\documentstyle [12pt] {article}
\begin{document}
\rightline{TAUP 1884 - 91}
\rightline{June 1991}
\vskip .5in
%\baselineskip=21pt
\def\eval#1{\left\langle #1 \right\rangle}

{%\baselineskip=14pt
\centerline{\large \bf PAIR PRODUCTION IN A STRONG ELECTRIC FIELD}
\vskip .25in
\centerline{Y. Kluger, J. M. Eisenberg, and B. Svetitsky}
\vskip .25in
{\it
\centerline{School of Physics and Astronomy}
\centerline{Raymond and Beverly Sackler Faculty}
\centerline{of Exact Sciences}
\centerline{Tel Aviv University}
\centerline{69978 Tel Aviv, Israel}  }
\vskip .25in
\centerline{F. Cooper and E. Mottola}
\vskip .25in
{\it
\centerline{Theoretical Division}
\centerline{Los Alamos National Laboratory}
\centerline{Los Alamos, New Mexico 87545 USA}  }

\vskip 0.5in
\noindent {\it ABSTRACT.\/}
We investigate the mechanism of pair creation in scalar QED
from spatially homogeneous strong electric fields in 1+1 dimensions. 
Solution of the semiclassical field equations shows particle creation followed
by plasma oscillations.
We compare our results with a model based on a relativistic Boltzmann--Vlasov 
equation with a pair-creation source term related to the Schwinger 
mechanism.  
The time evolution of the electric field and the 
current obtained from the Boltzmann--Vlasov model is surprisingly
similar to that found in the 
semiclassical calculation. 
  
} % end of \baselineskip 14pt

\newpage

The problem of spontaneous pair production in the presence of an external
electric field has been investigated by many authors [1--6].
The most commonly used formula for the  (boson) pair creation 
rate is based on Schwinger's one-loop calculation [2]
for a constant and homogeneous  electric field $E$, 
\begin{equation}
w = \frac{m^4}{(2\pi)^3}\left(\frac{E}{E_0}\right)^2
\sum_{n=1}^{\infty}\frac{1}{n^2}\exp (-n{\pi}E_0/E)\quad,\quad E_0=m^2/e
\end{equation}
(in units such that $\hbar=c=1$).
In this expression the mutual interactions of the particles produced and
the effect of these particles on the original electric field 
(back reaction) are not taken into account. 

This formula has been used in modeling particle production in the
central rapidity region in high-energy nucleus--nucleus collisions
[7--9].
The production process is viewed as the tunneling
of quark-antiquark and gluon pairs in the presence of a background 
color-electric field, formed after the collision in the region between the
two nuclei
that have been color-charged by the exchange of soft gluons.  
One usually studies only an Abelian version of the theory;
equation (1) is used as if the electric field were constant, and
the time-dependence of $E$  is put in by hand.
    	
A scheme for solving the quantum back-reaction problem in 
scalar QED has been offered by Cooper and Mottola [10].
We present here the results of numerical analysis based on this formalism.
One would expect substantial pair production
if the initial electric field is strong
enough (of order $E_0$), and at large times one should see plasma
oscillations.
Studying this requires a numerical
calculation that proves quite challenging.
It is useful to investigate this problem in 1+1 dimensions 
as a first step in testing numerical procedures since the requirements of
renormalization
are relatively trivial in that case [11].

We treat the electric field as  
a semiclassical or mean field, which
is justified if the electric field is strong enough, 
or if we take the matter field to be a scalar field in the large-$N$ limit [10].
With a view towards implementing an adiabatic renormalization scheme,
we restrict ourselves to a spatially uniform field.
The full scheme is needed when one solves the
(3+1)-dimensional problem, but infinities must be subtracted
even in the (1+1)-dimensional case.

We also assume a special, albeit natural,
initial configuration of the charged-matter field---the adiabatic vacuum.
This corresponds to the initial configuration assumed by Schwinger [2]
which results in (1) for a constant electric field.

