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

<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.