intuitive understanding <strong>of</strong> the
reason why B/Bc becomes the order parameter <strong>of</strong> the power-law expansion
in the weak-field limit.
Figure 1: Electron propagator in a weak-magnetic field
limit. The magnitude <strong>of</strong> the next-order term is obtained
by multiplying the order parameter B/Bc. See a simple
pro<strong>of</strong> shown in APPENDIX A.
If the strength <strong>of</strong> the magnetic field is larger than the critical
value, we cannot make use <strong>of</strong> the known techniques in
the perturbation theory. Then we have to perform a fullorder
<strong>of</strong> the calculation nonperturbatively, which means
that we have to sum up all <strong>of</strong> the diagrams. It is notable
that about this kind <strong>of</strong> nonperturbative calculations including
the full-order <strong>of</strong> the coupling <strong>of</strong> the external magnetic
fields, the leading term comes only from the tree level, not
from the higher-loop levels (See Fig. 1).
Under these circumstances, some significant modifications
from the normal linear QED are possible in
• Energy gaps <strong>of</strong> Landau levels larger than me, which
induce anisotropic electron pressure
• Non-zero vacuum polarization, which induces anomalous
refractive indices
• Possible “photon splitting” as a new class <strong>of</strong> cooling
process for high-energy photons
Next we will discuss the details <strong>of</strong> those modifications and
their effects on phenomena in astrophysics.
LANDAU LEVELS IN STRONG
MAGNETIC FIELD AND ANISOTROPIC
ELECTRON PRESSURE
First <strong>of</strong> all, it is notable that the energy gap in the Landau
levels in the supercritical strong magnetic field can be
larger than electron rest mass. The electron energy in the
magnetized plasma with a uniform magnetic field along the
z-axis B = Bˆz is given by
E = √ m 2 e + p 2 z + eB(2n + 1 − α), (1)
where pz means the z-component <strong>of</strong> the momentum, n denotes
the index <strong>of</strong> a Landau level, and α is the spin index

<strong>of</strong> electron. When the magnetic field is larger than the critical
value, i.e., eB ≫ m 2 e, the x- and y- components <strong>of</strong>
electron momentums are no longer distributed uniformly
and isotropicly because they are quantized in the x-y plane
which is perpendicular to B.
This anisotropic momentum induces the corresponding
anisotropic dynamic pressure ˜ Pi = (n + 1/2 − α/2)eB/E
with i = x and y. By averaging the dynamic pressure in the
grand canonical ensemble, we get the pressure, Pi which
corresponds to a diagonal component <strong>of</strong> the stress-energy
tensor and is still anisotropic. This should be the pressure
which can be used in the fluid dynamics [3].
It was reported that this anisotropic pressure can induce
a nonstandard result in supernovae. In the neutrino-driven
wind which appears after a supernova, especially an additional
entropy <strong>of</strong> the plasma can be produced by the
anisotropic pressure <strong>of</strong> electron [3]. For a successful rprocess
nucleosynthesis which is believed to occur after
supernovae, we need the additional entropy per baryon <strong>of</strong>
the order <strong>of</strong> ∆S/kB ∼ 200. As is shown in Fig. 2, the
production <strong>of</strong> the entropy per baryon with being the order
<strong>of</strong> O(200) can be possible in case <strong>of</strong> the strong magnetic
fields such as B ∼ 10 16 G. See Ref. [3] for the further
details. We expect that future developments in numerical
simulations <strong>of</strong> the supernovae explosion will reveal the details
<strong>of</strong> the current scenario with its dynamics coupled to
the supercritical magnetic field.
Figure 2: Produced entropy per baryon in neutrino-driven
wind in supernovae. We need ∆S/kB ∼ 200 for a successful
r-process nucleosynthesis. Further details are shown in
Ref. [3].
Figure 3: Diagrams <strong>of</strong> the vacuum polarization tensors. We
could have expanded it as a series <strong>of</strong> the order parameter
B/Bc in the perturbation theory in a weak magnetic field
limit. However, if the magnetic field is larger than the critical
one, we have to add all <strong>of</strong> the diagram nonperturbatively.
The double line denotes the fully-dressed propagator
<strong>of</strong> electron.
NON-ZERO VACUUM POLARIZATION
AND ANOMALOUS REFRACTIVE
INDICES
In the supercritical magnetic field, it is expected that the
regularized polarization tensor Π µν can have a non-zero
value. By adding the all <strong>of</strong> the external field lines, we
can calculate the fully one-loop vacuum polarization tensor
(Fig. 3). By using Schwinger’s proper time method
(See Appendix B for a brief review), the sum <strong>of</strong> the external
fields can be replaced by the integral <strong>of</strong> the proper
time. In the full calculation <strong>of</strong> the vacuum polarization tensor,
we should have to perform 2-dim integral by proper
time.
If there were non-zero vacuum polarization tensor, we
expected that the dispersion relation <strong>of</strong> photon can be modified.
The refractive indices are defined by
µ 2 = |k|2
, (2)
ω2 where k denotes the spatial 3-vector <strong>of</strong> the 4-dim photon
momentum k µ with µ = 0, 1, 2, 3, and ω = k 0 = Eγ
means the photon energy.
Then the wave equation <strong>of</strong> photon can be expressed by
[ k 2 g µν k µ k ν + regΠ µν (k) ] Aµ(k) = 0. (3)
In this case, the polarization tensor can be written by 2-dim
integral to be
Π αβ (x, x ′ )
∝ Tr [ γ α G(x, x ′ )γ β G(x ′ , x) ]
∝
∫ ∞
0
ds
s
∫ ∞
0
(4)
ds ′
s ′ [· · ·] , (5)
with the two spatial points, x, and x ′ .
To have nontrivial solutions for Aµ(k), the determinant
<strong>of</strong> the operator [· · ·] in the left-hand side should vanishes [5,
6]. Then we obtain the solutions <strong>of</strong> µ’s.
In Fig. 4, we plotted refractive indices µ1 which corresponds
to the eigen vector <strong>of</strong> the polarization (0, 1, 0), and
µ2 which corresponds to the other mode. The angle between
B and k is taken to be cos 2 θ = 1/2.

Figure 4: Refractive indices <strong>of</strong> two polarization modes as
a function <strong>of</strong> B/Bc. µ1 corresponds to the eigen vector <strong>of</strong>
the polarization (0, 1, 0), and µ2 corresponds to the other
mode. This is shown in Ref. [4].
PHOTON SPLITTING
It has been said that the most impressive phenomenon in
the supercritical magnetic field should be photon splitting.
In the weak magnetic field limit, a photon can not split into
two photons. On the other hand, in the supercritical magnetic
field, a photon can scatter <strong>of</strong>f the external magnetic
field and produce two photons by one loop effects <strong>of</strong> the
dressed internal electron.
More precisely, γ → γ + γ might be allowed only kinematically
through an one-loop diagram with the normal undressed
propagator. However, Furry’s theorem [7] does
not permit odd number <strong>of</strong> vertices. Thus, the actual diagram
should start from the four-point box diagram which
is shown in Fig. 5. Therefore the photon splitting is purely
non-linear QED effect through the dressed electron propagator.
In this study, thus it is essential to calculate the
full-orders <strong>of</strong> the dressed electron propagator.
Figure 5: Diagrams <strong>of</strong> triangle tensor for photon splitting.
Note that Furry’s theorem does not permit odd number <strong>of</strong>
vertices.
We have known that it is possible to write down the fullexpression
<strong>of</strong> the decay rate or the triangle tensor <strong>of</strong> photo
splitting by using Schwinger’s proper time method as well
as the case <strong>of</strong> the polarization tensor:
Π αβδ (x, x ′ , x ′′ )
∝ Tr [ γ α G(x, x ′ )γ β G(x ′ , x ′′ )γ δ G(x ′′ , x) ]
∝
∫ ∞
0
ds
s
∫ ∞
0
ds ′
s ′
∫ ∞
0
(6)
ds ′′
s ′′ [· · ·] , (7)
with the three spatial points, x, x ′ , x ′′ . In this case, the dimension
<strong>of</strong> the integral is inevitably three, which makes the
calculation to obtain a concrete numerical number much
more difficult. So far no groups have succeeded to get concrete
values <strong>of</strong> the rate for arbitrary physical variables, such
as for B or ω by performing the integrations in the 3-dim
proper time [8, 9].
On the other hand, only solutions with both low-energy
ω ≪ me and weak-field limits B ≪ Bc have been
known [8]. According to those solutions, for example
the inverse <strong>of</strong> the photon attenuation-length due to photon
splitting can be approximately proportional to ∝ ω 5 B 6 .
Under this circumstance, Baring and Harding
(2001) [10] extrapolated those approximate solutions
to larger values <strong>of</strong> B and ω and applied them to energyloss
processes <strong>of</strong> electron and photon which exist around
magnetars. It is astrophysically-impressive that since
the magnetic field line and the magnetosphere have
nontrivial structures around pulsars, the rate <strong>of</strong> the
e + e − -pair creation by photon scattering <strong>of</strong>f the magnetic
field (γ → e + + e − ) could be smaller than the photon
splitting rate (γ → γ + γ). That is because the paircreation
rate depends on the direction <strong>of</strong> the beam photon
Γ ∝ B sin θkB/Bc with θkB being an angle between the
magnetic-field line and the photon momentum k although
the photon splitting rate does not depend on the direction
under the uniform magnetic field. It might be intuitive,
for example, the former rate can become large in head-on
collision, or on the other hand it is negligible with a
smaller energy than the threshold <strong>of</strong> the pair creation
Eγ < me/ sin θkB depending on a value <strong>of</strong> θkB. It should
be the astonishing point that the authors <strong>of</strong> [10] proposed
that the reason why the synchrotron radio emission has
not been observed from magnetars is that high-energy
photons emitted by curvature radiation can loose their
energy mainly through photon splitting, not through the
e + e − -pair creation in the supercritical magnetic field. To
check this quite an attractive scenario, we would have to
succeed the full calculation <strong>of</strong> the photon-splitting rate
to obtain the concrete numerical numbers as functions <strong>of</strong>
magnetic field and photon energies.
CONCLUSION
In this talk, we have reviewed various astrophysical phenomena
induced by the non-linear QED effect such as large
Landau levels, non-zero vacuum polarization tensor <strong>of</strong> photons
and possible photon splitting. About the rate <strong>of</strong> photon

splitting, no one has succeeded to fully calculate it and obtain
the concrete numerical values as functions <strong>of</strong> arbitrary
values <strong>of</strong> B and Eγ by performing the 3-dim integration.
Recently hard power-low emissions with the photon index
Γ ∼ 1 from magnetars has been reported by X-ray
observation by Suzaku [11, 12]. So far we have not known
the origin <strong>of</strong> such a hard component from normal pulsars.
They might relate with new emission mechanisms induced
by the non-linear QED effects in the supercritical magnetic
field (See also Refs. [13, 14]).
ACKNOWLEDGEMENT
The author would like to thank Shoichi Yamada for longterm
collaborations and continuous discussions. He also
thanks Kazuo Makishima and Teruaki Enoto for fruitful
discussions.
APPENDIX
Appendix A: Analytical understanding <strong>of</strong> the order
<strong>of</strong> radiative correction
In this section, we show that the first-order radiative correction
to the undressed electron propagator in an external
magnetic field is the undressed term multiplied by a factor
<strong>of</strong> the order <strong>of</strong> O(eB/m 2 e). This simply means that the order
parameter in terms <strong>of</strong> the perturbation due to the weak
external-magnetic field could be eB/m 2 e.
In Fig. 6 we showed the diagram <strong>of</strong> the coupling between
the free electron field (solid line) and an external static
magnetic field Aµ (wavy line) with a coupling g = −|e|.
The space-time points, X, Y, Z are being considered in the
X3 ∼ Y3 ∼ Z3 ∼ 0 plane.
Figure 6: diagram <strong>of</strong> the coupling between the free electron
field (solid line) and an external static magnetic field Aµ
(wavy line) with a coupling g = −|e|. The space-time
points, X, Y, Z are being considered in the X3 ∼ Y3 ∼
Z3 ∼ 0 plane.
We consider the first-order propagator <strong>of</strong> an electron
which propagates from X to Y in real space.
∫
˜G(X, Y ) = dZ igγ µ ∫
i
Aµ(Z) dp e
̸ p − me
−ip(X−Z)
∫
i −iq(Z−Y )
× dq e
̸ q − me
= +igZ3Bγ 2
∫ ∫
1
dZ dp e
̸ p − me
−ip(X−Z)
∫
1
× dq
̸ q − me
e −iq(Z−Y ) , (8)
where we took Aµ(Z) = (0, 0, −Z3B, 0) with a constant
magnetic field B, and we used γ µ Aµ(Z) = γ 2 (−Z3B) in
the last line.
By integration by parts, the last integral is estimated to
be ∫
1 −iq(Z−Y )
dq e
̸ q − me
∫
1 ∂
= dq
̸ q − me ∂q3
∫
1
= [· · ·] +
dq
i(Z3 − Y3)
−iq(Z−Y ) 1
e
−i(Z3 − Y3)
γ 3
(̸ q − me) 2 e−iq(Z−Y ) ,(9)
where we can omit the first term because <strong>of</strong> a natural
boundary condition. Then we get
˜G(X, Y ) =
Z3
gBγ
Z3 − Y3
2 γ 3
∫ ∫
dZ dp e−ipX
̸
×
∫
p − me
∼
∫
dq e−iZ(q−p)
e+iqY
(̸ q − me) 2
i
dp
̸ p − me
e −ip(X−Y ) eBγ2 γ 3
,
(̸ p − me) 2
(10)
where we have assumed that Z3/(Z3 − Y3) ∼ O(1), and
made use <strong>of</strong> a delta function, δ(q − p). Because the momentum
̸ p <strong>of</strong> the internal line is <strong>of</strong>f-shell, its 4-dim norm
is smaller than the rest mass, √ p2 ≤ m. By using this
approximation, we obtain the final expression,
∫
˜G(X,
i
Y ) ∼ dp
̸ p − me
e −ip(X−Y ) × O( eB
m2 ). (11)
e
From this integrand, we find that the 1st-order correction
is given by the 0-th order term by multiplying the factor
O(eB/m 2 e) in a perturbation theory. This means that the
order parameter for the perturbation should be (eB/m 2 e).
Therefore a series <strong>of</strong> the above calculation could have given
us a brief pro<strong>of</strong> <strong>of</strong> what the order parameter in the perturbation
theory is.
Appendix B: Schwinger’s proper-time method
Here we briefly introduce an outline <strong>of</strong> the Schwinger’s
proper-time method. In position representation, we can
write the Dirac’s delta function to be
δ 4 (x − x ′ ) = ⟨x |1| x ′ ⟩ = ⟨x|x ′ ⟩, (12)
with the propagator in position representation,
G(x, x ′ ) = ⟨ x ˆ G x ′⟩ . (13)
Originally the operator <strong>of</strong> the green function can be expressed
by definition as
[γ µ Πµ − m] ˆ G = 1, (14)

with the canonical momentum Πµ ≡ i∂µ + eAµ. Here the
operator <strong>of</strong> the green function is rewritten to be
ˆG =
=
=
1
γ µ Πµ − m
γ µ Πµ + m
(γ µ Πµ) 2 − m2 ∫ ∞
ds
0 i e−im2 s −iHs µ
e (γ Πµ + m) ,
(15)
with H ≡ − (γ µ Πµ) 2 . In the transformation to the
last line, the integral representation <strong>of</strong> the operator was
adopted.
Eq. (15) means that the operator can time-evolve through
the unitary matrix U(s) = e −iHs as if that would be governed
by an effective time s and an effective Hamiltonian
H in quantum mechanics. Then it is notable that their
mass dimensions should be two and minus two for H and
s, respectively. Therefor H and s do not denote the usual
Hamiltonian and time. In this situation, this effective time s
is called the “Schwinger’s proper time” [2]. After we have
formulated quantum mechanics in this system, we will be
able to use this unitary matrix to discuss the time evolution
<strong>of</strong> operators.
REFERENCES
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due to the QED effect in strong magnetic fields and
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volume.

Abstract
WIDE-BAND X-RAY OBSERVATIONS OF MAGNETARS ∗
K. Makishima, Dept. Physics † and RESCEU, University <strong>of</strong> Tokyo
and the Institute <strong>of</strong> Physical and Chemical Research (RIKEN)
After a brief review <strong>of</strong> physics <strong>of</strong> neutron stars, this article
focuses on so called magnetars, a special subset <strong>of</strong> neutron
stars, which are though to have magnetic fields raching
10 14−15 G. Recent observational results <strong>of</strong> these objects,
made with the Japanese X-ray satellite Suzaku, suggest
that their X-ray to gamma-ray emission results from
photon splitting process in the strong magnetic field.
INTRODUCTION
As a brief introduction to neutron stars, a paragraph may
be spent on stellar evolution. As illustrated in Fig. 1, stars,
first born as protostars, take different evolutionary paths
depending on their initial mass. The lightest ones end up
with planets, where Coulombic repulsion among ions supports
the gravity. Somewhat more massive ones become
brown dwarfs, which employ electron degenerate pressure
instead. Objects with > 0.08 times the solar mass become
normal stars, where classical gas pressure generated by nuclear
fusion balances the gravity. After 10 6−10 years depending
on their initial mass, these stars come to their evolutionary
endpoints. Lightest normal stars leave at their
centers white dwarfs, which are similar to brown dwarfs
except their lack <strong>of</strong> hydrogen. More massive ones experience
gravitational-collapse supernova explosions, and their
cores collapse into neutron stars which are supported by degenerate
neutron pressure. If the initial mass is even higher,
the final gravity is too strong for a neutron star to support,
and the object becomes a black hole.
This article uses the MKSA units, except that magnetic
field B is expressed in Gauss (G); 1 G = 10 −4 T.
DEGENERATE STELLAR OBJECTS
Stellar Equilibrium
The balance between the gravity and pressure p <strong>of</strong> a
spherical star can be expressed at each radius r as
dp/dr = GρMr(r)/r , (1)
where G, ρ and Mr(r) = ∫ r
0 4πr2 ρ(r)dr are the gravitational
constant, mass density, and “mass coordinate”, respectiveky.
Instead <strong>of</strong> exactly solving eq.(1), it is useful to
multiply its both sides by 4πr 3 and integrate from r = 0 to
the stellar surface r = R, to obtain a Viriar relation as
3 < p > V = −Φ ≡
∫ M
0
GMr
r dMr
2 5GM
∼ . (2)
3R
∗ Work supported by T. Enoto, Y.E. Nakagawa, and M. Nakajima
† maxima@phys.s.u-tokyo.ac.jp
Figure 1: A very schematic illustration <strong>of</strong> stellar evolution.
Abscissa at the top indicates initial stellar mass, while that
at the bottom final mass. Time goes from top to bottom.
Here, V is the stellar volume, means volume average,
M ≡ Mr(R) is the stellar mass, Φ is self-gravitational
energy, and the last approximation assumes a constant ρ.
Degenerate Stars
Consider a star which is supported by degenerate pressure
pd <strong>of</strong> some Fermions, <strong>of</strong> which the mass is mf and
density is nf. We may substite p in eq.(2) with the nonrelativistic
degenerate pressure <strong>of</strong> an ideal Fermi gas, i.e.,
pd = A¯hn 5/3
f /mf, where ¯h is the Dirac constant and A is a
numerical constant <strong>of</strong> order unity. Performing elementary
calculations, and using another numerical constant A ′ , we
obtain a mass-radius relation <strong>of</strong> this degenerate star as
R/λf = A ′ α −1
g η −5/3 (M/mn) −1/3 . (3)
Here, mn is the nucleon mass, η is the nucleon-to-Fermion
number ratio, λf ≡ 2π¯h/mfc is the Fermion’s Compton
wavelength, and αg ≡ Gm 2 n/¯hc = 5.9 × 10 −39
is a dimensionless constant representing gravitational interaction.
Thus, the stellar radius is proportional to the
Fermion’s Compton wavelength. If we use the extreme
relativistic expression for pd, an upper limit mass (Chandrasekhar
mass) is derived.
Brown and White Dwarfs
If the Fermions are electrons with λe = 2.4 × 10 −12
m, the star becomes a brown dwarf (η = 1.2) or a white
dwarf (η = 2.0). Normalizing M to the solar mass M⊙ ≡
2.0 × 10 30 kg, and faithfully calculating A ′ , we obtain
R = 1.0 × 10 7 (M/M⊙) −1/3 (2/η) −5/3 m (4)
which implies an object <strong>of</strong> the Earth’s size.

Basic Concepts
NEUTRON STARS
When the stellar interior is mostly “neutronized” and the
neutrons’ degenerate pressure supports the gravity, the object
becomes a neutron star (NS). From eq.(3), we find
that a NS with M ∼ M⊙ is by 3 orders <strong>of</strong> magnitude
(∼ λe/λn = mn/me) smaller in radius than a white dwarf
<strong>of</strong> the same mass; approximately ∼ 10 km. However, more
accurate estimates <strong>of</strong> the mass-radius relation <strong>of</strong> NSs are
a subject <strong>of</strong> yet unsolved studies, because it is affected by
general relativity and the nuclear equation <strong>of</strong> state.
Classification
NSs may be classified in several aspects. One is their
surface magnetic fields (MFs) B, which range from < 10 9
G to ∼ 10 15 . Another is whether the object is isolated or in
a binary system with another star. The classification from
these two dimensions is presented in Table 1.
Shown with [ ] in Table 1 is yet another classification
axis, the source energy <strong>of</strong> their radiation. Isolated NSs with
long rotation periods (e.g.,> 10 s) can utilize only their
internal energies [i], to emit blackbody radiation. Fastrotating
magnetized ones can spend their huge rotational
energies [r], while those in binaries can alternatively utilize
gravitational energies [g] <strong>of</strong> accreting materials. A
subset <strong>of</strong> these accreting NSs can also use nuclear energies
[n], when the accreted material intermittently make
nuclear fusion on the NS surfaces (a phenomenon called
X-ray bursts). Finally, those with the strongest B, magnetars,
are thought to be magnetically powered [m].
Table 1: Classification <strong>of</strong> neutron stars. ∗,#
B (G) isolated binary
< 10 10 isolated NS [i](R) X-ray bursters [g,n](R)
10 11−13 radio pulsars [r](B) X-ray pulsars [g](M, B)
10 14−15 magnetars [m](B) —
* : [ ] indicate radiation energy sources: [i]=internal,
[r]=rotational, [n]=nuclear, [g]=gravitational, [m]=magnetic
# : M, R and B indicate that the mass, radius, and surface
magnetic fields are measurable, respectively.
Mass and Radius
NSs have three important parameters; mass M, radius
R, and MF B. In Table 1, we indiacate, with ( ), which <strong>of</strong>
them can be measured in each class <strong>of</strong> objects.
Like in other astrophysical contexts, the binary environment
provides opportunities to measure the NS mass. Consider
a NS in a binary system with five unknowns; the
NS mass Mns, the companion mass Mc, orbital separation
a, orbital angular frequency Ω, and orbital inclination
i. We can observe Ω, as well as the orbital velocity
Kns = aΩ sin i/(1 + q) (with q ≡ Mns/Mc) <strong>of</strong> the
NS via pulse arrival delays, and that <strong>of</strong> the companion star
Kc = aqΩ sin i/(1 + q) via optical Doppler spectroscopy.
Furthermore, we can utilize the Kepler’s law,
G(Mns + Mc) = a 3 Ω 2 . (5)
Then, if i is somehow estimated, the five variables can be
all uniquely determined. The results show a sharp concentration
at Mns = 1.4M⊙ [24].
The radius <strong>of</strong> an NS can be measured if it emits blackbody
radiation uniformly from its surface, like during the
X-ray bursts. Then, by observing the bolometric (=frequency
integrated) flux f and the surface temperature T ,
we can estimate R from the Stefan-Boltzmann law,
f = 4πR 2 σT 4 /4πD 2 , (6)
where D is the source distance and σ is the Stefan-
Boltzmann constant. Measurements give R ∼ 10 km, but
are not yet accurate enough to constrain nuclear equation
<strong>of</strong> state [19]. This is mainly because few subsets in Table 1
provide simultaneous measurements <strong>of</strong> M and R.
Radio Pulsar Number
250
Radio Pulsars
8
200 Accr. Pulsars
Magnetar
Magnetars
6&
150
4
100
Accreting
50
2
Pulsars
0
8
9 10 11 12 13 14 15
Log Magnetic Field (Gauss)
Figure 2: Distributions <strong>of</strong> surface magnetic fields <strong>of</strong> neutron
stars in our Galaxy. Blue represents rotation-powered
pulsars, and red magnetars, with their field strengths both
estimated using eq.(7). Green (with the number on the right
hand side) shows accretion-powered pulsars <strong>of</strong> which the
field intensity is measured with eq.(11).
Surface Magnetic Fields
MF strengths <strong>of</strong> rotation-powered NSs (radio pulsars in
Table 1) are estimated assuming that they lose their rotational
energies by emitting magnetic dipole radiation as
they rotates [9]. Then, the dipolar MF strength is expressed,
in terms <strong>of</strong> the period P and its derivative ˙ P , as
B = 1.0 × 10 12
√
(P/0.1s)( ˙ P /1 × 10−14ss−1 ) G , (7)
assuming a canonical moment <strong>of</strong> inertia. Similar results are
obtained if we assume that the rotational energies are spent
in particle acceleration [7].
The MF distributions <strong>of</strong> rotation-powered NSs (i.e., objects
with [r]), thus estimated, are given in Fig. 2 in blue
and red. The major peak at B ∼ 10 12 G encompasses ordinary
radio pulsars, a minor peak at 10 8−9 G millisecond
pulsars, while red are magnetars. Green data points are explained
in the next section.
0

Radiation from Rotation-Powered Pulsars
The Crab pulsar (borne in AD1054), a prototypical
rotation-powered NS, has B ∼ 3 × 10 12 G, and rotates
with an angular frequency <strong>of</strong> ω = 200 Hz. As a result, the
induced Lorentz electric field amounts to
E ∼ RωB ∼ 10 14 Vm −1 , (8)
and the electric potential to ER ∼ 6 × 10 18 V. This ultrahigh
electric field will accelerate electrons (plus probably
positrons), which then emit non-thermal radiation via synchrotron
radiation, inverse Compton scattering, and curvature
radiation [18]. Emergent spectra inevitably span many
orders <strong>of</strong> magnitude in frequency. To observe these objects,
without hampered by background stellar signals, we
may use either radio frequencies, or gamma-ray range as
demonstrated by the Fermi Gamma-ray Space Telescope
launched in 2008 June [1].
ACCRETING X-RAY PULSARS
X-ray Emission
Contrary to rotation-powered NSs, accretion-powered
NSs (those with [g] in Table 1) radiate predominantly in
the X-ray frequency, for the following reason. Consider an
accreting NS with weak MF, radiating spherically at a luminosity
L. Then, the photon momentum flux L/4πr 2 c exerts
an outward force Fout ≡ LσTne/4πr 2 c per unit volume <strong>of</strong>
the accreting matter, with σT being the Thomson cross section.
The maximum luminosity LE, called Eddington limit,
is realized when Fout balances the inward gravitational pull
[right hand side <strong>of</strong> eq.(1)] exerted on the same volume. Assuming
η = 1.2, we then have
LE = 4πGηmpcM/σT = 2.2 × 10 31 (M/1.4M⊙) W.
(9)
Equating this with the blackbody luminosity 4πR 2 σT 4 , the
blackbody temperature is obtained as
T = 2.3 × 10 7 (L/LE) K. (10)
This falls right on the X-ray energy band.
X-ray photons have high penetrating power, as evidenced
by medical diagnostics. However, they cannot penetrate
the atmosphere, which is ∼ 50 times thicker (in Oxygen
column density) than human body. As a result, a number
<strong>of</strong> X-ray astrophysics satellites have been launched. In
Japan, Hakucho (launched in 1979), Tenma (1983), Ginga
(1987), ASCA (1993)[21], and Suzaku (2005)[13] have
been contributing. We are now preparing for the next mission,
ASTRO-H, to be launched in 2014.
Cyclotron Resonances
While eq.(10) applies to spherical accretion onto weak-
FM NSs, the condition somewhat differs when the NS has
a strong MF (X-ray pulsars in Table 1), because the accretion
flow is channeled onto the two magnetic poles, and the
Thomson cross section is magnetically modified [10]. As
a result, the X-ray emission from X-ray pulsars attains a
higher temperature, ∼ 10 8 K, or ∼ 10 keV.
An outstanding property <strong>of</strong> X-ray pulsars is a clear spectral
feature due to electron cyclotron resonance, or Cyclotron
Resonance Scattering Feature (CRSF), which <strong>of</strong>ten
appears in their spectra at an energy <strong>of</strong>
Ea = ¯heB/me = 11.6(B/10 12 G) keV. (11)
So far, CRSFs have been observed, sometimes in multiple
harmonics, all in absorption, from ∼ 15 objects [12], about
half the known X-ray pulsars. Figure 3 gives one <strong>of</strong> the
most prominent examples <strong>of</strong> CRSFs [17].
Equation (11) have been used to directly measure the
surface MF strengths <strong>of</strong> X-ray pulsars. Obviously, this
method gives a much higher accuracy than eq.(7) in measuring
B. The green data points in Fig. 2 refer to these measurements
[12]. The results are concentrated over a rather
narrow range, B = (1−4)×10 12 G, although the region <strong>of</strong>
B > 5 × 10 12 G is yet to be explored with high-sensitivity
hard X-ray instruments, including the Hard X-ray Detector
[20] onboard Suzaku. This result argues against a view,
which was popular in the 1990’s, that the MF <strong>of</strong> strongly
magnetized NSs decay wit time [23].
In Fig. 2, the green data points appear to be rathe distinct
from the more weakly magnetized objects. Then, the
MF strengths may exhibit bimodal behavior. Based on this,
we further speculate that the NS magnetism is a manifestation
<strong>of</strong> ferromagnetism in nuclear matter [12], rather than
the more popular idea <strong>of</strong> permanent current. Those with
B = 10 8−9 G can be interpreted as entirely paramagnetic
objects. The stronger-field objects with B ∼ 10 12 G can be
explained if only ∼ 10 −4 <strong>of</strong> the total NS volume becomes
ferromagnetic. Furthermore, we expect B ∼ 10 16 G if the
entire volume <strong>of</strong> a NS becomes ferromagnetic. Magnetars,
to be described below, may be such objects.
Figure 3: Examples <strong>of</strong> CRSFs, measured from the transient
pulsar X0331+53. The fundamental and 2nd harmonic features
are indicated by arrows. Dashed lines indicate the
underlying thermal spectrum.

