
%\documentclass[twocolumn,showpacs,preprintnumbers,amsmath,amssymb]{revtex4}
\documentclass[prd,showpacs,preprintnumbers,amsmath,amssymb]{revtex4}
%%\documentstyle[aps,prd]{revtex}

% Some other (several out of many) possibilities
%\documentclass[preprint,aps]{revtex4}
%\documentclass[preprint,aps,draft]{revtex4}
%\documentclass[prb]{revtex4}% Physical Review B

% uncomment next 3 lines for revtex4 docs.
\usepackage{graphicx}% Include figure files
\usepackage{dcolumn}% Align table columns on decimal point
\usepackage{bm}% bold math

%\nofiles

%\draft command below not necessary for revtex4 docs.
%\draft
\begin{document}



\title{Schwinger Pair Production in Electric and Magnetic Fields}


\author{Sang Pyo Kim}\email{sangkim@kunsan.ac.kr;spkim@phys.ualberta.ca}

\affiliation{Department of Physics, Kunsan National University,
Kunsan 573-701, Korea}

\author{Don N. Page}\email{don@phys.ualberta.ca}

\affiliation{Theoretical Physics Institute, Department of Physics,
University of Alberta, Edmonton, Alberta, Canada T6G 2J1}


\date{\today}


\begin{abstract}
Charged particles in static electric and magnetic fields have Landau levels
and, in the Coulomb gauge, tunneling states from vacuum. 
Using the instanton method of Phys. Rev. D {\bf 65}, 105002 (2002), 
we obtain new formulae for
the pair-production rate in spinor and scalar QED, which sum over all Landau 
levels. The new formulae not only recover exactly the well-known results but 
also give rise to other expressions more rapidly converging 
in the strong electric field limit.
\end{abstract}
\pacs{PACS number(s): 12.20.-m, 13.40.-f}

\maketitle


Vacuum polarization and pair production are two physically important
phenomena of quantum electrodynamics (QED) in strong electromagetic 
fields \cite{heisenberg,schwinger}. The one-loop effective action in 
constant electric and magnetic fields has nonlinear contributions to 
the classical action and, thereby, leads to the nonlinear Maxwell
equations. Another interesting phenomenon is the Schwinger pair production
due to vacuum instability in the presence of electric fields above
the critical strength $E_c = m^2 c^3/e \hbar ~(2.2 \times 10^{15} V/cm)$
\cite{schwinger} (see Ref. \cite{kim-page} for references).

Strong QED has many physical applications \cite{greiner}. In particular,
electromagnetic fields of some neutron stars and black holes \cite{putten}
and high-intensity laser fields \cite{burke} above the critical strength
$E_c = B_c = m^2 c^3/e \hbar ~ (4.4 \times 10^{13} G)$ have revived recent 
interest and applications of strong QED. 
The one-loop QED effective action is known exactly in the 
background of constant electric and magnetic fields \cite{schwinger}. 
In the case in which ${\bf E}^2 - {\bf B}^2$ and ${\bf E} \cdot 
{\bf B}$ are not both zero, one can go to a frame in which ${\bf E}$
and ${\bf B}$ are parallel with magnitude $E$ and $B$, and obtain
the imaginary part of the one-loop effective action per four-volume 
for spinor QED given by \cite{schwinger,daugherty} 
\begin{equation}
{\rm Im} {\cal L}^{(1)}_{\rm spinor} = \frac{(qE)(qB)}{ 2 (2 \pi)^2} 
\sum_{n = 1} \frac{1}{n} {\rm coth} \Bigl(\frac{n \pi B}{E} \Bigr) 
\exp\Bigl(- \frac{n \pi m^2}{qE} \Bigr), \label{form2}
\end{equation}  
and that for scalar QED by \cite{itzykson,cho}
\begin{equation}
{\rm Im} {\cal L}^{(1)}_{\rm scalar} = \frac{(qE)(qB)}{4 (2 \pi)^2} 
\sum_{n = 1}^{\infty} \frac{(-1)^{n+1}}{n} 
{\rm csch} \Bigl( \frac{n \pi B}{E} \Bigr) 
\exp \Bigl(- \frac{n \pi m^2}{qE} \Bigr). \label{form1}
\end{equation}
    

