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

ϵ is the energy <strong>of</strong> the particle, m is the mass <strong>of</strong> the particle,
and Es = m2 /(e¯h) is the Schwinger field. This
can be understood by observing that the peak <strong>of</strong> the emitted
radiation energy <strong>of</strong> an energetic electron in a constant
magnetic field versus its energy is approximately given by
(¯hω0/ϵ) (ϵ/m) 3 = B⊥ϵ/(mEs), where ω0 = eB⊥/ϵ.
Further details can be found in [5]. In the lab frame this
parameter becomes the quantum efficiency parameters χ
χ = e¯h
√
− (F µν pν) 2
m3 , (3)
where pν is the electron or positron 4-momentum. In the
classical radiation realm the emitted photon energies are
much smaller than the charged particle energy. Hence,
χ ≪ 1 holds. In the quantum realm χ ≫ 1 is valid. The
transition rate for radiation emission depends on a second
quantum parameter κ given by
κ = e¯h
√
− (F µν kν) 2
m3 , (4)
where kν is the photon 4-momentum. The angle integrated
transition rate for photon emission is
where
dWγ
dω
(∫ ∞
m2
= −α
ϵ2 dz Ai(z) (5)
x
[
2
+
x + κ √ ] )
x ∂zAi(x) ,
[
κ
x =
χ (χ − κ)
] 2
3
, 0 ≤ κ < χ . (6)
The rate holds for radiation emission by electrons or
positrons. In the simulations we assume that
′
⃗k = ⃗p + ⃗p (7)
holds for the momenta. For the energy balance we find
q + ω = ϵ ′
+ ϵ . (8)
Making use <strong>of</strong> Eqns. (7) and (8) the energy taken from the
external laser field is
q =
√
ϵ 2 + 2ωϵ ′ (1 − v ′ cos θ) − ϵ ,
where θ is the angle between the emitted photon and the
outgoing electron or positron ⃗p ′
and v ′
= |⃗p ′
|/ϵ ′
is the
velocity <strong>of</strong> the electron or positron after the radiation process.
For electrons or positrons retaining large momenta ⃗p ′
along ⃗k after the emission process v ′
cos θ ≈ 1 holds. The
external field has to deliver only little energy, q ≈ 0, in that
case. Given the electron or positron 4-momentum and the
external field context χ can be calculated. In a second step
κ is obtained for permissible photon emission and hence x.
With the help <strong>of</strong> the crossing symmetry the angle integrated
transition rate for e + e−-pair creation is given by
where
dW e + e −
dϵ−
= α m2
ω2 (∫ ∞
y
+
[
κ
y =
χ (κ − χ)
[
2
y − κ √ ]
y
] 2
3
dz Ai(z) (9)
)
∂zAi(y) ,
, 0 ≤ χ < κ . (10)
The following kinematic relations are used when pairs are
created
⃗ k = ⃗p+ + ⃗p− , (11)
q + ω = ϵ+ + ϵ− . (12)
Making use <strong>of</strong> Eqns. (11) and (12) the energy taken from
the external laser field is
√
q = ϵ2 − + 2ωϵ+ (1 − v+ cos θ) − ϵ− ,
where θ is the angle between the photon and the emitted
positron, and v+ = |⃗p+|/ϵ+ is the velocity <strong>of</strong> the positron.
Given a photon 4-vector and the field context κ can be obtained.
Next it is possible to calculate χ for any permissible
electron energy and hence y.
If χ and κ are large enough we find that photons generate
electrons and positrons and the latter again photons.
The chain process leads to exponential growth <strong>of</strong> particles
in the simulation. The solution <strong>of</strong> the underlying transport
equations for uniform circular polarized light is approximately
N e + e −(t) = N e + e −(0) e Γ t , (13)
where the growth rate Γ can be estimated as
Γ = α µ 1
√
m ω c2 4 ,
¯h
µ = E
.
α Es
(14)
Here E is the external field.
Quantum efficiency becomes very large in circular
purely electric fields. Hence we consider two counterpropagating
circular laser beams in the center <strong>of</strong> which
there is a rotating electric field. Figure 1 shows the situation
and the simulation results. Along the purple line electrons
are injected into the laser focus at 600 MeV. Red and
green lines represent secondary electrons and positrons. As
a main result we find that the mean energy per particles is
not decreasing in the course <strong>of</strong> the cascade development.
The contrary could be expected since the number <strong>of</strong> secondary
particles grows exponentially. The finding implies
enhanced energy deposition. In fact, the empirical observation
<strong>of</strong> almost constant mean energy per particle represents
exponentially growing energy deposition in the evolving
plasma.