General behavior
MAGNETARS
About 15 NSs shown in Fig. 2 in red are called magnetars,
and possess the following common characteristics:
1. Rotation periods <strong>of</strong> P = 2 − 11 sec, and high spindown
rates as ˙
P 10 −11 s s −1 , yielding via via eq.(7)
B = 10 14−15 G. (12)
2. Young characteristic ages, τ ≡ P/2 ˙
P = 10 2−4 yr,
which is also supported by their occasional association
with supernova remnants.
3. Sporadically recurring active periods, when a large
number <strong>of</strong> short (< 1 s) bursts are emitted as in Fig. 4.
4. X-ray emission with a luminosity L ∼ 10 27 W, which
much exceeds the spin-down energy release.
5. No evidence <strong>of</strong> binary companions.
6. No radio emission unlike typical rotation driven NSs.
Figure 4: An example <strong>of</strong> magnetar short bursts. These
were observed with Suzaku, for one day, from the magnetar
1E1547-54 during its 2009 January activity.
From 4 and 5, magnetars can neither be rotation powered
nor accretion powered. Furthermore, eq.(12) implies
that the magnetic energy <strong>of</strong> these objects exceed their rotational
energies. Thus, magnetars are considered to be magnetically
powered objects (marked with [m] in Table 1),
producing persistent and burst X-ray radiation by somehow
releasing their magnetic energy [22]. This interpretation is
supported by Fig. 5, where B <strong>of</strong> magnetars measured with
eq.(7) decreases with τ. Since an NS with B = 10 15 G has
a magnetic energy <strong>of</strong> ∼ 3×10 40 J, its decay over ∼ 100 kyr
(Fig. 5) would afford L ∼ 1×10 28 W, which is sufficient to
explain item 4. Magnetars may have even stronger toroidal
MF, which is mostly confined with the stellar interior.
Although we believe magnetars to be a special subset
<strong>of</strong> NSs, their masses and radii remain totally unknown.
Therefore, we cannot tell at present whether magnetars are
different from other NSs in any aspect other than the MF.
Likewise, nothing is known, either, what kind <strong>of</strong> supernova
explosions lead to the formation <strong>of</strong> magnetars.
One fascinating aspect <strong>of</strong> magnetars is that their inferred
MF strengths [eq.(12)] exceeds the critical value,
Bcr = (mec) 2 /¯he = 4.4 × 10 13 G (13)
at which the CRSF energy [eq.(11)], or equivalently the
Landau level separation, reaches mec 2 . Therefore, we expect
various “strong field” effects to take place.
Figure 5: The estimated surface MF strengths <strong>of</strong> magnetars
(the same as in Fig. 2), shown as a function <strong>of</strong> τ.
Emission from Magnetars
In the 20th Century, magnetars were known to emit (except
short bursts) pulsating persistent s<strong>of</strong>t X-ray emission
<strong>of</strong> which the spectrum is approximated by a ∼ 0.5 keV
blackbody. Then, the European Gamma-ray observatory
INTEGRAL discovered [11, 8], from several magnetars,
a strongly pulsing hard X-ray component, extending to
∼ 100 keV with a very flat photon index <strong>of</strong> Γ ∼ 1 (photon
number flux scaling with energy E as ∝ E −Γ ). Such
a hard-slope emission has rarely been observed from other
types <strong>of</strong> cosmic X-ray sources, and is difficult to explain in
terms <strong>of</strong> know radiation processes (e.g., synchrotron emission).
Therefore, this component is considered to provide
an important clue to the nature <strong>of</strong> magnetars.
We have conducted extensive X-ray studies <strong>of</strong> magnetars,
using Suzaku [2, 3, 5, 4, 6, 16] which has a superior
wide-band capability realized by the HXD [20], and the
gamma-ray burst satellite HETE-2 [14, 15]. Our results,
covering both persistent and burst emissions, are summarized
as follows.
1. As exemplified by Fig. 6, about 8 magnetars we observed
all exhibit persistent spectra that are composed
<strong>of</strong> a s<strong>of</strong>t and a hard component [2, 16, 5, 4, 6]. This
reinforces the INTEGRAL discovery, and improves it.
2. The s<strong>of</strong>t component is approximated by two blackbodies,
with temperatures TL and TH which scales as
TH ≈ 3.5TL. The emission areas are roughly consistent
with the size <strong>of</strong> an NS.
3. The above two items hold for both the persistent emission,
and short bursts detected from a few objects [15].
4. As shown in Fig. 7 (top), the spectral hardness ratio
(1–60 keV flux ratios between the hard and s<strong>of</strong>t components)
has been discovered to anti-correlates with τ
[6]. This is a discovery <strong>of</strong> magnetar evolution.
5. The hard component has an extremely flat slope, Γ =
1.7−0.4, which becomes even harder (flatter) towards
more aged objects. This is visualized in Fig. 6, and
summarized in Fig. 7 (bottom).

2 -2 -1
keV (ph cm s keV -1 )
0.1
0.01
0.001
1
1E1547-54 (τ=1.4 kyr)
4U 0142+61 (τ=70 kyr)
10
Energy (keV)
100
Figure 6: Suzaku νFν spectra <strong>of</strong> two magnetars. Green is
1E1547−54 in activity [4], while red is 4U 0142+61 [6].
How Magnetars Work?
Based on the Suzaku and HETE-2 results, let us present
a speculative interpretation <strong>of</strong> magnetars. First, we believe
that they really have MF strengths exceeding Bcr
<strong>of</strong> eq.(13), even though the estimates <strong>of</strong> eq.(12) could be
very crude. This belief is based on the fact that their twocomponent
X-ray spectra and burst activity (Fig. 4) are really
unique among various celestial X-ray sources. Thus,
magnetars must be in a unique physical condition, and the
proposed strong-MF scenario [22] is most appropriate.
Next, we found many similarities between persistent and
burst emissions. Therefore, the persistent emission is likely
to be formed by a large number <strong>of</strong> micro-bursts [15], just
as the solar corona may be heated by micro-flares. The
bursts themselves could be a result <strong>of</strong> sporadic magnetic
reconnection [22], either in the magnetosphere or stellar
interior. Then, Fig. 7 (top) can be understood as a gradual
decline <strong>of</strong> magnetic activity, in agreement with Fig. 5.
Third, the s<strong>of</strong>t component can be interpreted as thermal
radiation from the NS surface, heated by the magnetic
energy release. The two blackbodies may represent two
photon polarizations (O-mode and X-mode) with respect
to the strong MF, which have different electron-scattering
cross sections [10, 3]. We expect the emission to be hence
strongly polarized; this will be tested by the wold’s first
X-ray polarimetric mission, GEMS, developed under US-
Japan collaboration and scheduled for launch in 2014.
Finally, the enigmatic hard component may be a result
<strong>of</strong> photon splitting in the super-critical MF. In the magnetosphere,
the strong Lorentz field [eq.(8)] will accelerate
electrons (and positrons), which will emit gamma-ray photons.
These photons cannot escape out, since they would
interact with the MF and split into two lower-energy photons
[10, 6]. A cascade <strong>of</strong> this process will produce a hard
X-ray continuum extending down to ∼ 10 keV with a very
flat Γ. In addition, this cascade will stop at relatively high
energies in older magnetars, because <strong>of</strong> their weaker MF
(Fig.5). This can explain Fig. 7 (bottom). Detailed investigation
<strong>of</strong> magnetars in the 100–600 keV region is an important
future task <strong>of</strong> our ASTRO-H.
Figure 7: (Top) Ratios <strong>of</strong> the 1–60 keV luminosities between
the hard and s<strong>of</strong>t components <strong>of</strong> magnetars, mesured
with Suzaku, shown as a function <strong>of</strong> their characteristic age
τ [6]. Red and blue indicate different subsets. (Bottom)
The photon index Γ <strong>of</strong> the hard component shown in the
same way [6]. Green points are take from literature.
REFERENCES
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[24] R. Valentim et al., astro-ph 1101.4872v1 (2011)

QCD ORIGIN OF STRONG MAGNETIC FIELD IN COMPACT STARS ∗
Abstract
T. Tatsumi † , Department <strong>of</strong> Physics, Kyoto University, Kyoto 606-8502, Japan
Some magnetic properties <strong>of</strong> quark matter and a microscopic
origin <strong>of</strong> the strong magnetic field in compact stars
are discussed; ferromagnetic order is discussed with the
Fermi liquid theory and possible appearance <strong>of</strong> spin density
wave is suggested within the NJL model. Implications
<strong>of</strong> these magnetic properties are briefly discussed for compact
stars.
INTRODUCTION AND MOTIVATION
Nowadays there have been many works about the QCD
phase diagram. Here, we are concentrated in magnetic
properties <strong>of</strong> quark matter and their implications on compact
star phenomena. Let’s begin with a simple question:
what is and where can we expect magnetism in QCD. At
low densities we may expect the appearance <strong>of</strong> pion condensation
in hadronic matter, where the classical pion field
develops , followed by the specific spin-isospin order <strong>of</strong>
nucleons [1]. In quark matter we shall see a non-uniform
phase (called dual chiral density wave (DCDW) phase),
accompanying the restoration <strong>of</strong> chiral symmetry, where
the pseudoscalar condensate ⟨¯qiγ5τ3q⟩ ̸= 0 as well as the
scalar condensate spatially oscillates [2]. Accordingly the
magnetization <strong>of</strong> quark matter also oscillates like spin density
wave (SDW) in condensed matter physics. Furthermore
we may expect a ferromagnetic phase at some density
region [3].
On the other hand such magnetic properties should have
some implications on the compact star phenomena. In particular
it has been well known that there is a strong magnetic
field in compact stars. The origin <strong>of</strong> such strong
field is not clear even now, and it has been a long-standing
problem since the first discovery <strong>of</strong> pulsars in early seventies.
The recent discovery <strong>of</strong> magnetars seems to revive
the problem again [4]. Their magnetic field amounts to
O(1015G) from the P − ˙ P diagram. At present many people
believe the inheritance <strong>of</strong> the magnetic field from the
progenitor main-sequence stars or dynamo scenario due to
the electron current. We consider here a microscopic origin
<strong>of</strong> magnetic field by examining a possibility <strong>of</strong> spontaneous
spin polarization in quark matter.
Next, we consider a possibility <strong>of</strong> a non-uniform state
in the vicinity <strong>of</strong> the chiral transition. Recently there have
done many works about the non-uniform state at moderate
∗ Work partially supported by the Grant-in-Aid for the Global COE
Program “The Next Generation <strong>of</strong> Physics, Spun from Universality and
Emergence” from the Ministry <strong>of</strong> Education, Culture, Sports, Science
and Technology (MEXT) <strong>of</strong> Japan and the Grant-in-Aid for Scientific Research
(C) (16540246, 20540267).
† tatsumi@ruby.scphys.kyoto-u.ac.jp
densities. We shall see that this phase exhibits an interesting
magnetic property like SDW.
FERROMAGNETIC TRANSITION
The first study about the ferromagnetism in quark matter
has been performed by using the Bloch idea about the
ferromagnetism <strong>of</strong> itinerant electrons [5]. The mechanism
is rather simple due to the Pauli principle: consider electrons
interacting with each other by the Coulomb interaction
in the background <strong>of</strong> the uniformly distributed positive
charge to compensate the electron charge. Then the
Fock exchange interaction gives a leading-order contribution.
Then the electron pair with the same spin can effectively
avoid the Coulomb interaction to give an attractive
contribution to the total energy due to the Pauli principle.
As the counter effect such polarized electron system costs
more kinetic energy increase. So when the former effect
becomes larger than the latter one, we can expect spontaneous
spin polarization. A perturbative calculation <strong>of</strong> quark
matter interacting with the one-gluon-exchange interaction
shows the transition to ferromagnetic phase at the order <strong>of</strong>
nuclear density. Applying this idea to a star with solar mass
and radius <strong>of</strong> 10km, we can roughly estimate the magnetic
field <strong>of</strong> O(10 13−17 G). Thus we can feel that quark matter
inside the core region may give an origin <strong>of</strong> magnetic field
in compact stars.
Recently we have studied the magnetic susceptibility
χM to get more insight about the properties <strong>of</strong> ferromagnetic
transition within the Fermi-liquid theory [6]. χM is
written in terms <strong>of</strong> the quasiparticle interactions,
χM =
( )2 ¯gDµq N(T )
2 1 + N(T ) ¯ , (1)
f a
where ¯gD ≡ ∫
|k|=kF dΩk/4πgD(k) is the effective gyromagnetic
ratio, N(T ) the effective density <strong>of</strong> states around
the Fermi surface and ¯ f a the spin dependent Landau-
Migdal parameter [7, 8]. The spin susceptibility can be
easily evaluated by using the OGE interaction. Then we
can see that the Landau-Migdal (LM) parameters involve
infrared (IR) divergences in gauge theories (QCD/QED).
So we must take into account the screening effect at least
to obtain the meaningful results. We have done it by calculating
the quark polarization operator by the hard-denseloop
(HDL) resummation. As the results, we can see that
the Debye screening for the longitudinal gluons surely improves
the IR behavior, while the transverse gluons only
receive the dynamic screening due to the Landau damping.

Zero temperature case
There are still left the divergences in the LM parameters
at T = 0, but they cancel each other to give a finite χM. Finally
magnetic susceptibility is given as a sum <strong>of</strong> the contributions
<strong>of</strong> the bare interaction and the static screening
effect,
(χM/χPauli) −1
0 = 1 − Cf g 2 µ
12π 2 E 2 F kF
− 1
2 (E2 F + 4mEF − 2m 2 )κ ln 2
κ
[
m(2EF + m) −
]
, (2)
with κ = m2 D /2k2 F in terms <strong>of</strong> the Debye mass, m2D ≡
g2 µkF /2π2 2
Nc , and Cf = −1
. Thus the screening effect
2Nc
gives the g4 ln g2 term.
To demonstrate the screening effect, we show in Fig. 4.3
the magnetic susceptibility. We assume a flavor-symmetric
quark matter, ρu = ρd = ρs = ρB/3, and take the QCD
coupling constant as αs ≡ g2 /4π = 2.2 and the strange
quark mass ms = 300MeV inferred from the MIT bag
model. Note that the screening effect is qualitatively dif-
χ M /χ Pauli
20
15
10
5
0
-5
-10
-15
-20
α S=2.2
ms=300MeV
mu=md=0
0 0.5 1 1.5 2
kF [1/fm]
Figure 1: Magnetic susceptibility at T = 0. Screening
effects are shown in comparison with the simple OGE
case: the solid curve shows the result with the simple OGE
without screening, while the dashed and dash-dotted ones
shows the screening effect with Nf = 1 (only s quark)and
Nf = 2 + 1 (u, d, s quarks), respectively.
ferent, depending on the number <strong>of</strong> flavor Nf . The Debye
mass is given by all the flavors,
m 2 D = ∑
flavors
g 2
2π 2 kF,f EF,f , (3)
so that the κ ln(2/κ) term changes its sign for κ =
m2 D /2k2 F > 2. Thus we can see the screening flavors spontaneous
magnetization in large Nf .
Non-Fermi-liquid effect at finite temperature
We consider the low temperature case, T/µ ≪ 1, but
usual low-temperature expansion can not be applied, since
the quasiparticles exhibits an anomalous behavior near the
Fermi surface. Since the longitudinal gluons are short
ranged due to the Debye screening, their contributions are
almost temperature independent. Thus the main contribution
to the temperature dependence comes from the transverse
gluons. Careful considerations about the quasiparticle
energy show that quark matter behaves like marginal
Fermi liquid, where the Fermi velocity and the renormalization
factor vanish at the Fermi surface [9]. Such behavior
is brought about by the transverse gluons. The magnetic
susceptibility is then given as
(χM /χPauli) −1 = (χM /χPauli) −1
0
+ π2
6k 4 F
(
2E 2 F − m 2 + m4
E 2 F
)
T 2
( )
Λ
T
+ Cf g2uF (2k
72
4 F + k2 F m2 + m4 )
k4 F E2 T
F
2 ln
+ O(g 2 T 2 ). (4)
with uF ≡ vF /EF , where we can see the T 2 ln T term appears
as a novel non-Fermi-liquid effect, besides the usual
T 2 term, It should be interesting to compare this term with
other ones in specific heat or the superconducting gap energy.
Furthermore, such logarithmic behavior also resembles
the one by the spin fluctuations or paramagnons. Finally
the phase diagram is presented in Fig. 2, where we
can also asses the importance <strong>of</strong> the non-Fermi-liquid effect.
T [MeV]
70
60
50
40
30
20
10
Full.
w/o dynamical scr.
w/o static scr.
w/o any scr.
0
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
kF [1/fm]
Figure 2: Magnetic phase diagram in the densitytemperature
plane. The open (filled) circle indicates the
Curie temperature at kF = 1.1(1.6) fm −1 while the
squares show those without the T 2 ln T term.
Finally we present a phase diagram in Fig. 2, where we
can estimate the Curie temperature <strong>of</strong> several tens <strong>of</strong> MeV.
We can also see how the non-Fermi-liquid effect works for
the ferromagnetic transition.
MAGNETISM AND CHIRAL SYMMETRY
Recently there have been appeared many studies about
the non-uniform states in QCD, stimulated by the development
<strong>of</strong> the studies about the exact solutions in 1+1 dimensional
models [10]. The formation <strong>of</strong> the non-uniform

states in quark matter have been studied in relation to chiral
transition [11]. The appearance <strong>of</strong> density waves or
crystalline structures has been an interesting possibility at
moderate densities. Note that the appearance <strong>of</strong> the nonuniform
phase is not special, but rather familiar in condensed
matter physics. In some cases it may exhibit an interesting
magnetic property; the spin density wave (SDW)
discussed by Overhauser is a typical example. In the previ-
ψψ
1
0
-1
0
Z
-1
0
1
ψiγ5τ3ψ
Figure 3: Sketch <strong>of</strong> DCDW, where pseudoscalar density as
well as scalar density oscillates along z direction.
ous paper [2] we have discussed the possibility <strong>of</strong> a density
wave, where pseudoscalar density as well as scalar density
oscillate in harmony along one direction, which is called
dual chiral density wave (DCDW).
⟨ ¯ ψψ⟩ = ∆ cos(θ(r)),
⟨ ¯ ψiγ5τ3ψ⟩ = ∆ sin(θ(r)). (5)
The chiral angle θ(r) is taken in the one dimensional form,
θ(r) = q · r. The amplitude ∆ generates the dynamical
mass M, M = −2G∆, while θ produces the axial-vector
field for quarks, τ3γ5γ · ∇θ/2 = τ3γ5γ · q/2. The singleparticle
(positive) energy is then given by
E ± p = [E 2 p + q 2 /4 ± q √ p 2 z + M 2 ] 1/2 , (6)
with Ep = √ p 2 + M 2 , depending on the spin degree <strong>of</strong>
freedom. Accordingly the Fermi sea is split into two deformed
ones: one is deformed in the prolate shape and the
other in the oblate shape.
DCDW enjoys many interesting features. First, the symmetry
breaking pattern is Tˆp × U Q 3 5 (1) → U ˆp+Q 3 5 , which
may be 1+1 dimensional analog <strong>of</strong> Skyrmion. Then the
Nambu-Goldstone boson (”phason”) has a hybrid nature <strong>of</strong>
”pion” and ”phonon”. Secondly, a direct evaluation <strong>of</strong> the
magnetization gives ⟨σ12⟩ ∝ cos(q · r), which means a
kind <strong>of</strong> SDW. In this case quark matter can be regarded as
a kind <strong>of</strong> liquid crystal endowed with two-dimensional ferromagnetic
order and one dimensional anti-ferromagnetic
order. Note that magnetic field is globally vanished in this
phase, but locally very strong.
”Nesting” mechanism
Here we discuss the mechanism for the formation <strong>of</strong>
DCDW. There seems to be some confusions about it. In
the references [12] authors emphasized the nesting effect
(or Overhauser effect) for the essential mechanism <strong>of</strong> chiral
4
3.5
3
2.5
2
1.5
1
0.5
0
-q/2
q/2+M
q/2
q/2-M
q/2
-0.5
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2
pz
E+(pz)
E-(pz)
Figure 4: Energy spectra for p⊥ = 0 for q/2 > M. Solid
(magenta) curves show the one for massive quarks, while
dashed (blue) curves for massless ones, |pz ± q/2|.
density waves, but there is little discussion about DCDW or
other inhomogeneous phases. For 1+1 dimensional case,
we can immediately see that q is given as q = 2µ for given
chemical potential µ [10]. This is simply because the energy
spectrum (6) is reduced to E ±
p → √ p2 z + M 2
± q/2
and q is decoupled from pz. Recall that the outstanding relation
q = 2pF is held in the usual density wave like CDW
or SDW in the one dimensional system, due to the nesting
effect <strong>of</strong> the Fermi surface. We can see that the similar
mechanism works in the case <strong>of</strong> DCDW, but in somewhat
different manner from the usual one. For the case,
M > q/2, E ± p are only shifted ±q/2 from the free particle
energy, so that formation <strong>of</strong> DCDW depends on the interaction
strength like in the Stoner model. However, numerical
calculation shows this is not the case: q/2 > M is always
held in the DCDW phase. In Fig. 4 we sketch the energy
levels <strong>of</strong> the single quark energy for the case, q/2 > M.
Note that mass is generated by the interaction with DCDW
in this case. So E ± p can be regarded as a consequence <strong>of</strong> the
switching <strong>of</strong> the interaction with DCDW between massless
quarks with relative momentum q. For massless quarks, the
two levels cross each other at pz = 0 for any q. Once the
interaction with DCDW is present, mass is generated and
two levels avoid the crossing with the energy gap, 2M, at
pz = 0 (magenta curves in Fig. 4). So if we choose q = 2µ
and fill the levels up to pF = µ, there is always the energy
gain due to the interaction with DCDW. In the three dimensional
case, the simple relation is no more held, but we can
expect some reminiscence. Actually we can numerically
check that the similar relation is held in the three dimensional
case. In the vicinity <strong>of</strong> the critical end point we have
seen that the chiral correlation function χ(q) diverges at finite
q <strong>of</strong> q ∼ 2pF , but the effective mass is almost vanished
in this situation.
Thus we can understand DCDW with q/2 > M in
terms <strong>of</strong> the ”nesting” effect <strong>of</strong> the FErmi surface, while
q smoothly increases from zero for RKC. Their situation is
very different from our case: the opposite relation, |q/2

M|, is realized in their case, and the level diagram looks
very different from Fig. (4).
Deformed DCDW
Here we generalize the original DCDW by taking into
account the symmetry breaking effect with the current
quark mass mc ∝ m 2 π. Using a variational method, we can
show that the functional form <strong>of</strong> the chiral angle in DCDW
is deformed, satisfying the sine-Gordon (SG) equation, and
thereby the allowed region <strong>of</strong> DCDW is extended. The stable
solution is then given in terms <strong>of</strong> the Jacobian elliptic
function with modulus k,
θ = π + 2am (m ∗ πz/k, k) , (7)
with the effective pion mass in medium, m ∗ π. To recover
the original DCDW in the chiral limit, we must require the
following relation,
qk = m ∗ ππ/K (8)
with the complete elliptic integral <strong>of</strong> the first kind K. Note
q/m* π
6
5
4
3
2
1
0
0 0.2 0.4 0.6 0.8 1
k
Figure 5: Relation <strong>of</strong> the modulus k and the parameter q.
k → 0 in the chiral limit.
that we can recover the SG equation in ref. [13] and m ∗ π →
mπ in the 1+1 dimensional case.
CONCLUDING REMARKS
We have discussed two kind <strong>of</strong> magnetic properties <strong>of</strong>
quark matter separately, but a unified or comprehensive
description <strong>of</strong> magnetism should be desired. Furthermore
when we are interested in magnetism at moderate densities,
we must take into account some non-perturbative effects
explicitly. It may be one way to use the effective models <strong>of</strong>
QCD, e.g. NJL model, to this end.
Ferromagnetic order may have a direct implication on
magnetic evolution <strong>of</strong> compact stars, but SDW order should
also have some implications. DCDW may catalyse the elementary
processes by providing extra momentum; for example
it allows the quark β-decay process as neutrino emission
during the thermal evolution. The magnetization is
globally vanished there, but its fluctuation, ⟨M 2 ⟩, becomes
large. Accordingly the local strong magnetic field may induce
new QED processes.
For the present it needs more studies about the properties
<strong>of</strong> the non-uniform states and relations among them.
In particular, the comparison <strong>of</strong> DCDW and the real kink
crystal in [11] is important, since they are typical structures
in QCD, reflecting the different symmetries, U(1) vs Z2.
More studies are needed including the symmetry breaking
effect, thermal effect or model dependence.
The existing region <strong>of</strong> magnetism may be overlapped
with color superconductivity.It is then interesting to elucidate
the mutual relation in quark matter. Some works
have been already done [14], but more studies are needed,
including unconventional mechanism <strong>of</strong> pairing.
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T. Tatsumi, T. Maruyama and E. Nakano, Superdense QCD
Matter and Compact Stars, p.241 (Springer, 2006).

Recent progress and prospects on laser-plasma
acceleration <strong>of</strong> charged particles*
Kazuhisa Nakajima #†
High Energy Accelerator Research Organization (KEK) 1-1 Oho, Tsukuba 305-0081, Japan
Abstract
Recent progress in laser-driven plasma-based
acceleration <strong>of</strong> charged particles such as electrons and
ions is overviewed in theoretical and experimental aspects.
In particular laser-plasma acceleration physics such as
laser wakefield acceleration (LWFA) <strong>of</strong> electrons and ion
acceleration mechanism is highlighted, showing recent
achievements <strong>of</strong> laser plasma accelerator technologies
that produce high-energy, high-quality beams required for
compact particle beam and radiation sources.
INTRODUCTION
In this decade, worldwide experimental and theoretical
researches on laser-driven plasma-based acceleration <strong>of</strong>
charged particles have achieved great progress in highenergy,
high-quality electron beams <strong>of</strong> the order <strong>of</strong> GeVclass
energy and a 1%-level energy spread [1-6], whereas
laser-driven production <strong>of</strong> ion beams such as protons and
carbons is underdeveloped, harnessing Petawatt-class
ultra-intense lasers and ultra-thin foil targets. These highenergy
high-quality particle beams make it possible to
open the door for a wide range <strong>of</strong> applications in research,
medical and industrial uses.
Here recent progress in laser-plasma acceleration <strong>of</strong>
charged particles including electron- and ion-acceleration
is overviewed from the aspects <strong>of</strong> injection or particle
generation, acceleration process and resultant beam
properties, which are strictly determined by acceleration
mechanism or laser-plasma interaction such as the bubble
mechanism for electrons and radiation pressure
acceleration for ions.
Although there is no practical application to date,
underdeveloped are various applications <strong>of</strong> laser plasma
accelerators such as a compact THz or coherent X-ray
radiation source and radiation therapy driven by laseraccelerated
electrons [7]. On the other hand, a promising
application project <strong>of</strong> laser-driven proton and ion beams
to the future hadron therapy is implemented worldwide.
In the future laser-plasma accelerators may come into
being as a novel versatile tool for developing fields such
as space science where a compact and cost-effective tool
is required as well as inherent application to energyfrontier
particle accelerators.
___________________________________________
* Work supported by Chinese Academy <strong>of</strong> Sciences Visiting
Pr<strong>of</strong>essorship for Senior <strong>International</strong> Scientists.
# Visiting affiliations : Shanghai Institute <strong>of</strong> Optics and Fine
Mechanics, Chinese Academy <strong>of</strong> Sciences, Shanghai 201800, P.R.
China; Shanghai Jiao Tong University, Shanghai 200240, P.R. China.
† nakajima@post.kek.jp
LASER WAKEFIELD ACCELERATION
OF ELECTRONS
LWFA in the linear regime
In underdense plasma an ultraintense laser pulse excites
a large-amplitude plasma wave with frequency p =
(4e 2 ne/me) 1/2 and electric field on the order <strong>of</strong> ~ ne 1/2
V/cm for the electron rest mass mec 2 and plasma density
ne cm -3 due to the ponderomotive force expelling plasma
electrons out <strong>of</strong> the laser pulse and the space charge force
<strong>of</strong> immovable plasma ions restoring expelled electrons on
the back <strong>of</strong> the ion column remaining behind the laser
pulse. Since the phase velocity <strong>of</strong> the plasma wave is
approximately equal to the group velocity <strong>of</strong> the laser
pulse vg/c = (1- p 2 /0 2 ) 1/2 ~1 for the laser frequency 0
and the accelerating field <strong>of</strong> ~ 1 GeV/cm for the plasma
density ~ 10 18 cm -3 , electrons trapped into the plasma
wave are likely to be accelerated up to ~ 1 GeV energy in
a 1 cm plasma. More accurately in the linear regime for
the normalized vector potential defined by
2 1
2 18 2
2
a 0 0. 85 I
/10 Wcm m
,
where I is the laser intensity and = 2c/0 the laser
wavelength, the energy gain is given by [8]
2 2
E 1.
3mec
a0
nc
ne
,
2
18 -3
1
P TWr
m
n 10 cm GeV
35
0
e
for the peak laser power P TW focused onto the spot
radius r0 m, assuming that the plasma wave is efficiently
excited at p ~ cL for the pulse duration L, and that
electrons are accelerated over the dephasing length given
by Ldp ~ p(p 2 /0 2 ) = p(nc/ne), where nc = /(re 2 ) ⋍
1.115 × 10 21 cm -3 (/m) -2 is the cut<strong>of</strong>f density, re =
e 2 /mec 2 the classical electron radius. The accelerated
electrons overrun the accelerating field toward the
decelerating field beyond the dephasing length.
Quasi-monoenergetic acceleration in the
nonlinear regime
The leading experiments [9] that successfully
demonstrated the production <strong>of</strong> quasi-monoenergetic
electron beams with narrow energy spread have been
elucidated in terms <strong>of</strong> self-injection and acceleration
mechanism in the bubble regime [10,11]. In these
experiments, electrons are self-injected into a nonlinear
wake, referred to as a “bubble”, i.e. a cavity <strong>of</strong> plasma
electrons consisting <strong>of</strong> a spherical ion column surrounded
with a narrow electron sheath, formed behind the laser
pulse instead <strong>of</strong> a periodic plasma wave in the linear

egime. As analogous to a conventional RF cavity inside
which electromagnetic energy is resonantly confined at
the matched frequency to accelerate externally injected
particles, inducing a current flow in a skin depth <strong>of</strong> a
metal surface, plasma electrons radially expelled by the
radiation pressure <strong>of</strong> the laser form a sheath with
thickness <strong>of</strong> the order <strong>of</strong> the plasma skin depth 1/kp =
c/p outside the ion sphere remaining “unshielded”
behind the laser pulse moving at relativistic velocity so
that the cavity shape should be determined by balancing
the Lorentz force <strong>of</strong> the ion sphere exerted on the electron
sheath with the ponderomotive force <strong>of</strong> the laser pulse.
This estimates the bubble radius RB matched to the laser
spot radius w0 , approximately as ,
for which a best spherical shape <strong>of</strong> the bubble is created.
This condition is reformulated as
where Pc = 17(0/p) 2 GW is the critical power for the
relativistic self-focusing[11].
The longitudinal electric field inside the bubble is
obtained as
, where =z-vBt is
the coordinate in the frame <strong>of</strong> the bubble moving at the
velocity vB [10]. One can see that the maximum
accelerating field is given by e|Ez|max = (1/2)mec 2 kp 2 RB at
the back <strong>of</strong> the bubble and the focusing force is acting on
an electron inside the bubble. Assuming the bubble phase
velocity is given by vB ~ vg-vetch~c[1-(1/2+1)(p/0) 2 ],
where vetch ~c(p/0) 2 is the velocity at which the laser
front etches back due to the local pump depletion, the
dephasing length leads to
Ldp ~ c/(c-vB)RB ~ (2/3) (0/p) 2 RB = (2/3) (nc/ne)RB.
Hence the electron injected at the back <strong>of</strong> the bubble can
be accelerated up to the energy
1
2 2 nc
E e Ez
Ldp
mec
a
max
0 .
2
3 ne
Using the matched bubble radius condition, the energy
gain is approximately given by
2
P nc
E mec
,
Pr
ne
where Pr = me 2 c 5 /e 2 = 8.72 GW [12].
The 2D or 3D particle-in-cell simulations confirm that
quasi-monoenergetic electron beams are produced due to
self-injection <strong>of</strong> plasma electrons at the back <strong>of</strong> the
bubble from the electron sheath outside the ion sphere as
the laser intensity increases to the injection threshold. As
expelled electrons flowing the sheath are initially
decelerated backward in a front half <strong>of</strong> the bubble and
then accelerated in a back half <strong>of</strong> it toward the
propagation axis by the accelerating and focusing forces
<strong>of</strong> the bubble ions, their trajectories concentrate at the
back <strong>of</strong> the bubble to form a strong local density peak in
the electron sheath and a spiky accelerating field.
Eventually the electron is trapped into the bubble when its
velocity reaches the group velocity vg <strong>of</strong> the laser pulse.
Theoretical analysis on the trapping threshold gives kpRB
≥ (2nc/ne) 1/2 [13]. This trapping condition leads to
, while the trapping cross section ≃
(2/kp 3 d)(ln kpRB/8 1/2 ) -1 [10] with the sheath width d
1 3
2 3
,
imposes kpRB ≥ 2.8, i.e. for the matched bubble
radius. Once an electron bunch is trapped in the bubble,
loading <strong>of</strong> trapped electrons reduces the wakefield
amplitude below the trapping threshold and stops further
injection. Consequently the trapped electrons undergo
acceleration and bunching process within a separatrix on
the phase space <strong>of</strong> the bubble wakefield. This is a simple
scenario for producing high-quality monoenergetic
electron beams in the bubble regime. However, in most <strong>of</strong>
laser-plasma experiments aforementioned conditions and
scenarios are not always fulfilled.
In the experiment for the plasma density ne =(1 ~ 2)×
10 19 cm -3 , observation <strong>of</strong> the self-injection threshold on
the normalized laser intensity gives after
accounting for self-focusing and self-compression that
occur during laser pulse propagation in the plasma. In
terms <strong>of</strong> the laser peak power
)2, the self-injection threshold for the power (P/Pc )th
≈ 12.6 as the laser spot size reduces to the plasma
wavelength due to the relativistic self-focusing[14]. In the
experiment at ne =(3 ~ 5)×10 18 cm -3 , the self-injection
threshold is (P/Pc )th = 3, corresponding to [6].
Our 2-D PIC simulations on the self-injection threshold
show that for the uniform density plasma such as a gas jet
or a gas cell <strong>of</strong> ne =(1.7 ~ 5)×10 18 cm -3 , the self-injection
occurs at and for the preformed plasma channel
such as discharge capillary <strong>of</strong> the plasma density ne = 2×
10 18 cm -3 with the density depth nch/ne = 0.3, the
threshold is .
Controlled laser wakefield accelerator
For many applications <strong>of</strong> laser wakefield accelerators,
stability and controllability <strong>of</strong> the beam performance such
as energy, energy spread, emittance and charge are
indispensable as well as compact and robust features <strong>of</strong>
the system. In contrast to the conventional accelerators
composed <strong>of</strong> various complex-functioned systems, the
performance <strong>of</strong> laser plasma accelerators is strongly
correlated to the injection mechanism <strong>of</strong> electron beams
as well as the laser performance. To date, the external
injection into laser wakefields from the conventional RF
injector [15] or the staging concept, which is conceivable
on the analogy <strong>of</strong> the high-energy RF accelerators, has not
been always successful for generating intense highquality
electron beams that could be useful for
applications. Hence, besides the self-injection, the optical
injection scheme with two colliding pulses is highlighted.
The optical injection scheme for manipulating electron
beams in a phase space <strong>of</strong> laser wakefield acceleration
with fs-synchronization and MeV-energy response utilize
an injection pulse split from the same drive pulse with fs
duration, crossing the drive pulse at some angle in the
plasma. When crossing each other, the phase space <strong>of</strong>
wakefields excited by the drive pulse overlaps with the
phase space <strong>of</strong> beat waves generated by mixing the drive
pulse and the injection pulse. As a result <strong>of</strong> the
ponderomotive force <strong>of</strong> the beat wave boosts plasma
electrons and locally injects them into the separatrix <strong>of</strong>