The main purpose of this paper is to apply the recently 
introduced instanton method 
\cite{kim-page} to find new formulae for the pair-production rate
in constant electric and magnetic fields in spinor and scalar QED.
The idea of Ref. \cite{kim-page}, the elaboration of
the role of tunneling in pair production 
\cite{brezin}, is that for static electric fields, 
fermions and bosons have tunneling states from the vacuum in the 
Coulomb (space-dependent) gauge, and their pair-production rates are
determined by these instanton actions for tunneling. 
In the presence of static
magnetic fields, charged fermions and bosons have discrete spectrum of
Landau levels. Taking into account both Landau levels and instanton 
actions for tunneling, we obtain new but equivalent 
formulae for pair-production
rates in spinor and scalar QED. These formulae are given as sums over
all Landau levels. Furthermore, we recover exactly the well-known 
results and find the leading terms for a strong electric field.  



In scalar QED the charged scalar paricles (spin 0) 
obey the Klein-Gordon equation
(in units with $\hbar = c =1$ and with metric signature $(+, -, -, -)$),
\begin{equation}
\Bigl[ \eta^{\mu \nu} (\partial_{\mu} + i
q A_{\mu}) (\partial_{\nu} + i q
A_{\nu})  + m^2 \Bigr] \Phi (t, {\bf x}) = 0, \label{kg
eq}
\end{equation}
where $q~ (q > 0)$ and $m$ are the charge 
and mass of the scalar particle. In spinor QED 
the fermions (spin 1/2) satisfy the Dirac equation
\begin{equation}
\Bigl[ i \slash{\hspace{-0.2cm}}{\partial}_{\mu} - 
q \slash{\hspace{-0.25cm}}{A}_{\mu}  - m 
\Bigr] \Psi (t, {\bf x}) = 0. \label{dirac eq}
\end{equation}
Classically charged particles in a uniform pure magnetic field 
undergo circular motion with Larmor frequency. In quantum theory
these particles have a discrete spectrum of states known as Landau levels.


In a pure magnetic field ${\bf B} = B {\bf e}_z$ along the z-direction, 
${\bf A} = (0, Bx, 0)$, and Eq. (\ref{kg eq}) has a solution of
the form
\begin{equation}
\Phi = e^{ - i \omega t + i (k_y y + k_z z)} f_{\omega k_y k_z}(x),
\end{equation}  
which leads to the equation
\begin{equation}
\Bigl[- \partial_x^2 + (q Bx + k_y)^2 
 + m^2 + k_z^2 - \omega^2 \Bigr] f_{\omega k_y k_z}(x) = 0. \label{ks osc}
\end{equation}
The mode equation (\ref{ks osc}) has bound states given by
harmonic wavefunctions with the energy spectrum
\begin{equation}
\omega^2 = m^2 + k_z^2 + qB (2j+ 1), \quad (j = 0,1, \cdots). \label{sc lan}
\end{equation}
The discrete spectrum due to the magnetic field is the Landau levels for the
scalar particles. Note that all the Landau levels are non-degenerate for 
the scalar particles. On the other hand, the fermion field
\begin{eqnarray}
\Psi =  \left(\begin{array}{c}
\phi\\
\chi 
\end{array}
 \right)
\end{eqnarray}
has a solution to Eq. (\ref{dirac eq}) of the form
\begin{equation}
\phi_{\sigma} = e^{ - i \omega t + i (k_y y + k_z z)} 
f_{\omega k_y k_z}(x) \chi_{\sigma},
\end{equation}   
where $\chi_{\sigma}$ is a unit spinor. The energy spectrum 
is known to take the form \cite{greiner2}
\begin{equation}
\omega^2 = m^2 + k_z^2 + qB (2 j + 1 - \sigma), \quad (j = 0, 1, \cdots ),
\end{equation}
where the spin projection $\sigma$ takes the value $\pm 1$. Note that $j = 0$ 
and $\sigma = 1$ gives the unique lowest Landau level, and 
all the other states
are doubly degenerate as $j - 1, \sigma = +1$ and $j, \sigma = - 1$ give 
the same energy.  



