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

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