# Pair Photoproduction in Constant and Homogeneous Electromagnetic Fields

V.M. Katkov
Budker Institute of Nuclear Physics,
Novosibirsk, 630090, Russia
e-mail<EMAIL_ADDRESS>
###### Abstract

The process of pair creation by a photon in a constant and homogeneous
electromagnetic field of an arbitrary configuration is investigating. At high
energy the correction to the standard quasiclassical approximation (SQA) has
been calculated. In the region of intermediate photon energies where SQA is
inapplicable the new approximation, developed recently by authors, is used.
The influence of weak electric field on the process in a magnetic field is
considered. In particular, in the presence of electric field the root
divergence in the probability of pair creation on the Landau energy levels is
vanished. For smaller photon energies the low energy approximation is used.
The found probability describes the absorption of soft photon by particles
created by field. At low photon energy the electric field action dominates and
the influence of magnetic field on the process is connected with the
interaction of it and the magnetic moment of creating particles.

##  1  Introduction

The pair photoproduction in an electromagnetic field is the basic QED reaction
which can play the significant role in many processes.

This process was considered first in a magnetic field. The investigation was
started in 1952 independently by Klepikov and Toll [1, 2]. In Klepikov’s paper
[3], which was based on the solution of the Dirac equation, the probability of
photoproduction had been obtained on the mass shell ( $k^{2}=0,k$ is the
4-momentum of photon. We use the system of units with $\hbar=c=1$ and the
metric $ab=a^{\mu}b_{\mu}=a^{0}b^{0}-\boldsymbol{ab}$). In 1971 Adler [4] had
calculated the photon polarization operator in a magnetic field using the
proper-time technique developed by Schwinger [5] and Batalin and Shabad [6]
had calculated this operator in an electromagnetic field using the Green
function found by Schwinger [5]. In 1975 the contribution of charged-particles
loop in an electromagnetic field with $n$ external photon lines had been
calculated in [7]. For $n=2$ the explicit expressions for the contribution of
scalar and spinor particles to the polarization operator of photon were given
in this work. Making use of the imaginary part of this operator for spinor
particles the pair photoproduction probability was analyzed in the pure
magnetic [8] and the pure electric [9] field.

The probability of pair photoproduction in a constant and homogeneous electric
field in the quasiclassical approximation had been found by Narozhny [10]
using the solution of the Dirac equation in the Sauter potential [11].
Nikishov [12] had obtained the differential distribution of this process also
using the solution of the Dirac equation in the indicated field.

In the present paper we consider the integral probability of pair creation in
a constant and homogeneous electromagnetic field of an arbitrary configuration
basing on the polarization operator [7]. In Sec.2 the exact expression for
this probability has been received for the general case $k^{2}\neq 0.$In Sec.3
the standard quasiclassical approximation (SQA) is outlined for the high-
energy photon $\omega\gg m$ ( $m$ is the electron mass). The corrections to
SQA, determined also the applicability region of SQA, have been calculated.
The found expressions, given in the Lorentz invariant form, contain two
invariant parameters. In Sec.4 the new approach has been developed for the
relatively low energies where SQA is not applicable. This approach is based on
the method proposed in [8]. The obtained probability is valid in the wide
interval of photon energy, which is overlapped with SQA. In Sec.5 the case of
the ”nonrelativistic” photon $\omega\ll m$ is analyzed. In particular, in the
energy region $\omega\lesssim$ $eE/m$ where the previous approach is
inapplicable, the low energy approximation has been developed basing on the
analysis in [9]. In tern the found results have an overlapping region of
applicability with the previous approach. So just as in [9] we have three
overlapping approximations which include all photon energies. At the photon
energy $\omega\ll$ $eE/m$ the probability has been found for arbitrary values
of fields $E$ and $H.$

##  2  General expressions for the probability of process

Our analysis is based on the general expression for the contribution of spinor
particles to the polarization operator obtained in a diagonal form in [7] (see
Eqs. (3.19), (3.33)). The imaginary part of the eigenvalue $\kappa_{i}$ of
this operator on the mass shell $(k^{2}=0)$ determines the probability per
unit length $W_{i}$ of $e^{-}e^{+}$ pair creation by the real photon with the
polarization $e_{i}$ directed along the corresponding eigenvector:

$W_{i}=-\frac{\mathrm{\operatorname{Im}}\kappa_{i}}{\omega};\ \
e_{i}^{\mu}=\frac{b_{i}^{\mu}}{\sqrt{-b_{i}^{2}}},~{}\ b_{2}^{\mu}=\
\left(Bk\right)^{\mu}+\frac{2\Omega_{4}}{\Omega}\left(Ck\right)^{\mu},\ $ (1)

$\
b_{3}^{\mu}=\left(Ck\right)^{\mu}-\frac{2\Omega_{4}}{\Omega}\left(Bk\right)^{\mu};$

