VACUUM PARTICLE CREATION IN PLASMA
ABSTRACT Influence of some initial particleantiparticle distributions (equilibrium and nonequilibrium) on the following evolution of the plasma created from vacuum under action of a strong timedependent electric field is considered.
 Citations (1)
 Cited In (0)
 Vacuum quantum effects in strong external fields. A A Grib, S Grib, V Mamaev, Mostepanenko .
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Теоретическая Физика, 8, 2007 г.
101
VACUUM PARTICLE CREATION IN PLASMA
c ? 2007A.V. Filatov1, A.G. Lavkin1, S.A. Smolyansky1,
L.V. Bravina2, E.E. Zabrodin2
Abstract
Influence of some initial particleantiparticle distributions (equilibrium and non
equilibrium) on the following evolution of the plasma created from vacuum under
action of a strong timedependent electric field is considered.
1. Introduction. At present, the problem of initial conditions remains actual for
dynamical description of nonequilibrium quantumfield systems (e.g., [1][4]), and, espe
cially, for the strong nonequilibrium systems, for example, at presence of strong quasi
classical external fields compared with the critical ones. At last years a considerable
progress in the kinetic description of such systems can achieved [5][9]. The peculiarity
of these methods are the nonperturbative dynamics with respect to the quasiclassical
fields and nontriviality of vacuum states under description of quasiparticle excitations
[10, 11]. These approaches was used for investigation of vacuum particle creation and
annihilation processes under action of strong fields of different natures: in the ultra rel
ativistic heavy ion collision physics [5, 7, 12], the vacuum boiling up in the focus spot of
superpower laser radiation [13, 14], matter creation in the early Universe [9, 15] etc. The
kinetics of vacuum particle production was considered here with the vacuum initial con
ditions, which don’t contain massive particles. However the situations with some initial
particleantiparticle distributions are quite real. For example, the quarkgluon plasma
(QGP) vacuum production at a fireball expansion takes place with some initial distri
butions of valent quarks and gluons (e.g., [16, 17]). The presence of initial plasma can
be actual also for the vacuum creation of electronpositron plasma (EPP) in strong laser
field [18], in magnetar magnetosphere [19] and so on.
The present work devotes to research of the influence of some initial distributions of
particles and antiparticles (by condition of general electroneutrality of the system) on the
consequent vacuum plasma creation under action of space homogeneous time dependent
electric field of the linear polarization. Some relevant results was obtained here earlier. So,
influence of the initial state occupation by n particles and m antiparticles on the process
of vacuum creation was investigated in the work [20] (see also [10]). Besides expected
effects such as the bose enhancement and the fermi suppression, the phenomenon of
transformation of the quarkantiquark plasma annihilation energy into the external field
was investigated. In the work [21] was showed that vacuum particle creation violates
the thermal character of the initial distribution. Some other aspects of this problem was
discussed in the literature (see [22] and having then references).
The present work extends the investigations begun in the works [20][25]. We use the
nonperturbative approach offered in the works [5] and next used in the application to
different physical problems with zero initial conditions (e.g., generation of QGP under
collisions of ultrarelativistic heavy ions [5, 7, 12], vacuum EPP creation in the fields of
superpower optical and Xray lasers [13, 14], the early cosmology [15] etc.). The relevant
kinetic equations (KE) for the scalar and spinor (spin 1/2) brought in Sec. 2. In the
1Filatov Andrey Victorovich, Lavkin Alexander Grigorievich, Smolyansky Stanislav Alexandrovich,
email: smol@sgu.ru, Saratov State University, Saratov, 410026, Russia.
2Bravina Larisa Vladimirovna, Zabrodin Eugene Evgenievich, email: eugen.zabrodin@fys.uio.no, De
partment of Physics, University of Oslo, Oslo, N0316, Norway; Institute of Nuclear Physics, Moscow
State University, Moscow, 119899, Russia.