Now we apply the formulae \cite{kim-page} for pair production based 
on the instanton calculation to the static uniform electric and magnetic 
fields. For the electric and magnetic fields parallel 
to each other along the z-direction, the 4-potential in the Coulomb gauge 
is given by
\begin{equation}
A_{\mu} = (-Ez, 0, Bx, 0).
\end{equation}
First, the scalar field equation, 
\begin{equation}
[(\partial_t - i q E z)^2 - \partial_x^2 - (\partial_y + i qBx)^2 
- \partial_z^2 + m^2 ] \Psi = 0,
\end{equation}  
has the solution of the form
\begin{equation}
\Phi = e^{- i \omega t + k_y y} f_{\omega k_y} (x,z).
\end{equation}
Then the above equation becomes
\begin{equation}
[\{- \partial_x^2 + (qBx+ k_y)^2\}  + \{- \partial_z^2
- (\omega +  q E z)^2 \} + m^2 ] f_{\omega k_y} (x, z) = 0.
\end{equation}  
The first parenthesis has the harmonic wavefunctions as eigenfunctions, 
so the remaining equation becomes
\begin{equation}
[- \partial_z^2 -
(\omega +  q E z)^2 \} + m^2 + q B (2j+1) ] f_{\omega k_y j} (z) = 0, 
\quad (j = 0, 1, \cdots, ).
\end{equation} 
Therefore, the scalar particles, exactly described by 
the inverted harmonic potential, have, as 
the instanton action for tunneling states,
\begin{equation}
S^{s}_j = \pi \frac{m^2 + qB (2 j+1)}{2qE}. \label{inst1}
\end{equation}
Similarly, the Dirac equation has, for the tunneling states of fermions,
the instanton action
\begin{equation}
S^{f}_{j,\sigma} = \pi \frac{m^2 + qB (2 j+ 1 - \sigma)}{2qE}, \label{inst2}
\end{equation}
where $\sigma = \pm 1$. The effect of the magnetic field is the same as 
shifting the effective mass to $m^2_* = m^2 + qB (2j+1)$ for scalar particles
and to $m^2_* = m^2 + qB (2 j + 1 - \sigma)$ 
for fermions for each Landau level.


The essential idea of  Ref. \cite{kim-page} is that the 
production-rate per each mode in static electric fields is given by
\begin{equation}
w_{\pm} = \pm \ln \Bigl(1 \pm e^{-2 S} \Bigr),
\end{equation}
where the upper (lower) sign is for scalar particles 
(fermions) and $S$ is the instanton
action for the corresponding mode, 
here given in Eqs. (\ref{inst1}) and (\ref{inst2}). The main result 
of this paper is then the pair-production rate for scalar particles
\begin{equation}
w_{\rm scalar} = 2 {\rm Im} {\cal L}^{(1)}_{\rm scalar}
= \Bigl(\frac{qE}{2 \pi} \Bigr) \Bigl(\frac{qB}{2 \pi} \Bigr)
 \sum_{j = 0}^{\infty}  
\ln \Biggl\{ 1 + \exp \Bigl(-  \pi \frac{m^2 
+ qB (2 j+1)}{qE} \Bigr) \Biggr\}, \label{b-p}
\end{equation}
and for fermions
\begin{equation}
w_{\rm spinor} = 2 {\rm Im} {\cal L}^{(1)}_{\rm spinor}
= - \Bigl(\frac{qE}{2 \pi} \Bigr) \Bigl(\frac{qB}{2 \pi} \Bigr) 
\sum_{j = 0}^{\infty} \sum_{\sigma = \pm 1} 
\ln \Biggl\{1 -  \exp \Bigl(- \pi \frac{m^2 
+ qB (2 j+1 - \sigma)}{qE} \Bigr) \Biggr\}. \label{f-p}
\end{equation}
Here $(qE)/(2 \pi)$ is the number of states from the $\omega$-integration
and $(qB)/(2 \pi)$ is that available for each Landau level.
We can show that our production-rate formulae 
reduce to Eqs. (\ref{form2}) and 
(\ref{form1}). First expanding the logarithm and 
then summing over $j$ in Eq. (\ref{b-p}), we obtain Eq. (\ref{form1}):
\begin{eqnarray}
{\rm Im} {\cal L}^{(1)}_{\rm scalar}
&=& \frac{(qE)(qB)}{2(2 \pi)^2}
 \sum_{j = 0}^{\infty} \sum_{n = 1}^{\infty} \frac{(-1)^{n+1}}{n} 
\exp \Bigl(- \frac{n (2j+1) \pi B}{E} \Bigr) 
\exp \Bigl(- \frac{n \pi m^2}{qE} \Bigr) \nonumber\\
&=& \frac{(qE)(qB)}{4(2 \pi)^2}
 \sum_{n = 1}^{\infty} \frac{(-1)^{n+1}}{n} 
{\rm csch} \Bigl(\frac{n \pi B}{E} \Bigr) 
\exp \Bigl(- \frac{n \pi m^2}{qE} \Bigr),
\end{eqnarray} 
where we have used the identity
\begin{equation}
{\rm csch} (x) = 2 e^{-x} \frac{1}{1 - e^{-2x}} = 2 \sum_{j = 0}^{\infty}
e^{- (2j +1) x}.
\end{equation}
Similarly, first expanding the logarithm and then summing over
$j$ and $\sigma$ in Eq. (\ref{f-p}), we obtain Eq. (\ref{form2}): 
\begin{eqnarray}
{\rm Im} {\cal L}^{(1)}_{\rm spinor}
&=& \frac{(qE)(qB)}{2(2 \pi)^2}
 \sum_{j = 0}^{\infty} \sum_{\sigma = \pm 1} \sum_{n = 1}^{\infty} \frac{1}{n} 
\exp \Bigl(- \frac{n (2j+1 - \sigma) \pi B}{E} \Bigr) 
\exp \Bigl(- \frac{n \pi m^2}{qE} \Bigr) \nonumber\\
&=& \frac{(qE)(qB)}{4(2 \pi)^2}
 \sum_{n = 1}^{\infty} \frac{1}{n} 
\coth \Bigl(\frac{n \pi B}{E} \Bigr) 
\exp \Bigl(- \frac{n \pi m^2}{qE} \Bigr),
\end{eqnarray} 
where we have used the identity
\begin{equation}
{\rm coth} (x) = (1+  e^{-2x}) \frac{1}{1 - e^{-2x}} = 
\sum_{j = 0}^{\infty} \Bigl[e^{- 2j x} + e^{- 2 (j+1)x} \Bigr].
\end{equation}