$\displaystyle-\mathrm{\operatorname{Im}}\kappa_{2}$
$\displaystyle=r\left(\Omega_{2}-\frac{2\Omega_{4}^{2}}{\Omega}\right),\ \ \
-\mathrm{\operatorname{Im}}\
\kappa_{3}=r\left(\Omega_{3}+\frac{2\Omega_{4}^{2}}{\Omega}\right),\ \ $ (2)
$\displaystyle\ \Omega$
$\displaystyle=\Omega_{3}-\Omega_{2}+\sqrt{(\Omega_{3}-\Omega_{2})^{2}+4\Omega_{4}^{2}},\
\ r=\frac{\omega^{2}-k_{3}^{2}}{4m^{2}}.$

The consideration realizes in the frame where electric $\mathbf{E}$ and
magnetic $\mathbf{H}$ fields are parallel and directed along the axis 3\. In
this frame the tensor of electromagnetic field $F_{\mu\nu}$ and tensors
$F_{\mu\nu}^{\ast}$, $B_{\mu\nu}$ and $C_{\mu\nu}$ have a form

$\displaystyle F_{\mu\nu}$ $\displaystyle=C_{\mu\nu}E+B_{\mu\nu}H,\ \
F_{\mu\nu}^{\ast}=C_{\mu\nu}H-B_{\mu\nu}E,\ \
C_{\mu\nu}=g_{\mu}^{0}g_{\nu}^{3}-g_{\mu}^{3}g_{\nu}^{0},\ $ $\displaystyle\
B_{\mu\nu}$ $\displaystyle=g_{\mu}^{2}g_{\nu}^{1}-g_{\mu}^{1}g_{\nu}^{2};\ \
eE/m^{2}=E/E_{0}\equiv\nu,\ \ eH/m^{2}=H/H_{0}\equiv\mu;$ (3)
$\displaystyle\Omega_{i}$ $\displaystyle=\frac{\alpha
m^{2}}{2\pi\text{{i}}}\mu\nu\int\limits_{-1}^{1}\
dv\int\limits_{-\infty-\text{{i}0}}^{\infty-\text{{i}0 }}\
f_{i}(v,x)\exp(\text{{i}}\psi(v,x))xdx.$ (4)

Here

$\displaystyle f_{1}$ $\displaystyle=\frac{\cos(\mu xv)\cosh(\nu xv)}{\sin(\mu
x)\sinh(\nu x)}-\frac{\cos(\mu x)\cosh(\nu x)\sin(\mu xv)\sinh(\nu
xv)}{\sin^{2}(\mu x)\sinh^{2}(\nu x)},$ $\displaystyle f_{2}$
$\displaystyle=2\frac{\cosh(\nu x)(\cos(\mu x)-\cos(\mu xv))}{\sinh(\nu
x)\sin^{3}(\mu x)}+f_{1},\ $ $\displaystyle\ \ f_{3}$
$\displaystyle=2\frac{\cos(\mu x)(\cosh(\nu x)-\cosh(\nu xv))}{\sin(\mu
x)\sinh^{3}(\nu x)}-f_{1},$ $\displaystyle f_{4}$
$\displaystyle=\frac{\cos(\mu x)\cos(\mu xv)-1}{\sin^{2}(\mu
x)}\frac{\cosh(\nu x)\cosh(\nu xv)-1}{\sinh^{2}(\nu x)}$
$\displaystyle+\frac{\sin(\mu xv)\sinh(\nu xv)}{\sin(\mu x)\sinh(\nu x)};$ (5)
$\displaystyle\ \ \psi(v,x)$ $\displaystyle=2r\left(\frac{\cosh(\nu
x)-\cosh(\nu xv)}{\nu\sinh(\nu x)}+\frac{\cos(\mu x)-\cos(\mu xv)}{\mu\sin(\mu
x)}\right)-x.$ (6)

Let us note that the integration contour in Eq.(4) is passing slightly below
the real axis.

After all calculations have been fulfilled we can return to a covariant form
of the process description using the following expressions

$\displaystyle E^{2},H^{2}$
$\displaystyle=\left(\mathcal{F}^{2}+\mathcal{G}^{2}\right)^{1/2}\pm\mathcal{F,\
\ F=}\left(\mathbf{E}^{2}-\mathbf{H}^{2}\right)\diagup 2,\ \
\mathcal{G=}\mathbf{EH},\ $ $\displaystyle\left(C^{2}\right)_{\mu\nu}$
$\displaystyle=\left(F_{\mu\nu}^{2}+H^{2}g_{\mu\nu}\right)\diagup\left(E^{2}+H^{2}\right),\
\ \left(C^{2}\right)_{\mu\nu}-\left(B^{2}\right)_{\mu\nu}=g_{\mu\nu}.$ (7)

##  3  Quasiclassical approximation

The standard quasiclassical approximation (SQA) was developed first for a
magnetic field in [3], [13], [14]. The SQA is valid for ultrarelativistic
created particles ( $r\gg 1$) and can be derived from Eqs.(4)-(6) by expanding
the functions $f_{i}(v,x)$, $\psi(v,x)$ over $x$ powers. To get the correction
to the probability in SQA we shall keep leading to leading powers of $x$. We
have