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A.V. Filatov, A.G. Lavkin, S.A. Smolyansky, L.V. Bravina, E.E. Zabrodin
Sec. 3 these KE’s are solved numerically for different variants of the initial distributions
both equilibrium (BoseEinstein and FermiDirac) and nonequilibrium (e.g., the initial
distributions of valent quarks and gluons having place at the instant of fusion of two
ultrarelativistic heavy ions). We corroborate the results of previous works [20, 21] and
discussed some new aspects:
1. "the pressing out" of fermion plasma created from vacuum in a strong electric field
by the initial thermalized one of rather high density;
2. the nonmonotone entropy change (by condition of t  reversibility of the basic KE) as
a consequence of increase of degrees number of freedom (particle creation);
3. finally, we research some features of spectrum deformations of strong nonequilibrium
initial parton states.
Sec. 4 contains short sum of the work.
2. Statement of the problem.
Let us consider the problem of the kinetic de
scription of vacuum particle creation of the charged scalar and spinor particles under
action of the space homogeneous time dependent electric field collinear to the x3axis
(the Hamilton gauge)
Aµ(t) = (0,0,0,A3(t) = A(t)).
(1)
It is assumed that the electric field (1) is switchedon smoothly, when the system pos
sesses some initial distributions of particles f0
the electroneutrality condition means f0
sidered below as most probable. In the general case, the initial distribution corresponds
to some quantum state with the density matrix ρin.
The basic objects of kinetics theory of vacuum particle creation are the distribution
functions of particles and antiparticles, that are the average values of particle and antipar
ticle number operators over the density matrix ρin(it is assumed the electroneutrality
condition is valid always):
p(p) and antiparticles f0
p(p) = f0
a(p). In particular,
a(p) = f0(p). Just this case will be con
f(p,t) = Spρina†
p(t)ap(t) = Spρinb†
−p(t)b−p(t)
(2)
and anomalous correlators
f+(p,t) = Spρina†
p(t)b†
p(t),f−(p,t) = Spρinbp(t)ap(t).
(3)
In the fermion case the spin indexes are omitted because the spin effects do not play
role in the chosen field geometry [10]. If the particles are absent in the initial state, the
averaging procedure is fulfilled over the initial vacuum state, ρin= 0in>< 0in.
The closed KE is obtained in two stages: at first, the equation of motion for the
distribution function (2) follows directly from dynamics [5],
˙f(p,t) =1
2∆(p,t){f+(p,t) + f−(p,t)} =1
2∆(p,t)u(p,t).
(4)
Analogously, it can derive the equation of motion for the anomalous correlators (3), which
has the following integral form:
?t
In the considered geometry, the amplitudes of pair creation take the form:
f±(p,t) = f±
0(p) +1
2
0
dt
?∆(p,t
?)[1 ± 2f(p,t
?)]exp±iθ(p;t,t
?).
(5)
∆(p,t) =eE(t)P3
ω2(p,t)
?ε⊥
P3
?g−1
,
(6)
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Vacuum particle creation in plasma
103
where e is particle charge with its sign, E(t) = −˙A(t) is the electric field strength,
P3= p3− eA(t), ε⊥= (m2+ p2
the dispersion law of the particle in the field (1), g is the degeneracy factor (g = 1 for
the spinless bosons and g = 2 for the Dirac fermions). The high frequency phase in Eq.
(5) is equal
?t
Finally, the functions f±
usually, it is assumed
f±
(absence of the initial anomalous correlations). In this case, from Eqs. (4) and (5) follows
KE [5]
˙f(p,t) =1
2∆(p,t)
0
The corresponding system of ordinary differential equations has the form:
˙f =1
2∆u,
where v = i(−f++ f−). The system of equations (10) posses by the first integral of
motion
(1 ± 2f)2∓ (u2+ v2) = const.