the wakefields. In the case <strong>of</strong> head-on collision <strong>of</strong> two
counter-propagating laser pulses at the angle <strong>of</strong> the
ponderomotive force
<strong>of</strong>
the injection beat wave oscillating with the wavelength
0/2 locally accelerates the plasma electrons to be injected
into the wakefield bucket, where k0 = 2/0 is the laser
wave number, ɑ0, and ɑ1 the intensity <strong>of</strong> the drive pulse
and the injection pulse, respectively, and the Lorentz
factor <strong>of</strong> the plasma electron, i.e. 1 for the cold
plasma. On the contrary to the self-injection with a single
drive pulse, this force is independently controllable by
changing the injection pulse intensity and/or its
polarization with respect to that <strong>of</strong> the drive pulse as well
as the injection position, where two pulses collides.
Therefore the energy and the charge can be controlled
within some extent, associating with evolution <strong>of</strong> the
energy spread due to the beam loading and the injection
volume <strong>of</strong> the phase space.
These effects are successfully demonstrated with good
stability by the experiments carried out below the selfinjection
threshold <strong>of</strong> the drive pulse intensity. The
experiment <strong>of</strong> ref. [16] with , and the
pulse duration <strong>of</strong> 30 fs for both pulses shows an almost
linear control <strong>of</strong> the monoenergetic beam energy from 50
MeV to 250 MeV by changing the colliding position over
the 2-mm gas jet at the plasma density <strong>of</strong> ne = 7.5×10 18
cm -3 , consequently changing the acceleration length in
the average accelerating field <strong>of</strong> Ez = 270 GV/m, which is
close to an estimate <strong>of</strong> the wave breaking field for the
cold plasma, Ewb ≈ 0.96ne 1/2 ~ 263 GV/m. The experiment
<strong>of</strong> ref. [17] demonstrates the colliding optical injection at
the crossing angle <strong>of</strong> 135 °in the 1-mm gas jet with
(Pdrive = 3 TW), (Pinj = 0.14 TW) and
70 fs pulse duration, resulting in the energy E =15 MeV
and the energy spread E/E =7.8% at ne = 3.95×10 19 cm -
3 free from the self-injection as well as the head-on
colliding injection at 180° with (Pdrive = 10
TW), (Pinj = 0.6 TW) and 40 fs pulse duration,
resulting in the energy E =134 MeV and the energy
spread E/E =3.5% at ne = 1 × 10 19 cm -3 . These
experiments suggest a very compact system <strong>of</strong> the
electron beam source including the laser and the
accelerator on a table-top size with high quality and high
stability.
The trapped electrons inside the bubble generate
electromagnetic fields and modify the bubble wakefields.
As a result, the trailing electron bunch undergoes less
accelerated field that limits the charge and produces
energy spread. The analysis <strong>of</strong> the beam loading in the
bubble regime gives the energy absorbed per unit length
<strong>of</strong> the beam is given as
Q
eE
4 k
16 3
s s 10 cm
0.
047
pRB
1nC
mec
p
ne
where Qs is the total charge and Es the accelerating
wakefield at the phase position where the bunch charge
starts, assuming the density distribution <strong>of</strong> the bunch
charge has a trapezoidal shape so that the energy spread
inside the bunch is minimized [18]. This equation implies
the trade-<strong>of</strong>f between the total charge that can be
accelerated and the accelerating gradient, i.e. the
accelerated energy. With , the charge is
proportional to ne -1/2 .
Toward high-energy acceleration beyond GeV
Since the energy gain scales as , it would be
necessary for the multi-GeV acceleration to decrease the
operating plasma density from the 10 18 cm -3 range to the
10 17 cm -3 range and increase the accelerator length, i.e.
plasma channel length, up to several tens cm. Recent
experiments demonstrated GeV-class quasimonoenergetic
electron beams with a cm-scale gas jet or a capillary
plasma waveguide, relying on self-injection <strong>of</strong> plasma
electrons:
E=1 GeV, E/E=2.5%(rms), =1.6mrad, Q=35pC
for P= 40TW, L = 37 fs, a0 =1.4, ne = 4.3×10 18 cm -3
with 3.3-cm gas-fill capillary [1].
E = 0.5 GeV, E/E = 2.5 % (FWHM), = 0.3mrad,
Q > 0.3 pC for P=18TW, L = 42 fs, a0 = 0.84, ne =
8.4×10 18 cm -3 with 15-mm gas-fill capillary [2].
E = 0.59 GeV, E/E = 1.2 % (FWHM), = 0.59
mrad, Q > 10 fC for P=24TW, L = 27 fs, a0 = 1.7, ne
= 1.9×10 18 cm -3 with 4-cm ablative capillary [3].
E = 0.52 GeV, E/E = 5 % (FWHM), = 5.4 mrad,
Q=70 pC for P=32TW, L=80fs, a0=0.8, ne= 1.8×10 18
cm -3 with 3-cm gas-fill capillary [4].
E = 0.8 GeV, E/E = 12 % (FWHM), = 3.6 mrad,
Q=90pC in average for P=180TW, L =55fs, a0=3.9,
ne= 5.7×10 18 cm -3 with 8-mm gas jet [5].
E=0.72 GeV, E/E = 14 % (FWHM), = 2.9 mrad,
Q~100pC for P=65TW, L =60fs, a0=2.8, ne= 3×10 18
cm -3 with 8-mm gas jet [6].
One finds that high-energy and high-quality electron
beams with less charge are accelerated with cm-scale
capillary plasma waveguides in the quasi linear regime (a0
< 2), while high-energy and high-charge electron beams
with less quality are produced with gas jets relying on
self-guiding in the bubble regime (a0 > 2). Recent
progress <strong>of</strong> PW-class lasers boosts the acceleration energy
beyond 1 GeV. Two experiments are attempted with a cmscale
gas cell and a capillary, respectively, showing nonmonoenergetic
spectra with a clear cut<strong>of</strong>f energy Emax:
Emax=1.45 GeV for P= 110TW, L = 60 fs, a0 =3.8, ne
= 1.3×10 18 cm -3 with 1.3-cm gas cell containing 97%
He and 3% CO2 mixed gas [19].
Emax=1.8 GeV for P= 72TW, L = 50 fs, a0 =2.9, ne =
2.1×10 18 cm -3 with 4-cm ablative capillary made <strong>of</strong>
polycarbonate [20].
Although both cases correspond to the bubble regime for
a0 high enough to cause self-injection <strong>of</strong> electrons, the
ionization-induced trapping mechanism [21] due to
oxygen impurities enhances the electron injection into the
bubble continuously over the plasma length. For the
multi-GeV LWFA operating in the plasma density lower
than 10 18 cm -3 , a dedicated injector would be essential to
produce high-quality electron beams with high-stability.

With extremely high-peak power lasers beyond PW,
based on a naïve scaling <strong>of</strong> attainable energies in a single
stage, it is possible to design ultrahigh-energy laserplasma
accelerators beyond 10 GeV in a much smaller
size than that <strong>of</strong> conventional accelerators. However,
various requirements for applications such as the beam
qualities and the electrical power would strictly limit
acceptable parameters on the accelerator performance.
Recent 3D-PIC simulations [22] show that 10-PW and 3-
PW lasers cannot produce high-energy ( 10 23 W/cm 2 with a linearly
polarized laser pulse <strong>of</strong> 10 kJ energy that could accelerate
GeV quasi-monoenergetic ions with high conversion
efficiency (> 40%) [27], though such experiments are far
beyond current technology. Circularly polarized laser
pulses <strong>of</strong> which the ponderomotive force has no
oscillating component can push forward electrons steadily,
suppressing foil heating and expansion. The absence <strong>of</strong>
hot electrons suppresses TNSA and allows RPA even at
currently achievable laser intensities (I < 10 22 W/cm 2 ).
The one-dimensional (1D) model <strong>of</strong> RPA is modeled
by the following relativistic equation <strong>of</strong> motion for the
foil [25]:
2 2 2
dp 2I
1
2I
p c p
2 2 2
dt c 1
c p c p
where is the foil velocity, p the areal momentum
<strong>of</strong> the foil and = minil the areal mass <strong>of</strong> the foil with the
ion mass mi, the ion density ni, and the foil thickness l.
The solution is given by
with
. One obtains
the final velocity <strong>of</strong> the foil
, where
and
with
the initial electron density n0, the cut<strong>of</strong>f density nc, the
laser wavelength , the pulse duration in units <strong>of</strong> laser
period and the normalized vector potential
for a circularly polarized laser pulse. The final kinetic
energy results in
. Hence the ion final energy scales as and
. As an example, for a 100 fs circularly polarized laser

pulse with I = 10 21 Wcm -2 at = 0.8 m and a 150 nm
thick target with density, monoenergetic ions
with the kinetic energy ~ 230 MeV per nucleon may be
accelerated.
The optimal condition for producing ions with a narrow
spectral peak is obtained from the equilibrium between
the electrostatic and radiation pressure that cause a strong
expulsion <strong>of</strong> electrons and produce a density distribution
consisting <strong>of</strong> the electron depletion region with density
, thickness d and the compressed electron layer
with density np0, thickness ls where all electrons pile at the
rear surface <strong>of</strong> the foil with the reflectivity :
, where
is the total
radiation pressure and
is
the electrostatic pressure, obtained from the peak
electrostatic field and the charge
conservation . This condition
results in , which determines the optimal foil
thickness
.
The beam quality <strong>of</strong> ion beams produced by RPA is
expected to improve drastically on that <strong>of</strong> TNSAproduced
ions. The PIC simulation for the interaction
between the circularly polarized laser pulse <strong>of</strong>
, the pulse duration and a 150 nm
thick foil at the proton density <strong>of</strong> shows a
monoenergetic spectral peak <strong>of</strong> protons at 485 MeV and
an excellent longitudinal emittance <strong>of</strong>
, which is three orders magnitude
smaller than that <strong>of</strong> TNSA [25].
Despite <strong>of</strong> many simulation works, a few experiments
have attempted to demonstrate the ion acceleration driven
by RPA because <strong>of</strong> difficulty <strong>of</strong> producing laser pulses
with extremely high-contrast ratio <strong>of</strong> the main pulse
intensity over the intensity <strong>of</strong> ASE (amplified
spontaneous emission) pedestal and prepulses as well as
high intensities (I 2 > 10 20 Wcm -2 m 2 ) to avoid target
heating prior to arrival <strong>of</strong> the main pulse. The first pro<strong>of</strong><strong>of</strong>-principle
experiment <strong>of</strong> RPA was demonstrated by
irradiating a circularly polarized laser pulse <strong>of</strong>
(a0 =3.5) on diamondlike carbon (DLC)
foils <strong>of</strong> thickness 2.9 – 40 nm at normal incidence and
ultrahigh contrast (~10 11 ), produced by the double-plasma
mirror technique [29]. The results showed a peak <strong>of</strong> the
maximum ion energy per nucleon at the optimum
thickness <strong>of</strong> 5.3 nm that corresponds to the condition
and a broad quasimonoenergetic peak <strong>of</strong>
(2.5MeV/u) in the carbon C 6+ spectra for circular
polarization.
CONCLUSIONS
Recent progress in laser-plasma acceleration <strong>of</strong> charged
particles are overviewed from the aspects on laser
wakefield acceleration <strong>of</strong> electrons in the linear and
nonlinear regime, so-called “bubble” regime,
characterized by quasi monoenergetic beam production
due to the self- and the controlled injection as well as the
beam loading, and ion acceleration mechanisms such as
TNSA and RPA. Based on up-to-date achievements, the
design parameters <strong>of</strong> 10 GeV single-stage LWFA are
presented, aiming at a currently advocated goal.
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Abstract
Flying Mirror as a tool to access ultra-high fields ∗
M. Kando, A. S. Pirozhkov, T. Zh. Esirkepov, T. Nakamura, J. Koga, H. Kotaki
Y. Hayashi, S. V. Bulanov
JAEA, Kizugawa, Kyoto 619-0215, Japan
Thanks to the recent progress <strong>of</strong> laser technology, there
are growing interests to explore ultra-high fields (electromagnetic
fields) by focusing intense, ultra-short laser
pulses down to a few micron sizes. Presented here is a
study to possibility reach (or boost) such ultra-high fields
using a new concept employing the interaction <strong>of</strong> intense
laser pulses with plasma. The concept uses breaking waves
excited by ultra-short, intense laser pulses in plasma. We
present example parameters to reach the Schwinger field
and review the recent experimental progress <strong>of</strong> the flying
mirror concept.
INTRODUCTION
Since the innovation <strong>of</strong> the chirped pulse amplification
(CPA) technique [1], the peak power <strong>of</strong> lasers has been increasing
and the focused irradiance has reached as high as
10 22 W/cm 2 [2, 3]. The extreme light infrastructure (ELI)
project[4] is being promoted and the goal is to reach 10 26
W/cm 2 . The unprecedented irradiances allow us to explore
a new regime <strong>of</strong> physics, which has previously not been
accessible experimentally. Theoretically there are challenging
tasks which can be done in the high electromagnetic
fields. One example is to explore the so-called quantum
electrodynamics critical field or the Schwinger field,
at which vacuum breaks down and therefore non-virtual
electron-positron pairs are created from vacuum. Assuming
a direct-current (DC) electric field the QED critical
field Ec = 1.3 × 10 18 V/m is obtained. The associated
laser irradiance is Ic = 2.3 × 10 29 W/cm 2 , which is 7 orders
<strong>of</strong> magnitude higher than the world record and still
3 orders <strong>of</strong> magnitude higher than that at the planned ELI
project. Can we use the present laser technology to reach
the Schwinger field?
The answer might be ’Yes’. There is theoretical work indicating
lowering <strong>of</strong> the limit <strong>of</strong> electron-positron creation
from vacuum[5, 6, 7]. Here in addition to the work, we recall
the proposal that an plasma device –relativistic flying
mirror– can be used to intensify the laser pulse as shown in
Fig. 1[8]. Flying mirrors are electron density cusps propagating
almost at the speed <strong>of</strong> light in tenuous plasma.
Such electron density cusps are formed when plasma wake
waves excited by intense, ultrashort laser pulses are breaking.
The flying mirror reflects an incoming laser, and the
reflected pulse is upshifted due to the double Doppler effect
∗ Work in part supported by JAEA and KAKENHI No. 20244065
and is compressed as well. In addition, the flying mirror
may focus the laser pulse. Thus, the flying mirror can be
a novel device that enhances the laser focused irradiance
drastically. In this paper, we discuss the possibility to access
ultra-high electromagnetic fields employing the flying
mirror.
THEORY
We shall consider a mirror moving at the speed vM =
βMc, where βM denotes the ratio <strong>of</strong> the mirror speed to the
speed <strong>of</strong> light c. The reflected frequency in the laboratory
frame <strong>of</strong> reference is expressed as
ωr = ωs
1 + βM cos θ
, (1)
1 − βM cos θr
where ωs is the frequency <strong>of</strong> the incident source pulse and
π −θ and θr are angles in the laboratory frame <strong>of</strong> reference
between the wave propagation direction and the mirror velocity
for the incident and reflected waves, respectively. If
the incident angle satisfies the condition −π/2 < θ < π/2,
the frequency <strong>of</strong> the reflected pulse is upshifted. The maximum
upshift is achieved when θ = θr = 0. In this case the
reflected frequency is approximately equal to ωr ≈ 4γ 2 M ωs
for γM ≫ 1, where γM = 1/ √ 1 − β 2 M .
The electric field amplitude in the reflected wave is ex-
pressed as
Er = R 1/2 Es
( ωr
ωs
)
, (2)
where R is the reflectivity <strong>of</strong> the moving mirror in terms <strong>of</strong>
the photon number. For a mirror with a paraboloidal shape,
the reflected light can be focused down to a spot equal to
λs/(2γM). In this case the focused irradiance is given by
Ir = 64Rγ 6 M
( D
λs
) 2
Is. (3)
Here Is is the irradiance <strong>of</strong> the incident source pulse on
the mirror with a waist <strong>of</strong> D. As explained in the previous
section flying mirrors are created during the interaction <strong>of</strong>
intense, utra-short laser pulses with tenuous plasma. The
velocity <strong>of</strong> the flying mirror is equal to the phase velocity
<strong>of</strong> the plasma wave, i.e. βMc = βphc. The phase velocity
<strong>of</strong> the plasma wave is approximately equal to the group velocity
<strong>of</strong> the laser in the plasma βphc = c[1−(ωpe/ω) 2 ] 1/2 ,
where ωpe = (4πnee 2 /m) 1/2 , m and e are the electron
mass and charge, and ω is the laser angular frequency.

Plasma
Flying mirrors
Driver pulse
Reflected pulse
Source pulse
Figure 1: Conceptual scheme <strong>of</strong> the flying mirror. The driver pulse generates electron density modulations (wake waves).
The counter-propagating source pulse is reflected, frequency-upshifted, compressed, and focused by the flying mirror.
The dependence <strong>of</strong> the reflectivity on the parameters <strong>of</strong>
the laser-plasma interaction is <strong>of</strong> the key interest, and has
been analyzed in Refs. [8] and [9] for different configurations
<strong>of</strong> breaking nonlinear waves. In the case <strong>of</strong> a strong
singularity, the electron density modulation in the breaking
wave may be described by the Dirac delta function. In this
case the reflectivity in terms <strong>of</strong> photon number is
(
ωpe
Rδ ≈
ωs cos2 ) 2
1
. (4)
(θ/2)
2γph
Using the approximation γph ≈ ωd/ωpe, where ne is the
plasma density, and assuming that ωd = ωs and θ = 0, the
reflectivity is simplified to
Rδ ≈ 1
2γ3 . (5)
ph
When the electron density has a cusp form in the vicinity
<strong>of</strong> a singularity expressed as n(X) ∝ X −2/3 (cusp), the
reflectivity is
R2/3 = 4 −7/3 3 −1/3 Γ 2 (
(1/3)
ωpe
ωs cos 2 (θ/2)
) 8/3
1
γ 4/3
ph
(6)
where Γ(x) is the Euler gamma function. Substituting
γph ≈ ωd/ωpe, ωd = ωs and θ = 0, Eq. (6) reduces to
R2/3 ≈ 0.2
γ4 . (7)
ph
By using the best reflectivity described in Eq. (5), the
irradiance obtained in the flying mirror scheme, Eq. (3) is
expressed as
Ir = 32γ 3 ph
( D
λs
EXAMPLES
) 2
Is. (8)
Here we show numerical examples to obtain the
Schwinger limit (10 29 W/cm 2 ) with the flying mirror concept.
We consider a laser system that does not exist but
is possible to construct within existing laser technologies.
The assumptions used here are as follows. A source laser
irradiance is set to be weakly relativistic Is = 10 17 W/cm 2
in order to avoid possible modifications to the flying mirrors.
Both driver and source laser pulses have the same
,
wavelength <strong>of</strong> λd = λs=0.8 µm and the same pulse duration
<strong>of</strong> τ=20 fs. We assume that the driver irradiance is
equal to the one-dimensional wave breaking limit a 2 0/2 =
γph, where a0 = eE/(mcω)=0.86×10 −9 λ[µm]I[W/cm 2 ]
is the normalized field amplitude <strong>of</strong> a laser with the electric
field <strong>of</strong> E , the angular frequency <strong>of</strong> ω, the wavelength <strong>of</strong>
λ, and the irradiance <strong>of</strong> I.
From Eq. (3) the free parameters seem to be γph or
D. We choose γph (plasma density), therefore D is determined.
Shown in Table 1 are examples when γph =
100, 200.
Table 1: Example parameters to achieve a focused irradiance
<strong>of</strong> 10 29 W/cm 2 . ne is the plasma density, Is and Id are
irradiances <strong>of</strong> the source and driver laser pulses. D and wd
are diameters <strong>of</strong> the flying mirror and the driver laser. If
is the focused irradiance <strong>of</strong> the reflected pulse. Ex denotes
the pulse energy <strong>of</strong> the driver (source) pulse.
Parameters Units Case I Case II
γph 100 200
ne cm −3 2×10 17 4×10 16
Is W/cm 2 1×10 17 1×10 17
D µm 140 50
Id W/cm 2 2×10 20 2×10 18
wd µm 280 100
Ir W/cm 2 1×10 29 1×10 29
Ed kJ 5.4 1.3
Es mJ 320 40
DISCUSSIONS
In the previous section we assume that all the functions
<strong>of</strong> the flying mirror work very well as described in Eq. (8).
We notice this assumption is too optimistic and need further
investigation to implement it in a device.
First, let us check the experimental progress <strong>of</strong> the flying
mirror. A basic concept that flying mirrors can reflect laser
pulses and upshift the incident source laser frequency was
verified in an experiment [10, 11]. In the experiment, 180
mJ, 76 fs, 0.8 µm laser pulses were focused onto helium
gas-jet targets and 10 mJ, ∼ 100 fs laser pulses were used as
source laser pulses at the angle <strong>of</strong> 135 ◦ . The reflected light

Reflected light intensity (μJ/sr/nm)
0.5
0.4
0.3
0.2
0.1
0
12 14 16 18 20 22
Wavelength (nm)
Figure 2: Observed spectrum <strong>of</strong> the reflected laser light in
the flying mirror experiment [12].
was measured with a grazing-incidence flat-field spectrograph
in the forward (0 ◦ ) direction. The observed spectra
distributed in the range <strong>of</strong> 7–15 nm. The upshift factors
were 50–114.
Although the experiment showed a somewhat coherent
effect, the reflected photon number was smaller than the
theoretical expectation. In the 2nd experiment[12], a headon
collisions setup <strong>of</strong> the driver and source pulses was employed
thanks to the improvement <strong>of</strong> the laser system and
a more powerful laser was used. The driver and source
laser had peak powers <strong>of</strong> 15 TW(400 mJ/27 fs) and 1.2
TW(42 mJ/ 34 fs), respectively. Reflected source pulses
were observed with an imaging spectrograph that covered
the wavelength range <strong>of</strong> 12-25 nm at the angle range <strong>of</strong>
9 ◦ − 17 ◦ . The result is shown in Fig. 2. The reflected photon
number increased to half <strong>of</strong> the theoretical cusp model
(see Table 2).
To increase the reflectivity further, we can use a variant
<strong>of</strong> Eq. (5) assuming ωd ̸= ωs and θ = 0,
Rδ ≈
( ) 2
ωd 1
ωs 2γ3 ph
. (9)
If ωd > ωs, we obtain a gain <strong>of</strong> (ωd/ωs) 2 . For example,
we can use a frequency doubling crystal for the driver laser.
Table 2: Comparison <strong>of</strong> reflectivities and photon numbers
between the theoretical model and the experiment.
Reflectivity Photon number
Theory 4×10 −5 1.5×10 10
Experiment 3×10 −6 1.1×10 9
Exp. (corrected) 2×10 −5 7.9×10 9
There are no measurement so far on the pulse duration
<strong>of</strong> a reflected pulse from flying mirrors. However, the large
photon number obtained in the experiment implies that the
coherent effect <strong>of</strong> the electrons exists. Therefore, we may
Intensity (arb. u.)
-3
-3
0
300 400 500
Figure 3: Observed spectrum in a high-resolution particlein-cell
simulation.
be allowed to assume that the pulse duration is compressed
down to the theoretical model prediction.
Most serious and to be confirmed is the spot size <strong>of</strong> the
reflected pulse. There are many points that degrades the
focus spot such as a surface error from an ideal parabola,
surface roughness, alignment error to the flying mirror,
etc. These points have not yet been addressed both experimentally
and theoretically. We have just started to estimate
the roughness <strong>of</strong> the flying mirror. As a preliminary
result we have obtained that the surface roughness
in two-dimensional particle-in-cell simulations is smaller
than the simulation resolution 0.02 µm. Further analysis
is needed to determine the focusability <strong>of</strong> the reflected
pulse, especially at low densities (high γph) in Table 1. We
also obtained the upshift factor <strong>of</strong> ωr/ωs ∼500 in a highresolution
particle-in-cell simulation as shown in Fig. 3.
Here the driver laser has an energy <strong>of</strong> 2.3 J and a pulse
duration <strong>of</strong> 20 fs, and the focused irradiance <strong>of</strong> 3.7×10 19
W/cm 2 (a0 = 4.1). The source laser has a wavelength <strong>of</strong>
2.4 µm in this simulation. The plasma density is 1.5×10 19
cm −3 and mesh sizes are ∆x =0.8 nm and ∆y=20 nm. Although
the resolved wavelength is larger than the requirement
in Table 1, water-window or shorter wavelength Xrays
are generated in the simulation.
CONCLUSION
A relativistic flying mirror is examined as a tool to access
ultra-high electromagnetic field irradiance. We carved
out laser and plasma parameters assuming plasma and laser
parameters that are not far from present-day laser technologies.
We reviewed the recent experimental progress <strong>of</strong> the
flying mirror. The concept is in progress showing that the
reflectivity is close to the theoretical estimate using a cusp
model. Still further study is necessary to intensify the focused
laser.

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4-mirror laser stacking cavity for high intensity polarized photon generation
T. Akagi, M. Kuriki, T. Takahashi, S. Miyoshi : Hiroshima Univ.
S. Araki, J. Urakawa, T. Omori, T. Okugi, H. Shimizu, N. Terunuma, Y. Funahashi, Y. Honda : KEK
K. Sakaue, T. Hirose, M. Washio : Waseda Univ.
Abstract
We are developing a compact light source based on the
laser-Compton scattering. We have performed a photon
generation experiment at the KEK-ATF using a Fabry-
Perot type 2-mirror laser pulse stacking cavity[1]. The laser
pulses are accumulated and their power was enhanced by
up to 760 times in the 2-mirror Fabry-Perot cavity. In order
further improve performance <strong>of</strong> the laser power enhancement,
a new three dimensional 4 mirror cavity is being designed.
In this article, we report status and prospect <strong>of</strong> the
photon generation experiment, and 3D 4-mirrorcavity design.
INTRODUCTION
X-ray have been used in various scientific industrial and
medical applications. Compact and bright light source is
highly desirable for these applications. We are developing
compact light source based on the laser-Compton scattering.
In this scheme, for example, tens <strong>of</strong> MeV photons can
be generated by collisions <strong>of</strong> laser photons with about 1
GeV electron beam, as shown in Fig. 1. To increase the intensity
<strong>of</strong> generated photons by laser-Compton scattering,
increasing intensity <strong>of</strong> laser pulses and focusing at collision
point by accumulating them in an optical cavity is an
attractive method.
We already achieved laser intensity enhancement <strong>of</strong> 760
and laser waist size is 30µm (1σ) by the 2-mirror Fabry-
Perot cavity. To increase the intensity <strong>of</strong> generated photons
more, it is necessary to use high reflectivity mirrors and
to make lasar waist size smaller. However, it is difficult
to achieve the improvement <strong>of</strong> enhancement and focusing
performance at the same time in the 2-mirror Fabry-Peror
cavity. Thus, to achieve the two requirement, now we are
developing three dimensional 4-mirror optical cavity.
Figure 1: laser-Compton scattering
OPTICAL CAVITY
We succeeded to generate gamma-rays by laser-
Compton scattering with the 2-mirror Fabry-Perot cavity at
the KEK-ATF. This section describes our 2-mirror Fabry-
Perot cavity. We use a 357 MHz mode-locked laser, its
repetition rate is the same as electron bunch spacing in the
ATF. The wavelength and pulse width <strong>of</strong> the laser are 1064
nm and 5 ps in the root mean square.
In order to accumulate laser pulses in the optical cavity,
the optical cavity has to be on-resonance. The length (L) <strong>of</strong>
the optical cavity has to be
L = n λ
2
where λ is the wavelength <strong>of</strong> the laser, and n is a positive
integer. In addition, accmulation <strong>of</strong> laser pulses from
a mode-locked pulsed laser requires that the length <strong>of</strong> the
optical cavity is integer times <strong>of</strong> the cavity inside the laser
oscillator.
L = mL ′
(2)
where L ′ is the length <strong>of</strong> the cavity inside the mode-locked
laser oscillator, and m is a positive integer. In this experiment,
the length <strong>of</strong> optical cavities are L = L ′ = 420mm.
The waist size <strong>of</strong> laser pulses is 30µm (1σ) in the optical
cavity, because the curvature radius <strong>of</strong> mirrors is 210.5mm.
The performance <strong>of</strong> the optical cavity is expressed by the
finesse (F), and can be expressed as
F = π√ R
1 − R
R = √ R1R2
and the power enhancement factor (S) <strong>of</strong> laser pulses in the
optical cavity can be estimated as
S =
T1
(1 − R) 2
where R1 and T1 are the reflectivity and transmissivity <strong>of</strong>
the entrance mirror, R2 is the reflectivity <strong>of</strong> the other mirror
respectively. The resonant condition <strong>of</strong> the optical cavity
is monitored by the intensity <strong>of</strong> transmitted light from the
cavity, which is at the maximum when the cavity is onresonance.
The intensity <strong>of</strong> the transmitted light obtained
by changing the length <strong>of</strong> the cavity is shown in Fig. 2.
The width <strong>of</strong> resonant peak is 0.36 nm with our 2-mirror
optical cavity, which indicates that the length <strong>of</strong> the optical
cavity has to be controlled with precision <strong>of</strong> smaller enough
than one tens <strong>of</strong> a nanometer. The optical cavity has to
be high finesse for high enhancement factor. However, the
(1)
(3)
(4)
(5)

Figure 6: To reduce the laser waist size, 2-mirror cavity has to be concentric type. On the other hand, 4-mirror cavity is
confocal type.
However, in the 4-mirror cavity, effective focal length
(ft, fs) are different in tangential plane and sagittal plane,
and the difference causes astigmatism at the focal point. ft
and fs are expressed as
ft = ρ
cos θ (6)
2
fs = ρ
2 cos θ
where θ is reflection half angle <strong>of</strong> concave mirror. Because
<strong>of</strong> this astigmatism, laser pr<strong>of</strong>ile inside the 4-mirror cavity
will be ellipse.
To avoid the astigmatism, the cavity has to be three dimensional
configuration. 3D 4-mirror optical cavity generally
have a circular-polarization dependent property due to
the rotation <strong>of</strong> the image in the three-dimensional optical
path. A new method utilizing this property to obtain a differential
signal from the cavity resonance was proposed[2].
The differential signal can be used to lock an optical cavity
at a resonance peak.
We conducted experiment to obtain differential signal
using a 3D 4-mirror optical cavity testbench. The setup
is shown in Fig. 7. A linear polarization wave was injected
to the 4-mirror cavity. Then we measured the output <strong>of</strong> the
differential amplifier while scanning the cavity length. The
signal observed is shown in Fig. 8. The bottom line is the
signal <strong>of</strong> the photo-diode that measurred the transmission.
And it shows the points that the cavity’s resonance <strong>of</strong> right
and left-handed polarized light. The top and middle lines
are output <strong>of</strong> the differential amplifier. The signal crossed
zero at the point <strong>of</strong> resonace. It provides a good differential
signal for locking the cavity to one <strong>of</strong> the peaks <strong>of</strong>
resonance. Actually, we have succeeded to lock the optical
cavity on the resonance peak and switch the circularpolarization
peak. It means the polarization <strong>of</strong> the generated
photons by laser-Compton can be switched quickly.
Now, a 3D 4-mirror cavity are being designed in order to
be installed in the ATF during summer 2011. Then we will
cunduct the experiment <strong>of</strong> gamma-rays generation.
CONCULSION
In order to increase the number <strong>of</strong> generated photons by
laser-Compton scattering, it is necessary to reduce the laser
(7)
Figure 7: Setup to obtain the differential signal.
Figure 8: Signal <strong>of</strong> transmission and difference.
waist size. We performed gamma-rays generation experiment
using a 2-mirror cavity and observed 10.8 ± 0.1 photons/bunch.
The laser power enhanced by up to 760 times
and laser waist size is 30µm (1σ).
As the next step, a 4-mirror ring cavity with a threedimensional
(non-planar) configration is being constructed.
The cavity has unique property such that it only resonate
with circular polarized light. It allows us to utilize new
method to obtain feed back signal for the cavity stabilization
as well as fast polarization switching.
REFERENCES
[1] S. Miyoshi et al., Nucl. Inst. Meth. A 623 (2010) 576.
[2] Y. Honda et al., Opt. Commun. 282 (2009) 3108.