$~{}\ b_{2}^{\mu}=\
\left(Bk\right)^{\mu}+\frac{\nu}{\mu}\left(Ck\right)^{\mu}\propto
F^{\mu\nu}k_{\nu},\ \
b_{3}^{\mu}=\left(Ck\right)^{\mu}-\frac{\nu}{\mu}\left(Bk\right)^{\mu}\propto
F^{\ast\mu\nu}k_{\nu};$ (8)

$\displaystyle-\mathrm{\operatorname{Im}}\kappa_{i}$
$\displaystyle=\mathrm{i}\frac{\alpha
m^{2}}{12\pi}r(\mu^{2}+\nu^{2})\int\limits_{-1}^{1}dv\left(1-v^{2}\right)\int\limits_{-\infty}^{\infty}h_{i}(v,x)\left[-\mathrm{i}\gamma\left(v,x\right)\right]xdx;$
$\displaystyle\gamma\left(v,x\right)$
$\displaystyle=x+\frac{x^{3}}{12}r\left(1-v^{2}\right)^{2}\left(\nu^{2}+\mu^{2}\right),$
(9) $\displaystyle h_{2}(v,x)$
$\displaystyle=\frac{3+v^{2}}{2}+\frac{1}{30}\left(15-6v^{2}-v^{4}\right)\left(\mu^{2}-\nu^{2}\right)x^{2}$
$\displaystyle-\frac{\mathrm{i}}{720}r(\mu^{2}+\nu^{2})\left(1-v^{2}\right)^{2}\left(9-v^{2}\right)\left(\mu^{2}-\nu^{2}\right)x^{5},$
$\displaystyle h_{3}(v,x)$
$\displaystyle=3-v^{2}+\frac{1}{60}\left(15-2v^{2}+3v^{4}\right)\left(\mu^{2}-\nu^{2}\right)x^{2}$
$\displaystyle-\frac{\mathrm{i}}{360}r(\mu^{2}+\nu^{2})\left(1-v^{2}\right)^{2}\left(3-v^{2}\right)^{2}\left(\mu^{2}-\nu^{2}\right)x^{5}.$
(10)

We are using the known integrals:

$\displaystyle\int\limits_{-\infty}^{\infty}\cos\left(x+\frac{ax^{3}}{3}\right)dx$
$\displaystyle=\frac{2}{\sqrt{3a}}K_{1/3}\left(\frac{2}{3\sqrt{a}}\right),\ \
$
$\displaystyle\int\limits_{-\infty}^{\infty}x\sin\left(x+\frac{ax^{3}}{3}\right)dx$
$\displaystyle=\frac{2}{\sqrt{3}a}K_{2/3}\left(\frac{2}{3\sqrt{a}}\right).$
(11)

Conserving the first (independent on $x)$ terms in Eq.(10) we obtain the
probabilities in SQA

$\displaystyle W_{i}^{(SQA)}$
$\displaystyle=-\frac{\mathrm{\operatorname{Im}}\kappa_{i}}{\omega}=\frac{\alpha
m^{2}}{3\sqrt{3}\pi\omega}\int\limits_{-1}^{1}\frac{s_{i}}{1-v^{2}}K_{2/3}\left(z\right)dv,\
\ z=\frac{8}{3\left(1-v^{2}\right)\kappa},\ $ $\displaystyle s_{2}$
$\displaystyle=2(3-v^{2}),\ \ \ s_{3}=3+v^{2},\ \ \
\kappa^{2}=4r(\mu^{2}+\nu^{2})=-\frac{e^{2}}{m^{6}}\left(F^{\mu\nu}k_{\nu}\right)^{2}.$
(12)

The correction to SQA has a form

$W_{i}^{(1)}=-\frac{\alpha
m^{2}\widetilde{\mathcal{F}}}{15\sqrt{3}\pi\omega\kappa}\int\limits_{-1}^{1}\frac{dv}{1-v^{2}}G_{i}\left(v,z\right),\
\
\widetilde{\mathcal{F}}=\frac{e^{2}\mathcal{F}}{m^{4}}=\frac{\nu^{2}-\mu^{2}}{2},$
(13)

where

$\displaystyle G_{2}\left(v,z\right)$
$\displaystyle=\left(36+4v^{2}-18z^{2}\right)K_{1/3}\left(z\right)+\left(3v^{2}-57\right)zK_{2/3}\left(z\right),$
$\displaystyle G_{3}\left(v,z\right)$
$\displaystyle=-\left(34+2v^{2}+36z^{2}\right)K_{1/3}\left(z\right)+\left(78-6v^{2}\right)zK_{2/3}\left(z\right).$
(14)

The mathematical transformations of integrals can be found in Appendix C [8].
It is seen that in this order of decomposition the correction does not depend
on the invariant parameter $\mathcal{G}$, because $\mathcal{G}$ is the
pseudoscalar. The asimptotic of the integrals incoming in the correction terms
have been given in the mentioned Appendix C. The asymptotic at $\kappa\ll 1$
will become necessary further

$W_{2}^{(1)}=\frac{4\alpha
m^{2}\widetilde{\mathcal{F}}}{5\omega\kappa^{2}}\sqrt{\frac{2}{3}}\exp\left(-\frac{8}{3\kappa}\right),\
\ W_{3}^{(1)}=2W_{2}^{(1)},\ \ \frac{\
W_{i}^{(1)}}{W_{i}^{(SQA)}}=\frac{64\widetilde{\mathcal{F}}}{15\kappa^{3}}.$
(15)