The scalar QED coupled equations of motion in the mean field approximation (and without
the $\phi^4$ self-interaction term of Ref.~10) consist of the
Klein-Gordon equation,
\begin{equation}
[(\partial ^\mu +ieA^\mu )(\partial _\mu +ieA_\mu )+ m^2]\Phi (x)=0
\quad,
\label{eq:KG}
\end{equation}
and the semiclassical Maxwell equations,
\begin {equation}
\partial_\mu F^{\mu \nu}=\eval{0| j^\nu | 0}  \quad,
\label{eq:Maxwell}
\end{equation}
where $|0\rangle$ is the adiabatic vacuum
and $j^\nu$ is the current of the charged scalar field.
Spatial homogeneity  
causes the charge density 
$j^0$ to vanish everywhere. 
The Maxwell equations in the gauge $A_0=0$ 
consist of the single equation 
\begin{equation}
\ddot{A}= \eval{0|j|0}       \quad,
\end{equation}
where $A\equiv A_1$ and $j\equiv j_1$.
Homogeneity 
also enables us to expand the boson field operator $\Phi $ 
in terms of plane waves, defining the operators $a_k$~and~$b_k^{\dagger}$
via
\begin{equation}
\Phi (x,t) = \int\frac{dk}{2\pi}\,\left[f_k(t)a_k+
f_k^{\ast} (t) b_{-k}^{\dagger}\right]e^{i{k}{x}}  \quad,
\end{equation}
where to satisfy (2) we demand
\begin{equation}
\frac{d^2 f_k(t)}{dt^2}+\omega _k^2 (t)f_k(t)=0 \quad,\quad
\omega _k (t)^2 \equiv [k-eA(t)]^2 +m^2    \quad.
\end{equation}
The canonical commutation relations imply that
the $f_k(t)$ can be expressed in terms of the real and 
positive functions $\Omega _k(t)$,
\begin{equation}
f_{{k}} (t)=\frac{1}{\sqrt {2 \Omega_{{k}}(t)}}\,e^{-i\int ^{t} \Omega_{{k}}(t')dt'}\quad,
\end{equation}
where the $\Omega_{{k}}(t)$ obey the equation of motion
\begin{equation}
\Omega^{2}_{k}(t) =-\ddot{\Omega}_{k}/(2\Omega_{k})
+\frac{3}{4}{(\dot{\Omega}_{k}/{\Omega_k})}^2 + {\omega}^{2}_{k}(t)\quad.  
\end{equation}
Upon subtracting the divergent part of the current, Eq.~(4) becomes
\begin{equation}   
\ddot{A}=\eval{0| j | 0}=e\int\frac{dk}{2\pi}(k-eA)
\left[\frac{1}{\Omega_k(t)}-\frac{1}{\omega _k(t)}\right]
\quad.
\end{equation}
Eqs.~(8) and (9) define the numerical problem.

We show in Fig.~1 the time evolution of the scaled electric 
field $\tilde E\equiv eE/m^2$ and induced current $\tilde \jmath\equiv ej/m^3$,
as functions of $\tau\equiv mt$.
Stability was attained for these results, as well as for those presented below,
with a time step $d\tau=10^{-4}$ and
a momentum grid with $d\tilde k \equiv dk/m=0.004$, forcing very
long running times.
With strong initial electric fields we find that the induced current
increases rapidly and becomes saturated at a constant value for some time,
after which clear
plasma oscillations are seen.

The saturation of the first oscillation is easy to understand.
In a classical kinetic picture, we have $j=2en\eval v$, where $n$ is the
density of particles (or antiparticles) and $v$ is their velocity.
$j$ saturates as $v$ is driven to the speed of light by the strong
electric field.
In subsequent oscillations, $n$ is larger and $E$ is weaker, so the
particles remain nonrelativistic even at the peak of the current.
This analysis gives us a method for estimating the number of pairs
created in the first rise of $j$, allowing us to evade the ambiguities
in the definition of particle number which are inherent in the
adiabatic approach.
We display a graph of the peak current in the first oscillation,
as a function of the initial field, in Fig.~2.

As time progresses, there develops
a highly oscillatory behavior (as a function of $k$)
for the integrand in Eq.~(9).
We see this behavior also in
the expectation value of the number operator for large times
\begin{eqnarray}
{1\over V}{dN\over dk}&=&{1\over2\pi}
\eval{0\left|a_k^{{\rm out}\dagger}a_k^{\rm out}
\right|0}
\nonumber\\
&= &\lim_{t\to\infty}{1\over8\pi\Omega_k\omega_k}
\left[(\Omega_k-\omega_k)^2+{1\over4}\left(
(\dot\Omega_k/\Omega_k) - (\dot\omega_k/\omega_k)\right)^2\right]\quad,
\end{eqnarray}
shown in Fig.~3.  
These oscillations are the result of oscillations in {\it time} with the
momentum-dependent frequency $\Omega_k(t)$, as seen in  (7).
They disappoint the hope of a simple
power-law fall off at large $k$,
and hint at some likely difficulties 
in the implementation of the adiabatic renormalization scheme in 3+1 dimensions.

