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

(a) longitudinal momentum distribution with fixed
transverse momentum p⊥ = 0
(b) transverse momentum distribution with fixed
longitudinal momentum √ eEpz = 1
Figure 3: Time evolution <strong>of</strong> the momentum distribution <strong>of</strong> created particles under the constant electric field. m2
2eE = 0.1.
These features are consistent with the semi-classical tunneling
description <strong>of</strong> pair creation [10]. However, the longitudinal
momenta that particles bring when they are created
are not exactly zero but broadened around zero due
to quantum fluctuation. After created, particles are accelerated
by the electric field according to the classical equation
<strong>of</strong> motion: pz = eEt + const. In contrast, there is no
acceleration in the transverse directions and the transverse
momentum distributions exhibits a Gaussian-like form.
Back reaction
Previously, we have treated the pair creation in the constant
electric field. However, if charged particles are created,
they generate further electromagnetic fields and the
original field should be modified. This effect, called back
reaction, is not negligible under a strong field where pair
creation happens intensively. To take account <strong>of</strong> the back
reaction, we treat an electromagnetic field as a dynamical
variable obeying the Maxwell equations. The uniformity
<strong>of</strong> the system reduces the Maxwell equations into the single
equation
dEz
dt = −d2 A 3
dt 2 = −jz, (7)
Figure 4: Time evolution <strong>of</strong> the longitudinal momentum
distribution. The effect <strong>of</strong> the back reaction is taken into
account. m2 = 0.01 and e = 1.
2eE0
where jz is the charge current generated by created particles
and antiparticles.
We have numerically solved the coupled equations <strong>of</strong> (1)
and (7). The results are shown in Figs. 4 and 5. The longitudinal
momentum distribution (Fig. 4) oscillates in momentum
space, and also the current and the electric field
(Fig. 5) show oscillations. These oscillations can be understood
as usual plasma oscillation.
Other than the plasma oscillation, which is a classical
dynamics, also quantum effects such as the Pauli blocking
and interference between matter fields play remarkable
roles in the time evolution. Because <strong>of</strong> the interference, the
momentum distribution shows fine oscillations at later time
(Fig. 4).
EFFECTS OF MAGNETIC FIELDS
Since the formation <strong>of</strong> longitudinal color-magnetic fields
as well as color-electric fields is predicted by the framework
<strong>of</strong> the CGC [6], we study the effects <strong>of</strong> a longitudinal
magnetic field on the pair creation. As illustrated in Fig. 6,
under the longitudinal magnetic field, transverse momentum
p⊥ is discretized into the Landau levels as
p 2 ⊥ −→ (2n + 1)eB (n = 0, 1, 2, . . . ). (8)
Notice that even the lowest Landau level depends on the
magnetic field strength B for the case <strong>of</strong> scalar particles.
Since the transverse momentum acts as an effective mass
m2 ⊥ = m2 + p2 ⊥ , scalar particles become effectively heavy
under a strong magnetic field, so that their pair production
(a) current density (b) electric field
Figure 5: Time evolution <strong>of</strong> the charge current density and
the electric field. e = 1. (a = m2
2eE0 ).