##  4  Region of intermediate photon energies

In the field which is weak comparing with the critical field $E/E_{0}=\nu\ll
1$ ( $E_{0}=1.32\cdot 10^{16}\operatorname{V}/\operatorname{cm}$), $H/H_{0}$ =
$\mu\ll 1$ $(H_{0}=4.41\cdot 10^{13}\operatorname{G})$ and at the relatively
low photon energies $r\lesssim\nu^{-2/3}$ the standard quasiclassical
approximation Eq.(12) is non-applicable. This follows from the last equality
in Eq.(15). For these energies, if the condition $r\gg\nu^{2}$ is fulfilled,
the method of stationary phase can be applied at integration over $x$ in
Eq.(4). In this case the small values of $v$ contribute to the integral over
$v$. So one can expand the phase $\psi(v,x)$ over $v$ and extend the
integration limit to the infinity. We get

$\Omega_{i}=\frac{\alpha
m^{2}}{2\pi\text{{i}}}\mu\nu\int\limits_{-\infty}^{\infty}\
dv\int\limits_{-\infty}^{\infty\text{ }}\
f_{i}(0,x)\exp\left\\{-\text{{i}}\left[\varphi\left(x\right)+v^{2}\chi\left(x\right)\right]\right\\}xdx,$
(16)

where

$\displaystyle\varphi\left(x\right)$
$\displaystyle=2r\left(\frac{1}{\mu}\tan\frac{\mu
x}{2}-\frac{1}{\nu}\tanh\frac{\nu x}{2}\right)+x,\ $ $\displaystyle\
\chi\left(x\right)$ $\displaystyle=rx^{2}\left(\frac{\nu}{\sinh(\nu
x)}-\frac{\mu}{\sin(\mu x)}\right).$ (17)

From the equation $\varphi^{\prime}(x_{0})=0$ we find the saddle point $x_{0}$

$\tan^{2}\frac{\nu s}{2}+\tanh^{2}\frac{\mu s}{2}=\frac{1}{r},\ \
x_{0}=-\mathrm{i}s.$ (18)

Substituting this value of $\rule{7.22743pt}{7.22743pt}_{0}$ in the
expressions determined the integrals in Eq.(16) we have

$\displaystyle\text{{i}}\varphi\left(x_{0}\right)$
$\displaystyle=2r\left(\frac{1}{\mu}\tanh\frac{\mu
s}{2}-\frac{1}{\nu}\tan\frac{\nu s}{2}\right)+s\equiv b(s),\ $ (19)
$\displaystyle\ \text{{i}}\chi\left(x_{0}\right)$
$\displaystyle=rs^{2}\left(\frac{\nu}{\sin(\nu s)}-\frac{\mu}{\sinh(\mu
s)}\right)\equiv\frac{1}{2}rs^{2}A(s),$ (20) $\displaystyle\
\text{{i}}\varphi^{\prime\prime}(x_{0})$
$\displaystyle=r\left[\nu\sin\frac{\nu s}{2}\diagup\cos^{3}\frac{\nu
s}{2}+\mu\sinh\frac{\mu s}{2}\diagup\cosh^{3}\frac{\mu s}{2}\right]\equiv
rD\left(s\right),$ (21)

$\displaystyle f_{2}(0,x_{0})$ $\displaystyle=\frac{1}{\sinh(\mu s)\sin(\nu
s)}\left[\cos(\nu s)\diagup\cosh^{2}\frac{\mu s}{2}-1\right]\equiv-a_{2}(s),\
$ $\displaystyle f_{3}(0,x_{0})$ $\displaystyle=\frac{1}{\sinh(\mu s)\sin(\nu
s)}\left[1-\cosh\mu s\diagup\cos^{2}\frac{\nu s}{2}\right]\equiv-a_{3}(s),\ $
$\displaystyle\ f_{4}(0,x_{0})$ $\displaystyle=-\left(4\cos^{2}\frac{\nu
s}{2}\cosh^{2}\frac{\mu s}{2}\right)^{-1}\equiv-a_{4}(s).$ (22)

Performing the standard procedure of the stationary phase method and using
Eqs.(1)-(2) one obtains the following expressions

$\displaystyle\Omega_{i}$ $\displaystyle=a_{i}\frac{\alpha
m^{2}\mu\nu}{r\sqrt{AB}}\exp\left(-b\right),\ \ W_{i}=\lambda_{i}\frac{\alpha
m^{2}\mu\nu}{\omega\sqrt{AB}}\exp\left(-b\right);$ (23)
$\displaystyle\lambda_{2}$ $\displaystyle=a_{2}-\frac{2a_{4}^{2}}{a},\ \
\lambda_{3}=a_{3}+\frac{2a_{4}^{2}}{a},\ \
a=a_{3}-a_{2}+\sqrt{\left(a_{3}-a_{2}\right)^{2}+4a_{4}^{2}};$ $\displaystyle
b_{2}^{\mu}$ $\displaystyle=\
\left(Bk\right)^{\mu}+\frac{2a_{4}}{a}\left(Ck\right)^{\mu},\ \
b_{3}^{\mu}=\left(Ck\right)^{\mu}-\frac{2a_{4}}{a}\left(Bk\right)^{\mu}$ (24)

