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an external electromagnetic field strongly modifies known physical processes and allows new ones to occur when its strength approaches the critical value @xmath0 , @xmath1  g. an area of research where the processes in strong magnetic fields play an important role is the study of neutron star atmospheres and their radiation . predicted values of neutron star magnetic field vary from @xmath2  g for radiopulsars to @xmath3  g for magnetars and grb s . for instance , developing of electron - positron pair cascades in the magnetosphere of a pulsar has long been considered as an important part of the pulsar s emission mechanism @xcite@xcite .    in laboratory conditions the strong quasi - static magnetic field up to about 30  mg
can be obtained by utilization of exploding generators @xcite .
laser assisted magnetic field generation attracts great interest as well .
irradiation of a solid with a short - pulse laser generates picosecond - duration pulses of giga - gauss magnetic field at laser intensities of @xmath4  w/@xmath5 @xcite .
petawatt class laser facilities are capable of delivering ultra - high focused intensities greater than @xmath6  w/@xmath5 corresponding to field strength of the order of @xmath7  g. nonlinear effects of qed in strong electromagnetic field were observed for the first time at slac national accelerator laboratory in experiments with terawatt laser @xcite@xcite . in particular ,
positron production in collisions of electron beam with intense laser pulses was reported @xcite .
the effect was explained as two step process where the first a high energy @xmath8-photon is generated by compton backscattering off the electron beam , which afterwards creates the pair in a photon - multiphoton collision @xcite@xcite .
this process , however , may be treated as a resonant case of the laser - dressed trident pair creation , @xmath9 a non - perturbative qed calculations of this process are provided in @xcite .
nevertheless , it was discussed in refs .
@xcite , @xcite , that indirect treatment of the process ( [ react ] ) is possible using nikishov - ritus theorem @xcite , @xcite . according to this theorem the rates of some reaction in different field configurations
are closely connected .
this allows to describe the laser - dressed trident production ( [ react ] ) using relatively simple analytic expressions for the case of an external magnetic field .
a cascade of photon emission process followed by photoproduction in a magnetic field was first studied in ref .
the estimation of the number of produced positrons in the slac experiment was made in ref .
@xcite , that showed a reasonable agreement with the experimental results .
however , in ref .
@xcite the simplest case of pair production to ground levels was studied . in the present work the process of magnetic pair production by an electron
is studied in the general case when particles can be produced on excited energy levels .
the feynman diagrams of the magnetic @xmath10 pair production by an electron is shown in fig .
[ diagrams ] .
it is known that an electron in a magnetic field occupies discrete energy levels .
if @xmath11-axis is directed along the field then the energy eigenvalues are @xmath12 where @xmath13 is magnetic field strength in the units of the critical one and @xmath14 is the energy level number .        at the same time
, the initial electron energy @xmath20 should exceed the threshold value .
the threshold condition is @xcite @xmath21 where the subscript @xmath22 denotes the initial electron quantities .
we assume the longitudinal momenta of the final particles to be small , @xmath23 .
this condition is fulfilled near the process threshold when @xmath24 and , consequently , @xmath25 =     4 + o(b).\end{aligned}\ ] ]    the probability amplitude of the given process has the following form : @xmath26 .
\end{array}\ ] ] the solutions of the dirac equation in a magnetic field in a cartesian coordinates were used as electron wave functions , @xmath27\psi_{l\sigma}(x)},\ ] ] where the coordinate wave functions @xmath28 depend on electron polarization @xmath29 .
they can be expressed via normalized functions @xmath30 where @xmath31 are the hermite polynomials .
the choice of the wave functions ( [ psi ] ) allows simple integration of the amplitude over the 4-radius vectors @xmath35 and @xmath36 as well as over the @xmath37- and @xmath11- components of the virtual photon momentum .
it results in appearance of delta functions expressing conservation laws of energy and corresponding momentum components .
the probability amplitude describing the first feynman diagram takes on the form @xmath38
@xmath45 are the known functions @xcite resulting from the integration over the `` quantized '' coordinate @xmath35 , @xmath46 where @xmath47 with @xmath48 for electrons and @xmath49 for positrons respectively .
the explicit form of the functions ( [ i_gen ] ) is @xmath50 here , @xmath51 note that in the considered case of pair production by an electron the conservation laws give @xmath52    when kinematics allows the denominator of the photon propagator ( [ dmn ] ) to vanish , @xmath53 then resonant divergences occur in the amplitude and in the process rate , which are common for two - vertex qed processes in an external field . to eliminate the divergences it is necessary to introduce the state width @xmath54 to the denominator in eq .
( [ dmn ] ) according to the breit - wigner prescription  @xcite , @xmath55 .
it has been shown in ref .
@xcite that the process rate is determined by the parameter region where the integrand in eq .
( [ sfi1 ] ) contains a pole and the condition eq .
( [ res_case ] ) is satisfied . in the notation defined by eqs .
( [ xi_a ] ) , ( [ q_kappa ] ) , it takes on the form @xmath56 where @xmath57 it will be shown below that the differential rate contains a singularity located at @xmath58 .
taking into account that in the considered case @xmath59 , @xmath60 and @xmath61 , one can estimate @xmath62 as @xmath63 .
hence , for the vicinity of the resonance it follows that @xmath64 and @xmath65 , and one can neglect @xmath66 compared to @xmath67 in polynomials in eq .
( [ i_expl ] ) to simplify the integral in @xmath68 . as a result
, the amplitude @xmath69 can be expressed in terms of the known integrals @xmath70 , @xmath71 where @xmath72      with the adopted definitions eqs .