Next, we obtain another expression for pair-production rate from Eqs. 
(\ref{b-p}) and (\ref{f-p}). Using the summation formula of Euler
and MacLaurin \cite{e-m sum}
\begin{equation}
\sum_{n = 0}^{\infty} F(n) = \frac{1}{2} \Bigl(F(0) + F(\infty) \Bigr)
+ \int_{0}^{\infty} dx F(x) + \sum_{k = 1}^{\infty} 
\frac{B_{2k}}{(2k)!} \Bigl(F^{(2k-1)} (\infty) - F^{(2k - 1)} (0) \Bigr),
\end{equation}
where $B_{k}$ is Bernoulli number and $F^{(k)}$ denotes 
the $k$th derivative, we obtain
\begin{equation}
w_{\rm scalar} = \frac{(qE)^2}{(2 \pi)^3} \sum_{k = 1}^{\infty}
\frac{(-1)^{k+1}}{k^2} e^{-k a_s} +
\frac{(qE)(qB)}{2(2 \pi)^2} \ln (1 + e^{- a_s} ) - 
\frac{(qE)(qB)}{(2 \pi)^2} \sum_{k = 1}^{\infty} 
\frac{B_{2k}}{(2k)!} F_{\rm scalar}^{(2k-1)} (0). \label{b-p2} 
\end{equation} 
Here, we have introduced dimensionless parameters
\begin{equation}
a_s =  \pi \frac{m^2 + qB}{qE}, \quad b = \frac{2 \pi qB}{qE},
\end{equation}
and
\begin{equation}
F_{\rm scalar} (x) = \ln ( 1 + e^{-a_s} e^{-b x}),
\end{equation}
and $F^{(k)}_{\rm scalar} (\infty) = 0$ for all $k$, and we have used
\begin{equation}
\int_{0}^{\infty} dx F_{\rm scalar} (x) = \frac{1}{b}
\int_{0}^{1} dt \frac{\ln (1 +  e^{- a_s} t) }{t}, 
\end{equation}
and the integral formula \cite{pbm}
\begin{equation}
\int dx \frac{\ln (\beta + \alpha x)}{x} = \ln \vert \beta \vert
\ln \vert x \vert + \sum_{k = 1}^{\infty}
\frac{(-1)^{k+1}}{k^2} \Bigl(\frac{\alpha x}{\beta} \Bigr)^k. \label{int}
\end{equation}
Similarly, for the spinor case we obtain
\begin{equation}
w_{\rm spinor} = \frac{2(qE)^2}{(2 \pi)^3} \sum_{k = 1}^{\infty}
\frac{1}{k^2} e^{- k a_f} + 
\frac{2(qE)(qB)}{(2 \pi)^2} \sum_{k = 1}^{\infty} 
\frac{B_{2k}}{(2k)!} F_{\rm spinor}^{(2k-1)} (0), \label{f-p2} 
\end{equation} 
where
\begin{equation}
 a_f = \pi \frac{m^2}{qE},
\end{equation}
and
\begin{equation}
F_{\rm spinor}(x) = \ln ( 1 - e^{- a_f} e^{- bx}).
\end{equation}
Here, we have used again $F^{(2k-1)} (\infty) = 0$ 
and the integral (\ref{int}).
Note that the first series in Eq. (\ref{b-p2}) is identical to 
to the result in a  pure electric field only with the mass shift 
$m^2 \rightarrow m^2 + qB$, and so is the 
first series in Eq. (\ref{f-p2}). 
In the limit of strong electric fields, $(qE > \pi( m^2 + qB))$, we can
use  Eq. (28) of Ref. \cite{kim-page} to obtain
\begin{eqnarray}
w_{\rm scalar}  &=& 
\frac{(qE)^2}{(2\pi)^3} \Biggl\{\frac{\pi^2}{12}  - (\ln 2) 
\Bigl( a_s - \frac{b}{2} \Bigr) + \frac{1}{4} \Bigl( a_s^2 - a_s b 
+  \frac{1}{6} b^2 \Bigr) - \frac{1}{8} \Bigl(\frac{1}{3} a_s^3 
- \frac{1}{2} a_s^2 b + \frac{1}{6} a_s b^2 \Bigr) 
+ {\cal O} (a_s^4, b^4) \Biggr\}, \label{pure1}
\end{eqnarray}
and, in the limit of  $qE > \pi m^2 > \pi qB$,  
Eq. (32) of \cite{kim-page} gives us 
\begin{eqnarray}
w_{\rm spinor} &=& \frac{2(qE)^2}{(2\pi)^3} \Biggl\{\frac{\pi^2}{6}
 - a_f (1 - \ln a_f) - \frac{1}{4} a_f^2 + \frac{1}{72} a_f^3
+ \frac{1}{12} \frac{b}{a_f} \Bigl(1 - \frac{1}{2}a_f b 
+ \frac{1}{12} a_f^2 b \Bigr) - \frac{1}{360} \Bigl(\frac{b}{a_f} \Bigr)^3
b + {\cal O} (a_f^4, b^4)\Biggr\}. \label{pure2}
\end{eqnarray}
The 2 in Eq. (\ref{pure2}) is the spin factor for fermions.
In the pure electric field $({\bf B} =0)$, $a_s = a_f$ and $b = 0$,
and Eq. (\ref{pure1}) reduces to the result of Ref. \cite{kim-page}, and  
Eq. (\ref{pure2}) recovers the result in Ref. \cite{dittrich}.