CURRENT STATUS OF LFEX LASER AND EXA-WATT LASER CONCEPT
AT ILE/OSAKA
J. Kawanaka, LFEX-Team, EXA-Team, and H. Azechi
Institute for Laser Engineering, Osaka University, Yamadaoka, Suita 565-0871 Japan
Abstract
The LFEX laser has been developed for the basic
research <strong>of</strong> plasma heating in fast ignition scheme.
Its demonstration ensures the high potential <strong>of</strong> the LFEX
laser and various novel application fields has been
strongly discussed, such as lab-astrophysics, particle
physics and so on. We are planning the “Gekko-EXA”
laser, which is a sub-exa-watt ultrahigh peak power laser
where various advanced laser technologies developed in
the LFEX laser project are used. In this letter, the recent
status <strong>of</strong> the LFEX laser and the rough conceptual design
<strong>of</strong> “Gekko-EXA” are mentioned.
INTRODUCTION
The central ignition for the laser fusion has been
studied. The National Ignition Facility (NIF) in Lawrence
Liver More National Laboratory (LLNL) will
demonstrate it with a mega-joules class laser [1]. On the
other hand, the fast-ignition has been actively researched
in our institute. It reduces a total laser power to one tenth,
which improves not only a compactness <strong>of</strong> the laser
system but also a repeatability <strong>of</strong> laser operation and a
stability <strong>of</strong> the laser operation. A high peak power laser
with short pulse duration is necessary as a heating laser
for the fast ignition. The LFEX (Laser for Fusion
EXperiments)-Laser system has been developed to clarify
the heating mechanism [2]. The plasma heating
experiments have started by using the LFEX-Laser. In
Figure 1: Block diagram <strong>of</strong> the LFEX-Laser.
addition, the “Gekko-EXA” Laser with a higher peak
power has been under a conceptual design to improve the
fusion researches and to open the new high field
researches in the plasma physics.
LFEX-LASER
The LFEX laser generates 10 kJ pico-seconds pulse
energy with four beams by using a chirped-pulse
amplification (CPA) technique, which corresponds
to several peta-watts peak power. The laser system
consists <strong>of</strong> a front end, rod amplifier chains, a main
amplifier, and rear end (pulse compression), shown in fig.
1. A front end is a combination <strong>of</strong> a mode-locked fibre
oscillator, two pulse stretchers and three optical
parametric chirped-pulse amplification (OPCPA) stages.
The OPCPA stages are used instead <strong>of</strong> a conventional
regenerative amplifier to improve the temporal pulse
contrast. Much care <strong>of</strong> a beam transport with image relays
are taken not to make the beam quality inferior. The
maximum pulse energy is obtained up to 100 mJ in fig. 2.
The typical pulse duration is 3 ns with a 6 nm spectral
width at a 1053 nm centre wavelength. After four-pass
Nd:glass rod (φ50mm)-amplification (RA), the amplified
beam is divided into four beams and each beam
experiences two rod-amplifiers to obtain about 10 J pulse
energy. The main amplifier is a four-pass amplifier with a
Nd:glass slab amplifier and a long image relay. Eight
large slab glasses (46 cm x 81 cm x t4cm) are used for

Amplified Pulse Energy (mJ)
100
Horizontal (μrad)
10
1
0
12
10
8
6
4
2
0
-2
100
Figure 2: The obtained pulse energy at the threestages
OPCPA.
PV : 4.144!
RMS : 0.937
200
3rd Path
2nd Path
DFM75
each beam and the slab glass series are set side by side to
arrange 2 x 2 beams spatially. The obtained pulse energy
is 1.77 kJ/beam. The obtained spectral width is gainnarrowed
to 3.1 nm and the pulse duration is shortened at
1.45 ns. A highly excellent focusibility on the target is
strongly required for a heating laser. Two pairs <strong>of</strong> largeaperture
bimorph deformable mirror (DFM) and Shack-
Hartmann sensor are used for the beam wave front control
in the main amplifier chain, shown in fig. 3. A loop
program is used for automatic optimization <strong>of</strong> the wave
front correction. The obtained wave front after four-pass
amplification is 4.56λ (peak-valley) and 0.94λ (rms)
without the deformable mirrors. Using DFMs, they were
considerably improved at 0.95λ and 0.18λ, respectively,
1st Path 4th Path
4th Path
M0 DA400
SF400
FR400
M1 IMAP
AA
SF50
RA50
OS50 RA50 SF75 OS75
Numerical Cal.!
&!
Apply PZT voltage
OS125
PCS125
DFM125
Sensor Image
Shack-Haltmann
Numerical Cal.!
&!
Apply PZT voltage
Shack-Haltmann
Sensor Image
Figure 3: A schematic diagram <strong>of</strong> a deformable mirror system and the measured wavefront (a) without and
(b) with deformable-mirror compensation.
!"μ#$%
-4
Menu
-6
1st
-8
2nd
-10
-12
3rd
-12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12
Vertical (μrad)
300
(a) (b)
400
Pump Energy (mJ)
500
PV : 0.952!
RMS : 0.177
Figure 4: Pointing stability <strong>of</strong> the LFEX
laser before compression.
600
2x2
700
Focusing Optics
(2F)
Pulse Compressor (1F)
Beam Transfer
(2F ->1F)
1x4
s
2x2
From the Front
Figure 5: Diamond-trace pulse compressor with 16 meter-size
segmented gratings for the LFEX laser.

which were comparable to those before the main
amplification. The encircled energy was estimated to be
more than 70% at Fλ=2.5 with the observed far-field
patterns when 10% without the DFM correction. In
addition, all shots with kilo-joule pulse energy were
below 10 μrad in the pointing beam stability before the
compressor in fig. 4. The amplified pulse beams go into
Figure 6: Block diagram <strong>of</strong> the “Gekko-EXA” laser.
Figure 7: Arrangement plan <strong>of</strong> the “Gekko-EXA” laser.
the pulse compressor to be arranged with 1 x 4 beams
from 2 x 2. The pulse compressor is our original diamond
trace compressor with four large (91 cm x 42 cm) gratings
for one-beam diffraction, which results in sixteen gratings
for one-beam compression, shown in fig. 5. The obtained
minimum pulse duration is 2.2 ps. The LFEX laser is
gradually moving into the plasma experiments. The laser

demonstration encourages four-beams full operation for
the plasma heating.
GEKKO-EXA LASER
The “Gekko-EXA” laser is our great challenge for the
next generation <strong>of</strong> the high field science, the energetic
beam generation and their applications, and is under
conceptual design. The “Gekko-EXA” has two kinds <strong>of</strong>
operation mode, 1 PW at 100 Hz with a single beam and
0.2 EW as a single shot laser with two bundle beams,
respectively, shown in fig. 6. These output power is ultrashort
high-peak power laser with pulse duration <strong>of</strong> 10 fs.
An effective laser gain over the extremely wide spectral
width up to 500 nm is required for such short pulse
amplification. The laser system is, therefore, based on the
large-aperture optical parametric chirped-pulse
amplification (LA-OPCPA).
In the front end, the femto-second oscillator supplies
seed pulses for both the pump laser and the white light
generation in the LA-OPCPA. Because a temporal jitter
between the pump beam and the seed pulse should be
reduced as possible. The oscillator generates few cycle
pulses with the considerably broad spectral width. A seed
pulse for the pump laser is temporally stretched and
amplified by using fibre amplifiers after a pulse picker
and a pulse shaper.
A part <strong>of</strong> the amplified pulse is used as a seed pulse <strong>of</strong>
a repeatable amplification system, which based on highpower
diode-pumped solid-state lasers (DPSSL) with
cryogenic Yb:YAG ceramics[3] as a laser material. The
repeatable pump source generates green laser pulses with
70 J pulse energy at 100 Hz repetition rate after second
harmonic generation. The other part <strong>of</strong> the amplified
pulse is divided into four beams and each <strong>of</strong> them seeded
into the single-shot-based Nd:glass amplifier system.
Pulse energy increases up to 3 kJ per beam. A bundle <strong>of</strong>
four beams supplies 12 kJ (ω) in 1 ns and is frequencydoubled
at 8.4 kJ. Two bundles are prepared as pump
sources.
On the other hand, the generated white coherent light is
used as a seed pulse for the OPCPA system. Several
hundreds mili-joules pulse energy is obtained after fibre
and small-power (SP-) OPCPA stages. Then, in the LA-
OPCPA with partially deuterated DKDP crystals, kilojoule
class pulse energy is obtained with a more than 200
nm spectral width (FWHM). Using high-damagethreshold
gratings and chirped mirrors in large aperture,
0.1 EW (1 kJ/ 10 fs) pulse beam is generated and two set
<strong>of</strong> the pump beam and the seed pulse results in 0.2 EW.
There are some key issues in “Gekko-EXA” laser
design. Spectral dispersion compensation is considerably
important to improve the temporal pulse contrast ratio up
to 10 12 to reduce pre-pulses and a pedestal.
Technologically, there are two significant optics to be
developed, a grating and a large aperture chirped mirror,
which have both high optical damage strength in J/cm 2 in
an ultra-broadband up to 500 nm with a large aperture.
Figure 7 shows an arrangement plan for the “Gekko-
EXA” laser. The front end and the 1 PW/ 100 Hz laser
system are in another room, which is not shown here. The
kJ-pump-laser is at the next to the “Gekko” laser system.
In the additional planed building, the booster amplifiers
for 12 kJ, the second harmonic generator and the kJ-class
OPCPA system are on the same floor. Two pulse
compressors are in front <strong>of</strong> the target chamber.
REFERENCE
[1] G. M. Heestand, C. A. Haynam, P. J. Wegner, M. W.
Bowers, S. N. Dixit, G. V. Erbert, M. A. Henesian, M.
R. Hermann, K. S. Jancaitis, K. Knittel, T. Kohut, J. D.
Lindl, K. R. Manes, C. D. Marshall, N. C. Mehta, J.
Menapace, E. Moses, J. R. Murray, M. C. Nostrand, C.
D. Orth, R. Patterson, R. A. Sacks, R. Saunders, M. J.
Shaw, M. Spaeth, S. B. Sutton, W. H. Williams, C. C.
Widmayer, R. K. White, P. K. Whitman, S. T. Yang,
and B. M. Van Wonterghem, “Demonstration <strong>of</strong> highenergy
2ω (526.5 nm) operation on the National
Ignition Facility Laser System,” Applied Optics, Vol.
47 Issue 19, pp.3494-3499 (2008).
[2] N. Miyanaga, H. Azechi, K. A. Tanaka, T. Kanabe, T.
Jitsuno, J. Kawanaka, Y. Fujimoto, R. Kodama, H.
Shiraga, K. Knodo, K. Tsubakimoto, H. Habara, K.
Sueda, H. Murakami, N. Morio, S. Matsuo, N.
Sarukura, Y. Izawa, and K. Mima, “Technological
Challenge and Activation <strong>of</strong> 10-kJ PW Laser LFEX
for Fast Ignition at ILE,” Frontiers in Optics (FiO)
2008 paper: FWQ1.
[3] J. Kawanaka, Y. Takeuchi, A. Yoshida, S. J. Pearce, R.
Yasuhara, T. Kawashima, and H. Kan, “Highly
Efficient Cryogenically_Cooled Yb:YAG Laser”,
Laser Physics vol. 20, No. 1079-1084 (2010).

Abstract
X-ray Emission from Magnetars and Its Physical Interpretation
T. Enoto ∗ , KIPAC, Stanford University, CA, 94305-4085, USA †
K. Makishima, Dep. <strong>of</strong> Physics, The University <strong>of</strong> Tokyo ‡ , and Suzaku Magnetar Team
Recent astronomy has been accumulating evidence that a
subset <strong>of</strong> neutron stars has an ultra strong magnetic filed, ∼
10 10−11 T. These enigmatic sources are called magnetars,
and their field is believed to be stronger than those (∼ 10 8
T) <strong>of</strong> normal pulsars by 2–3 orders <strong>of</strong> magnitudes. Since
such a field exceeds the QED critical field, 4.4 × 10 9 T,
magnetars are a promising laboratory in the universe for
the high-field physics. Using the Japanese astrophysical
satellite Suzaku, we performed X-ray observations <strong>of</strong> ∼9
magnetars. We found a systematic trend <strong>of</strong> the wide-band
X-ray spectra, which is considered to be related with the
strong field physics (e.g., photon splitting).
MAGNETIC FIELD OF NEUTRON STARS
Celestial objects, especially compact objects such as
black holes and neutron stars (NSs), are ideal laboratories
to examine the extreme physics, which cannot be attained
through the ground experiments. For example, astronomical
observations have been revealed that NSs exhibit a
strong magnetic field (B ∼ 10 8 T), an intense photon field
(its maximum luminosity at L ∼ 10 38 erg s −1 ), and a high
gravity field (its gravitational redshift at z ∼ 0.2). Thanks
to recent sensitive detectors, we acquired an observational
approach to study these exotic physics.
Figure 1: An X-ray image <strong>of</strong> the Crab Nebula obtained by
the Chandra observatory [1]. The Crab pulsar locates at
the center <strong>of</strong> the Nebula.
NSs are born after a gravitational collapse <strong>of</strong> a massive
star. Since their mass and radii are estimated to be
M ∼ 1.4 − 2.1M⊙ and R ∼ 10 km, this compact star is
∗ JSPS (Japan Society for the Promotion <strong>of</strong> Science) Fellow, e-mail:
enoto@stanford.edu
† Address: Stanford University, Kavli Institute for Particle Astrophysics
& Cosmology, Physics Astrophysics Building, 452 Lomita Mall,
MC 4085, Stanford, CA 94305-4085, USA
‡ Address: Department <strong>of</strong> Physics,School <strong>of</strong> Science, The University
<strong>of</strong> Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo, 113-0033, Japan
believed to be supported by a degeneracy pressure <strong>of</strong> neutrons.
Electromagnetic waves from NSs have been detected
by various wavelengths with their stellar rotations. Therefore,
most NSs are called “pulsars”. For example, Figure
1 shows an X-ray image <strong>of</strong> the Crab Nebula, which was
formed at a historically recorded supernova in 1054. At the
center <strong>of</strong> this nebula, the Crab pulsar is rotating at its period
<strong>of</strong> P ∼ 33 ms and its period derivative <strong>of</strong> ˙ P ∼ 4.2×10−13 s s−1 . So far, pulsations and their derivatives have been
measured from more than 1500 normal NSs, which are
plotted on the P - ˙ P diagram in Figure 2, mainly distributed
around P ∼ 0.4 s and ˙ P ∼ 10−15 s s−1 . Since most
<strong>of</strong> normal pulsars exhibit a long-term timing stability, they
are though to be isolated NSs1 Pdot (s/s)
10 −8
10 −9
10 −10
10 −11
10 −12
10 −13
10 −14
10 −15
10 −16
10 −17
10 −18
10 −19
10 −20
10 −21
10 13 G
10 12 G
10 11 G
10 10 G
10 9 G
10 14 G
Bcr
10 15 G
10−3 10
0.01 0.1 1 10
−22
Period (s)
Figure 2: (a) A P - ˙
P diagram <strong>of</strong> pulsars, including
rotation-powered pulsars (black), SGRs (red stars),
and AXPs (blue circles) after [2], http://www.atnf.csiro.
au/research/pulsar/psrcat/. Dotted lines <strong>of</strong> negative slopes
represent constant magnetic field grids.
Following a standard pulsar model [3], emission from
normal isolated NSs can be powered by their rotational
energy loss. Let us assume the loss <strong>of</strong> the rotation energy,
d/dt(0.5IP 2 ) is eventually converted into a magnetic
dipole radiation (e.g., electromagnetic wave), where
I = 10 45 g cm 2 is a momentum <strong>of</strong> inertia <strong>of</strong> the NS. Then,
1 “Isolated” means that the NS is not a binary system (accretionpowered
pulsars), where mass accretion from a companion star make the
pulsation fluctuate.

Radio Pulsar Number
250
200
150
100
50
0
0
7 8 9 10 11 12 13 14 15 16
Magnetic Field Strength [Log B(Gauss)]
Figure 3: Magnetic field distribution <strong>of</strong> normal (radio) pulsars
(blue, peak at ∼ 10 12 G) from [2], accretion-powered
pulsars (green), and magnetars (red, peak at ∼ 10 15 G) [4].
1 T = 10 4 G.
the surface magnetic field <strong>of</strong> the NS can be estimated to be
B = 3.2 × 10 15√ (P ˙
P ) T. (1)
The evaluated fields are shown in Figure 3 and Fig. 2 as
dashed lines. Typical value <strong>of</strong> normal NSs, ∼ 10 8 T, is a
basis <strong>of</strong> the current emission model from pulsars, and the
model explains the observations so far.
ULTRA-STRONG FIELD OF MAGNETARS
During a last few decades, a new subclass <strong>of</strong> isolated
NSs is emerging on the upper right corner on the P - ˙ P
diagram, mostly observed only in the X-ray band. From
their peculiar properties, the one <strong>of</strong> them are named S<strong>of</strong>t
Gamma Repeaters (SGRs) and the others are Anomalous
X-ray Pulsars (AXPs). Both <strong>of</strong> them are slowly rotating
X-ray pulsars (P ∼ 2 − 10 s), showing large period derivatives
( ˙ P ∼ 10−12 − 10−9 s s−1 ). Using the above estimation
<strong>of</strong> the magnetic field, SGRs and AXPs should have an
ultra-strong magnetic field, B ∼ 1010−11 T (Fig. 3). This
field strength is stronger than those <strong>of</strong> normal pulsars by 2–
3 orders <strong>of</strong> magnitude, and even exceeds the QED critical
field, Bcr = m2 ec3 /¯he = 4.4 × 109 T, where the Landau
level <strong>of</strong> electrons in the magnetic field becomes equivalent
with the rest mass <strong>of</strong> an electron. If SGRs and AXPs have
indeed the ultra-strong field, they are promising targets to
examine the QED physics. As shown in Fig. 3, SGRs and
AXPs have a different field distribution from those <strong>of</strong> normal
rotation powered pulsars, theay are correctively called
“magnetars” [5, 6, 7].
There are accumulated evidence that magnetars have an
ultra-strong field. First <strong>of</strong> all, bright X-ray luminosity
(Lx ∼ 1035 erg s−1 ) <strong>of</strong> magnetars cannot be explained
by their spin-down power (L ∼ 1033 erg s−1 ). This is
a prominent difference from normal pulsar, and thus, the
emission <strong>of</strong> magnetars are believed to be sustained by their
huge stored magnetic energy. Secondly, magnetars exhibit
various burst activities, which cannot be seen from normal
pulsars so far. Sporadic burst emission occurres with a
8
6
4
2
Magnetar & Accretion Pulsar Number
Figure 4: A light curve <strong>of</strong> the giant flare from SGR 1806-20
in the 20–100 keV energy range, recorded by the RHESSI
satellite on 2004 December 27. The inset shows a precursor
which was recorded 142 s before the giant flare [8].
keV 2 cm -2 s -1 keV -1
1 mCrab
1806-20 (2007 Oct.)
1900+14 (2006 Apr.)
1547-54 (2009 (2009 Jan.) Jan.)
1841-04 (2006 (2006 Apr.) Apr.)
1708-40 (2009 (2009 Aug.) Aug.)
0501+45 (2008 (2008 Aug.) Aug.)
0142+61 (2007 (2007 Aug.) Aug.)
2259+58 2259+58 (2009 (2009 (2009 (2009 (2009 May) May) May) May) May)
1 10 100
Energy (keV)
Figure 5: X-ray spectra (νFν form) <strong>of</strong> the magnetars observed
by Suzaku with abbreviated names [12]. Individual
spectra are shown with <strong>of</strong>fsets, and are arranged in order
<strong>of</strong> increasing magnetic field from bottom (weak) to top
(strong). Blue horizontal lines indicate a 1 mCrab intensity.
short time scale (a few hundread ms – a few s). One prominent
example is a giant flare from SGRs, as demonstrated in
Figure 4. A current hypothesis speculates that these bursts
are emission from trapped fire balls on the magnetar surface,
presumably related with magnetic activities (e.g., reconnections).
Interestingly, the luminosities <strong>of</strong> these burst

sometimes exceed the Eddington luminosity (a maximum
permitted luminosity), and this excess is considered to originate
from a suppression <strong>of</strong> the Thomson cross section in
the high magnetic field. Thirdly, there is marginal evidence
<strong>of</strong> proton cyclotrons in the magnetar X-ray spectra, which
suggests B ∼ 10 15 G [9].
X-RAY EMISSION OF MAGNETARS
Persistent X-ray emission from magnetars have been extensively
observed in the ∼0.2–10 keV. In this energy band,
a thermal emission with its temperature <strong>of</strong> kT ∼ 0.5 keV is
considered to be emitted from the stellar surface. However,
through a new hard X-ray imaging with the INTEGRAL
satellite, some persistently bright magnetars were discovered
to emit a distinct hard-tail component which emerges
above ∼10 keV with an extremely hard photon index <strong>of</strong>
Γh ∼ 1 [10]. This unusual new component, though not yet
observed from all magnetars, is expected to provide a hint
to the high field physics in magnetars.
Using the Suzaku X-ray satellite [11], we performed
broad-band (0.8–70 keV) spectral analyses <strong>of</strong> the persistent
X-ray emission from 9 magnetars [12]. As shown in
Figure 5, the s<strong>of</strong>t thermal component was detected from all
<strong>of</strong> them (green lines), while the hard-tail component, dominating
above ∼10 keV, was detected at ∼1 mCrab 2 intensity
from 7 <strong>of</strong> them (red lines). In addition, as shown in this
figure, the hard-tail component has a tendency to become
stronger but s<strong>of</strong>ter towards sources with larger magnetic
fields. To quantitatively evaluate this trend, we employ the
1–60 keV fluxes <strong>of</strong> s<strong>of</strong>t-thermal and hard-tail components,
Fs and Fh, respectively, and then, calculate the hardness
ratio (HR) between these two components, ξ ≡ Fh/Fs. As
shown in Figure 6 (top), the HR is found to be tightly correlated
with the magnetic field B as
ξ = (0.09 ± 0.07) × (B/Bcr) 1.2±0.2
with a correlation coefficient <strong>of</strong> 0.873, over the range from
ξ ∼ 10 to ξ ∼ 0.1. On the other hand, as shown in Figure 6
(bottom), the photon index becomes s<strong>of</strong>ter toward stronger
field pulsars with ξ becoming larger.
Although several scenarios have been proposed [13, 14,
15, 16], the emission mechanism <strong>of</strong> the hard X-rays has
not yet been resolved. One <strong>of</strong> the biggest difficulties is
how to explain the extremely hard Γh ∼ 1 with its spectral
trend depending on B. A possible candidate <strong>of</strong> the emission
process is photon-splittings [17, 18]. In ultra-strong
magnetic fields exceeding Bcr, electron-positron pair cascades
are suppressed, while the photon splitting may be
dominant. In this case, gamma-rays from the surface may
repeatedly split into lower energy photons. This process
can also explain the differences in Γh among magnetars, in
such a way that higher fields objects will allow the photonsplitting
cascade to proceed down to lower energies, and
hence to make the continuum s<strong>of</strong>ter.
2 1 mCrab is one-thousandth <strong>of</strong> the Crab Nebula intensity, which is a
standard candle <strong>of</strong> the astronomy.
(2)
Hardness Ratio ξ = F h /F s
Hardness Ratio ξ = F h /F s
10
1
0.1
10
1
0.1
2259+58
(b)
0142+61
0501+45
(2009)
0501+45
(2008)
0142+61
1547-54
1900+14
1708-40
1841-04
10
Magnetic Field (G)
14 1015 0501+45
1806-20
1841-04
1900+14
1806-20
0 0.5 1 1.5 2
Photon index Γ h
1708-40
1547-54
Figure 6: (top) A correlation between the HR ξ and the
magnetic field B [12]. Green solid line represents the best
fit <strong>of</strong> equation (2). SGRs and AXPs are shown in red and
blue, respectively. (bottom) The HR ξ as a function <strong>of</strong> photon
indices Γh <strong>of</strong> the hard-tail component [12].
REFERENCES
[1] Hester, J. J., et al. 2002, Astrophys. J. Let., 577, L49
[2] Manchester, R. N., et al., 2005, VizieR Online Data Catalog
[3] Goldreich, P., & Julian, W. H. 1969, Astrophys. J., 157, 869
[4] Enoto T., et al., 2009, Suzaku <strong>Conference</strong> 2009 Proceedng
[5] Duncan, R. C., & Thompson, C. 1992, ApJL, 392, L9
[6] Woods, P. M., & Thompson, C. 2006, Compact stellar X-ray
sources, 547
[7] Mereghetti, S. 2008, Astron. and Astrophy. Rev., 15, 225
[8] Hurley, K., et al. 2005, Nature, 434, 1098
[9] Ibrahim, A. I., et al., 2003, ApJL, 584, L17
[10] Kuiper, L., et al., 2006, Astrophysical Journal, 645, 556
[11] Mitsuda, K., et al. 2007, PASJ, 59, 1
[12] Enoto, T., et al., 2010, ApJL, 722, L162
[13] Heyl, J. S., & Hernquist, L. 2005, MNRAS, 362, 777
[14] Thompson, C., & Beloborodov, A. M. 2005, ApJ, 634, 565
[15] Beloborodov, A. M., & Thompson, C. 2007, ApJ, 657, 967
[16] Baring, M. G., & Harding, A. K. 2001, ApJ, 547, 929
[17] Baring, M. G., & Harding, A. K. 2001, ApJ., 547, 929
[18] Enoto, T., et al. 2010, Ph.D thesis, The University <strong>of</strong> Tokyo

Abstract
THE NIELSEN-OLESEN INSTABILITIES IN THE GLASMA
H. Fujii, Institute <strong>of</strong> Physics, University <strong>of</strong> Tokyo, Komaba, Tokyo
K. Itakura, Institute <strong>of</strong> Particle and Nuclear Studies (IPNS), KEK, Ibaraki
A. Iwazaki, Int’l Economics and Politics, Nishogakusha University, Kashiwa, Chiba
In the framework <strong>of</strong> the Color Glass Condensate, strong
color electric and magnetic fields are expected to appear in
the early transient stage <strong>of</strong> ultrarelativistic heavy-ion collisions.
We show that this configuration with longitudinally
polarized strong gauge fields has the Nielsen-Olesen
instability[1], and discuss its relevance to the initial stage
<strong>of</strong> the heavy ion collisions[2, 3, 4].
STRONG COLOR FIELD IN GLASMA
In ultrarelativistic heavy-ion collisions (HIC), two heavy
nuclei at nearly the light speed smash into each other to
generate a dense medium <strong>of</strong> liberated quarks and gluons.
Studying properties <strong>of</strong> this extremely dense medium is one
<strong>of</strong> the main subjects in subnuclear physics.
The Color Glass Condensate picture
Each <strong>of</strong> incident nuclei is seen as an assembly <strong>of</strong> partons
(quarks and gluons) with longitudinal momentum fraction
x = p/( √ s/2) in HIC at very large center-<strong>of</strong>-mass energy
√ s. At high energies, particle productions are dominated
by collisions <strong>of</strong> the partons <strong>of</strong> small x such as x √ s/2 ∼
mπ with mπ the pion mass because these small-x partons,
predominantly gluons, are abundant through QCD
bremsstrahlung processes from the larger-x colored partons
within the nuclear wavefunction and because the QCD
cross section is larger at the smaller momentum scale.
Such dense small-x gluon components <strong>of</strong> the nuclear
wavefunction should be described as classical gauge fields
generated from larger-x color source, and the effective
theory for small-x degrees <strong>of</strong> hadronic wavefunction is
known as the Color Glass Condensate (CGC) framework.
The field strength is characterized by a momentum scale,
Q 2 s(x), called saturation scale, which emerges from nonlinear
effects <strong>of</strong> QCD and estimated empirically as about
1 (GeV/c) 2 in HIC at √ s = 200 GeV. Keep in mind that
any hadrons (nuclei) are color singlet objects and the (transversely
polarized) color fields exist only within the hadron
as fluctuations whose lifetime is Lorentz-elongated.
Since the particle productions are dominated by the
small-x partons in the high-energy limit, nucleus-nucleus
collisions then may be effectively modeled as CGC-CGC
collisions where the classical fields coupled with the charge
sources on the light-cone intersect with each other. It is
shown that crossing the two CGCs with each other results
in longitudinally polarized color electric and magnetic
fields due to non-Abelian nature <strong>of</strong> QCD. The term
x –
τ = const
t
η=const
A µ
=pure gauge Aµ =pure gauge
(1) (2)
CGC (1)
QGP
Glasma
CGC (2)
Figure 1: CGC-CGC collision. Strong fields confined in
each nucleus before the collision, and boost-invariantly extend
between the two receding nuclei after the collision.
τ = √ t 2 − z 2 and η = (1/2) ln((t + z)/(t − z)).
Glasma[5] was recently coined for this transient configuration
<strong>of</strong> strong fields produced from the CGC, which evolves
toward the QCD plasma. This is a longitudinal-boost invariant
configuration <strong>of</strong> very strong classical fields. See
Fig. 1, where incoming nuclei are modeled as two CGCs.
Expanding flux tube
With a simple Abelian Ansatz for a purely magnetic
flux tube with longitudinal polarization, one can find an
analytic solution[3], which stretches along in the z direction
and expands transversely, as shown in Fig. 2. There
2
1
-2 -1 1 2
-1
-2
Qsr
x +
z
Qsz
Figure 2: Magnetic flux tube pr<strong>of</strong>ile at Qsτ = 1, 2.
are only two nonvanishing components <strong>of</strong> field strengths,
longitudinal Bz and transverse ET . The time evolution
<strong>of</strong> the field strengths integrated over the transverse plane
at z = 0 is plotted in Fig. 3. The stress tensor at

1
0.8
0.6
0.4
0.2
B 2 z
E 2 T
0.5 1 1.5 2 2.5 3
Qsτ
Figure 3: Evolution <strong>of</strong> the field strength as a function <strong>of</strong>
Qsτ. Evolution <strong>of</strong> a purely electric flux tube is obtained by
exchanging E and B.
τ = 0+ is found as diag(E, E, E, −E) with energy density
E, which is quite different from the equilibrium form
diag(E, E/3, E/3, E/3). Note that the longitudinal pressure
∝ E 2 T − B2 z is always negative and only approaches
to zero at later times, while the transverse pressure ∝ B 2 z is
positive.
How will this highly anisotropic configuration <strong>of</strong> strong
fields evolve into locally thermalized plasma — quark
gluon plasma (QGP)? Here we argue that the initial
glasma configuration with color magnetic fields has unstable
modes, which play a role in the system evolution.
NIELSEN-OLESEN INSTABILITY
Since late 70s it has been known that a uniform magnetic
field configuration in SU(2) gauge theory is unstable — the
Nielsen-Olesen (N-O) instability[1], which we review here.
We decompose the SU(2) gauge fields into diagonal and
<strong>of</strong>f-diagonal parts, and assume a homogeneous magnetic
field in the 3rd color direction:
L = − 1
4 fµνf µν − 1
2 |Dµφν − Dνφµ| 2 + igf µν φ ∗ µφν
+ 1
4 g2 (φµφ ∗ ν − φνφ ∗ µ) 2 , (1)
where Aµ ≡ A 3 µ, φµ ≡ (A 1 µ + iA 2 µ)/ √ 2 and fµν =
∂µAν − ∂νAµ. D = ∂µ + igAµ is the covariant derivative
w.r.t. Aµ. Regarding Aµ gauge theory, the <strong>of</strong>f-diagonal
parts φµ behave as charged massless vector fields. They
form the Landau levels EN = (2N +1)gB (N = 0, 1, · · · )
in a constant magnetic field background B within the linear
approximation with φ being small fluctuations. However,
the most important observation is that the third term
in Eq. (1) gives the Zeeman splitting ±2gB for spin ±, resulting
in the mode energy
ω 2 = p 2 z + (2N + 1)gB ± 2gB , (2)
where we see that the lowest mode has the mass term with
a wrong sign −gB. This is the N-O instability. The growth
rate <strong>of</strong> the instability depends on the longitudinal momentum
pz as γ = √ gB − p 2 z. The instability appears even at
pz = 0, which is qualitative difference from the pz dependence
<strong>of</strong> the Weibel instability known in plasma physics.
Figure 4: Schematic picture for glasma with fluctuations,
between two receding nuclei.
GLASMA INSTABILITIES
As previously mentioned, the boost-invariant background
never acquires positive longitudinal pressure,
which is necessary for system isotropization. We consider
here small fluctuations φ around the background field in the
τ-η coordinates. For simplicity, we neglect the transverse
structure <strong>of</strong> the collision event and assume transversely a
uniform magnetic background B, even though, more realistically,
the background field should have transverse structures
characterized by the scale Qs, and an electric field in
the background is also allowed.
By combining the fluctuating fields φµ(τ, η, x⊥) as
φ ± ≡ 1
√ 2 (φ1 ± iφ2) , (3)
we find the mode equation for φ ± [2, 3],
(
)
˜φ ± = 0 , (4)
1
τ ∂τ (τ∂τ ˜ φ ± ) +
EN ± 2gB + ν2
τ 2
where ˜ φ is the mode with N specifying the Landau level
and with ν the momentum conjugate to η. There is another
equation for φ ∗ which has the opposite charge to φ.
• First we note that the spin-+ mode with ν = 0 in the
lowest Landau level N = 0 has a negative mass −gB
just as in the previous section. There exists the N-O
instability in the glasma.
• Nonzero ν modes are stabilized by the term (ν/τ) 2 .
For ν is dimensionless, physical longitudinal momentum
corresponds to pz ∼ ν/τ[6], which is large at
small τ with fixed ν. The maximum value νmax for
the instability is determined by
νmax = τ √ gB ∼ τQs . (5)
In fact, these points were observed in a numerical simulation
for the glasma[7], although the proportionality constant
in the second point was found very small there. At
the same time, we notice that the expansion generally delays
the appearance <strong>of</strong> the instability.