These equations is valid at $r\gg 1$ if the condition $b\gg 1$ is fulfilled.
The first two terms of the decomposition of the functions $s\left(r\right)$
Eq.(18)) and $b\left(s\left(r\right)\right)$ Eq.(19) over $1/r$ are

$s(r)\simeq\frac{4}{\kappa}\left(1-\frac{8\widetilde{\mathcal{F}}}{3\kappa^{2}}\right),\
\ b(r)\simeq\frac{8}{3\kappa}-\frac{64\widetilde{\mathcal{F}}}{15\kappa^{3}},\
\ \kappa^{2}=4(\mu^{2}+\nu^{2})r.$ (25)

It is follows from this formula that the applicability of Eq.(23) is limited
by the condition $\kappa\ll 1$. The main values of the rest terms in
Eqs.(23),(24) have a form

$\displaystyle A$ $\displaystyle=\frac{1}{3}\left(\mu^{2}+\nu^{2}\right)s,\
D=\frac{3}{2}A;\ a_{2}=\frac{\mu^{2}+2\nu^{2}}{4\mu\nu},\
a_{3}=\frac{2\mu^{2}+\nu^{2}}{4\mu\nu},$ $\displaystyle\ \ \ a_{4}$
$\displaystyle=\frac{1}{4},\ \ a=\frac{\mu}{2\nu},\ \
\lambda_{2}=\frac{\mu^{2}+\nu^{2}}{4\mu\nu},\ \ \lambda_{3}=2\lambda_{2},\ $
(26)

and the vectors of polarization are given by Eq.(8). Substituting this values
into equation for $W_{i}$ we have

$W_{2}=\frac{\alpha
m^{2}\kappa}{8\omega}\sqrt{\frac{3}{2}}\exp\left(-\frac{8}{3\kappa}+\frac{64\widetilde{\mathcal{F}}}{15\kappa^{3}}\right),\
\ W_{3}=2W_{2}.$ (27)

In the region of the SQA applicability and for $\kappa\ll 1$ this probability
coincides with the results of the previous section and so the overlapping
region of both approximations exists.

It is interesting to consider the photon energy region $|r-1|\ \ll 1$ in the
presence of a weak electric field $(\nu\ll\mu)$ where in the absence of an
electric field the approach under consideration is valid if the condition
$r-1\gg\mu$ is fulfilled [8]. In this case Eq.(18) and its solutions are given
by the following approximate equations

$\displaystyle\frac{\xi^{2}y_{0}^{2}}{16}$
$\displaystyle\simeq\exp(-y_{0})+\frac{1-r}{4},\ \ y_{0}=\mu s,\
\xi=\frac{\nu}{\mu};$ (28) $\displaystyle\ y_{0}$ $\displaystyle\simeq
2\ln\frac{2}{\xi\ln\frac{4}{\xi}}\left(1-\frac{r-1}{2\xi^{2}\ln\frac{2}{\xi}\ln^{3}\frac{4}{\xi}}\right),\
|r-1|\ \lesssim\xi^{2};$ (29) $\displaystyle y_{0}$
$\displaystyle\simeq\ln\frac{4}{r-1}\left(1-\frac{\xi^{2}}{4\left(r-1\right)}\ln\frac{4}{r-1}\right),\
\ r-1\gg\xi^{2};$ (30) $\displaystyle\xi y_{0}$ $\displaystyle=\nu s\simeq
2\sqrt{1-r},\ \ 1-r\gg\xi^{2}.$ (31)

The applicability of the using saddle-point method is connected with the large
value of the coefficient to the second power $(y-y_{0})^{2}$ of the
decomposition in the phase Eq.(17). In the energy region under consideration
we have

$\mathrm{i}\varphi^{\prime\prime}(x_{0})(x-x_{0})^{2}/2\simeq\frac{\xi^{2}}{4\mu}\left[y_{0}+\frac{y_{0}^{2}}{2}+\frac{2\left(r-1\right)}{\xi^{2}}\right](y-y_{0})^{2}.$
(32)

So, we have from the upper equations that in the case $\nu/\mu=\xi\ll 1,\
|r-1|\ \lesssim\xi^{2}$ Eq.(23) is valid if the condition $\xi^{2}/\mu\gg 1$
is fulfilled. In the case $1\gg r-1\ \gg\xi^{2}\ $the condition $r-1\ \gg\mu$
has to be available for that. And in the case $1\gg 1-r$ $\gg\xi^{2}$ the
condition
$\sqrt{1-r}\xi/\mu=\sqrt{\left(\xi^{2}/\mu\right)\left(1-r\right)/\mu}\gg 1$
is necessary for the applicability of the approach under consideration.