For the initial conditions (8) we have
⊥)1/2is the transversal energy, ω(p,t) =?ε2
⊥+ (P3)2is
θ(p;t,t
?) = 2
t?dτω(p,τ).
(7)
0(p) are the initial values of the anomalous correlators (3). As
0(p) = 0
(8)
?t
dt
?∆(p,t
?)[1 ± 2f(p,t
?)]cosθ(p;t,t
?).
(9)
˙ u = ∆(1 ± 2f) − 2ωv,
˙ v = 2ωu,
(10)
(11)
(1 ± 2f)2∓ (u2+ v2) = (1 ± 2f±
0)2.
(12)
The multiplier 2 in the statistical factors is a consequence of the electroneutrality condi
tion (2).
It is follows from (12) that the influence of electric field pulse on the particles with
integer spin brings to increase of initial density, i.e.
f(p,t) ≥ f0(p),
∀p ∪ ∀t,
(bosons).
(13)
The other behavior shows the fermi particles, where the final value of distribution function
can be less than initial one:
f(p,t)
≥
=
f0(p),
if
f0(p) ≤ 1/2,
f0(p) = 1/2
f(p,t)
f0(p),
if (stable point),
(fermions)
f(p,t)
≤
f0(p),
if
f0(p) ≥ 1/2.
(14)
The basic characteristics of particleantiparticle plasma are the densities of total par
ticle number, the energy and the entropy
?
ε(t)=2(2π)−3gd3pω(p,t)f(p,t),
?
(−1)g(1 + (−1)g+1f(p,t))ln(1 + (−1)g+1f(p,t)).
n(t)=2(2π)−3gd3pf(p,t),
?
S(t)=
−2(2π)−3gd3p(f(p,t)lnf(p,t) +
(15)
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A.V. Filatov, A.G. Lavkin, S.A. Smolyansky, L.V. Bravina, E.E. Zabrodin
0
1
2
3
5
2.5
0
p/m
2.5
5
0
2
4
6
f, x103
f, x103
pT/m
0
1
2
3
5
2.5
0
p/m
2.5
5
0
10
20
30
f, x103
f, x103
pT/m
Figure 1. The final plasma state with the (a)fermion and (b)boson momentum distribution
functions during the field pulse passing from the vacuum initial state
5 05
0,0
0,1
0,2
0,3
0,4
f ( 0, p3 )
p / m
in
out q
out q

+
505
0,0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
f ( 0, p3 )
p / m
in
out q
out q

+
Figure 2. The initial and final longitudinal momentum distributions for the boson (left) and
fermion (right) cases at the initial thermoequilibrium state (17) with β = 1/m and µ = 0:
dashed line  for particles and dot line for antiparticles
Other macroscopical characteristics (pressure, currents) will not discussed below.
3. Role of initial distributions We chose the Souter potential [26] as the test one
A(t) = −E0τ tanh(t/τ),E(t) = E0cosh−2(t/τ),
(16)
with parameters E0 = 2Ecr and τ = 3 fm/c. A Sautertype electric field provides an
analytical model, which acts effectively for −τ ≤ t ≤ τ. In scalar quantum electrody
namics, the pair production rate at finite temperature was calculated in the work [25]. It
was found that the number of pairs produced by the electric field is proportional to the
initial BoseEinstein distribution. From this scalar QED model the authors of [25] have
concluded that the Schwinger pairproduction rate by a timedependent electric field is
enhanced by a thermal factor of the initial BoseEinstein distribution.
Let us consider several cases of the initial states.
1) The popular initial distributions for bosons (−) and fermions (+) are the relativistic
BoseEinstein distributions [27]
f0(p) = {expβ[ω0(p) − µ] ± 1}−1,
(17)
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Vacuum particle creation in plasma
105
where µ is the chemical potential, ω0(p) =?ε2
e=0.07. As an other example of the initial condition it would be interesting to consider
the ideal electron gas on the positive charged compensated background, and also the
influence of a monoenergetic beam of charged particles.