( [ xi_a ] ) , ( [ v_def ] ) , the process rate may be written as @xmath76 here , @xmath77 is the exchange amplitude obtained from @xmath78 by replacing subscripts @xmath79 , @xmath80 is the normalizing area and @xmath81 .
note that the interference term in the rate eq .
( [ wfi_ab ] ) may be neglected due to the presence of rapidly oscillating factors  @xcite .
the expansion of the amplitude into power series in @xmath82 allows to derive the rate ( [ wfi_ab ] ) in a simple analytical form .
to find the approximate expressions for the confluent hypergeometric functions @xmath83 entering ( [ i_expl ] ) note that in the resonant case the argument is much greater than unity , @xmath84 .
at the same time , the level numbers @xmath85 and @xmath86 are considered to be small , @xmath87 .
thus , for the functions that depend on @xmath85 and @xmath86 it is possible to use the known expansion @xcite , @xmath88,\ ] ] where @xmath89 and @xmath90 , @xmath82 are limited .
inserting here @xmath91 and @xmath92 one can readily find @xmath93 here , @xmath94 , @xmath95 and the residual term is @xmath96    the asymptotic formula ( [ f - pp ] ) can not be used for the functions @xmath97 since the threshold condition ( [ lh ] ) requires the inequality @xmath98 to be fulfilled .
consequently , @xmath99 when @xmath100 .
however , considering the presence of a sharp maximum at @xmath101 in the amplitude ( [ sfi1 ] ) , the relatively slow hypergeometric function can be replaced by its value at the singularity point .
it can be shown , that its resonant value is @xmath102 in the threshold case when @xmath103 .
note the absence of the factor @xmath104 compared to eq .
( [ f - pp ] ) .
after substitution the approximate form of the hypergeometric functions ( [ f - pp ] ) , ( [ f - rad ] ) to the rate ( [ sfi1 ] ) the dependency on @xmath67 and @xmath105 factors out and enters the rate in the form of the integrals @xcite @xmath106 when the number @xmath107 is large .
the rates for the other spin states divided by @xmath113 are @xmath114    \displaystyle      w^+_{--- } = \frac{\delta}{3 } \ : l_+ b,\\[2ex ]    \displaystyle      w^+_{-+- } = \frac{1}{8 }   \ :       l_2 l_+ b^2,\\[2ex ]    \displaystyle      w^+_{+-- } = \frac{1}{8 }    \ :      l_1 l_+ b^2 ,    \end{array } \quad %    \begin{array}{l }    \displaystyle       w^+_{+-+ } = \frac{5\delta}{3}\ : l_1 b,\\[2ex ]    \displaystyle       w^+_{-++ } = \frac{5\delta}{3}\ : l_2 b,\\[2ex ]    \displaystyle       w^+_{+++ } = \frac{1}{4 }        \ : l_1 l_2 b^2,\\[2ex ]    \displaystyle       w^+_{++- } = \frac{\delta}{4 } \ :
l_1   l_2 l_+ b^3 ,    \end{array}\ ] ] @xmath115    \displaystyle       w^-_{--- } =                      l_+ b,\\[2ex ]    \displaystyle       w^-_{-+- } = \frac{2\delta}{3}\ : l_2 l_+ b^2,\\[2ex ]    \displaystyle       w^-_{+-- } = \frac{2\delta}{3}\ : l_1 l_+ b^2 ,    \end{array } \quad %    \begin{array}{l }    \displaystyle       w^-_{+-+ } = \frac{1}{2 } \ :        l_1 b,\\[2ex ]    \displaystyle       w^-_{-++ } = \frac{1}{2 } \ :        l_2 b,\\[2ex ]    \displaystyle       w^-_{+++ } = \frac{\delta}{3 } \ : l_1 l_2 b^2,\\[2ex ]    \displaystyle       w^-_{++- } = \frac{\delta^2}{6 } l_1 l_2 l_+ b^3 .
\end{array}\ ] ] here , the superscript denotes the initial electron polarization and the subscripts denote polarizations of the final electrons and a positron respectively .
the quantity @xmath116 is defined in eq .  ( [ de ] ) .
let us consider the spin dependence of the obtained rates ( [ w1])([w2 ] ) .
apparently , the process rate substantially depends on spin projections of the final particles .
the rate has the greatest order of magnitude in the spin state defined by eq .
( [ sigma_main ] ) when magnetic moments of the final particles are oriented along the field . in this case
the energy of dipole interaction with magnetic field has the minimum value .
the change of spin orientation of each particle results in appearance of the factor @xmath117 in the rate .
the similar effect can be seen in one - photon pair production @xcite .    on the other hand , when the polarizations of the final particles are fixed then changing the initial electron spin projection does not affect the power of the small parameter @xmath82 .
weak influence of the initial electron polarization on the rate seems to be unexpected .
indeed , in the resonant case the process decomposes to photon emission followed by pair production .
it is known that emission of a photon is less probable when the change of the electron spin is involved . however , the exception is the case of near - ground transitions from relativistic initial state @xcite .
when a relativistic electron transits to the lowest levels emitting a hard photon , then the rate of spin - flip process approaches the rate of radiation without change of the spin projection .
apparently , conditions of the lll approximation ( [ lll ] ) together with the treshold requirement @xmath118 require such near - ground transition , which is the reason for the weak influence of initial electron spin on the rate of pair production by an electron .
the resonant process of electron - positron pairproduction by an electron in a subcritical magnetic field has been studied when the pair is produced to exited landau levels . 
 the spin dependency of the process rate has been analyzed . in the spin state with the greatest rate 
 the virtual photon is emitted with a flip of electron spin . 
 this behavior is not suppressed for radiative transitions from a relativistic initial state to low energy levels .    _ 
 2015 phys . 
 scr .  * 90 * 085305 ; _ * * doi:**10.1088/0031 - 8949/90/8/085305