At low photon energy $r\ll 1\ $($\nu^{2}\ll r\ll\nu^{2/3})$ we have

$\displaystyle\nu s$
$\displaystyle\simeq\pi-2\sqrt{r}+r^{3/2}\left(\frac{2}{3}-\tanh^{2}\frac{\pi\eta}{2}\right),$
$\displaystyle b$
$\displaystyle\simeq\frac{1}{\nu}\left(\pi-4\sqrt{r}+\frac{2r}{\eta}\tanh\frac{\pi\eta}{2}\right);$
(33) $\displaystyle a_{2}$
$\displaystyle=\frac{1}{\sqrt{r}\sinh(\pi\eta)}\left(1-\frac{1}{2}\tanh^{2}\frac{\eta\pi}{2}+\frac{\mu}{4r}\coth\pi\eta\right),\
$ $\displaystyle\ a_{3}$
$\displaystyle=\frac{\coth(\pi\eta)}{2r^{3/2}}\left(1+\frac{4\eta\sqrt{r}}{\sinh(2\pi\eta)}\right)\simeq
a,\ \ a_{4}=\left(4r\cosh^{2}\frac{\eta\pi}{2}\right)^{-1},$ (34)
$\displaystyle\ \ \lambda_{2}$
$\displaystyle=\frac{1}{\sqrt{r}\sinh(\pi\eta)}\left[1-\left(\frac{1}{2}+\frac{1}{\cosh(\pi\eta)}\right)\tanh^{2}\frac{\eta\pi}{2}+\frac{\mu}{4r}\coth(\pi\eta)\right],$
$\displaystyle\lambda_{3}$ $\displaystyle\simeq a_{3},\ \
A=\frac{\nu}{\sqrt{r}}\left(1-\frac{2\eta\sqrt{r}}{\sinh(\pi\eta)}\right),\ \
D=\frac{\nu}{r^{3/2}},\ \eta=\frac{\mu}{\nu}.$ (35)

Here we have retained the leading and the leading to leading terms of
decomposition. The term $\propto\mu$ in $a_{2}$ has appeared as the
contribution of the second term in $f_{1}$ ($\propto v^{2}$) in Eq.(5).
Substituting these values into Eq.(23) one obtains the following expression
for the probability of the process

$\displaystyle W_{3}$ $\displaystyle=\frac{\alpha
m^{2}\mu}{2\omega\sqrt{r}}\coth\left(\pi\eta\right)\left(1+\frac{\eta\sqrt{r}}{\sinh(\pi\eta)}+\frac{4\eta\sqrt{r}}{\sinh(2\pi\eta)}\right)\exp\left(-b\right),$
$\displaystyle W_{2}$ $\displaystyle=\frac{\alpha
m^{2}\mu\sqrt{r}}{\omega\sinh(\pi\eta)}\left[1-\frac{2+\cosh(\pi\eta)}{2\cosh(\pi\eta)}\tanh^{2}\frac{\eta\pi}{2}+\frac{\mu}{4r}\coth(\pi\eta)\right]\exp\left(-b\right),$
(36)

where $b$ is given by Eq.(33). One can see out of this equation that $W_{2}\ll
W_{3}.$At $\eta\gg 1$ the probability $W_{3}$ has been increased by the factor
$\eta\pi\exp\left(\pi r/\nu\right)$ in comparison with the case of the absence
of magnetic field. The probability $W_{2}$ has been reduced by the additional
factor $(\exp\left(-\pi\mu/\nu\right))$ and becomes non-applicable at
$\mu\gtrsim\sqrt{r}\gg\nu$. In that case for the probability $W_{2}$ one can
use Eq.(40) which will be get below.

##  5  Approximation at low photon energy

At $r\sim\nu^{2}$ the above approximation becomes non-applicable and another
approach has to be. We close the integration over $x$ contour in Eq.(4) in the
lower half-plane and represent this equation in the following form

$\Omega_{i}=\frac{\alpha m^{2}}{2\pi\text{{i}}}\mu\nu\int\limits_{-1}^{1}\
dv\sum\limits_{n=1}^{\infty}\oint f_{i}(v,x)\exp(\text{{i}}\psi(v,x))xdx,$
(37)

where the path of integration is any simple closed contour around the point
$-$i$\pi n\diagup\nu.$ Let us choose the contour near this point in the
following way $\nu x=-$i$\pi n+\xi_{n},\ \ |\xi_{n}|\ \sim\sqrt{r}\sim\nu$ and
expand the function entering in over the variables $\xi_{n}$. In the case
$\nu\ll 1,$ because of appearance of the factor $\exp\left(-\mathrm{i}\pi
n\diagup\nu\right),$ the main contribution to the sum gives the term $n=1$.
Near the point $-$i$\pi\diagup\nu$ the main terms of expansion such as
$(\xi\equiv\xi_{1})$

$\displaystyle f_{3}$
$\displaystyle=\frac{4\text{{i}}}{\xi^{3}}\coth(\pi\eta)\cos^{2}\frac{\pi
v}{2},\ \ f_{2}=-\frac{1}{\xi^{2}}\frac{\coth(\pi\eta)\text{
}}{\sinh(\pi\eta)}\sinh(v\pi\eta)\ \sin(v\pi),\ $ $\displaystyle\ f_{4}$
$\displaystyle=\frac{2}{\xi^{2}}\frac{\cosh(\pi\eta)-\cosh(v\pi\eta)}{\sinh^{2}(\pi\eta)}\cos^{2}\frac{\pi
v}{2},\ \ \psi=\frac{4r}{\xi\nu}\cos^{2}\frac{\pi
v}{2}-\frac{\xi}{\nu}+\frac{\mathrm{i}\pi}{\nu}.$ (38)