GLASMA INSTABILITIES IN A BOX
Recently, classical statistical simulations <strong>of</strong> non-Abelian
gauge theories were performed in a box without expansion
but with very anisotropic initial conditions[8], motivated by
the idea <strong>of</strong> the glasma. They found the unstable behavior
whose growth rate is shown in Fig. 5 as “primary.” Furthermore,
they claimed that there is a secondary instability
which is triggered by the first (primary) instability.
The pz dependence <strong>of</strong> the growth rate γ <strong>of</strong> the primary
instability looks consistent with the N-O instability; γ is
nonzero at zero pz and decreases as pz increases. On the
other hand, the growth rate γ <strong>of</strong> the secondary increases
with pz, and extends to the larger pz region. In Ref. [8], the
secondary was discussed as nonlinear effects <strong>of</strong> the primary
instability in terms <strong>of</strong> the re-summed self-energy diagrams.
Here, we attempt to interpret their findings from the viewpoint
<strong>of</strong> the N-O instability scenario[4].
The numerical simulation starts with the initial condition
very smooth in the z direction but randomized in the x-y directions
with the scale ∆ ∼ Qs. The color electric fields
are set to zero at the initial time. This configuration will
allow tube structures <strong>of</strong> the color magnetic fields along the
z direction, and therefore the N-O instability is expected
there as the primary[4]. If the magnetic field point to the
3rd direction in the SU(2) color space, the <strong>of</strong>f-diagonal
components φ ± w.r.t. the magnetic field are amplified in
time due to the N-O instability.
Because φ ± are charged fields, they can also induce
the color electric current along the magnetic field. Although
the direction and magnitude <strong>of</strong> the electric current
cannot be determined within the linear analysis, one
may roughly estimate the strength <strong>of</strong> the current as J z ∼
O(g · √ gB · B/g) = O((gB) 3/2 /g) ∼ O(Q3 s/g) with
the momentum scale √ gB ∼ Qs and the field amplitude
φ ± ∼ √ B/g. According to the Ampère law, the color
magnetic field <strong>of</strong> order O(Q2 s/g) will be generated around
this current. Let us examine the consequence <strong>of</strong> this azimuthal
magnetic field.
Following the same analysis as the N-O instability but in
the cylindrical coordinates with a constant background Bθ ,
we find the mode equation for the <strong>of</strong>f-diagonal component
ϕ ± = (φz ± iφr )/ √ 2 as
{
− 1 d d
r
r dr dr + ( pz − gB θ r )2 1
+ − 2gBθ} ϕ
2r2 − (r)
=ω 2 ϕ − (r) , (6)
assuming that ϕ + is small compared to ϕ − and setting
ϕ + = 0. The parameter pz shifts the potential minimum
outward. Thus, for larger pz the N-O instability appears as
in the original case, while 1/r 2 term cannot be neglected
for smaller pz — the growth rate is approximately
−ω 2 ≡ γ 2 ∼ gB θ
(
1 − gBθ
4p2 )
z
. (7)
This gives rise to qualitatively the same behavior as γ for
the secondary shown in Fig. 5.
γ / ε 1/4
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
0.0 0.5 1.0 1.5 2.0 2.5
pz / ε 1/4
prim. 96 3
prim. 128 3
sec. 96 3
sec. 128 3
Figure 5: Growth rates <strong>of</strong> the primary and secondary instabilities
vs longitudinal momentum pz, observed in Ref. [8]
(in units <strong>of</strong> the initial energy density ε).
OUTLOOK
In ultrarelativistic HIC, a system <strong>of</strong> extremely strong
color fields, glasma, is generated. Non-equilibrium dynamics
<strong>of</strong> the glasma itself is very interesting and at the same
time is quite important to understand the production mechanism
<strong>of</strong> QGP. We have discussed the Nielsen-Olesen instabilities
in the context <strong>of</strong> HIC, and shown that results <strong>of</strong>
the numerical simulations[7, 8] can be understood nicely
from this viewpoint.
The Nielsen-Olesen instability certainly exists in non-
Abelian gauge theories. Its relevance to “thermalization”
mechanism in HIC, however, is under debate. The Weibel
instability is another possibility among others. Towards understanding
the early stage <strong>of</strong> HIC, one needs obviously to
go beyond the linear analysis and has to study the nonlinear
dynamics <strong>of</strong> gauge field. To this end, it is indispensable to
pursue more detailed numerical simulations together with
theoretical physics insights.
REFERENCES
[1] N.K. Nielsen and P. Olesen, Nucl. Phys. B 144 376 (1978).
[2] A. Iwazaki, Prog. Theor. Phys. 121 809 (2009).
[3] H. Fujii and K. Itakura, Nucl. Phys. A 809 88 (2008).
[4] H. Fujii, K. Itakura and A. Iwazaki, Nucl. Phys. A 828 178
(2009).
[5] T. Lappi and L. McLerran, Nucl. Phys. A 772 200 (2006).
[6] N. Tanji, arXiv:arXiv:1010.4516 [hep-ph].
[7] P. Romatschke and R. Venugopalan, Phys. Rev. Lett. 96
045011 (2006).
[8] J. Berges, S. Scheffler and D. Sexty, Phys. Rev. D 77 034504
(2008).

First order quantum correction to the Larmor radiation ∗
Gen Nakamura
Department <strong>of</strong> Physical Science, Hiroshima University, Higashi-Hiroshima 739-8526, Japan
Abstract
First-order quantum correction to the Larmor radiation is
investigated on the basis <strong>of</strong> the scalar QED on a homogeneous
background <strong>of</strong> a time-dependent electric field, which
is a generalization <strong>of</strong> a recent work by Higuchi and Walker
so as to be extended for an accelerated charged particle in
a relativistic motion. We obtain a simple approximate formula
for the quantum correction in the limit <strong>of</strong> the relativistic
motion when the direction <strong>of</strong> the particle motion is
parallel to that <strong>of</strong> the electric field.
INTRODUCTION
The Larmor radiation is the classical radiation from a
charged particle in an accelerated motion [2]. In the recent
paper by Higuchi and Walker [3], the quantum correction
to the Larmor radiation is investigated on the basis
<strong>of</strong> the scalar quantum electrodynamics (QED). In their
approach, the mode function for the complex scalar field
is constructed with the Wentzel-Kramers-Brillouin (WKB)
approximation, in a form expanded with respect to ¯h. In a
series <strong>of</strong> Higuchi and Martin’s work [4, 5, 6] (see also references
therein), it has been well understood that the mode
function reproduces the classical Larmor formula when the
radiation energy is evaluated at the order <strong>of</strong> ¯h 0 . The firstorder
quantum correction to the classical Larmor radiation
is evaluated at the order <strong>of</strong> ¯h in Ref. [3], though the investigation
is limited to the non-relativistic motion <strong>of</strong> the
charged particle.
In our work, we consider a simple generalization <strong>of</strong>
Higuchi and Walker’s work [3], in order to investigate the
case a relativistic motion <strong>of</strong> an accelerated charge. Assuming
a homogeneous but time-varying background <strong>of</strong> electric
field, we derive a formula for the radiation energy <strong>of</strong>
the order <strong>of</strong> ¯h, the first-order correction due to the quantum
effect. This generalized formula is applicable to the
accelerated charge in a relativistic motion, and we focus
our investigation on the first-order quantum correction to
the Larmor radiation in the limit <strong>of</strong> the relativistic motion.
FORMULATION
We consider the scalar QED with the action,
∫
S = dtd 3 x
×
[
(Dµφ) † D µ φ − m2
¯h 2 φ† φ − 1
∗ This presentation is based on Ref.[1]
4µ0
FµνF µν
]
, (1)
φ
Pi
k
Pf
Figure 1: Feynman Diagram for the process.
where Dµ = (∂/∂x µ + ieAµ/¯h), e and m are the charge
and the mass <strong>of</strong> the massive scalar field, respectively, and
µ0 is the magnetic permeability <strong>of</strong> vacuum. We work in
the Minkowski spacetime, but consider the homogeneous
electric background field E(t), which is related to the vector
potential by Aµ = (0, A(t)) and ˙ A(t) = −E(t), where
the dot denotes the differentiation with respect to the time.
The equation <strong>of</strong> motion <strong>of</strong> the free scalar field yields
( ∂ 2
∂t 2 + (p − eA(t))2 + m 2
¯h 2
γ
φ
)
ϕp(t) = 0, (2)
where ϕp(t) is the coefficient <strong>of</strong> the Fourier expansion <strong>of</strong>
the field, i.e., the mode function. Using the mode function,
which is normalized so as to be ˙ϕ ∗ pϕp − ϕ ∗ p ˙ϕp = i, the
quantized field is constructed as
φ(x) =
√ ¯h
L 3
∑
p
(
ϕp(t)bp + ϕ ∗ −p(t)c †
)
−p e ip·x/¯h , (3)
where L 3 is the volume <strong>of</strong> the space, the creation and annihilation
operators satisfy the commutation relations,
[bp, b †
p ′] = δp,p ′, [bp, bp ′] = [b† p, b †
′] = 0, (4)
and the same relations hold for cp and c † p. We also quantize
the free electromagnetic field as,
Aµ =
√ µ0¯h
L 3
∑
λ=1,2
∑
k
ɛ λ µ
p
( −ikt e
√ a
2k λ )
k + h.c. e ik·x , (5)
where ɛ λ µ denotes the polarization vector, and a λ†
k and aλ k
are the creation and annihilation operators which satisfy the
following commutation relation,
[a λ k, a λ′ †
k ′ ] = δ λλ′
δk,k ′. (6)
We consider the process, in which one photon is emitted
from a charged particle, as shown in Fig. 1. Note that
this process is prohibited without the background electric

field because <strong>of</strong> the Lorentz invariance <strong>of</strong> the Minkowski
spacetime, which ensures existence <strong>of</strong> the frame that the
charged particle is at rest. However, on the electric field
background, we have the radiation energy from the process,
which can be evaluated, as follows. Using the in-in
formalism [7, 8], we may compute the radiation energy at
the lowest order <strong>of</strong> the coupling constant,
E = ∑
∫
λ
∑
∫
−2
= ¯h
λ
d 3 k¯hk〈a λ†
k aλ k〉
d 3 ∫ ∞ ∫ ∞
k¯hkRe dt2
−∞
× in|HI(t1)a λ†
k aλkHI(t2)|in dt1
−∞
, (7)
where we adopted the range <strong>of</strong> the integration from the infinite
past to the infinite future, and |in〉 denotes the initial
state, which we choose as one charged particle state with
the momentum pi, i.e., |in〉 = b † pi |0〉, and
HI(t) = − ie
∫
d
¯h
3 xA µ
{(
× ∂µ − ie
¯h Āµ
)
φ † φ − φ †
(
∂µ + ie
¯h Āµ
) }
φ .(8)
In order to evaluate the quantum correction, we consider
the expansion in terms <strong>of</strong> a power series <strong>of</strong> ¯h. Up to the
order <strong>of</strong> O(¯h), we have
where we defined
E = E (0) + E (1) + O(¯h 2 ), (9)
E (0) =
(( 2 d x
×
dξ2 e2
(4π) 2ɛ0 )2
∫ ∫
dΩˆ k dξ
(
− ˆk · d2x dξ2 )2)
. (10)
The expression (10) yields the classical formula <strong>of</strong> the Larmor
radiation from a charged particle. The first-order quantum
correction <strong>of</strong> the order <strong>of</strong> ¯h is described by
E (1) =
e2¯h (4π) 3 ∫ ∫ ∫
dΩˆ k dξ dξ
ɛ0
′ 1
ξ − ξ ′
×
{ (
d d
−
dξ dξ ′
)
d d
dξ dξ ′
[( (ˆk
dx
)(
· ˆk
dx
·
dξ
′
dξ ′
)
− dx dx′
·
dξ dξ ′
)(
ˆk · dx dτ
dt dt + ˆ k · dx′
dt ′
dτ ′
dt ′
+
)]
2 d2
dξ2 d2 dξ ′2
[( (ˆk
dx
)(
· ˆk
dx
·
dξ
′
dξ ′
)
− dx dx′
·
dξ dξ ′
×
)
∫ ξ(t) ′′
′′ dτ
dξ
dξ ′′
( (
1 − ˆk · dx′′
dt ′′
) 2)] }
, (11)
ξ ′ (t ′ )
where we follow the notations in Ref.[1]
APPROXIMATE FORMULAS
In the non-relativistic limit, where the velocity v =
dx/dt is small enough compared with the velocity <strong>of</strong> light,
|v| ≪ 1, Eqs. (10) and (11) reduce to
E (0) e
=
2 ∫
dt ˙v(t) · ˙v(t), (12)
6πɛ0
E (1) =
e 2 ¯h
6π 2 ɛ0m
∫ ∫
dt dt ′
× ¨v(t) · ˙v(t′ ) − ˙v(t) · ¨v(t ′ )
t − t ′ , (13)
respectively. Eq. (13) was found for the first time by
Higuchi and Walker in Ref. [3]. In the case <strong>of</strong> the periodic
electric field, |E| = E0 sin ωt, where E0 is a constant,
we have the periodic acceleration, | ˙v| = (eE0/m) sin ωt.
Then,
dE (0)
dt = e4E2 0
m2 dE (1)
dt = −¯he4 E2 0
m2 sin 2 ωt
,
6πɛ0
(14)
ω
.
12πɛ0m
(15)
After taking an average over a long time-duration, we have
E (1)
E
¯hω
= − , (16)
(0) mc2 where c is the light velocity, which is restored here. The
quantum effect becomes important when the time scale <strong>of</strong>
the acceleration multiplied by c is comparable to the Compton
wavelength, namely, when the wave-like feature <strong>of</strong> the
particle appears.
Next, let us consider the relativistic limit, |pi| ≫
|eA|, m. For simplicity, we consider the case when the
direction <strong>of</strong> the particle motion is always parallel to that <strong>of</strong>
the background electric field, i.e., v ∝ A. Namely, we consider
the case when the directions <strong>of</strong> the particle’s motion
and the background electric field are parallel at any moment,
and adopt this direction as the z axis. Then, we may
write A = (0, 0, A(t)), A ˙ = (0, 0, −E(t)), v = (0, 0, v),
and pi = (0, 0, pi). In this case, we have
E (0) = 1 m
6πɛ0
4e4 p6 i
∫
dt
˙
A 2 (t)
(1 − v2 . (17)
) 3
We consider the case, pi ≫ |eA|, m. We also assume
|A| ∼ | ˙ A/ω| ∼ | Ä/ω2 |, where 1/ω is a time-scale <strong>of</strong> timevarying
background electric field. In this relativistic limit,
we have
E (1) − e4¯h 3(2π) 2 m
ɛ0
2
p5 ∫ ∫
dt dt
i
′
×
1
(1 − ¯v 2 ) 3
Ä(t) ˙ A(t ′ ) − ˙ A(t) Ä(t′ )
t − t ′ . (18)
In the case <strong>of</strong> the periodic background <strong>of</strong> the electric
field, ˙
A(t) = −E0 sin ωt, where E0 is a constant, we have
dE (0)
dt = e4 m 4
6πɛ0p 6 i
E2 0 cos2 ωt
(1 − v2 , (19)
) 3

dE (1)
dt = ¯he4 m 2
12πɛ0p 5 i
E2 0ω
(1 − v2 . (20)
) 3
After averaging over sufficiently long time-duration, we
have
E (1)
pi
=
E (0) mc
¯hω
. (21)
mc2 Note that the quantum correction E (1) is positive, which is
a contrast to the non-relativistic case.
For the radiation from an electron in a periodic electric
field, e.g., by a laser field, Eq. (21) is estimated as
E (1)
∼ 2.6 × 10−3
E (0)
×
(
pic
)
GeV
( mc2 )−2 (
0.5MeV
ω
10 15 s −1
)
,(22)
where ω ∼ 10 15 s −1 corresponds to an X-ray laser. The
quantum effect becomes significant when the electron kinetic
energy reaches TeV scale. The above formula is
derived under the condition, pi ≫ |eA|, m. For a periodic
electric field <strong>of</strong> large amplitude, pi ∼ |eA|, the
condition <strong>of</strong> the relativistic motion cannot be always guaranteed,
because the physical momentum might become
|pi − eA| ∼ m. In this case, it is difficult to express the
quantum correction in a simple analytic form. We need a
more general treatment including fully numerical calculation,
because the non-locality plays an important role. Potentially,
there is a lot <strong>of</strong> room for discussion about how to
detect the quantum effect <strong>of</strong> the Larmor radiation experimentally,
but this is out <strong>of</strong> scope <strong>of</strong> the present work.
SUMMARY
In this work, we obtained the general formula for the
first-order quantum correction to the Larmor radiation from
a charged particle moving in spatially homogeneous timedependent
electric field. This formula reproduces the same
result as that in Ref. [3], in the limit <strong>of</strong> a non-relativistic
motion <strong>of</strong> the charged particle. Our result is useful to investigate
the case <strong>of</strong> a relativistic motion.
By applying the formula to the cases <strong>of</strong> a periodic acceleration,
it was demonstrated that the leading quantum
effect enhances the radiation in the relativistic limit and
that it decreases in the non-relativistic limit. This quantum
effect will become important when the incident kinetic
electron energy approaches TeV scale with an x-ray laser.
This situation is the same in some possible function <strong>of</strong> acceleration
cases. The details <strong>of</strong> our work can be found in
Ref.[1].
REFERENCES
[1] K. Yamamoto, G. Nakamura, arXiv:1012.5182, accepted for
publication in Phys. Rev. D
[2] J. D. Jackson, Classical Electrodynamics (Wiley, 1998)
[3] A. Higuchi and P. J. Walker, Phys. Rev. D 80 105019 (2009)
[4] A. Higuchi and G. D. R. Martin, Found. Phys. 35 1149
(2005)
[5] A. Higuchi and G. D. R. Martin, Phys. Rev. D 73 025019
(2006)
[6] A. Higuchi and G. D. R. Martin, Phys. Rev. D 74 125002
(2006)
[7] S. Weinberg, Phys. Rev. D 72 043514 (2005)
[8] P. Adshead, R. Easther and E. A. Lim, Phys. Rev. D 80
083521 (2009)
[9] H. Nomura, M. Sasaki and K. Yamamoto JCAP 0611 013
(2006)
[10] P. Chen, T. Tajima, Phys. Rev. Lett. 83 256 (1999)
[11] R. Schutzhold, G. Schaller, D. Habs, Phys. Rev. Lett. 97,
121302 (2006) ;Erratum-ibid. 97 139902 (2006)
[12] L.C. B. Crispino, A. Higuchi, and G. E. A. Matsas, Rev.
Mod. Phys. 80 787 (2008)
[13] S. Iso, Y. Yamamoto, S. Zhang, arXiv:1011.4191
[14] A. Higuchi and P. J. Walker, Phys. Rev. D 79 105023 (2009)
[15] R. Kimura, G. Nakamura, and K. Yamamoto,
arXiv:1101.4699, accepted for publication in Phys.
Rev. D
[16] N. D. Birrell and P. C. W. Davies, Quantum fields in curved
space (Cambridge University Press ,1982)

FAST VACUUM DECAY INTO PARTICLE PAIRS
IN STRONG ELECTRIC AND MAGNETIC FIELDS ∗
Y. Hidaka, T. Iritani, and H. Suganuma,
Department <strong>of</strong> Physics, Kyoto University, Kitashirakawa Oiwakecho, Sakyo, Kyoto 606-8502, Japan
Abstract
We discuss fermion pair productions in strong electric
and magnetic fields. We point out that, in the case <strong>of</strong> massless
fermions, the vacuum persistency probability per unit
time and volume is zero in the strong electric and magnetic
fields, while it is finite when the magnetic field is absent.
The contribution from the lowest Landau level (LLL) dominates
this phenomenon. We also discuss dynamics <strong>of</strong> the
vacuum decay, using an effective theory <strong>of</strong> the LLL projection,
taking into account the back reaction.
INTRODUCTION
Dynamics in strong fields has been an interesting subject
in theoretical physics. Recently, this subject is being
paid attention also in the experimental physics <strong>of</strong> creation
<strong>of</strong> the quark gluon plasma. In high-energy heavy-ion collision
experiments, at the so-called Glasma stage [1] just
after the collision, longitudinal color electric and magnetic
fields are expected to be produced in the context <strong>of</strong> the
color glass condensate <strong>of</strong> order 1–2 GeV in RHIC and 5
GeV in LHC. In the peripheral collision, a strong magnetic
field <strong>of</strong> order 100 MeV would be induced. The question is
how the strong fields decay and the system is thermalized.
In this work, we concentrate on how the strong fields decay
into particles. For this purpose, we first briefly review
the Schwinger mechanism in the coexistence <strong>of</strong> electric and
magnetic fields. We will point out that the vacuum immediately
decays in the case <strong>of</strong> massless fermion and nonzero
E and B. For simplicity, we consider the case that the
electric and magnetic fields are covariantly constant [2],
i.e., [Dµ, E] = [Dµ, B] = 0, where Dµ = ∂µ − igAµ
is the covariant derivative with the gauge field Aµ. The
electric and magnetic fields are defined as E i = F i0 and
B i = −ɛ ijk Fjk/2 with Fµν = i[Dµ, Dν]/g. This is a generalization
<strong>of</strong> constant fields in QED, ∂µE = ∂µB = 0, to
the non-Abelian fields. For the covariantly constant fields,
all the components <strong>of</strong> E and B can be diagonalized to be
constant matrices in color space by a gauge transformation.
Without loss <strong>of</strong> generality, one can also set E = (0, 0, E)
and B = (0, 0, B) by choosing an appropriate Lorentz
frame and the coordinate axis.
∗ This work was supported by the Grant-in-Aid for the Global COE
Program “The Next Generation <strong>of</strong> Physics, Spun from Universality and
Emergence” from the Ministry <strong>of</strong> Education, Culture, Sports, Science and
Technology (MEXT) <strong>of</strong> Japan.
SCHWINGER MECHANISM
The vacuum decay in an electric field was discussed by
[3, 4]. Consider the vacuum persistency probability, which
is defined by
|〈Ωout|Ωin〉| 2 = exp(−V T w), (1)
where V and T are infinite space volume and time length.
|Ωin〉 and |Ωout〉 are the in-vacuum and the out-vacuum defined
at t = −T/2 and t = T/2, respectively. If the vacuum
is unstable, w has a nonzero value, while, if the vacuum
is stable, w vanishes. Therefore, w denotes magnitude
<strong>of</strong> the vacuum decay per unit volume and time. When w is
small, |〈Ωout|Ωin〉| 2 ≈ 1 − V T w, so that w is regarded as
the pair production probability per unit volume and time.
For QCD, the analytic formula <strong>of</strong> w for the quark-pair
creation in the covariantly constant is given by [2]
w =
∞∑
n=1
tr g2 EB
4π 2
1
n coth(πnBE−1 )e −nπm2 / √ g 2 E 2
,
(2)
where m denotes the quark-mass matrix and the trace is
taken over the indices <strong>of</strong> color and flavor. This is a non-
Abelian extension <strong>of</strong> the following formula for QED [5]:
w =
∞∑
n=1
e 2 EB
4π 2
1
n coth(πnBE−1 )e −nπm2 / √ e 2 E 2
, (3)
with the QED coupling constant e(> 0). Note that the
fermion pair creation formalism in the covariantly constant
fields in QCD is similar to that in QED, so that we hereafter
give the formula for QED, where we set E ≥ 0 and B ≥ 0
by a suitable axis choice and the parity transformation.
In the absence <strong>of</strong> the magnetic field, this formula reduces
to the well-known result,
w =
∞∑
n=1
e 2 E 2
4π 3
1
n 2 e−nπm2 /(eE) . (4)
If the masses are zero, w has a finite value <strong>of</strong> w =
e 2 E 2 /(24π). The situation changes if the magnetic field
exists. From Eq. (3), w diverges in the presence <strong>of</strong> the magnetic
field. To see this, summing over all modes in Eq. (3),
we obtain for small m as
w e2EB 4π2 ln eE
. (5)
πm2 As m → 0, w logarithmically diverges as
w ∝ − ln m → ∞. (6)

Next, let us consider the origin <strong>of</strong> the divergence <strong>of</strong> w in
terms <strong>of</strong> effective dimensional reduction in a strong magnetic
field. When a magnetic field exists, the spectrum <strong>of</strong>
the transverse direction is discretized by Landau quantization.
Actually, the energy spectrum for E=0 is given by
ε = ± √ p 2 z + 2eB(n + 1/2 ∓ sz) + m 2 , (7)
where n = 0, 1, · · · correspond to the Landau levels,
and sz = ±1/2 is the spin. The system effectively becomes
1+1 dimensional system with infinite tower <strong>of</strong> massive
state: m 2 n,eff ≡ 2eBn + m2 . For the lowest Landau
level (LLL), n = 0 and s = +1/2, the energy is
ε = ± √ p 2 z + m 2 . This is the spectrum in 1+1 dimensions.
This LLL causes the divergence <strong>of</strong> w as will be shown below.
The divergence <strong>of</strong> w does not mean the divergence <strong>of</strong>
the infinite pair production per unit space-time. The divergence
<strong>of</strong> w rather implies that the vacuum always decays
and produces pairs <strong>of</strong> fermion. The question is where the
vacuum goes. In the coexistence <strong>of</strong> B and E, one can obtain
the probability <strong>of</strong> the n pairs <strong>of</strong> fermion with LLL as
|〈n pairs|Ωin〉| 2
= exp
[
V eB
−
4π2 ( ∫
eET −
dpznpz
)
ln eE
πm2 ]
.
The vacuum persistency probability corresponds to all
npz’s being zero in Eq. (8), and w is equal to Eq. (5), so
that w diverges at m = 0. At m = 0, this probability is
finite only if the following equation is satisfied:
∫
eET − dpznpz = 0. (9)
Therefore, the number <strong>of</strong> the particle with the LLL is restricted
by Eq. (9), and linearly increases with time. The
higher Landau levels give heavy effective masses <strong>of</strong> order
eB, so that all the contributions to the pair productions
from such modes are suppressed. The total number <strong>of</strong> the
particle pairs can be calculated:
N = V T e2 E 2
4π 3
πB
E
(8)
πB
coth . (10)
E
At B = 0, N = V T e 2 E 2 /(4π 3 ). The contribution <strong>of</strong> LLL
is obtained as
N = V T e2 E 2
4π 3
πB
, (11)
E
which is equal to taking coth(πB/E) → 1 in Eq. (10). In
Fig. 1, the total number <strong>of</strong> the particle for the full contribution
and LLL contribution are shown. The LLL dominates
for B > E, so that the effective model for the LLL works
well for B > E.
THEORY OF STRONG MAGNETIC FIELD
In this section, we study particle productions coming
from the LLL for QED taken into account the back reaction.
For this purpose, we consider LLL projected theory,
N(E,B)/N(E,B=0)
5
4
3
2
1
0
LLL
Full
0 0.5 1 1.5
B/E
Figure 1: Ratio <strong>of</strong> the particle number to that at B = 0.
The solid line denotes the contribution from the LLL, and
the dotted line denotes the contribution from all modes.
that is, the wave function <strong>of</strong> the fermion is projected to the
LLL state. The wave function <strong>of</strong> the LLL is
√ ( ) l
2
eB eB
φl(x, y) =
(x + iy)
2πl! 2
l
(
× exp − eB
4 (x2 + y 2 ) (12)
) ,
where l denotes the angular momentum in z direction for
the LLL, and the energy is degenerate for l. One can decompose
the fermion field into the longitudinal mode and
the transverse mode in a suitable representation as
(∑
ψ(x) = l φl(x,
)
y)ϕl(t, z)
, (13)
0
where ϕl(t, z) is the two component Dirac field in 1+1 dimensions.
Then the fermion action <strong>of</strong> QED in 3+1 dimensions
reduces to that <strong>of</strong> non-Abelian gauge theory in 1 + 1
dimensions:
∫
S = d 4 x ¯ ψ(x)iγ µ Dµψ(x)
∑
∫
dtdz ¯ϕl ′(t, z)i˜γµ D˜ l ′ l
µ ϕl(t, z),
(14)
l,l ′
where ˜γ t and ˜γ z are the gamma matrices in 1 + 1 dimensions
and ˜γ x = ˜γ y = 0. The covariant derivative is defined
by ˜ Dl′ l
µ = δl′ l∂µ − ieÃl′ l
µ with
à l′ l
µ (t, z) =
∫
dxdyφ ∗ l ′(x, y)φl(x, y)Aµ(x, y, z, t). (15)
à l′ l
µ (t, z) corresponds to the gauge field in U(∞) gauge
theory, since Ãl′ l
µ (t, z) is an Hermite matrix, Ã∗l′ l
µ (t, z) =
à ll′
µ (t, z), and the indices l and l ′ run from 0 to ∞. To
simplify the situation, we assume that the At and Az do
not depend on the transverse directions, x and y. Then the l
dependence can be factorized out: Ãl′ l
µ (t, z) = δll ′õ(t, z)
and ϕl(t, z) = ϕ(t, z). The action in Eq. (14) becomes
S eBV⊥
2π
∫
dtdz ¯ϕ(t, z)i˜γ µ Dµϕ(t, ˜ z), (16)

E/E 0
N/N max
1
0.5
0
-0.5
-1
-6 -4 -2 0
ωt
2 4 6
1
0.5
0
-6 -4 -2 0
ωt
2 4 6
Figure 2: The electric field (upper) and the number <strong>of</strong> pairs
(lower) at z = 0 plotted against time. Both values are normalized
by that at maximum values.
where V⊥ is the volume <strong>of</strong> the transverse directions. This
action is nothing but that in 1+1 QED, i.e., the Schwinger
model, except for the overall factor eBV⊥/(2π). The exact
solution <strong>of</strong> the effective action for the fermion is known as
Γ(Aµ) = − m2 ∫
γV⊥
dtdz
2
õ(t, z)
×
(
g µν
− ∂µ
∂ν
∂ 2
)
Ãν(t, z),
(17)
where denotes for longitudinal directions, t and z. mγ
denotes the effective photon mass, m 2 γ ≡ e 3 B/(2π 2 ).
Equation (17) is manifestly gauge invariant. In the Lorenz
gauge, it reduces to the result by [6]. The mass mγ is induced
by the axial anomaly, <strong>of</strong> which effect is called “dynamical
Higgs effect.” This mass generation is related to
the fact w → ∞ as m → 0. Using this form, we can
calculate the fermion and axial currents,
j µ (x) = δΓ(A)
δ(eAµ) = −e2 B
2π 2 õ (t, z), (18)
j µ
5 (x) = −ɛµν jν(x), (19)
where we choose the Lorenz gauge, ∂µ õ = 0. The divergence
<strong>of</strong> the axial current leads the axial anomaly in 1 + 1
dimensions except for the overall factor eB/(2π):
∂µj µ
5 (x) = e2 B
2π2 ɛµν∂µ Ãν(t, z) = e2
BE. (20)
2π2 This relation is nothing but axial anomaly in 3 + 1 dimensions.
Since the effective action in Eq. (17) has a quadratic
form in õ, the equation <strong>of</strong> motion for the photon can be
solved. For example, the electric field <strong>of</strong> z direction is
E = E0 cos(ωt − kzz), (21)
where ω =
√
k2 z + m2 γ. The currents satisfy ej µ =
−ɛ µν∂νE and ej µ
5 = ∂µ E. We show the electric field
and the number density for the spatially homogeneous case,
kz = 0, as a function <strong>of</strong> time in Fig. 2. The electric field
oscillates with a frequency ω. In this case, jt = 0, but
jt 5 = 0. The number density <strong>of</strong> pairs is equal to |jt 5|/2.
These results agree with the previous works [5, 7].
The generalization to non-Abelian theories is straight
forward if the magnetic field is enough strong. The fermion
determinant becomes Wess-Zumino-Witten action.
SUMMARY AND OUTLOOK
In this work, we have discussed the vacuum decay in
strong electric and magnetic fields. When the fermion is
massless, the vacuum persistency probability per unit time
and volume becomes zero, and hence w diverges. The origin
<strong>of</strong> the divergence is from discretized spectra <strong>of</strong> transverse
directions and the lowest Landau level. The LLL
level dominates for B > E. With the LLL projection,
we have analytically calculated the effective action in this
situation, and reproduced the previous numerical results.
Since the effective theory <strong>of</strong> the LLL is solvable, there is
no chaotic behavior nor thermalization. The thermalization
does not happen in the LLL-dominant process.
In the case <strong>of</strong> QCD, the gluon is more interesting because
<strong>of</strong> its self-interaction. The Landau quantization
causes instability because the helicity <strong>of</strong> gluon is one. Inserting
sz = 1 in Eq. (7), the energy becomes imaginary
when pz < eB, <strong>of</strong> which instability is known as Nielsen-
Olesen instability. The situation is similar to quenching
phenomenon in general phase transition, where a temperature
suddenly changes. In such a situation, a phase separation
occurs. The same phenomena would happen in relativistic
heavy-ion collisions, because the electric and magnetic
fields are suddenly induced by the collision, and the
perturbative vacuum <strong>of</strong> gluon is unstable. Although dynamics
<strong>of</strong> unstable gluon vacuum is very interesting, we
leave this topic in the future work.
REFERENCES
[1] T. Lappi and L. McLerran, Nucl. Phys. A 772, 200 (2006).
[2] H. Suganuma, T. Tatsumi, Prog. Theor. Phys. 90, 379 (1993).
[3] W. Heisenberg and H. Euler, Z. Phys. 98, 714 (1936).
[4] J. Schwinger, Phys. Rev. 82, 664 (1951).
[5] N. Tanji, Ann. Phys. 324, 1691 (2009); ibid 325, 2018 (2010).
[6] J. Schwinger Phys. Rev. 128, 2425 (1962); in Theoretical
Physics, Trieste Lectures 1962 (IAEA, Vienna, 1963) p.89.
[7] A. Iwazaki, Phys. Rev. C 80, 7 (2009).

