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

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