Using the integrals Eq.(7.3.1) and Eq.(7.7.1 (11)) in [15] and substituting
the result in Eqs.(1)-(2) we find

$\displaystyle W_{3}$ $\displaystyle=2\frac{\alpha
m^{2}}{\omega}\eta\pi\coth(\pi\eta)\text{
}\exp\left(-\frac{\pi}{\nu}\right)I_{1}^{2}\left(z\right),\ \ \
z=\frac{2\sqrt{r}}{\nu},$ (39) $\displaystyle W_{2}$
$\displaystyle=\frac{\alpha
m^{2}}{\omega}\mu\coth(\pi\eta)\exp\left(-\frac{\pi}{\nu}\right)\left[\frac{\pi\eta}{\sinh(\pi\eta)}\int\limits_{0}^{1}\cosh(v\pi\eta)I_{0}\left(2z\cos\frac{\pi
v}{2}\right)dv-1\right],$ (40)

where $\mathrm{I}_{n}\left(z\right)$ is the Bessel function of imaginary
argument. At calculation $W_{2}$ the integration by parts over $v$ has been
performed. For $\eta\ll 1$ one obtains

$W_{2}=\frac{\alpha
m^{2}}{\omega}\frac{\nu}{\pi}\exp\left(-\frac{\pi}{\nu}\right)\left(I_{0}^{2}\left(z\right)-1\right).$
(41)

The found probability is applicable for $r\ll\nu.$ Here we have kept the main
terms in $W_{i}$ only.

For $r\gg\nu^{2}$ the asymptotic representation
$\mathrm{I}_{n}\left(z\right)\simeq\exp\left(z\right)\diagup\sqrt{2\pi z}$ can
be used. As a result one obtains the probability Eq.(36) where the leading
terms have to be retained. At very low photon energy $r\ll\nu^{2},$ using the
expansion of the Bessel functions for the small value of argument, we have

$W_{3}=2\frac{\alpha
m^{2}r}{\omega\nu^{2}}\eta\pi\coth(\pi\eta)\exp\left(-\frac{\pi}{\nu}\right),\
\ W_{2}=\frac{\nu}{\pi\left(1+\eta^{2}\right)}W_{3}.$ (42)

The probability under consideration is of interest of theoretics for arbitrary
values $\mu$ and $\nu$. For $r\ll\nu^{2}\diagup\left(1+\nu^{2}\right)$ one can
conserve in the phase $\psi(v,x)$ the term $-x$ only. After integrating over
$v$ we get the following equation for the probability averaged over the photon
polarizations

$\displaystyle W$ $\displaystyle=\frac{W_{2}+W_{3}}{2}=\frac{\alpha
m^{2}r}{\mathrm{i}\pi\omega}\sum\limits_{n=1}^{\infty}\oint
F(y_{n})\exp\left(-\mathrm{i}\frac{y_{n}}{\nu}\right)dy_{n},$ $\displaystyle
F(y)$ $\displaystyle=\frac{\cosh(y)\left(\eta y\cos\left(\eta
y\right)-\sin\left(\eta y\right)\right)}{\sinh y\sin^{3}\eta
y}+\frac{\eta\cos(\eta y)\left(y\cosh y-\sinh y\right)}{\sinh^{3}y\sin(\eta
y)}.$ (43)

Summing the residues in the points $y_{n}=-\mathrm{i}n\pi$ one obtains

$\displaystyle W$ $\displaystyle=\frac{\alpha
m^{2}r}{\omega}\sum\limits_{n=1}^{\infty}\exp\left(-\frac{\pi
n}{\nu}\right)\Phi\left(z_{n}\right),\ \ z_{n}=\eta\pi n,$ (44)
$\displaystyle\Phi\left(z_{n}\right)$
$\displaystyle=\frac{z_{n}}{\nu^{2}}\coth
z_{n}+\frac{2}{\sinh^{2}z_{n}}\left[\frac{\eta
z_{n}}{\nu}+\left(1+\eta^{2}\right)z_{n}\coth z_{n}-1\right].$ (45)

In the absence of magnetic field ( $\eta\rightarrow 0,$ $z_{n}\rightarrow 0$ )
we have

$\displaystyle\Phi$ $\displaystyle=\frac{1}{\nu^{2}}+\frac{2}{\nu\pi
n}+\frac{2}{\pi^{2}n^{2}}+\frac{2}{3},$ $\displaystyle W$
$\displaystyle=\frac{\alpha
m^{2}r}{\omega}\left[\left(\frac{1}{\nu^{2}}+\frac{2}{3}\right)\frac{1}{e^{\pi/\nu}-1}-\frac{2}{\pi\nu}\ln\left(1-e^{-\pi/\nu}\right)+\frac{2}{\pi^{2}}\mathrm{Li}_{2}\left(e^{-\pi/\nu}\right)\right],$
(46)

where $\mathrm{Li}_{2}\left(z\right)$ is the Euler dilogarithm. In the
opposite case $\eta\gg 1$ one obtains