Abstract
NONCANONICAL LIE PERTURBATION ANALYSIS FOR THE
RELATIVISTIC PONDEROMOTIVE FORCE ∗
N. Iwata † , Y. Kishimoto and K. Imadera, Kyoto University, Kyoto, Japan
An analysis <strong>of</strong> a relativistic particle motion in a nonuniform
high intensity laser field with linear polarization
is presented by using the noncanonical Lie perturbation
method, which is based on the perturbation theory <strong>of</strong> phase
space Lagrangian. Introducing a smallness parameter ϵ by
the ratio between the excursion length l and the scale length
<strong>of</strong> the laser field amplitude L, the relativistic ponderomotive
force and the corresponding particle motion are derived
up to the second order with respect to ϵ, which is a nonlocal
extension <strong>of</strong> the conventional ponderomotive force. Specifically,
the particle is found to exhibit a betatron-like oscillation
with a long period characterized by the curvature <strong>of</strong>
the laser field amplitude.
INTRODUCTION
Recently, the intensity <strong>of</strong> ultra-short high power lasers
has reached at the level <strong>of</strong> 10 22 W/cm 2 . In this regime,
electrons irradiated by the lasers exhibit highly relativistic
characters. In order to realize such high intensities, the reduction
<strong>of</strong> the pulse width and/or the spot size is necessary.
In such spatially localized laser fields, the ponderomotive
force (light pressure) exists inevitably and plays an important
role in the particle dynamics [1, 2].
The relativistic ponderomotive force, which is proportional
to the field gradient, has been investigated using the
averaging method to the equation <strong>of</strong> motion assuming that
the ratio between the excursion length l and the scale length
<strong>of</strong> the laser field amplitude L is small, i.e. ϵ ∼ l/L ≪ 1.
However, as the laser field is tightly focused, higher order
perturbations such as the spatial curvature become important.
Namely, in the non-uniform laser fields, besides the
force that simply ejects charged particles from the region <strong>of</strong>
the strong field, the particles suffer an additional force originating
from the curvature <strong>of</strong> the field. Such a higher-order
force may be utilized to confine the particle by carefully
controlling the laser field pattern.
In order to investigate the particle motion in complicated
electromagnetic fields, we have introduced the noncanonical
Lie perturbation method based on the perturbation theory
<strong>of</strong> phase space Lagrangian [3, 4, 5], and derived the
ponderomotive force up to the first order <strong>of</strong> ϵ [6]. The
method is found to be efficient and powerful in determining
the particle motion systematically keeping the Hamiltonian
structure. Motivated by these studies, here, we extend the
analysis to the higher order particle dynamics including the
curvature effect.
∗ Work supported by a Grant-in-Aid from JSPS (No. 21340171)
† iwata@center.iae.kyoto-u.ac.jp
NONCANONICAL TRANSFORMATION
We consider the motion <strong>of</strong> a particle with charge q in
vacuum irradiated by a linearly-polarized high-intensity
laser field. The field is assumed to propagate in the zdirection
and be localized in the transverse x- and ydirections.
As discussed in introduction, we define a smallness
parameter, ϵ ∼ l/L, where l and L (see later) are the
transverse excursion length <strong>of</strong> the particle and the transverse
scale length <strong>of</strong> the laser field amplitude, respectively.
Here, we normalize the vector potential <strong>of</strong> the laser field A
as a ≡ qA/mc 2 , where m is the rest mass <strong>of</strong> the particle
and c is the speed <strong>of</strong> light, and express a as
a(x, y, η) = ax(x, y, η)êx + ϵaz(η)êz, (1)
where ax(x, y, η) ≡ a0x(x, y) sin η, az(η) ≡ a0z cos η, êx
and êz are the unit vectors in the x- and z-directions, respectively,
and η ≡ ωt − kzz. Note that az is necessary to
satisfy the Maxwell equation in the order <strong>of</strong> ϵ. As we discuss
later, it is found that az does not affect on the secular
motion in the first order, whereas it does in the second order
where we neglect the influence in the present analysis for
simplicity. We expand the amplitude <strong>of</strong> the vector potential
around the initial particle position (x, y, z) = (x0, y0, 0) as
a0x (x⊥) = a0x0 + ϵ˜x⊥ · ∂x⊥ a0x (x⊥0)
+ ϵ2
[ ˜x 2 ∂ 2 xa0x (x⊥0) + ˜y 2 ∂ 2 ya0x (x⊥0) ]
2
+ ϵ 2 ˜x˜y∂x∂ya0x (x⊥0) + O ( ϵ 3) , (2)
where a0x0 ≡ a0x(x⊥0), ˜x⊥ ≡ x⊥ − x⊥0 and
∂x⊥a0x (x⊥0) = ∂a0x (x⊥) /∂x⊥|x⊥=x⊥0 .
Here, we introduce the extended phase space expressed
by the canonical variables as z µ = (t; q, pc) =
(t; qx, qy, qz, pcx, pcy, pcz), where the time t is the independent
variable. The corresponding covariant vector is given
by γµ = (−h; pc, 0), where h is the relativistic Hamiltonian
expressed as
√
h(q, pc, t) = m2c4 + c2 (pc − mca) 2 . (3)
In this paper, we use Latin indices that run from 1 to 6
whereas Greek from 0 to 6. Using these notations, the variational
principle is expressed as δ ∫ γµdz µ = 0. We call
ˆγ ≡ γµdz µ a fundamental 1-form. The general transformation
law from γµ to the new covariant vector Γµ under
arbitrary coordinate transformation z µ → Z µ can be obtained
by the relation γµdz µ = ΓµdZ µ . As a preparatory
transformation, we first introduce a noncanonical coordinate,
z µ = (η; x, y, z, px, py, pη), (4)

where p = pc − mca is the mechanical momentum, x =
q, and pη ≡ pz − γmc where γ is the relativistic factor.
Here, we take η as the independent variable to move to
the oscillation-center coordinate in the later analysis. The
corresponding covariant vector is then calculated as
γµ = (−K; px + mcax(x⊥, η), py,
pη + ϵmcaz(η), 0, 0, 0), (5)
where K = − (2kpη) −1 [ m2c2 + p2 ⊥ + p2 ]
η is the new
Hamiltonian. By taking pη as one <strong>of</strong> the coordinate variables,
we can simplify the zeroth-order Poisson tensor
which determines the structure <strong>of</strong> the equations <strong>of</strong> motion
[3] as that in the canonical coordinate. Note that the field a
does not explicitly appear in the new Hamiltonian but in the
first component <strong>of</strong> γµ, that also simplifies the perturbation
analysis.
ORBIT ANALYSIS IN LASER FIELDS
In the coordinate given by Eq. (4), the unperturbed particle
orbit z (0)i is obtained by solving the zeroth-order equations
<strong>of</strong> motion, which are derived by the variational prin-
ciple to the 1-form ˆγ (0) = γ (0)
µ dz (0)µ , as
z (0)i =
( a0x0
kzζ0
1
2kzζ 2 0
(cos η − 1) + x0, y0,
[ a 2 0x0
2
(
η − 1
2
− mca0x0 sin η, 0, pη0
)
sin 2η
+ ( 1 − ζ 2 ]
)
0 η ,
)
, (6)
under the initial condition (x, p, pη) = (x⊥0, 0, 0, 0, pη0)
and pz = pz0 at η = 0. Here, we defined ζ0 as pη0 ≡
−mcζ0. In this notation, ζ0 = 1 when the initial momentum
<strong>of</strong> the particle is zero, i.e. pz0 = 0. The particle
exhibits the well-known figure-eight orbit with the
drift motion in the z-direction [7]. The excursion length
l is obtained from the first component <strong>of</strong> Eq. (6) as l =
a0x0/kzζ0.
Next, to investigate the secular motion, we transform the
coordinate z µ to that <strong>of</strong> the oscillation-center <strong>of</strong> the zerothorder
oscillatory motion, Z µ = (η; X, Y, Z, Px, Py, pη).
The relationship between the old and new coordinates is
given by zi = Zi + z i(0)
os. , where z i(0)
os. is the oscillatory part
<strong>of</strong> the zeroth-order orbit. Then, the new covariant vector is
obtained as
(
Γµ = − κ; Px + mc (ax(X⊥, η) − ax0(η)) ,
)
Py, pη + ϵmcaz(η), 0, 0, 0 , (7)
where ax0(η) ≡ ax(X⊥0, η). Here, κ is the new Hamiltonian
calculated using the relationship between the old and
new coordinates, which yields
κ =K + l [Px + mc (ax(X⊥, η) − ax0(η))] sin η
+ a0x0
l [pη + ϵmcaz(η)] cos 2η. (8)
4ζ0
The old Hamiotonian K is now written in terms <strong>of</strong>
the new coordinate variables Px⊥ , pη [
and η as K =
− (mc) 2 + (Px − mcax0(η)) 2 + P 2 y + p2 ]
η /2kzpη. In
the zeroth order, the trajectory is found to be consistent
with that given by Eq. (6).
To analyze the first-order motion, we perform a
near-identity Lie transformation from the oscillationcenter
coordinate Z µ to a new one, Z ′µ , as Z ′µ =
exp ( ϵL (1)) Z µ by the operator defined to act as L (n) f =
g (n)µ ∂µf for a scalar function f, where g (n)µ is the
nth order Lie generator <strong>of</strong> the transformation. In
the new coordinate, the first-order covariant vector
is simplified as Γ ′(1)
(
µ = V (0)µ Γ (1)
)
µ ; 0, 0 , where
( )
˜X ′ + Y ˜ ′ /2kzp ′ ηL. Here, L =
V (0)µ Γ (1)
µ = m 2 c 2 a 2 0x0
(∂x [log a0x(X⊥0)]) −1 = (∂y [log a0x(X⊥0)]) −1 , ˜ X ′ ≡
X ′ − x0, ˜ Y ′ ≡ Y ′ − y0, and V (0)µ is the unperturbed
flow vector defined by V (0)0 = 1 and V (0)i (Z µ ) =
dZ (0)i /dZ 0 , respectively. The overline indicates the average
over one cycle <strong>of</strong> the fast oscillatory motion with period
η. Note that all the terms including az are removed
out in the averaging process. The new covariant vector up
to the first order is given by Γ ′ µ = Γ (0)
µ +ϵΓ ′(1)
µ . Since Γ ′(1)
µ
contains variables X ′ , Y ′ and p ′ η, the equations <strong>of</strong> motion
for P ′ x, P ′ y and Z ′ contain the terms <strong>of</strong> O(ϵ) as
dZ ′
dη
dP ′ ⊥
dη
= dZ(0)
dη
Z ′µ + m2c2 a2 0x0
p ′2
η
2kz
[ ]
˜X ′ + Y ˜ ′
, (9)
L
mc
=
p ′ mca
η
2 0x0
2kzL ê⊥, (10)
where ê⊥ is the unit vector perpendicular to the z-axis. The
expression dZ (0) /dη|Z ′µ denotes to replace the coordinate
variables Z (0)µ in the zeroth-order equation <strong>of</strong> motion with
those <strong>of</strong> Z ′µ . Note that since the backward Lie transformation,
Z µ = exp ( −ϵL (1)) Z ′µ , adds only the oscillatory
components <strong>of</strong> the motion, we have the relation ¯ Z ′i = ¯ Zi .
Therefore, the right-hand side <strong>of</strong> Eq. (10) is the ponderomotive
force in the original oscillation-center coordinate.
From Eq. (10), we can see that az does not affect the firstorder
ponderomotive force. We have also confirmed that
the terms proportional to az appear in the first-order oscillatory
part in both the x- and z-directions. This result is
physically reasonable since the first order oscillation in the
z-direction generated by the z-component <strong>of</strong> the electric
field affects on the oscillation in the x-direction through
the v × B force.
Next, we analyze the second-order particle motion.
Here, we consider the case where the field is uniform in the
y-direction, i.e. ∂ya = 0, for simplicity. We also neglect
the z-component <strong>of</strong> the vector potential in order to see only
the curvature effect on the particle motion. We transform
the coordinate to a new one, Z ′′µ . In the second order Lie
transformation, the new covariant vector is given by Γ ′′ µ =
Γ (0)
µ + ϵΓ ′(1)
µ + ϵ2Γ ′′(2)
µ , where Γ ′′(2)
µ =
(
V (0)µ C (2)
µ ; 0, 0
)

with C (2)
µ = Γ (2)
µ − ˆ L (1) Γ (1)
µ + ˆ L (1)2 Γ (0)
µ /2. Then, we have
V (0)µ C (2)
µ = mca2 0x0
4kz
mc
p ′′
[ ( ) (
1 1
+
η L2 R
+ 1
L 2
3 a0x0
l
16 kz
˜X ′′2 + l2
4
)
mc
p ′′
]
η
. (11)
Here, R ≡ ([ ∂2 xa0x(x0) ] ) −1
/a0x0 is the scale length <strong>of</strong>
the field curvature. Since the new Hamiltonian, −Γ ′′
0, does
not contain the variable Z ′′ , the corresponding component
p ′′
η is found to be constant. Then, the equations <strong>of</strong> motion
in the x-direction are reduced to
dX ′′
dη
dP ′′
x
dη
′′ P x
= − , (12)
kzpη0
= −mca0x0 l
2
[
1
L +
( )
1 1
+
L2 R
˜X ′′
]
. (13)
These equations determine the particle motion up to the
second order <strong>of</strong> ϵ, which varies slowly compared with the
period <strong>of</strong> the fast oscillation appeared in the zeroth-order
orbit.
In the case 1/L 2 +1/R ≥ 0, we obtain a slow oscillatory
motion given by
P ′′
x = α sin θη + P (2)
x0
X ′′ = − α 1
(cos θη − 1) +
mc θζ0kz
cos θη, (14)
P (2)
x0
mca0x0
l
sin θη + X′′ 0 ,
θ
(15)
where θ = l √ (1/L 2 + 1/R) /2, α is a con-
stant determined by the initial condition as α =
mca0x0θ ( 1 − l/ ( 2Lθ 2) − 7l/ (8L) ) , X ′′
0 is the initial
particle position and P (2)
x0 is the second-order initial value
<strong>of</strong> P ′′
x calculated by the second-order backward Lie transformation.
It is remarkably noted that the unbounded secular
motion originating from the first-order ponderomotive
force given in Eq. (10) is changed to the bounded solution,
Eqs. (14) and (15), by taking into account the second order
curvature terms. This motion corresponds to a betatron
oscillation by which the particle is confined in the finite
radial region. Note that since the amplitude factor α and
the period θ are defined by the local gradient and curvature
at the initial particle position, they are valid only in the
region where the variation <strong>of</strong> the curvature is sufficiently
small during one cycle <strong>of</strong> the long period oscillations.
In the case 1/L2 + 1/R < 0, Eqs. (12) and (13) yield to
the solution
P ′′
x =
(2)
α + P x0
e
2
θη +
(2)
−α + P x0
e
2
−θη . (16)
This solution indicates that the particle is rapidly ejected
from the region <strong>of</strong> large laser field amplitude. Taking the
expansion <strong>of</strong> the right-hand side <strong>of</strong> Eq. (16) assuming θη ∼
O (ϵ), Eq. (16) leads to
P ′′
x = αθη + P (2)
x0
X ′′ = α
mc
1
kzζ0
, (17)
θ
2 η2 +
P (2)
x0
mc
1
kzζ0
η + X ′′
0 . (18)
This solution is consistent with that obtained in Eq. (10)
up to the first order, though the second order collection is
included in Eqs. (17) and (18).
Here, we have neglected the z-component <strong>of</strong> the vector
potential, az, for simplicity. The inclusion <strong>of</strong> az may cause
modulation to the amplitude factor α and/or the period θ,
which will be discussed separately.
SUMMARY
We derived a equation system describing the relativistic
ponderomotive force and the related particle dynamics
in a transversely-focused linearly-polarized laser field
up to the second order with respect to ϵ. In the first order,
we obtained the ponderomotive force proportional to
the field gradient in the x- and y-directions that is essentially
the same as the result in Ref. [6]. In the second order,
we found that the particle can exhibit a slow period
betatron-like oscillatory motion characterized by the curvature
<strong>of</strong> the laser field amplitude. This suggests that the
control <strong>of</strong> the curvature is important in confining the particle
and keeping the laser-particle interaction in transversely
localized high-intensity laser fields. The betatron-like oscillation
may cause intense radiation that will be discussed
in a future paper. The present result up to the first order
and the expansion form, Eqs. (17) and (18), up to the second
order are consistent with those obtained by performing
the perturbation expansion directly to the equation <strong>of</strong> motion.
However, in the present analysis, the nonlocal solutions,
Eqs. (14), (15) and (16) are obtained for the first time
through the Lie perturbation approach.
REFERENCES
[1] E. A. Startsev and C. J. McKinstrie, Phys. Rev. E 55 (1996)
7527.
[2] P. Gibbon, ”Short Pulse Laser Interactions with Matter”, Imperial
College Press, London, p. 36 (2005).
[3] J. R. Cary and R. G. Littlejohn, Ann. Phys. 151 (1983) 1.
[4] Y. Kishimoto, S. Tokuda and K. Sakamoto, Phys. Plasmas 2
(1995) 1316.
[5] K. Imadera and Y. Kishimoto, accepted for publication in
Plasma Fusion Res..
[6] N. Iwata, K. Imadera and Y. Kishimoto, Plasma Fusion Res.
5 (2010) 028.
[7] E. S. Sarachik and G. T. Schappert, Phys. Rev. D 1 (1970)
2738.

PARTICLE BASED INTEGRATED CODE EPIC3D
FOR LASER-MATTER INTERACTION
Y. Kishimoto, Graduate School <strong>of</strong> Energy Science, Kyoto University, Uji, Kyoto 611-0011, Japan
Abstract
A complex plasma state, where neutral atoms and
molecules, ions with different charge states, free
electrons and positrons, various wavelength radiations
including X-rays and γ-rays, etc. coexist is established in
nature and laboratory through complex atomic and
nuclear processes. In such plasmas, the level <strong>of</strong>
complexity is <strong>of</strong> especially high due to the synergetic
interplays among spatio-temporally different scale
dynamics inside and outside the Debye sphere. We refer
to this kind <strong>of</strong> plasma state as synergetic complexity in
distinction from that used in conventional ideal plasmas.
Such plasma can be established in high-power laser
matter interaction. In order to investigate such plasmas,
we have developed a comprehensive particle based
integrated code EPIC3D, which includes various atomic
and collisional relaxation processes self-consistently.
Based on the EPIC3D, we performed the simulations <strong>of</strong>
laser-cluster interaction in parameter regimes relevant to
the experiment and also complex discharge/lightning
process. Through these simulations, we successfully
reproduced prominent structure formations, which lead to
the key physical understandings <strong>of</strong> complex plasma state.
INTRODUCTION
Interaction among various material states, charged
particles, energetic photons, etc. leads to a complex
plasma state, in which multiply charged ions, electrons
and positrons, neutral atoms and molecules, etc coexist.
Such plasmas exhibit the characteristics as an active
medium which is highly non-liner, non-equilibrium and
non-stationary, and can be seen not only in laboratory, but
also in space and universe, such as aurora, lightning, solar
flares, inter-stellar medium, accretion disks, etc. We refer
to this kind <strong>of</strong> plasma state as synergetic complexity in
distinction from that used in conventional ideal plasmas.
Such plasma can be also established in high-power laser
matter interaction.
The particle-in-cell (PIC) method that solves the
nonlinear wave–particle interaction has been widely used
in simulating complex plasma dynamics. However, most
<strong>of</strong> them a priori made an assumption that the plasma is
fully ionized in the initial state. For applications utilizing
relatively high-Z materials, complex atomic and
relaxation processes play an important role in determining
the interaction and subsequent dynamics. Here, in order to
investigate such complex plasmas, we have developed a
three-dimensional particle based integrated code, EPIC3D,
in which complex atomic and relaxation processes are
self-consistently included [1].
Based on the EPIC3D, we performed the simulation <strong>of</strong>
laser-cluster interaction in parameter regimes relevant to
the experiment. We also performed the simulation <strong>of</strong> the
lightning/discharge process <strong>of</strong> a compressed neon gas and
successfully reproduced key physical processes such as
streamer formation and sprite events. Lightning and
discharges are a well known process, but the details have
not been fully clarified. The understanding <strong>of</strong> the process
is crucially important not only from the academic
viewpoint but also for various industrial applications.
SIMULATION MODEL OF EPIC3D
Here we describe EPIC3D, which was originally based
on a fully relativistic three-dimensional electromagnetic
PIC technique. The photon field is classically treated by
the Maxwell equations in terms <strong>of</strong> electric and magnetic
field (E,B) or its potential form (A, φ). The charge density
ρ and current density j are assigned on each spatial grid
using the cloud-in-cell (CIC) method. For the (A, φ)
version, the Poisson equation as well as the continuity
equation are imposed, while for the (E,B) version, a local
solver technique is employed [2]. We extended the model
by incorporating the ionization and relaxation processes
described in the following.
2.1. Collision and relaxation process
We here introduced a relativistic pairing method by
successive binary collisions <strong>of</strong> particle pairs, which
precisely conserves the momentum and energy before and
after the collision event in a relativistic regime. It has
been certified in [3] that the method successfully resolves
non-local electron heat transport where the temperature
scale length becomes comparable to that <strong>of</strong> the electron
mean free path and the assumption <strong>of</strong> the Fick’s law (or
diffusion approximation) breaks up.
2.2. Ionization process due to photon field (Optical Field
Ionization-process)
We introduced an additional dimension in each atom
that represents the internal electronic state <strong>of</strong> the atom [4].
The ionization rate W (k) j (E), where the suffix denotes the
k-th electronic state <strong>of</strong> j-th atom, is calculated from the
local electric field E. Here we employed a cycle-averaged
tunneling ionization rate <strong>of</strong> the Ammosov–Debre-Krainov
(ADK) formula [5]. Then, the ionization index defined in
the range {0,1} is obtained as
R (k) j (t) = 1 – exp[−W (k) j (E)Δt ]. (2.1)

log N (PIC particle)
When the condition Rj>R[0, 1] is satisfied, a free electron
with the zero kinetic energy (Ee =0) is created and then
the ion charge state is increased by one. Here, R[0,1]
represents the random number defined in the range {0,1}.
2.3. Ionization process due to electron impact
Here we utilize the electron–ion pairs introduced in Sec.
2.1 for the Coulomb collisions inside the computational
mesh. After the Coulomb scattering takes place ( p (i) e →
p (f) e), the ionization index is obtained from the cross
section <strong>of</strong> electron impact ionization σ (e) i (Ee). Thus, the
ionization event is successively determined for every
electron–ion pair. In the case where the ionization is
switched on, a free electron with Ee =0 is created from the
ion. The ionization energy is subtracted from that <strong>of</strong> the
Coulomb-scattered electron to keep the total momentum
and energy conservations.
LASER-CLUSTER INTERACTION
In Advance Photon Research Center <strong>of</strong> JAEA,
interaction experiments using Xe and Ar clusters and an
ultra-short pulse laser in the range <strong>of</strong> 10 17 ~10 18 W/cm 2
were performed and the energy distribution <strong>of</strong> multiply
charged ions and control <strong>of</strong> the energy distribution
utilizing a pulse shaping <strong>of</strong> the laser have been studied [2].
In order to clarify the underlying physical process <strong>of</strong>
interactions, we performed simulations using the EPIC3D.
The parameters are given as follows: cluster radius: a =
24nm, electron density: ne = 2.44 × 10 22 cm −3 , laser
wavelength: λ1 = 820nm, pulse length: τ1 = 20fsec,
maximum laser amplitude: IL =1.0×10 18 W/cm 2 .
Figure 1 illustrates the ion charge state distribution at
t = 61fsec. The Ne-like charge state (Ar +8 ) is almost
ionized and the peak appears at Ar +9 . Figure 2 shows the
ion energy distribution for different charge state. The
maximum ion energy increases with the ion charge state
and the energy distribution with higher charge state (Ar +12
− Ar +16 ) exhibits an inverted structure that is typically
seen in cluster Coulomb explosion. Figure 3 illustrates
the density distribution <strong>of</strong> Ar +9 (a) and Ar +16 (b),
respectively. The ions with Ar +9 distributes dominantly in
the core region, while those <strong>of</strong> Ar +16 distribute around the
2.5 10 5
2 10 5
1.5 10 5
1 10 5
5 10 4
Ar +9
Ar +10
Ar +11
Ar +16
0
0 5 10
charge stae
15 20
Fig. 1. Ion charge
state distribution at
t = 61fsec after the
interaction. Field
ionization and
electron impact
ionization are taken
into account in the
simulation.
log{f(E +n )}
10 -1
10 -2
10 -3
10 -4
10 -5
10 -6
10 -1
Ar +9 Ar +8
10 0
Ar +10 Ar +11
10 1
10 2
log{E +n (keV)}
Fig.2 (a)
Ar +16
(a) Z=9
(b) Z=16
Fig. 2 Energy
distribution <strong>of</strong> ions
with different charge
state at t = 61fsec.
Different symbols,
□ for Z=8, + for Z=9,
△ for Z=15, ○ for
Z=16, are used.
Fig. 3. Ion density distribution in (x,y) domain for different
charge state, (a) Z = 9 and (b) Z = 16, at t = 61 fsec. The
polarization <strong>of</strong> laser electric field is x-direction.
expansion front. This is due to the fact that ions with
higher charge state are produced by strong polarization
fields generated near the cluster surface [4]. Such
polarization field is found to reach the value
approximately 5 times larger than that <strong>of</strong> the original laser
field. Ions having large acceleration rate provide a
maximum energy around MeV.
DISCHARGE AND LIGHTNING PROCESS
Recently, discharge/lightning phenomena, which are
observed in the earth atmosphere and also in the
ionosphere, have attracted considerable attention [5]. For
example, high energy relativistic electrons and even γrays
were observed during such lightning event. Here, we
performed discharge simulations for high pressure neon
gas with the density <strong>of</strong> 4.6×10 20 cm −3 (17 times the ideal
gas), where high voltage electric field, E = 10 7 V/cm, is
uniformly applied. A tiny ionization spot with Ne +2 is
initially set to trigger discharge.
Figure 4 (a)-(c) show the dynamics <strong>of</strong> ion charge
density and Fig. 5 shows the time history <strong>of</strong> ion density
with different charge state. The density <strong>of</strong> Ne +1 ion is
found to slowly increase, but suddenly exhibits
exponential growth with fast time scale around t = 43psec.
After the explosive growth <strong>of</strong> Ne +1 ions, the ion density
with charge state higher than Ne +1 , i.e. Ne +σ with σ ≥ 2 ,
also explosively increases with the growth rate larger than
that <strong>of</strong> Ne +1 ion (Fig.5). Branch-like structures referred as
10 3
Ar +14
Ar +15
Ar +13
Ar +12
10 4