Many calculations in the realm of nucleus--nucleus collisions have
been based on a phenomenological model employing a relativistic
Boltzmann-Vlasov equation with a source term to describe particle creation
[12--14].
In order to gauge the validity of this model, let us compare its results
with those of our semiclassical analysis.
The relativistic kinetic equation in the presence of a homogeneous 
electric field is 
\begin{equation}
\frac{\partial f}{\partial t}+eE\frac{\partial f}{\partial p} =
\frac{dN}{dt\,dx\,dp}=\frac{| eE(t)| }{2\pi}
\log \left[1+\exp \left(-\frac {\pi m^2}{| eE(t) |}\right)\right]\, \delta (p)
\quad,
\end{equation}
where
$f(p,t)$ is the ($x$-independent) classical phase space distribution
and
the right-hand side is the boson pair-production 
rate in $1+1$ dimensions.
(We assume that the
particles are produced at rest, i.e., the source term is proportional to
$\delta (p)$.)
The solution of Eq.~(11) is given by
\begin {eqnarray}
f(p,t)&=&\int_{0}^{t}dt'\,\frac{|eE(t')|}{2\pi}
\,\log\left[1+\exp\left(-\frac{\pi m^2}{|eE(t')|}\right)\right]\,
\delta\bigl(p-eA(t')+eA(t)\bigr)\nonumber\\
&=&\frac{1}{2\pi}\sum_{i} \log \left[1+\exp\left(-\frac{\pi m^2}{|eE(t_i)|}
\right)\right]\quad ,
\end{eqnarray}
where the $t_i$'s fulfill $p+eA(t)-eA(t_i)=0.$  The field equation for 
$A$ is
\begin{equation}
\frac{d^2 A}{dt^2}=j_{total}=j_{cond}+j_{pol}
\end{equation}
where the conduction current is
\begin{equation}
j_{cond}=2e\int\frac{dp}{\epsilon_p}\,p\,f(p,t)  \quad,
\end{equation}
with $\epsilon_p\equiv\sqrt {p^2 + m^2}$, and the polarization current is
[13]
\begin{equation}
j_{pol}=\frac{2}{E}\int dp\,\epsilon_p\frac{dN}{dt\,dx\,dp} \quad.
\end{equation} 
(The factors of 2 in (14) and (15) account for the contributions of the \break
antiparticles.) 
Inserting Eq.~(12) 
into Eq.~(13) reduces the system to a single equation 
\begin{equation}
\frac{d^2 \tilde{A}}{d\tau ^2}=\frac{e^2}{\pi m^2}\int_{0}^{\tau}\,d\tau'
\frac{\tilde{A}(\tau')-\tilde{A}(\tau)}{\sqrt{\left[\tilde{A}(\tau')-\tilde{A}
(\tau)\right]^2+1}}
\left| \tilde{E}(\tau')\right| 
\log \left[1+\exp \left(-\frac {\pi}{\left|\tilde{E}(\tau')
\right|}\right)\right]
\end{equation}
\begin{eqnarray*}
+\frac{e^2}{\pi m^2}\,{\rm sign}\left(\tilde{E}(\tau)\right)
\log \left[1+\exp\left(-\frac {\pi}{\left|\tilde{E}(\tau)\right|}
\right)\right]\quad . 
\end{eqnarray*}
in terms of the dimensionless variables $\tilde{A}\equiv eA/m$,
$\tilde{E}$, and $\tau$.

The time evolution of $\tilde{E}$ and $\tilde\jmath$ is shown in the dashed
curves of Fig.~1, where we see that for an initial field $\tilde{E}_{t=0}=1$ 
there is good quantitative
agreement between the results obtained with the two very different methods. 
The oscillations are faster and the electric fields decay
more rapidly in the semiclassical calculation than in the
Boltzmann-Vlasov model.
This is because in addition to spontaneous 
pair creation, the quantum theory takes into account pair creation via
bremsstrahlung (``induced'' pair creation) and Bose-Einstein effects, all of
which are neglected in the kinetic
theory. 
Fig.~2 reflects this feature
by comparing the plateau currents (the peak values in the first oscillation)
as functions of the initial electric field.
Decay of the oscillations would presumably be accelerated by the inclusion
of collisions, which can be done by including a $\phi^4$ term in the
semiclassical equation (2) and the
corresponding collision integral in the
Boltzmann-Vlasov equation (11).

The distribution function $f(p,t)$, measured after several plasma oscillations,
may be compared to the quantum theory's
$V^{-1}dN/dk$ after the latter is smoothed, as shown in Fig.~3.
Naturally, the curves have different normalizations and a relative
displacement due to the slightly different values of $j$ and $A$.
Because of the similarity between the results 
in Fig.~3(b),
one can use the kinetic theory model
to explain various features of the particle distribution,
such as the sharp edges and tails. 