$\Phi=\frac{\pi\eta n}{\nu^{2}},\ \ W=\frac{\alpha
m^{2}r}{\omega\nu^{2}}\frac{\pi\eta}{4}\sinh^{-2}\frac{\pi}{2\nu}.$ (47)

##  6 Conclusion

We have considered the process of pair creation in constant and homogeneous
electromagnetic fields with a real photon taking part in. The probability of
the process has been calculated using three different overlapping
approximation. In the region of SQA applicability the created by a photon
particles have ultrarelativistic energies. The role of fields in this case is
to transfer the required transverse momentum and the electric and magnetic
field actions are equivalent. But even in this case it is necessary to note a
special significance of a weak electric field $E=\xi H$ $(\xi\ll 1)$ in the
removal of the root divergence of the probability when the particles of pair
are created on the Landau levels with the electron and positron momentum
$p_{3}=0$ [8]. The frame is used where $k_{3}=0.$

Generally speaking, at $\xi\ll 1$ the formation time $t_{c}$ of the process
under consideration is $1/\mu.$ Here we use units $\hbar=c=m=1.$ At this time
the particles of creating pair gets the momentum $\delta p_{3}\sim\xi.$ If the
value $\xi^{2}$ becomes more larger than the distance apart Landau levels
$2\mu$ $(\nu^{2}\gg\mu^{3})$ all levels have been overlapped. Under this
condition the divergence of the probability is vanished and the new
quasiclassical approach is valid even in the energy region $r-1\lesssim\mu$
where it has been inapplicable in the absence of electric field [8]. In the
opposite case $\nu^{2}\ll\mu^{3}$ for the small value of $p_{3}\ll\sqrt{\mu},$
in the region where the influence of electric field is negligible, the
formation time of the process $t_{f}$ is $1/p_{3}^{2}$ and $\delta
p_{3}\sim\nu/p_{3}^{2}$ $\ll p_{3}$ . It is follows from above that
$\nu^{1/3}\ll p_{3}\ll\sqrt{\mu}$ . At this condition the value of
discontinuity is $\sqrt{t_{f}/t_{c}}\sim\sqrt{\mu}/p_{3}.$For $\nu^{1/3}\gg
p_{3}$ the time $t_{f}$ is determined by the self-consistent equation
$\delta\varepsilon^{2}\sim 1/t_{f}\sim\nu^{2}t_{f}^{2},$ $t_{f}\sim\nu^{-2/3}$
and the value of discontinuity becomes $\sqrt{\mu
t_{f}}\sim(\mu^{3}/\nu^{2})^{1/6}$ instead of $\sqrt{\mu}/p_{3}.$

In the region $\omega\lesssim 2m$ $(r\lesssim 1)$ the energy transfer from
electric field to the created particles becomes appreciable and for
$\omega\ll$ $m$ it determines the probability of the process mainly. At
$\omega\ll eE/m$ the photon assistance in the pair creation comes to the end
and the probability under consideration defines the probability of photon
absorption by the particles created by electromagnetic fields. The influence
of a magnetic field on the process is connected with the interaction of the
magnetic moment of the created particles and magnetic field. This interaction,
in particular, has appeared in the distinction of the pair creation
probability by field for scalar and spinor particles [5].

## References

  * [1] N.P.Klepikov, Ph.D. dissertation, Moscow State University, 1952 (unpublished).
  * [2] J.S.Toll, Ph.D. dissertation, Prinston University, 1952 (unpublished).
  * [3] N.P.Klepikov, Zh. Eksp. Teor. Fiz., 26, 19 (1954).
  * [4] S.L.Adler, Ann. Phys. (N.Y.), 67, 599 (1971).
  * [5] J.Schwinger, Phys. Rev., 82, 664 (1951).
  * [6] I.A.Batalin and A.E.Shabad, Sov. Phys. JETP, 33, 483 (1971).
  * [7] V.N.Baier, V.M.Katkov and V.M.Strakhovenko, Sov. Phys. JETP, 41, 198, (1975).
  * [8] V.N.Baier and V.M.Katkov, Phys.Rev., D 75, 073009 (2007).
  * [9] V.N.Baier and V.M.Katkov, arXiv:0912.5250v1 [hep-ph]; Preprint BINP 2009-38, Novosibirsk, 2009.
  * [10] N.B.Narozhny, Zh. Eksp.Teor.Fiz., 54, 676 (1968).
  * [11] F.Sauter, Z.Phys. 69, 742 (1931).
  * [12] A.I.Nikishov, Zh.Eksp.Teor.Fiz., 59, 1262 (1970).
  * [13] V.N.Baier and V.M.Katkov, Sov. Phys. JETP, 26, 854 (1968).
  * [14] W.Tsai and T.Erber, Phys. Rev. D 10, 492 (1974)
  * [15] H.Bateman, A.Erdélyi, Higher Transcendental Functions, v.II, McGraw-Hill Book Co, New York, 1953.