P(k x )
(a) 44.3psec (b) 45.3pec (c) 45.9psec
(d) ionization spots
at t = 42psec
Fig. 4. Ion density distribution at three different times, (a) 44.3,
(b) 45.3, (c) 45.9psec, after prominent streamer formation and
avalanches takes place. Fig. 4(d) illustrates the ion spot
distribution prior to the avalanche.
log(N)
10 8
10 7
10 6
10 5
10 4
1000
100
0 1 10 4
2 10 4
3 10 4
4 10 4
t(fsec)
Fig. 5. Time history <strong>of</strong> ion density for different charge state.
Avalanche <strong>of</strong> Ne +1 ion is triggered around t = 43psec. The
region <strong>of</strong> t = 40-47psec is shown in order to see the details <strong>of</strong>
avalanches.
10 -3
10 -4
10 -5
10 -6
10 -7
10 -2
10 -1
k x (wave number)
+4
Ne +1
Ne +2
Ne +3
Ne Spectrum <strong>of</strong> B z Fig. 6.
Wave number spectrum
-1.86
kx “streamer” develop from the initial ionization spot
[Fig.4(a)]. However, after the exponential growth,
neighboring streamers connect each other (b) and develop
to a complex net-like structure (c), which contains
enormous branches with different spatial scales. This
structure may correspond to the so-called “sprite”.
10 0
<strong>of</strong> self- induced B z field
during the avalanche
process at t = 45.3psec.
The power law
dependence with k x -1.86
is obtained.
Figure 4 (d) illustrates the spatial distribution <strong>of</strong> ion
charge state in early time before the explosive event takes
place (t = 42psec). Many tiny ionization spots are found
to appear in the entire system. When the number <strong>of</strong> spots
(or equivalently “packing fraction” <strong>of</strong> the spot) exceeds a
certain value, micro-scale discharges are triggered
between neighboring ionization spots. Such a local event
simultaneously propagates over the wide spatial region,
leading to explosive “sprite” phenomenon. This process is
similar to that <strong>of</strong> “forest burning” and/or “percolation”
dynamics. Furthermore, since the electron current is
driven along ionization branches constituting sprites,
electro-magnetic signals are emitted from the system.
Figure 6 illustrates the wave number spectrum <strong>of</strong> induced
magnetic fields obtained from the data at t = 45.9psec. A
clear power low spectrum is found, suggesting that the
sprite shows a fractal nature that exhibits no special scales.
It is interesting to note that similar spectrum was observed
in low frequency electromagnetic signals during lightning
events in the atmosphere [5].
CONCLUSION
In order to investigate the complex plasmas dominated
by atomic and relaxation processes, we have developed a
three-dimensional particle based integrated code, EPIC3D.
Using the developed code, we performed the simulation
<strong>of</strong> laser-cluster interaction and also lightning process and
found the prominent ionization dynamics. Complex
ionization dynamics and ion accelerations <strong>of</strong> different
charge state were clarified. We found the complex
structures during lightning/discharge such as streamer and
sprite. A mechanism leading to the sudden appearance <strong>of</strong>
the lightning/discharge was investigated. Through these
simulations, we successfully reproduced prominent
structure formation, which lead to the key physical
understandings <strong>of</strong> complex plasma state dominated by
atomic and relaxation processes.
REFERENCES
[1] Y. Kishimoto, Annual Report <strong>of</strong> the Earth Simulator
Center, April 2002-Marcg 2003 (ISSN 1348-5822),
Chapter 4 Epoch-Making Simulation, pp.201-205.
[2] Y. Fukuda, K. Yamakawa, et al., Phys. Rev. A67,
061201(R) (2003). Also, Y. Fukuda and K.
Yamakawa, private communication, 2003.
[3] Y. Kishimoto, T. Masaki, and T. Tajima, Phys.
Plasmas 9, pp.589-601, February, 2002
[4] V.P. Pasko, M.A. Stanley, J.D. Mathews, U.S. Inan,
and T.W. Wood, Nature 416, pp.152-154, March,
2002, H.T. Su, R.R. Hsu, A. BB. Chen et al., Nature
423, pp.974-976, June, 2003
[5] M.A. Uman, The lightning discharge, Academic, San
Diego, 1987

Abstract
X-RAY GENERATION VIA LASER COMPTON SCATTERING BY
LASER-ACCELERATED ELECTRON BEAM ∗
E. Miura † , R. Kuroda, H. Toyokawa, AIST, Tsukuba, Ibaraki 3058568, Japan
S. Ishii, K. Tanaka, Tokyo University <strong>of</strong> Science, Noda, Chiba 2788510, Japan
X-ray generation by laser Compton scattering using a
quasi-monoenergetic electron beam with a narrow energy
spread obtained by laser-driven plasma-based acceleration
is reported. X-rays are produced by the collision <strong>of</strong> a
femtosecond laser pulse (140 mJ, 100 fs) with a quasimonoenergetic
electron beam with a peak energy <strong>of</strong> 50
MeV and a charge in the monoenergetic peak <strong>of</strong> 30 pC produced
by focusing an intense laser pulse (700 mJ, 40 fs) on
a He gas jet. A well-collimated X-ray beam with a divergence
angle <strong>of</strong> 5 mrad is obtained. The maximum photon
energy and the yield <strong>of</strong> the X-rays are estimated to be 60
keV and 10 5 photons/pulse.
INTRODUCTION
In laser-driven plasma-based acceleration, electrons are
accelerated by the electric field <strong>of</strong> a plasma wave driven
by an intense laser pulse[1]. To realize next-generation
electron accelerators, the research has been intensively
conducted over a few decades. Since 2004, the generation
<strong>of</strong> a well-collimated electron beam with a narrow energy
spread, that is quasi-monoenergetic electron (QME)
beam, has been demonstrated by several groups[2, 3, 4, 5].
The generation <strong>of</strong> a QME beam with a peak energy <strong>of</strong> 1
GeV [6], and a QME beam containing 0.5 nC electrons in
the monoenergetic peak [5], which is comparable to that
achieved with radio-frequency (rf) accelerators, has been
also demonstrated. The road toward the realization <strong>of</strong> a
practical laser electron accelerator is now open.
In laser-driven plasma-based acceleration, a high acceleration
field more than 100 GV/m, which corresponds to
thousands <strong>of</strong> those achieved by rf accelerators, is obtained.
A compact electron accelerator can be realized using such
a high acceleration field. Furthermore, the wavelength <strong>of</strong>
the acceleration field, that is the wavelength <strong>of</strong> the plasma
wave, is short, <strong>of</strong> the order <strong>of</strong> tens <strong>of</strong> micrometers. Then,
the electron pulse duration is extremely short, <strong>of</strong> the order
<strong>of</strong> a few tens <strong>of</strong> femtoseconds. The set <strong>of</strong> such unique characteristics
<strong>of</strong> laser-driven plasma-based acceleration enables
us to realize a compact, all-optical, ultrashort X-ray
source based on laser Compton scattering, that is scattering
<strong>of</strong> photons by energetic electrons.
∗ A part <strong>of</strong> this study was financially supported by the Budget for Nuclear
Research <strong>of</strong> the Ministry <strong>of</strong> Education, Culture, Sports, Science, and
Technology, Japan, based on the screening and counseling by the Atomic
Energy Commission.
† e-miura@aist.go.jp
So far, generation <strong>of</strong> an ultrashort X-ray pulse by laser
Compton scattering has been demonstrated by using a femtosecond
laser pulse and a picosecond electron pulse from
rf accelerators[7, 8]. The X-ray pulse duration is determined
by the interaction time between the laser and electron
pulses. To obtain a femtosecond X-ray pulse, 90 ◦
scattering geometry should be adopted for a picosecond
electron pulse. In contrast, 180 ◦ scattering (head-on collision)
geometry is available for a femtosecond electron
pulse. There are some advantages <strong>of</strong> using 180 ◦ scattering
geometry. Even though the electron energy and the charge
<strong>of</strong> an electron beam are the same, the photon energy and the
yield <strong>of</strong> X-rays are higher than those for 90 ◦ scattering geometry.
Recently, X-ray generation has been demonstrated
using a laser-accelerated electron beam[9]. However, the
X-ray energy was around 1 keV and the yield was not so
high, because an electron beam with Maxwell-like energy
distribution was used. To obtain a bright X-ray source with
higher photon energy, a QME beam with a higher energy
and a larger charge is necessary.
In this paper, we report X-ray generation by laser Compton
scattering using a QME beam obtained by laser-driven
plasma-based acceleration.
EXPERIMENTAL CONDITIONS
Figure 1 shows the experimental setup. In the following
part, laser pulses for electron acceleration and laser Compton
scattering are called ”main laser pulse” and ”colliding
laser pulse”, respectively. A p-polarized main laser pulse
(700 mJ, 40 fs, 800 nm) was focused on the edge <strong>of</strong> a He
gas jet using an <strong>of</strong>f-axis parabolic mirror with a focal length
<strong>of</strong> 720 mm. The laser spot diameter in vacuum was 13 µm
at full width at half maximum (FWHM). The peak intensity
was 4.7 × 10 18 W/cm 2 . The gas jet was ejected from
a supersonic nozzle with a conical shape. The diameter <strong>of</strong>
the nozzle exit was 1.6 mm. The focal position was set at
1 mm above the nozzle exit. A p-polarized colliding pulse
(140 mJ, 100 fs, 800 nm) was focused around the exit <strong>of</strong> the
main laser pulse from the gas jet using an <strong>of</strong>f-axis parabolic
mirror with a focal length <strong>of</strong> 300 mm. The laser spot diameter
in vacuum was 9 µm at FWHM. The incident angle <strong>of</strong>
the colliding laser pulse was 20 ◦ to the propagation axis <strong>of</strong>
the main laser pulse.
X-rays produced by laser Compton scattering were emitted
on the coaxial direction <strong>of</strong> an electron beam. The
electron beam was bended by a magnetic field and spatially
separated from the X-rays. Both X-rays and electrons
were incident on a phosphor screen (DRZ-HGH, Mit-

the position <strong>of</strong> the nozzle exit with 1.6-mm diameter. In
Fig. 3(d), the main pulse propagated from top to bottom
and the colliding pulse propagated from lower right to upper
left in the direction <strong>of</strong> 20 ◦ to the main laser propagation
axis. As shown by the arrow, a bright spot was observed,
only the synchronized collision <strong>of</strong> the two laser pulses was
achieved. It is supposed that the bright spot indicates the
collision point <strong>of</strong> the two laser pulses. The collision point
was set near the edge <strong>of</strong> the nozzle exit, which was near the
extraction position <strong>of</strong> an electron beam from a plasma.
(a)
(b)
(c)
Colliding Main Main
(d)
Collision
point
1 mm
Colliding
Collision
point
Figure 3: (a)-(c) Shadowgraph images observed for different
delay times <strong>of</strong> a probe pulse to a main pulse: (a) -1.33
ps, (b) -0.67 ps, and (c) 0 ps. (d) Thomson-scattered light
image when the synchronized collision <strong>of</strong> the main and colliding
laser pulses is achieved. The bright spot indicates the
collision point <strong>of</strong> the two laser pulse.
X-ray generation
X-rays produced by laser Compton scattering were observed,
when a QME beam with a considerably high charge
was obtained. Figure 4(a) shows a image <strong>of</strong> X-rays. The
image was obtained in a single shot. From the energyresolved
electron image simultaneously observed with the
image shown in Fig. 4(a), the peak energy and the charge in
the monoenergetic peak <strong>of</strong> the QME beam was 50 MeV and
30 pC, respectively. Figure 4(b) shows the intensity pr<strong>of</strong>ile
<strong>of</strong> the image in the vertical direction. The divergence angle
in vertical direction was 5 mrad at FWHM. The divergence
angle in horizontal direction was 7 mrad at FWHM. In laser
Compton scattering, a collimated X-ray beam can be obtained.
The divergence angle <strong>of</strong> the X-ray beam is given by
∼ 1/γ. Here, γ is the Lorentz factor <strong>of</strong> an electron energy.
The divergence <strong>of</strong> an X-ray beam is estimated to be approximately
10 mrad from the observed peak energy <strong>of</strong> 50 MeV.
The observed divergence angle was close to the predicted
value from the electron energy. The maximum photon energy
<strong>of</strong> the X-rays was estimated to be 60 keV from the
peak energy <strong>of</strong> the QME beam and the interaction angle.
The X-ray yield was also estimated to be approximately
10 5 photons/pulse from the charge <strong>of</strong> the QME beam and
the irradiation conditions <strong>of</strong> the colliding laser pulse by including
the dependence <strong>of</strong> the differential cross section <strong>of</strong>
scattering on the scattered angle.
The allowance range <strong>of</strong> the delay between the main and
colliding laser pulses for X-ray generation was investigated.
The allowance range was approximately 100 fs, corresponding
to the duration <strong>of</strong> the colliding pulse. This result
suggests that a pulse duration <strong>of</strong> a QME beam is nearly
equal to or less than 100 fs. The generation <strong>of</strong> an ultrashort
electron pulse by laser-drive plasma-based acceleration has
been demonstrated at the same time.
70
(a) (b)
Vertical position [pixel]
60
50
40
30
20
10
0
0 200 400 600 800
Intensity [arb. unit]
Figure 4: (a) Image <strong>of</strong> X-rays produced by laser Compton
scattering and (b) the vertical pr<strong>of</strong>ile <strong>of</strong> the image. A wellcollimated
X-ray beam with a divergence angle <strong>of</strong> 5 mrad
at FWHM is observed.
SUMMARY
X-ray generation by laser Compton scattering has been
demonstrated using a QME beam with a peak energy <strong>of</strong> 50
MeV and a charge in the monoenergetic peak <strong>of</strong> 30 pC obtained
by laser-driven plasma-based acceleration. A wellcollimated
X-ray beam with a divergence angle <strong>of</strong> 5 mrad
at FWHM was obtained. The maximum photon energy and
the yield <strong>of</strong> the X-rays were estimated to be 60 keV and
10 5 photons/pulse from the peak energy and the charge <strong>of</strong>
the QME beam.
REFERENCES
[1] T. Tajima and J. M. Dawson, Phys. Rev. Lett. 43 (1979) 267.
[2] E. Miura et al., Appl. Phys. Lett. 86 (2005) 251501.
[3] S. P. D. Mangles et al., Nature 431 (2004) 535.
[4] C. G. R. Geddes et al., Nature 431 (2004) 538.
[5] J. Faure et al., Nature 431 (2004) 541.
[6] W. P. Leemans et al., Nature Physics 2 (2006) 696.
[7] R. Schoenlein et al., Science 274 (1996) 236.
[8] M. Yorozu et al., Appl. Phys. B 74 (2002) 327.
[9] H. Schwoerer et al., Phys. Rev. Lett. 96 (2006) 014802.
[10] S. Masuda et al., Rev. Sci. Instrum. 79 (2008) 083301.
[11] S. Masuda and E. Miura, Appl. Phys. Express 1 (2008)
086002.
[12] E. Miura and S. Masuda, Appl. Phys. Express 2 (2009)
126003.

The electron beam is bent to the dump preventing it from
striking the detector.
Upgrade points from FFTB
We upgraded this monitor to measure the even smaller
beam sizes to be available at ATF2. The laser wavelength
has been modified from 1064 nm to 532 nm using a second
harmonics generator. The laser optics was newly designed
and constructed by implementing a laser wire scheme to
measure a larger horizontal beam size, and by enabling different
crossing angles <strong>of</strong> split laser beams to measure a
wide (diverse) range <strong>of</strong> vertical beam sizes. The gamma
detector for Shintake Monitor has also been newly developed.
MEASUREMENT SCHEME
The Shintake Monitor employs the interference pattern
created by splitting laser beams and crossing them at the
focal point <strong>of</strong> the electron beam. Laser interference plays
the role <strong>of</strong> changing the electron beam size in correspondence
to the gamma signal modulation. In their intersecting
region, the electromagnetic fields <strong>of</strong> the two laser beams
form a standing wave (interference fringe). The number <strong>of</strong>
Compton scattered photons varies according to the phase
<strong>of</strong> the standing wave where the electrons pass through.
With a smaller beam size, the number <strong>of</strong> photons variates
more significantly compared to a large beam size (Fig.3).
Figure 3: Different modulation for different beam size
Therefore the parameter called “Modulation Depth” can
be defined as
M ≡ N+ − N−
= | cos θ|exp(−2(kyσy)
N+ + N−
2 ) (1)
using the maximum number <strong>of</strong> photons N+ and the minimum
number <strong>of</strong> photons N− are
The beam size can be obtained from modulation depth.
Calculation formula is
σy = d
√
2π
| cos θ|
2ln( )
M
(2)
In eq.2, d is a distance between peak to peak <strong>of</strong> the
strength <strong>of</strong> the magnetic field. d = π
ky
ϕ is the half <strong>of</strong> θ.
= λ
2 sin(ϕ) .
SYSTEMATIC ERROR SOURCES
There are 2 types <strong>of</strong> errors related to the measurement
<strong>of</strong> the Shintake Monitor. The first type is related to the
strength <strong>of</strong> gamma signal an example <strong>of</strong> which is detector
resolution, Beam current jitter also belongs to this category.
This type <strong>of</strong> errors consequently lead to the error bars
at each gamma signal point and relate to the statistical error
and the fitting accuracy <strong>of</strong> the modulation. The second
type is related to the reduction <strong>of</strong> modulation. For example,
contrast <strong>of</strong> laser fringe, beam position jitter belongs to
this type <strong>of</strong> errors. These consequently become systematic
beam size measurement errors.
Using the parameters Cα, Cβ, which represents the reduction
factor <strong>of</strong> each sources, the measured modulation
Mmeas can be written as
Mmeas = CαCβ...Mideal
Table 1: Error sources
Source Value
Error bar Detector resolution 8 %
at each point Beam current jitter 1%(using ICT)
Laser power jitter 3 %
Total 9 %
Reduction BPM jitter 99.5 %
<strong>of</strong> modulation Spherical wavefront 99.4%
Laser temporalcoherence
99.7%
Laser alighnmentaccuracy
98.1%
Beamsize growth infringe
99.9%
Tilt <strong>of</strong> fringe 97.3%
Total 94 %
6 % <strong>of</strong> the reduction <strong>of</strong> modulaiton corresponds to 2.9
nm systematic error in the case <strong>of</strong> 37 nm measurement.
CROSSING ANGLE MODE
The Shintake Monitor for ATF2 has 3 crossing angle
modes for vertical beam size measurement and laser wire
mode for horizontal beam size measurement.
For the different crossing angle θ, the value <strong>of</strong> d and the
range <strong>of</strong> measurable beam size is shown in table 2
The continuous changable 2-8 degree mode was implemented
for the measurement down to several microns,
which overlaps with traditional wire scanner ranges.
(3)

Table 2: crossing angle and the range <strong>of</strong> measurable beam
size
θ d range beam size
2 deg 15.2µ m 1.4 - 6.0 µ m
8 deg 3.81µ m 0.36 -1.4 µ m
30 deg 1.28µ m 100 - 360 nm
174 deg 266nm 25 - 100 nm
A single laser path mode, without crossing is also implemented
for functioning as a laser wire. It enables to
measure σx beam size.
BEAM TIME RESULT
The measurement with 7.96 deg. mode was performed
until now. Fig.4 is the measurement in May <strong>of</strong> this year.
The size was 313 ± 31(stat.) +0
−40 (sys.)nm Corresponding
modulation depth was 0.85.
Signal Energy / ICT Charge [arb. units]
100
80
60
40
20
0
0 2 4 6 8 10 12 14
phase [rad]
Figure 4: Measurement <strong>of</strong> beam size
At that time, beam condition was as follows. The background
amount was 15 GeV. S/N was about 10, signal
amount was much more than background. The beta function(vertical)
at IP was 1mm, 10 larger times than the nominal
value. This was from the beam tuning issue. When
the beam is focused to 37nm, background is estimated to
increase because <strong>of</strong> strong focusing. But if the background
become 100 times larger, the resolution <strong>of</strong> the detetctor is
less than 10 %.
UPGRADE POINTS IN THE SUMMER
SHUTDOWN 2010
Laser position stabilization
In order to stabilize the laser position at IP, an actuator
followd by a PSD have been newly installed in front <strong>of</strong> the
20 m transport line
Focal point scan
In the case <strong>of</strong> <strong>of</strong>fset <strong>of</strong> the beam from the laser focus
point, spherical wavefront effect would cause systematic
errors. To prevent this we added a system for scanning the
laser focus position.
Tilt monitor
Tilt <strong>of</strong> the interfrence fringe relative to the beam axis
would also bring about systematic errors. To counter this
effect, 2 PSDs have been installed to monitor the tilt <strong>of</strong> the
beam.
SUMMARY AND NEAR FUTURE PLAN
The study reported in this paper can be concluded based
on the following three major points.
1. Shintake monitor at ATF2 is desgined to measure vertical
beam sizes from several micron down to 25 nm
2. About 300 nm beam size measurements were performed
with 2 degree laser crossing angle mode
3. 37 nm beam size measurement is planned to be
achieved and verified during the next run period(Nov
2010 May 2011)
ACKNOWLEDGMENT
The authors are very grateful for the researchers who
took part in the comissioning <strong>of</strong> the electron beam at ATF2.
We would like to express our thanks for the supporting fund
by KEK.
REFERENCES
[1] C. Petit-Jean-Genaz and J. Poole, “JACoW, A service to
the Accelerator Community”, EPAC’04, Lucerne, July 2004,
THZCH03, p. 249, http://www.JACoW.org.
[2] A. Name and D. Person, Phys. Rev. Lett. 25 (1997) 56.
[3] A.N. Other, “A Very Interesting Paper”, EPAC’96,
Sitges, June 1996, MOPCH31, p. 7984 (1996),
http://www.JACoW.org.
[4] T.Suehara, et al. Nucl. Instrum. Methods Phys. Res., Sect. A
616, 1 (2010)

INVESTIGATING THE ONE-PHOTON ANNIHILATION CHANNEL IN AN
e − e + PLASMA CREATED FROM VACUUM IN STRONG LASER FIELDS
D.B. Blaschke, University <strong>of</strong> Wroclaw, 50-204 Wroclaw, Poland; JINR, 141980 Dubna, Russia
G. Röpke, Institut für Physik, Universität Rostock, D-18051 Rostock, Germany
V.V. Dmitriev, S.A. Smolyansky ∗ , A.V. Tarakanov, Saratov State University, 410026 Saratov, Russia
Abstract
It is well known that in the presence <strong>of</strong> strong external
electromagnetic fields many processes forbidden in standard
QED become possible. One example is the onephoton
annihilation process considered recently by the
present authors in the framework <strong>of</strong> a kinetic approach to
the quasiparticle e − e + γ plasma created from vacuum in
the focal spot <strong>of</strong> two counter-propagating laser beams. In
these works the domain <strong>of</strong> large values <strong>of</strong> the adiabaticity
parameter γ ≫ 1 (corresponding to multiphoton processes)
was considered. In the present work we estimate the intensity
<strong>of</strong> the radiation stemming from photon annihilation in
the framework <strong>of</strong> the effective mass model where γ 1,
corresponding to large electric fields E Ec = m 2 /e and
high ”laser” field frequencies ν m (the domain characteristic
for X-ray lasers <strong>of</strong> the next generation). Under such
limiting conditions the resulting effect is sufficiently large
to be accessible to experimental observation.
INTRODUCTION
The planned experiments [1] for the observation <strong>of</strong> an
e − e + plasma created from the vacuum in the focal spot <strong>of</strong>
two counter-propagating optical laser beams with the intensity
I 10 21 W/cm 2 raises the problem <strong>of</strong> an accurate theoretical
description <strong>of</strong> the experimental manifestations <strong>of</strong>
the dynamical Schwinger effect [2], see also Refs. [3, 4, 5].
The existing prediction [6] in the domain <strong>of</strong> strongly subcritical
fields E ≪ Ec = m 2 /e <strong>of</strong> a considerable number
<strong>of</strong> secondary annihilation photons is not rather convincing
because it is based on the S-matrix approach for
the description <strong>of</strong> quasiparticle excitations in the presence
<strong>of</strong> a strong external electric field. In particular, this approach
does not take into account vacuum polarization effects.
Apparently, an adequate approach for the description
<strong>of</strong> vacuum excitations in strong electromagnetic fields
is a kinetic theory in the quasiparticle representation. The
simplest kinetic equation (KE) <strong>of</strong> such type for the e − e +
subsystem has been obtained for the case <strong>of</strong> linearly polarized,
time dependent and spatially homogeneous electric
fields [2]. Some generalizations <strong>of</strong> the KE in the fermion
sector have been worked out in Refs. [7, 8, 9]. It can be expected,
that electromagnetic field fluctuations <strong>of</strong> the e − e +
plasma are accompanied by the generation <strong>of</strong> real photons
∗ smol@sgu.ru
which can be registered far from the focal spot. The first
two equations <strong>of</strong> the BBGKY chain for the photon sector
<strong>of</strong> the e − e + plasma were obtained in [10, 11]. This level is
sufficient for the kinetic description <strong>of</strong> the one-photon annihilation.
In the presence <strong>of</strong> an external field such process
is not forbidden [12]. In the works [10, 11] it was shown
that the spectrum <strong>of</strong> the secondary photons in the low frequency
domain k ≪ m has the character <strong>of</strong> the flicker
noise. In the work [13] the inclusion <strong>of</strong> vacuum polarization
effects in the one-photon radiation spectrum led to an
essential change <strong>of</strong> the photon KE which was investigated
in a broad spectral band including the annihilation domain
ν ∼ 2m. First we have considered the domain <strong>of</strong> large
adiabaticity parameters γ ≫ 1, where the photon radiation
from the focal spot turns out to be very small. However,
the tendency <strong>of</strong> the effect to grow for γ → 1 has been discovered.
This is just the domain <strong>of</strong> practical interest for
parameters <strong>of</strong> modern lasers.
In the present work the effective mass model is considered
which allows to investigate the photon radiation in the
domain <strong>of</strong> rather strong fields not restricted to specific values
<strong>of</strong> the adiabacity parameter. Some crude estimations in
the framework <strong>of</strong> this model [12] lead to an unexpectedly
large total photon production intensity.
EFFECTIVE MASS MODEL
We will proceed from the photon kinetic equation
F ˙ ( e
k, t) =
2
4k(2π) 3
t
dt ′
d 3 pe −iθ(p,p+ k, k;t ′ ,t) ×
×K(p, p + k, k; t ′ , t)f(p, t ′ )f(p + k, t ′ ) + c.c. (1)
for the one-photon annihilation mechanism taking into account
vacuum polarization effects in the low density approximation
[13]. In Eq. (1) F ( k, t) and f(p, t) are the
photon and electron (positron) distribution functions, respectively,
k is the wave vector <strong>of</strong> the radiated photon and
θ(p1, p2, k; t ′ , t) =
t
t ′
dτ [ω(p1, τ) + ω(p2, τ) − k] (2)
is the high frequency phase.
The two-time convolution K(p, p+ k, k; t ′ , t) <strong>of</strong> the fourspinors
is a slowly varying function <strong>of</strong> its variables and can

e replaced by its average K → K0 ∼ 1, which is sufficient
for coarse estimations.
The effective mass model [12] is based on the approximation
ω(p, t) =
m 2 +
p − e 2 A(t) →
→ ω∗(p) = m 2 ∗ + p 2 , (3)
with the effective mass defined by the relation
m 2 ∗ = m 2 + e 2
A 2
(t) = m 2 + e 2 E 2 0/2ν 2 =
= m 2 (1 + 1/2γ 2 ), (4)
where 〈...〉 denotes the time averaging operation, ν is the
frequency <strong>of</strong> the periodic ”laser” field and E0 is its field
strength amplitude, γ = (Ec/E0)(ν/m) is the adiabaticity
parameter.
In this approximation the phase (2) becomes monochromatic
θ(p1, p2, k; t ′ , t) = Ω∗(p1, p2, k)(t − t ′ ), (5)
Ω∗(p1, p2, k) = ω∗(p1) + ω∗(p2) − k, (6)
i.e. the approximation (3) leads to a suppression <strong>of</strong> multiphoton
processes (it corresponds to the large values <strong>of</strong> the
adiabacity parameter γ ≫ 1) and the mismatch (6) can be
compensated by the harmonics <strong>of</strong> the fermion distribution
functions in Eq. (1) only.
The inspection <strong>of</strong> the fermion distribution function
shows, in particular, that it oscillates basically with twice
the laser frequency
f(p, t) = 1
2 ¯ f(p) [1 − cos(2νt)] . (7)
The substitution <strong>of</strong> Eqs. (5) and (7) into the kinetic equation
(1) allows to perform the time integration, leading to
the appearance <strong>of</strong> two harmonics in the radiation spectrum
only (the 2 nd and the 4 th ),
˙
F ( k, t) = −A (2) ( k) cos(2νt) + A (4) ( k) cos(4νt), (8)
A (2) ( k) = π2 K0α
2k
A (4) ( k) = π2 K0α
8k
d 3 p
(2π) 3 ¯ f(p) ¯ f(p + k)δ (2ν − Ω∗) , (9)
d 3 p
(2π) 3 ¯ f(p) ¯ f(p + k)δ (4ν − Ω∗) . (10)
It is important that a constant component is absent here,
because the mismatch (6) could not be compensated in this
case by other sources <strong>of</strong> the time dependence on the r.h.s.
<strong>of</strong> Eq. (1). 1
Thus, in the case <strong>of</strong> the infinite system the solution (8)
can be interpreted as ”breathing” <strong>of</strong> the photon subsystem.
However, the situation is changed, when the generation <strong>of</strong>
the e − e + γ plasma is considered in a small spatial domain
1 This is in contrast to the case γ ≫ 1, where accounting for multiphoton
processes in the phase (2) leads to a constant component [13].
<strong>of</strong> the focal spot with volume ∼ λ 3 due to the vacuum
condition <strong>of</strong> the absence <strong>of</strong> the e − e + γ plasma in the initial
moment <strong>of</strong> switching on the laser field. In this case
one can expect, that all annihilation photons generated in
the first half-period <strong>of</strong> the field will leave the volume <strong>of</strong>
the system and therefore in the next half-period the reverse
process (photon transformation to e − e + plasma) will be
impossible. Such a mechanism leads to a pulsation pattern
for the photon radiation from the focal spot. It corresponds
to introducing the condition <strong>of</strong> a positive definite photon
production rate on the r.h.s <strong>of</strong> Eq. (8).
For estimations <strong>of</strong> the amplitudes (9), (10) let us introduce
the additional model approximation in the spirit <strong>of</strong> the
model (3)
ω∗(p + k) → ω∗(p, k) = ω 2 ∗(p) + k 2 (11)
and the isotropisation condition ¯ f(p + k) → ¯ f(p + k). The
integrals on the r.h.s <strong>of</strong> Eqs. (9), (10) can then be calculated.
For example,
A (2) ( k) = K0α
4k ¯ f(p0) ¯ f(p0 + k) ω∗(p0)ω∗(p0, k)
ω∗(p0) + ω∗(p0, k) p0,
where
p0 =
(12)
4ν 2 (k + ν) 2
(k + 2ν) 2 − m2 ∗ (13)
is the root <strong>of</strong> the equation Ω∗ − 2ν = 0. From Eq. (13) it
is follows the threshold condition 2
2ν(k + ν)
k + 2ν m∗ . (14)
This condition is rather nontrivial because the effective
mass (4) depends also on ν. The minimal permissible value
ν = 2m∗ corresponds to k = 0. For the 4 th harmonic the
threshold value falls to ν = m∗, which is close to the parameters
<strong>of</strong> the XFEL [15].
The 1/k - dependence on the r.h.s. <strong>of</strong> Eq. (12) corresponds
to the flicker noise. This feature in the spectrum
<strong>of</strong> radiated annihilation photons has been found first in
Ref. [10].
The number <strong>of</strong> photons with the frequency k lying in the
interval [k, k + dk] and radiated from the focal spot with
the volume λ 3 = ν −3 per time interval is
d 2 N
dtdk
= 8πk2
ν 3
F ˙ ( k, t). (15)
The fraction on the r.h.s. <strong>of</strong> Eq. (12) is a slow function
<strong>of</strong> the frequencies k and ν and for the sake <strong>of</strong> a preliminary
estimation it can be replaced by m∗/2. For the 2 nd
harmonic we then obtain from Eqs. (12) and (15)
d 2 N (2)
dtdk
= 2παK0km∗
ν 3
¯f(p0) ¯ f(p0 + k)p0 . (16)
2 A similar effect was found first in the theory describing the absorption
<strong>of</strong> a weak signal by the e − e + plasma created from vacuum [14].

