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

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).