The Figs. 1 demonstrates the typical distribution functions of fermions and bosons in
the action period of external electric field in the case of absence of the primordial plasma:
the boson distribution has the valley in the neighborhood P3= 0 while the fermion
distribution reaches the maximal value. The next figures (e.g., Fig. 2) demonstrate the
deformations of these distributions at presence of thermal plasma for moderate density
with β = 1/m and µ = 0 in the initial state. The set of Figs. 3 shows the known effect
⊥+ (p3)2and β = 1/T . Below charge and
mass correspond to an LHC conditions [28] independently from statistics: m=0.27 Gev,
10 505 10
0,0
0,3
0,6
0,9
f ( 0, p3 )
p / m
in
out q
out q

+
1050510
6,70
6,75
6,80
6,85
6,90
6,95
7,00
n(t) [ fm
3 ]
t*m
Figure 3. Left: The initial and final longitudinal momentum distributions for the fermion at the
initial thermoequilibrium state (17) with β = 1/2m and µ = 2m: dashed line  for particles and
dot line for antiparticles. Right: The time dependence of fermion density
[10, 20] of partial annihilation of the primordial fermion plasma and transformation of its
energy into external field. This effect is possible under high concentrations of the initial
EPP (f0> 1/2) and field strengthes.
As some quantitative characteristic for comparison the different initial conditions can
be used the parameter
ς =nout(nin?= 0) − nin
nout(nin= 0)
The dependence of this parameter from the field amplitude and temperature is presented
of Fig. 5.
As it can see from Fig. 4, the entropy of the outstate (after passage of the external
field pulse) is increased relatively of the instate. The entropy is not monotonic In this
interval (the system is open) in spite of the corresponding equation system (10) are time
reversible because they are an direct nonperturbative consequence of the dynamical
equations. The particle creation increase the entropy, whereas the annihilation one leads
to its diminution. The entropy behaviour correlates with the field evolution (16). The
final value of entropy is defined by appearance of particles in final state (real residual
plasma).
The adduced results allow some speculations about of transformation of coherent and
incoherent states. Indeed, on the one hand, the states with positive and negative energies
are entanglement by means of the coherent connection via the correlators (3), which are
.
(18)
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A.V. Filatov, A.G. Lavkin, S.A. Smolyansky, L.V. Bravina, E.E. Zabrodin
0.3
0.6
0.9
0 1 2 3 4 5 6 7 8 9 10
E0/Ec
0.5
0.6
0.7
0.8
0.9
1
ς
ς
T, m
64 20246
0,99
1,02
1,05
1,08
1,11
1,14
Entropy, S(t)/Sin
t*m
fermions
bosons
Figure 4. The enhancement effect for fermion
plasma from the low initial plasma density
Figure 5. The plasma entropy evolution from
the initial equilibrium state
not contain of elements of an external influence (8). On the other hand, the considered
initial conditions for particles and antiparticles are statistical independent. The Figs.
2,3 show reciprocal influence of primordial plasma and “coherent” vacuum plasma. This
expected conclusion is coordinated with the principle of particles indistinguishability.
Let us mark, that deformation of the thermodynamic equilibrium distribution of the
primordial plasma under action of an electric field pulse was showed already in the
work [21]. Thus, it can make the conclusion, that the primordial incoherent system can
transform later on in the entanglement system in an external field of enough long action
going through some period of the mixed phase. Such kind situations are discussed in
literature [29, 30]. An open question is remained in which state will be founded the
residual particleantiparticle plasma after switchoff of the external field pulse.
2) As the second example we will consider evolution of the initial distributions of
strong nonequilibrium parton gas in the flux tube model of ultrarelativistic heavy ion
collisions.