Finally, we make a few comments.
First, in the strong magnetic field limit $(B \gg E)$, the 
instanton actions (\ref{inst1}) and (\ref{inst2}) suppress
the pair-production rate by $e^{- 2S^s_j}$ and 
$e^{- 2 S^f_{j, \sigma}}$, so the series in Eqs. (\ref{b-p})
and (\ref{f-p}) are rapidly convergent.
Second, we now extend the analysis to inhomogenous electric fields with 
a constant magnetic field. The constant magnetic field still provides 
all Landau levels, but the instanton actions are not exactly given 
by Eqs. (\ref{inst1}) and (\ref{inst2}). Adopting Eq. (36) of Ref. 
\cite{kim-page}, the instanton actions now 
take the form
\begin{equation}
S_j = \sum_{n = 0}^{\infty} S^{(2n)}_j
\end{equation}  
where the dominant contribution comes from
\begin{equation}
S^{(0)}_j = \int_{x_-}^{x_+} dx \sqrt{m^2 + qB (2j + 1 - \sigma) - 
(\omega - q A_0 (x))^2},
\end{equation}
where $x_{\pm}$ are turning points of the integrand
and $\sigma = 0$ for scalar and $\sigma = \pm 1$ for spinor QED. 
Note that pair production is allowed only 
when $\omega - qA_0 (+ \infty) \geq m$ and
$\omega - qA_0 (- \infty) \leq - m$, so there is a finite discrete set of $j$
for a localized electric field that have real instantons. Therefore, the 
summations in Eqs. (\ref{b-p}) and (\ref{f-p}) should be restricted to 
those of $j$ with real instantons.  