The kinetic theory can be improved by use of a source
term that takes stimulated pair creation into account.
Thus, we replace the right hand side of (11) with
\begin{equation}
\frac{dN}{dt\,dx\,dp}=\bigl(2f(p,t)+1\bigr)\frac{|eE(t)|}{2\pi}
\log \left[1+\exp \left(-\frac{\pi m^2}{|eE(t)|}\right)\right]\,\delta(p)\quad.
\end{equation}
With this source term the agreement between the kinetic theory
and the quantum theory is even more striking, as demonstrated 
by the dashed-dotted line in Fig.~1(b).

A more extended comparison among the semiclassical and Boltzmann-Vlasov
models will be presented elsewhere.
We conclude that the
phenomenological model with the stimulated emission source term yields a 
surprisingly good 
approximation to the semiclassical QED results. 
\newpage
\centerline{\bf ACKNOWLEDGMENTS}
\vskip\baselineskip
Y.K. and B.S. thank the Theoretical Division of Los Alamos National Laboratory
for their hospitality. 
This work was partially supported by the German--Israel Foundation and by a
Wolfson Research Award administered by the Israel Academy of Sciences and
Humanities.
Partial support from the Ne'eman Chair in Theoretical Nuclear
Physics at Tel Aviv University is also appreciated.
\newpage
\centerline{\bf REFERENCES} {\frenchspacing
\begin{enumerate}
\item F.~Sauter, Z. Phys. {\bf 69}, 742 (1931).
\item J.~Schwinger, Phys. Rev. {\bf 82}, 664 (1951).
\item N.~B.~Narozhny and A.~I~ Nikishov, Yad. Fiz. {\bf11}, 1072 (1972)
[Sov. J. Nucl. Phys. {\bf11}, 596 (1970)].
\item M.~S.~Marinov and V.~S.~Popov, Fortsch. Phys. {\bf25}, 373 (1977).
\item E.~Brezin and C.~Itzykson, Phys. Rev. D {\bf2}, 1191 (1970).
\item A.~Casher, H.~Neuberger, and S.~Nussinov, Phys. Rev. D 
	{\bf28}, 179 (1979).
\item T.~S.~Biro, H.~B.~Nielsen, and J.~Knoll, Nucl. Phys. {\bf B245},
	449 (1984).
\item A.~Bia\l as and W.~Czy\.z, Acta Phys. Pol. B {\bf17}, 635 (1986).
\item A.~K.~Kerman, T.~Matsui, and B.~Svetitsky, Phys. Rev. Lett. {\bf 56},
	219 (1986).
\item F.~Cooper and E.~Mottola, Phys. Rev. D {\bf40}, 456 (1989).
\item For earlier results, see F.~Cooper, E.~Mottola, B.~Rogers,
and P.~Anderson, in {\it Intermittency in High Energy Collisions}, edited
by F.~Cooper, R.~C.~Hwa, and I.~Sarcevic (World Scientific, Singapore, 1991),
p.~399.
\item K.~Kajantie and T.~Matsui, Phys. Lett {\bf164B}, 373 (1985).
\item G.~Gatoff, A.~K~ Kerman, and T.~Matsui, Phys. Rev. D {\bf36},
	114 (1987).
\item  A.~Bia\l as, W.~Czy\.z, A.~Dyrek, and W.~Florkowski, Nucl. Phys.
{\bf B296}, 611 (1988).
\end{enumerate}}
\newpage
\centerline{\bf FIGURE CAPTIONS}
\vskip\baselineskip
\parindent0pt
\parskip14pt
FIGURE 1.
(a) Time evolution of scaled electric field $\tilde E$ and current
$\tilde \jmath$,
with initial value $\tilde E=1.0$ and coupling $e^2/m^2=0.1$.
Solid line is semiclassical scalar QED, and dashed line is
Boltzmann-Vlasov model.
(b) Same as (a), but for $\tilde E=4.0$.  The dashed-dotted curve is
the Boltzmann-Vlasov model with the stimulated pair creation correction,
Eq. (17).

FIGURE 2. Scaled current $\tilde \jmath$ at the peak of the first oscillation,
as a function of the initial scaled electric field $\tilde E$.
Crosses are for semiclassical scalar QED, and circles are for Boltzmann-Vlasov
model.

FIGURE 3. (a) Momentum distribution of produced pairs, for the evolution shown
in Fig. 1(a), at time $\tau=550.$  The abscissa is the scaled kinetic momentum
$\tilde p\equiv\tilde k-\tilde A$, with $\tilde k\equiv k/m$.
(b) Data of (a) after smoothing (solid curve), compared with 
Boltzmann-Vlasov model (dashed curve).

\end{document}