We use parameters of the system corresponding to the LHC case [31], E0 ≈ 10
GeV/fm which corresponds to E0 ∼ 2.64Ecr for q¯ q plasma with mq = 230 MeV. The
value of parameter b in (16) is qualitatively corresponds to the QGP formation time
τ0 ∼ (1 ÷ 3) fm/c. For gluons we will use the idealized initial condition proposed by
Mueller [32] in the framework of the McLerranVenugopalanKrasnitz model [33][36]:
fg
0(p) =
c
αsNct0θ(Q2
s− p2
tr)δ(p3).
(19)
Incoming here parameters depends from energy. For LHC we have : the saturation mo
mentum Qs= 2 GeV and t0= 0.65 GeV−1. The rest parameters are equal:
αs= 0.25, c = 1.3 for SU(2), Nc= 2.
Fig. 6 (left) shows the evolution of the initial distribution (19) under action of the
field (16). The scaling factor here is identified with the gluon mass mg= 0.5GeV .
For the quark case, we investigate “quench” and “tsunami” initial conditions used
to mimic idealized farfromequilibrium dynamics in the context of heavyion collisions
[1, 37, 38]:
fq
−
0(p) = exp
?
1
2σ2(p − pts)2
?
(20)
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Vacuum particle creation in plasma
107
0 1 2 3 4 540
pT/m
30
20
10
0
0
2
4
6
8
10
f
f
p/m
0
4
818
12
6
0
6
0
0.2
0.4
0.6
0.8
f
f
pT/m
p/m
Figure 6. Transformation of the distribution functions for gluons (left) with McLerran
VenugopalanKrasnitz initial conditions (19) and quarks (right) with with "Tsunami" initial
conditions (20)
peaked around p = pts= 5m with a width determined by σ = 0.5m. The first result
of numerical investigation of quark subsystem evolution with the initial distribution (20)
is showed on Fig.6. This initial condition has been termed “tsunami” in Ref. [37] and
is reminiscent of colliding wave packets moving with opposite and equal momentum.
A similar nonthermal and radially symmetric distribution of highly populated modes
may also be encountered in a “color glass condensate” at saturated gluon density with
typical momentum scale pts[33]. Of course, a sudden change in the twopoint function
of a previously equilibrated system or a peaked initial particle number distribution are
general enough to exhibit characteristic properties of nonequilibrium dynamics for a large
variety of physical situations.
4. Conclusion We have fulfilled here preliminary investigation of influence of differ
ent initial distributions on the following evolution of particleantiparticle plasma created
from vacuum under of strong external electric field. It is assumed in this connection that
the initial particle and the antiparticle distributions are statistical independent and the
electroneutrality condition is valid. The severe initial constraints (8) was introduced
also on the rather intuitive basis for the anomalous correlators (3). Apparently, the re
lations (8) are a symbol of statistical independence of the initial particleantiparticle
distributions. This requirement is not obvious in the cases of strong nonequilibrium ini
tial parton distributions in the theory of ultrarelativistic heavy ion collisions and other
similar physical situations.
The authors are grateful to A.V. Prozorkevich for the stimulating discussion.
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ВАКУУМНОЕ РОЖДЕНИЕ ЧАСТИЦ В ПЛАЗМЕ
c ? 2007А.В. Филатов1, А.Г. Лавкин1, С.А. Смолянский1,
Л.В. Бравина2, Е.Е. Забродин2
Аннотация
Рассматривается влияние начальных распределений (равновесных и нерав
новесных) на последующую эволюцию плазмы, создаваемой за счет вакуумного
рождения частиц в сильном, зависящем от времени электрическом поле.
1Филатов Андрей Викторович, Лавкин Александр Григорьевич, Смолянский Станислав Алек
сандрович, Саратовский государственный университет, Саратов, 410026, Россия.
2Бравина Лариса Владимировна, Забродин Евгений Евгеньевич, Физический факультет универ
ситета г. Осло, N0316, Норвегия; Институт ядерной физики, Московский государственный уни
верситет, Москва, 119899, Россия.
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 Available from E. E. Zabrodin · May 26, 2014