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

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.