As a representative characteristics <strong>of</strong> the effectiveness <strong>of</strong>
the radiation from the focal spot domain we will consider
the total photon number per time interval,
˙N (2) = 2παK0m∗
ν 3
kmax
0
dk k ¯ f(p0) ¯ f(p0 + k)p0 . (17)
The electron and positron distribution functions entering
here are defined as the solutions <strong>of</strong> the corresponding nonperturbative
kinetic equation [2, 7] describing vacuum creation
<strong>of</strong> e − e + pairs under the action <strong>of</strong> a strong, time dependent
electric field <strong>of</strong> a standing wave <strong>of</strong> two counter
propagating laser beams. The cut<strong>of</strong>f parameter kmax =
2m∗ is introduced in order to take into account the annihilation
photons in the radiation spectrum.
The fermion distribution function f(p, t) is a rapidly decreasing
function with its maximum in the point p = 0 [3].
On this basis for a rough estimation one can put p0 = 0 in
the arguments <strong>of</strong> these functions on the r.h.s. <strong>of</strong> Eq. (17),
˙N (2) = 2παK0m∗
ν 3
¯f(0)
kmax
0
dk k ¯ f(k)p0 , (18)
where according to Eq. (13)
m
p0(k) =
2
∗
48 + 56
k + 4m∗
k
+ 15
m∗
k2
m2
∗
m
48 + 56
4
k
, (19)
m∗
because the small kmax ≪ m∗ is effective in the integral
(18). As the result, we obtain the following order <strong>of</strong> magnitude
estimate
˙N (2) ∼ αm∗ ¯ f 2 (0) . (20)
For the XFEL parameters E0 = 0.24Ec and λ = 15 nm
[15] we have according to the kinetic theory in the e − e +
sector ¯ f(0) ∼ 10 −2 . From Eq. (20) then follows
˙N (2) ∼ 10 17 s −1 . (21)
For the 4 th harmonic with the oscillation amplitude (10)
the threshold for the generation <strong>of</strong> annihilation photons is
lowered (see discussion after Eq. (14)) but the intensity <strong>of</strong>
the photon radiation is also lowered so that the order <strong>of</strong>
magnitude <strong>of</strong> (21) remains unchanged.
SUMMARY
We have considered the effective mass model [12] which
allows a rather simple solution <strong>of</strong> the kinetic equation describing
(in the framework <strong>of</strong> the one-photon annihilation
mechanism) the photon radiation from the focal spot <strong>of</strong> two
counter-propagating laser beams. This simple model leads
to considerable depletion <strong>of</strong> the spectrum <strong>of</strong> parametric oscillations
<strong>of</strong> the e − e + γ plasma: only the 2 nd and 4 th harmonics
remain due to the condition <strong>of</strong> the absence <strong>of</strong> a
constant component in the photon production rate. Thus
a compensation <strong>of</strong> the mismatch (6) is possible by means
<strong>of</strong> these two harmonics only. The domain <strong>of</strong> applicability
<strong>of</strong> this model is limited to the X-ray domain <strong>of</strong> the laser
radiation. The model suggests a high integral luminosity <strong>of</strong>
∼ 10 15 photons per sec from the focal spot. Other features
<strong>of</strong> the model are: the 1/k-behavior in the infrared domain
k ≪ m (the flicker noise <strong>of</strong> electrodynamic nature) and the
threshold effect. These results are encouraging for a further
detailed study <strong>of</strong> the photon kinetics on the basis <strong>of</strong>
the one-photon annihilation mechanism in the domain <strong>of</strong>
small adiabacity parameters γ 1.
ACKNOWLEDGEMENTS: We thank our colleagues
G. Gregori, C.D. Murphy, A.V. Prozorkevich, C.D. Roberts
and S. Schmidt for their collaboration. A.M. Fedotov,
D. Habs, B. Kämpfer, H. Ruhl and R. Sauerbrey are acknowledged
for their continued interest in our work and
valuable discussions. A.V.T would like to thank the Organizing
Committee for the warm and stimulating atmosphere
<strong>of</strong> the PIF2010.
REFERENCES
[1] G. Gregori, D.B. Blaschke, P.P. Rajeev, H. Chen, R.J.
Clarke, T. Huffman, C.D. Murphy, A.V. Prozorkevich, C.D.
Roberts, G. Röpke, S.M. Schmidt, S.A. Smolyansky, S.
Wilks, R. Bingham, High Energy Dens. Phys. 6 (2010) 166.
[2] S.A. Smolyansky, A.V. Prozorkevich, S.M. Schmidt, D.
Blaschke, G. Röpke and V.D. Toneev, Int. J. Mod. Phys. E 7
(1998) 515.
[3] D.B. Blaschke, A.V. Prozorkevich, G. Röpke, C.D. Roberts,
S.M. Schmidt, D.S. Shkirmanov and S.A. Smolyansky, Eur.
Phys. J. D 55 (2009) 341.
[4] F. Hebenstreit, R. Alk<strong>of</strong>er and H. Gies, Phys. Rev. D 78
(2008) 061701.
[5] R. Yaresko, M.G. Mustafa and B. Kämpfer, Phys. Plasmas
17 (2010) 103302.
[6] D.B. Blaschke, A.V. Prozorkevich, C.D. Roberts, S.M.
Schmidt and S.A. Smolyansky, Phys. Rev. Lett. 96 (2006)
140402.
[7] A.V. Filatov, A.V. Prozorkevich and S.A. Smolyansky, Proc.
<strong>of</strong> SPIE v6165, (2006) 616509.
[8] V.N. Pervushin, V.V. Skokov, Int. J. Mod. Phys. A, 20,
(2005) 5689.
[9] S.A. Smolyansky, A.V. Reichel, D.V. Vinnik, S.M. Schmidt,
in: ”Progress in Nonequilibrium Green’s Functions 2”,
World Scientific, Singapore , p.384 (2003).
[10] D.B. Blaschke, S.M. Schmidt, S.A. Smolyansky and A.V.
Tarakanov, Phys. Part. Nucl. 41 (2010) 1004.
[11] D.B. Blaschke, G. Röpke, S.M. Schmidt, S.A. Smolyansky
and A.V. Tarakanov, arXiv:1006.1098 [physics.plasm-ph].
[12] V.I. Ritus, Trudy FIAN SSSR, 111 (1979) 5.
[13] S.A. Smolyansky D.B. Blaschke, A.V. Chertilin G. Röpke,
A.V. Tarakanov, arXiv:1012.0559 [physics.plasm-ph].
[14] D.B. Blaschke, S.V. Ilyin, A.D. Panferov, G. Röpke and S.A.
Smolyansky, Contrib. Plasma Phys. 49 (2009) 602.
[15] A. Ringwald, Phys. Lett. 510, (2001) 107.

Abstract
Unruh radiation and Interference effect ∗
Satoshi Iso † , Yasuhiro Yamamoto ‡ and Sen Zhang § , KEK, Tsukuba, Japan
A uniformly accelerated charged particle feels the vacuum
as thermally excited and fluctuates around the classical
trajectory. Then we may expect additional radiation besides
the Larmor radiation. It is called Unruh radiation. In
this report, we review the calculation <strong>of</strong> the Unruh radiation
with an emphasis on the interference effect between the
vacuum fluctuation and the radiation from the fluctuating
motion. Our calculation is based on a stochastic treatment
<strong>of</strong> the particle under a uniform acceleration. The basics <strong>of</strong>
the stochastic equation are reviewed in another report in the
same proceeding [2]. In this report, we mainly discuss the
radiation and the interference effect.
STOCHASTIC ALD EQUATION
The Unruh radiation is the additional radiation expected
to be emanated by a uniformly accelerated charged particle
[3]. A uniformly accelerated observer feels the quantum
vacuum as thermally excited with the Unruh temperature
TU = a/2πckB. Hence as the ordinary Unruh-de Wit detector,
a charged particle interacting with the radiation field
can be expected to fluctuate around the classical trajectory.
Is there additional radiation associated with this fluctuating
motion? It is the issue <strong>of</strong> the present report.
In order to formulate the dynamics <strong>of</strong> such fluctuating
motion, we make use <strong>of</strong> the stochastic technique. Namely,
we solve a set <strong>of</strong> equations <strong>of</strong> the accelerated particle and
the radiation field in a semiclassical approximation. By
semiclassical, we mean that the radiation field is treated
as a quantum field while the particle is treated classically.
Since the accelerated particle dissipates its energy
through the Larmor radiation, the equation <strong>of</strong> motion contains
the radiation damping term. This is the Abraham-
Lorentz-Dirac (ALD) equation. Furthermore, since the accelerated
particle feels the Minkowski vacuum as thermally
excited, a noise term is also induced in the equation <strong>of</strong> motion.
The stochastic equation <strong>of</strong> the accelerated charged
particle is called the stochastic ALD equation and derived
by [4].
We consider the scalar QED whose action is given by
∫
S[z, ϕ, h] = − m
∫
+
dτ √ ˙z µ ∫
˙zµ +
d 4 x 1 2
(∂µϕ)
2
d 4 x j(x; z)ϕ(x). (1)
∗ Based on a poster presentation by Y.Yamamoto and [1].
† satoshi.iso@kek.jp
‡ yamayasu@post.kek.jp
§ zhangsen@post.kek.jp
where
∫
j(x; z) = e
dτ √ ˙z µ ˙zµδ 4 (x − z(τ)), (2)
We choose the parametrization τ to satisfy ˙z 2 = 1.
By solving the Heisenberg equation for ϕ, we get the
stochastic ALD equation for the charged particle:
m ˙v µ − F µ − e2
12π (vµ ˙v 2 + ¨v µ ) = −e −→ ω µ ϕh(z) (3)
where v µ = ˙z µ . The dissipative term corresponds to loss
<strong>of</strong> energy through the radiation and it is called the radiation
damping term. On the other hand, the noise term comes
from the Unruh effect, namely, interaction <strong>of</strong> a uniformly
accelerated particle with the thermal bath <strong>of</strong> the radiation
field.
We can easily solve the dynamics <strong>of</strong> small fluctuations <strong>of</strong>
the transverse velocities v i = v i 0+δv i in terms <strong>of</strong> the quantum
fluctuations <strong>of</strong> the field ϕh (or its Fourier tranformed
field φ) as
where
δ˜v i (ω) = eh(ω)∂iφ(ω), (4)
δv i ∫
dω
(τ) =
2π δ˜vi (ω)e −iωτ ,
∫
dω
−iωτ
∂iϕh(τ) = ∂iφ(ω)e
2π
(5)
(6)
h(ω) =
1
. (7)
−imω + e2
12π (ω2 + a 2 )
In the following, as an ideal case we consider a uniformly
accelerated charged particle in the scalar QED, and investigate
the radiation from such a particle. The main issue is
the effect <strong>of</strong> interference.
RADIATION AND INTERFERENCE
Now we calculate the radiation emanated from the uniformly
accelerated charged particle. First let’s consider the
2-point function <strong>of</strong> the radiation field. Since the field is
written as a sum <strong>of</strong> the quantum fluctuation (a homogeneous
solution) ϕh and the inhomogeneous solution in the
presence <strong>of</strong> the charged particle ϕI, the 2-point function is
given by
⟨ϕ(x)ϕ(y)⟩ − ⟨ϕh(x)ϕh(y)⟩ (8)
= ⟨ϕI(x)ϕh(y)⟩ + ⟨ϕh(x)ϕI(y)⟩ + ⟨ϕI(x)ϕI(y)⟩.
The Unruh radiation estimated in [3] is contained in
⟨ϕIϕI⟩, which include the Larmor radiation. We need special
care <strong>of</strong> the interference terms. As discussed in [5],

the interference terms ⟨ϕIϕh⟩ + ⟨ϕhϕI⟩ may possibly cancel
the Unruh radiation in ⟨ϕIϕI⟩ after the thermalization
occurs. The cancellation is explicitly shown for an internal
detector, but it is not obvious whether the same cancellation
occurs for the case <strong>of</strong> a charged particle we are considering.
The inhomogeneous solution <strong>of</strong> the scalar field is written
as
∫
ϕI(x) = e
dτGR(x − z(τ)) =
e
. (9)
4πρ(x)
ρ(x) = ˙z(τ x −) · (x − z(τ x −)), (10)
where τ x − satisfies (x − z(τ x −)) 2 = 0, x0 > z0 (τ x −),
which is the proper time <strong>of</strong> the particle whose radiation
travels to the space-time point x. Hence, z(τ x −) lies on an
intersection between the particle’s world line and the light
cone extending from the observer’s position x (See Fig 1).
We write the superscript x to make the x dependence <strong>of</strong> τ
explicitly.
The particle’s trajectory is fluctuating and expressed as
z = z0 + δz + δ2z + · · · where we have expanded the
tragectory with respect to the interaction with the radiation
field (i.e. e). Then ρ is also expanded as ρ = ρ0 + δρ +
δ2ρ + · · · and (9) becomes
(
) )
2
ϕI = e
4πρ0
1 − δρ
+
ρ0
( δρ
ρ0
− δ2 ρ
ρ0
+ · · ·
. (11)
The first term is the classical potential, but since the particle’s
trajectory deviates from the classical one, the potential
also receives corrections.
Inhomogeneous part
By inserting the expansion (11), the correlator <strong>of</strong> the inhomogeneous
solution ϕI becomes
⟨ϕI(x)ϕI(y)⟩ (12)
(
e
) 2 1
=
4π ρ0(x)ρ0(y)
)
×
.
(
1 + ⟨δρ(x)δρ(y)⟩
ρ0(x)ρ0(y) + ⟨(δρ(x))2 ⟩
ρ2 0 (x) + ⟨(δρ(y))2 ⟩
ρ2 0 (y)
The first term gives the Larmor radiation. The other terms
correspond to the radiation induced by the fluctuations <strong>of</strong>
the particle’s motion.
The calculations <strong>of</strong> these terms are easy, because one can
write ⟨δρδρ⟩ in terms <strong>of</strong> ⟨δ ˙z i δ ˙z i ⟩ = ⟨δv i δv i ⟩, which can
be obtained by solving the dynamics <strong>of</strong> fluctuations in the
stochastic ALD equation [1, 2]. They become
⟨ϕI(x)ϕI(y)⟩ (13)
(
e
) [
2 1
=
1 + e
4π ρ0(x)ρ0(y)
2
∫
dω
2π |h(ω)|2IS(ω) )
×
]
.
( i i −iω(τ
x y e x y
−−τ− )
ρ0(x)ρ0(y) + xixi ρ2 0 (x) + yiyi ρ2 0 (y)
Since we are considering the fluctuating motion whose frequency
is smaller than the acceleration, IS can be approximately
given by a 3 /12π 2 .
Figure 1: The hyperbolic line in the right wedge denotes
the world line <strong>of</strong> the particle. The points OF and OR are
observers in the future and right wedges, respectively. For
an observer in the right wedge, the light-cone <strong>of</strong> the observer
has two intersections with the world line, and the
proper time <strong>of</strong> the intersections are given by τ R ± . For an
observer in the future wedge, there is only one intersection
on the particle’s real trajectory which corresponds to τ F − .
The other solution T F + = τ F + + iπ/a is complex. One may
interpret this complex proper time as the intersection between
the light-cone <strong>of</strong> the observer and the world line <strong>of</strong> a
virtual particle with a real proper time τ F + in the left wedge.
The superscript letters R or F are used to distinguish two
different observers, but we do not use them in the body <strong>of</strong>
the paper to leave the space for the observer’s position x.
Interference Term
The calculation <strong>of</strong> the interference terms is a bit more
involved. The inhomogeneous solution ϕI is expanded as
(11). Since the leading term which has a nonvanishing correlation
with the quantum fluctuation ϕh is the second term,
we have
⟨ϕI(x)ϕh(y)⟩ + ⟨ϕh(x)ϕI(y)⟩
= − e
(
⟨δρ(x)ϕh(y)⟩
4π
ρ 2 0 (x)
+ ⟨ϕh(x)δρ(y)⟩
ρ 2 0 (y)
)
. (14)
The fluctuation <strong>of</strong> the distance δρ is written in terms <strong>of</strong> δv i
which is the solution <strong>of</strong> the stochastic ALD equation (4),
and we obtain
⟨ϕh(x)δρ(y)⟩ = −ey i
∫
dω
y
e−iωτ−h(ω)⟨ϕh(x)∂iφ(ω)⟩. 2π

The integrand can be written as
∫
⟨ϕh(x)∂iφ(ω)⟩ = dτe iωτ
( )
∂
⟨ϕh(x)ϕh(y)⟩
∂yi y=z(τ)
∫
= − dτe iωτ
(
∂P (x, ω)
∂xi )
, (15)
where
∫
P (x, ω) = dτ
e iωτ
(x 0 − z 0 (τ) − iϵ) 2 − (x 1 − z 1 (τ)) 2 − x 2 ⊥
x 2 ⊥ = (x2 ) 2 + (x 3 ) 2 is the transverse distance. The τ integral
can be calculated by the contour integral. The residues
are located where the invariant length between the observed
point x and a point on the particle’s trajectory vanishes.
The condition is nothing but the condition that the radiation
field propagates on the light cone. Fig.1 shows such
a situation. It is interesting that the condition for residues
has a solution on an intersection <strong>of</strong> the light-cone <strong>of</strong> the observer
and the virtual path <strong>of</strong> a particle (dotted line in the
left wedge). We skip the calculations and show the final
results <strong>of</strong> the interference terms;
⟨ϕI(x)ϕh(y)⟩ + ⟨ϕh(x)ϕI(y)⟩ (16)
=
×
where
−iae2xiy i
(4π) 2ρ0(x) 2ρ0(y) 2
[
e
x y
−iω(τ−−τ e
+ e
− e
∫
dω 1
2π 1 − e−2πω/a − ) h(−ω) ( aL2 x
2ρ0(x)
x y
−iω(τ−−τ− ) − h(ω) ( aL2y 2ρ0(y)
x
−iω(τ+ −τ y
x
−iω(τ− − iω
a
)
iω )
+
a
− ) h(−ω) ( − aL2x iω )
− Zx(−ω)
2ρ0(x) a
y
−τ+ ) h(ω) ( − aL2y iω ) ]
+ Zy(−ω)
2ρ0(y) a
Zx(ω) = e πω/a θ(x 0 − x 1 ) + θ(x 1 − x 0 ) (17)
L 2 x = −x µ xµ + 1
a2 , L2y = −y µ yµ + 1
. (18)
a2 Partial Cancellation
The correlation function <strong>of</strong> the inhomogeneous terms
(13) depends only on τ−. The interference terms contain
both <strong>of</strong> terms depending on τ− and τ+; the first term in the
parenthesis <strong>of</strong> (17) depends only on τ−, so it is the term
that may cancel the inhomogeneous terms (i.e. the Unruh
radiation). Using the relation
h(ω) + h(−ω) = e2
6π (ω2 + a 2 )|h(ω)| 2 , (19)
one can show that a part <strong>of</strong> the interference terms
iae2xiy i
(4π) 2ρ0(x) 2ρ0(y) 2
×
∫ x y
−iω(τ
dω e −−τ− 2π
)
1 − e−2πω/a ( iω
h(−ω)
a
)
+ h(ω)iω
a
(20)
.
cancels the first correction term <strong>of</strong> the inhomogeneous part
in (13). This term was obtained by taking a derivative <strong>of</strong>
x
iωτ e − in P (x, ω). But note that the cancellation occurs only
partially. Furthermore, the τ+-dependent terms in the interference
terms cannot be canceled with the Unruh radiation.
The Energy Momentum Tensor
Given the 2-point function, we can calculate the energy
momentum tensor <strong>of</strong> the radiation
⟨Tµν(x)⟩ = ⟨: ∂µϕ∂νϕ − 1
2 gµν∂ α ϕ∂αϕ :⟩. (21)
It is a sum <strong>of</strong> the classical and the fluctuation parts; Tµν =
Tcl,µν + Tfluc,µν. The classical part is given by
Tcl,µν ∼ e2∂µρ0∂νρ0 (4π) 2ρ4 . (22)
0
It corresponds to the energy momentum tensor <strong>of</strong> the Larmor
radiation. From (10) it can be seen to be proportional
to a 2 /r 2 where a is the acceleration and r is the spacial
distance from the particle to the observer. Tfluc,µν is the
energy momentum <strong>of</strong> the additional radiation
Tfluc,µν = (xi ) 2
− e2 a 2 L 2 x
(4π) 2 ρ 3 0
(
ρ 2 0
[ (e 2
π Im − 6ma2 I1L 2 x
ρ0
)
Tcl,µν
mI3 ∂µτ x −∂ντ x − + 2mI1
ρ0L2 (xµ∂νρ0 + xν∂µρ0)
x
+ e2Im 12πL2 (xµ∂ντ
x
x − + xν∂µτ x −)
− e2Im (∂µτ
24πρ0
x −∂νρ0 + ∂ντ x −∂µρ0)
) ]
(23)
where I1 = 3
2mae2 , I3 ∼ Ω2 −I1 ≪ a2I1, Im = I3+a2 I1 ∼
a2I1. Hence, these terms originating from the fluctuating
motion <strong>of</strong> the particle is proportional to a3 , and smaller
by a factor <strong>of</strong> a compared to the above Larmor radiation.
Though they have different angular distribution, there is an
overall factor (x2 i ) in front and they vanish at the forward
direction. Together with the long relaxation time discussed
in [2], the detection <strong>of</strong> the Unruh radiation seems to be very
difficult experimentally.
REFERENCES
[1] S. Iso, Y. Yamamoto and S. Zhang, arXiv:1011.4191 [hep-th].
[2] S. Iso, Y. Yamamoto and S. Zhang, in the same proceeding,
“Can we detect ”Unruh radiation” in the high intensity laser?”
[3] P. Chen and T. Tajima, Phys. Rev. Lett. 83 (1999) 256.
[4] P. R. Johnson and B. L. Hu, arXiv:quant-ph/0012137.
Phys. Rev. D 65 (2002) 065015 [arXiv:quant-ph/0101001].
P. R. Johnson and B. L. Hu, Found. Phys. 35, 1117 (2005)
[arXiv:gr-qc/0501029].
[5] D. J. Raine, D. W. Sciama and P. G. Grove, Proc. R. Soc.
Lond. A (1991) 435, 205-215

P r o g r a m <strong>of</strong> PIF2010
th
N o v (Wednesday) . 2 4
08:30 Registration starts @ foyer in front <strong>of</strong> Kobayashi Hall
[Opening and tutorial-1 : chaired by T shiki o Tajima (LMU, Munich)]
09:00 Opening address by Satoshi Iso (KEK, the conference chair) [10]
09:10 Welcome speech by Fumihiko Takasaki (KEK) [10]
09:20 Gerard Mourou (Ecole Polytechnique) [60]
“Extreme Light for High Energy Physics”
10:20 break [15]
[Nonlinear QED : chaired by Toshiaki Tauchi (KEK)]
10:35 Thomas Heinzl (Plymouth U.) [45]
“QED in ultra-intense laser fields”
11:20 Kaoru Yokoya (KEK) [30]
“Beam-beam interaction in <strong>International</strong> Linear Collider”
11:50 Anthony Hartin (DESY) [25]
“Second Order QED processes in the Furry Picture and their Radiative Corrections”
12:15 lunch [75]
[Heavy-ion collisions and Quark-Gluon Plasma : chaired by Tetsuo Hatsuda (U. T o )] k y o
13:30 Kazunori Itakura (KEK) [30]
“Strong Field Dynamics in Heavy Ion Collisions”
14:00 Andreas Ipp (TU Vienna) [25]
“Yoctosecond photon pulse generation in heavy ion collisions”
14:25 Kenji Fukushima (Keio U.) [25]
“Fields, instantons, and currents”
14:50 Kenji Morita (GSI) [25]
“Critical behavior <strong>of</strong> charmonium : QCD second order Stark effect”

15:15 break [15]
[Unruh radiation : chaired by Kensuke Homma (Hiroshima U.)]
15:30 Ralf Schutzhold (Essen U.) [45]
“Fundamental Quantum Effects in Strong Lasers”
16:15 Sen Zhang (KEK) [25]
“Does an uniformly accelerated electron radiate Unruh radiation? ”
16:40 Frieder Lenz (Erlangen-Nuernberg & RIKEN) [25]
“Quantum fields in accelerated frames”
17:05 break [15]
[Axion-like particle searches : chaired by Hiroshi Azechi (ILE, Osaka U.)]
17:20 Axel Lindner (DESY) [45]
“Shining light through walls: enroute towards a new particle physics frontier”
18:05 Kensuke Homma (Hiroshima U.) [25]
“Probing extremely light fields via resonance scattering by laser focusing”
th
N o v (Thursday) . 2 5
[Tutorial-2 : chaired by Dietrich Habs (LMU, Munich)]
09:00 Gerald Dunne (Connecticut U.) [60]
“The Heisenberg-Schwinger Effect: Nonperturbative Pair Production from Vacuum”
10:00 break [15]
[Schwinger mechanism in QGP and condensed matter : chaired by Masayuki Asakawa (Osaka
U.)]
10:15 Naoto Tanji (Tokyo U. Komaba) [25]
“Dynamical view <strong>of</strong> pair creation via the Schwinger mechanism”
10:40 Aiichi Iwazaki (Nishogakusha U.) [25]
“Exact solutions <strong>of</strong> pair productions in strong electric fields with finite sizes”
11:05 Takashi Oka (Tokyo U.) [25]
“Strong field physics in condensed matter”
11:30 Shin Nakamura (Kyoto) [25]
“Non-linear charge transport in plasma under strong field”

11:55 lunch & group photo [85]
[Recent developments in Schwinger mechanism 1: chaired by Philip Bambade (LAL, Orsay)]
13:20 Dietrich Habs (LMU,Munich) [45]
“Vacuum Pair Creation”
14:05 Hartmut Ruhl (LMU,Munich) [25]
“QED cascading: A particle in cell model”
14:30 Nina Elkina (LMU, Munich) [25]
“Numerical simulations <strong>of</strong> QED cascades in circularly polarized laser fields”
14:55 break [15]
[Recent developments in Schwinger mechanism 2: chaired by Gerald Dunne (Connecticut U.)]
15:10 Sergei Bulanov (Kansai Photon Science Institute, JAEA) [45]
“On the Schwinger limit attainability with extreme power lasers”
15:55 Natalia Naumova (Ecole Polytechnique) [25]
“Pair Creation in QED-Strong Pulsed Laser Fields Interacting with Electron Beams”
16:20 Heinrich Hora (New South Wales) [25]
“Acceleration up to black hole conditions and B-meson decay”
16:45 POSTER SESSION at the foyer in front <strong>of</strong> Kobayashi Hall [90] (~ 18:15)
18:30 Buses leave for banquet
19:00 Banquet at Okura Frontier Hotel Tsukuba (~21:00)
th
N o v ( F r i . d a y ) 2 6
[Recent progress <strong>of</strong> ultra-intense lasers : chaired by Alex Borisov (U. Illinois at Chicago)]
09:00 Hideaki Takabe (ILE, Osaka U.) [45]
“Present Status <strong>of</strong> Ultra-intense Lasers and High-Field Physics in the World”
09:45 Baifei Shen (Shanghai Institute <strong>of</strong> Optics and Fine Mechanics) [25]
“Generation <strong>of</strong> ultra intense X ray approaching the Schwinger limit”
10:10 Charles Rhodes (U. Illinois at Chicago) [25]
“Reaching the Schwinger Limit with X-Rays”

10:35 break (20)
[Magnetars : chaired by David Salzmann (Weizmann Institute <strong>of</strong> Science)]
10:55 Kazunori Kohri (KEK) [30]
“Nonlinear QED effects by strong magnetic field in astrophysics”
11:25 Kazuo Makishima (Tokyo U.) [45]
“W i -Band d e X-ray Observations <strong>of</strong> Magnetars”
12:10 Toshitaka Tatsumi (Kyoto U.) [25]
“QCD origin <strong>of</strong> strong magnetic fields in compact stars”
12:35 lunch @ foyer [105]
[New technologies : chaired by Kiminori Kondo (JAEA)]
14:20 Kazuhisa Nakajima (KEK) [45]
“Recent progress and prospects on laser-plasma acceleration <strong>of</strong> charged particles”
15:05 Masaki Kando (JAEA) [25]
“Flying Mirror as a tool to access ultra-high field”
15:30 break [15]
[Overview : chaired by Tohru Takahashi (Hiroshima U.)]
15:45 Toshiki Tajima (LMU, Munich) [60]
“High field science”
16:45 Closing remarks by Tohru Takahashi (Hiroshima U., the conference chair) [10]

List <strong>of</strong> Participants
Name Institution Name Institution
AKAGI, Tomoya Hiroshima University NAKAMURA, Shin Department <strong>of</strong> Physics, Kyoto University
ARAKI, Sakae KEK NAKAMURA, Gen Hiroshima University
ASAKAWA, Masayuki Osaka University NARA, Yasushi Akita <strong>International</strong> University
AZECHI, Hiroshi Institute <strong>of</strong> Laser Engineering Osaka University NAUMOVA, Natalia Ecole Polytechnique
BAIOTTI, Luca Institute <strong>of</strong> Laser Engineering, Osaka University NISHIMURA, Hiroaki Institute <strong>of</strong> Laser Engineering, Osaka University
BAMBADE, S.Philip LAL-Orsay NISHIMURA, Jun KEK
BORISOV, B.Alex University <strong>of</strong> Illinois at Chicago NODA, Akira Institute for Chemical Research, Kyoto University
BULANOV, V.Sergei Kansai Photon Science Institute, JAEA NOZAKI, Mitsuaki KEK
DELERUE, Nicolas LAL, Orsay OHTA, Masahiro KEK
DUNNE, V.Gerald University <strong>of</strong> Connecticut OKA, Takashi University <strong>of</strong> Tokyo
ELKINA, Nina Ludwig-Maximilian University, Munich OKADA, Yasuhiro KEK
ENOTO, Teruaki KIPAC / Stanford University OKAZAWA, Susumu SOKENDAI, KEK
FUJII, Keisuke KEK OMORI, Tsunehiko KEK
FUJII, Hirotsugu University <strong>of</strong> Tokyo OROKU, Masahiro University <strong>of</strong> Tokyo
FUJII, Yasunori Waseda University POSCH, Paul KEK
FUJITSUKA, Masashi SOKENDAI, KEK RHODES, K. Charles University <strong>of</strong> Illinois at Chicago
FUJIWARA, Mamoru Research Center for Nuclear Physics, Osaka Univ. RUHL, Hartmut Ludwig-Maximilians-University Munich
FUKUSHIMA, Kenji Keio University SAEKI, Takayuki KEK
GOTO, Hajime Graduate University for Advanced Studies SALZMANN, DAVID Weizmann Institute <strong>of</strong> Science
HABS, Dietrich Faculty <strong>of</strong> Physics, LMU Munich SASAKI, Toshihiko University <strong>of</strong> Tokyo, KEK
HARTIN, Anthony DESY SCHUETZHOLD, Ralf Fakultät für Physik der Universität Duisburg-Essen
HATSUDA, Tetsuo Phys. Dep., Univ. Tokyo SEKINO, Yasuhiro Okayama Institute for Quantum Physics
HATTORI, Koichi KEK SHEN, Baifei Shanghai Institute <strong>of</strong> Optics and Fine Mechanics
HEINZL, Thomas University <strong>of</strong> Plymouth, Computing and Mathematics SHIMADA, Kengo SOKENDAI, KEK
HIDAKA, Yoshimasa Department <strong>of</strong> Physics, Kyoto University SHIMIZU, Katsuya KYOKUGEN, Osaka University
HOMMA, Kensuke Hiroshima University / LMU SHIMIZU, Hirotaka KEK
HONDA, Yosuke KEK SOUDA, Hikaru Institute for Chemical Research, Kyoto University
HORA, Heinrich University <strong>of</strong> New South Wales SUNAHARA, Atsushi Institute for Laser Technology
IPP, Andreas Vienna University <strong>of</strong> Technology SUZUKI, Atsuto KEK
ISO, Satoshi KEK TAJIMA, Toshiki LMU
ITAKURA, Kazunori KEK TAKABE, Hideaki ILE and School <strong>of</strong> Science, Osaka University
IWATA, Natsumi Kyoto University TAKAHASHI, Tohru Hiroshima University
IWAZAKI, Aiichi Nishogakusha Universeity TAKASAKI, Fumihiko KEK
KANDO, Masaki Japan Atomic Energy Agency TANJI, Naoto University <strong>of</strong> Tokyo
KISHIMOTO, Yausaki Kyoto University TARAKANOV, Alexander Saratov State University, Physics Department
KITAMOTO, Hiroyuki SOKENDAI, KEK TATSUMI, Toshitaka Kyoto University
KITAZAWA, Yoshihisa KEK TATSUMI, Daisuke National Astronomical Observatory <strong>of</strong> Japan
KOHRI, Kazunori KEK TAUCHI, Toshiaki KEK
KOYAMA, Kazuyoshi University <strong>of</strong> Tokyo TERUNUMA, Nobuhiro KEK
KUBO, Kiyoshi KEK TOMARU, TAKAYUKI KEK
KUBOTA, Hirohisa SOKENDAI, KEK UEDA, Ken-ichi University <strong>of</strong> Electro-Communications
KUMADA, Masayuki NIRS URAKAWA, Junji KEK
KUMITA, Tetsuro Tokyo Metropolitan University YABANA, Kazuhiro University <strong>of</strong> Tsukuba
LENZ, Frieder University <strong>of</strong> Erlangen-Nuernberg YAMAGUCHI, Yohei University <strong>of</strong> Tokyo
LINDNER, Axel DESY YAMAMOTO, Yasuhiro SOKENDAI/KEK
MAKISHIMA, Kazuo University <strong>of</strong> Tokyo YAN, Jacqueline University <strong>of</strong> Tokyo
MATSUBA, Shunya SOKENDAI, KEK YAZAKI, Koichi Hashimoto Math. Phys. Lab., Nishina Center, RIKEN
MINAKATA, Hisakazu Tokyo Metropolitan University YOKOYA, Kaoru KEK
MIURA, Eisuke AIST ZHANG, Sen KEK
MORITA, Kenji GSI
MOUROU, A. Gerard Institut Lumière Extrême/Ecole Polytechnique ENSTA
MURAKAMI, Masakatsu Institute <strong>of</strong> Laser Engineering, Osaka University Secretaries
MUROYA, Shin Matsumoto University IKEDA, Kimiyo KEK
NAKAI, Mitsuo Institute <strong>of</strong> Laser Engineering, Osaka University KUSAMA, Hitomi KEK
NAKAJIMA, Kazuhisa KEK SHISHIDO, Tamao KEK