In conclusion, using the instanton method in the Coulomb (space-dependent)
gauge for constant electric and magnetic fields, 
we have obtained new formulae (\ref{b-p}) 
for pair-production in scalar QED and (\ref{f-p}) in spinor QED. 
These formulae recover exactly the well-known results from the exact
one-loop QED effective 
action. Further, we are able to find other expressions, Eqs. (\ref{b-p2})
and (\ref{f-p2}), which are rapidly convergent for strong electric fields
even when the magnetic field is nonzero. 
Finally, we suggested a generalization of these formulae to a constant 
magnetic field and an inhomogenous electric field.  



\acknowledgements

S.P.K. thanks S.J. Rey
for useful discussions and expresses his appreciation for the warm
hospitality of the Theoretical Physics Institute, University of
Alberta. The work of S.P.K. was supported by
the Korea Science and Engineering Foundation under Grant No.
1999-2-112-003-5 and the work of D.N.P. by the Natural Sciences and
Engineering Research Council of Canada.

\begin{references}

\bibitem{heisenberg} F. Sauter, Z. Phys. {\bf 69}, 742 (1931); 
W. Heisenberg and H. Euler, Z. Physik {\bf
98}, 714 (1936); V. Weisskopf, K. Dan. Vidensk. Selsk.
Mat. Fys. Medd. {\bf 14}, No. 6 (1936).

\bibitem{schwinger} J. Schwinger, Phys. Rev. {\bf 82}, 664 (1951).


\bibitem{kim-page} S.P. Kim and D.N. Page, Phys. Rev. D {\bf 65}, 105002 (2002).

\bibitem{greiner} W. Greiner, B. M\'{u}ller, and J. Rafelski,
{\it Quantum Electrodynamics of Strong Fields} (Springer-Verlag, Berlin,
1985); W. Dittrich and H. Gies, {\it Probing the Quantum Vacuum:
Perturbative Effective Action Approach in Quantum Electrodynamics and
its Applications} (Springer-Verlag, Berlin, 2000).

\bibitem{putten} M.H.P.M. Putten, Phys. Rev. Lett. {\bf 84}, 3752 (2000); 
references therein.

\bibitem{burke} D.L. Burke {\it et al.}, Phys. Rev. Lett. 
{\bf 79}, 1626 (1997); R. Alkofer, M.B. Hecht, C.D. Roberts,
S.M. Schmidt, and D.V. Vinnik, {\it ibid.} {\bf 87}, 193902 (2001); 
C.D. Roberts, S.M. Schmidt, and D.V. Vinnik, {\it ibid.} {\bf 89}, 
153901 (2002); B. Shen and M.Y. Yu, {\it ibid.} {\bf 89}, 275004 (2002).

\bibitem{daugherty} J.K. Daugherty and I. Lerche, Phys. Rev. D {\bf 14},
340 (1976). 

\bibitem{itzykson} C. Itzykson and J.B. Zuber, {\it Quantum Field
Theory} (McGraw-Hill, New York, 1980).


\bibitem{cho} Y.M. Cho and D.G. Pak, Phys. Rev. Lett. {\bf 86}, 1947 (2001).

\bibitem{brezin}  E. Brezin and C. Itzykson, Phys. Rev. D {\bf 2},
1191 (1970);  A. I. Nikishov, Nucl. Phys. {\bf B21}, 346
(1970);  A. Casher, H. Neuberger, and S. Nussinov,
Phys. Rev. D {\bf 20}, 179 (1979); H. Neuberger, {\it ibid.} {\bf
20}, 2936 (1979).


\bibitem{greiner2} W. Greiner and J. Reinhardt, {\it Quantum Electrodynamics}
(Springer-Verlag, Berlin, 1992) p 280, pp 286-287.


\bibitem{e-m sum} M. Abramowitz and I. Stegun, {\it Handbook of
Mathematical Functions} (Dover Pub., New York, 1964).

\bibitem{pbm} A.P. Prudnikov, Yu.A. Brychkov, and O.I. Marichev,
{\it Integrals and Series} (Gordon and Breach, Amsterdam, 1986), 
1.6.5.10-11.

\bibitem{dittrich} W. Dittrich, W.-Y. Tsai, and K.-H. Zimmermann, 
Phys. Rev. D {\bf 19}, 2929 (1979); J.S. Heyl and L. Hernquist,
{\it ibid.} {\bf 55}, 2449 (1997); R. Soldati and L. Sorbo, 
Phys. Lett. B {\bf 426}, 82 (1998).

\end{references}

\end{document}

