Resonant Interference Effect in Scattering of an Electron by an Electron in the Field of Two Pulsed Laser Waves

Resonant scattering of an electron by an electron in the field of two unidirectional pulsed laser waves is studied theoretically. The interference kinematical region is considered. Stimulated emission and absorption of external waves’ photons by electrons is correlated in the interference region. Resonance condition in the interference region is specified. Analytical expressions for the amplitude and differential cross section of the process for waves’ circular polarization are obtained in the first order of the perturbation theory. The cross section contains resonant peaks, their altitude and width are determined by external waves parameters. Considerable effect of wave polarization on the character of stimulated emission and absorption of external waves’ photons is found. It is demonstrated that the differential interference cross section of scattering of nonrelativistic electrons by each other in the field of two pulsed light waves of the optical frequency is approximately ten times greater in the magnitude than the corresponding one in the external field absence. Obtained results can be verified experimentally, for example, at the SLAC National Accelerator Laboratory (Stanford, USA) or in the frame of the FAIR Project (Darmstadt, Germany).


Introduction
Widespread application of laser pulses [1] in both applied and fundamental research stimulates study of quantum electrodynamics (QED) processes in pulsed laser fields of different configurations. Nonlinear effects in force fields have been the object of scientific research for a long time already [2][3][4][5][6][7][8][9]. Experimental verification of QED effects in the laser field was carried out at the facility SLAC National Accelerator Laboratory (Stanford, USA) [10,11], and also is included into scientific program of FAIR international project based on the laser system PHELIX [12][13]. Development of new laser systems naturally results in improving of a model of description of modern lasers electromagnetic radiation [14][15][16][17]. The theoretical basis for QED processes study is the exact solution of the Dirac equation for an electron in the plane electromagnetic field, which is correct for a plane wave of arbitrary spectral composition [18,19].
Scattering of an electron by an electron is one of the fundamental QED processes. The cross section of this process in the external field absence was theoretically obtained by Möller [19,20]. In the presence of an external laser field this process is accompanied by stimulated emission and absorption of external field photons. Scattering of an electron by an electron in the field of a laser wave was studied in [5][6][7][8]17].
The theory of QED processes in the field of two laser waves was developed in [21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36], in the field of arbitrary number of plane monochromatic waves -in [37,38]. The interference quantum effect for scattering of an electron by an electron in the field of two or more plane monochromatic waves was discovered [24]. The essence of this effect is that emission and absorption of photons of external waves are correlated at a certain kinematics of the process. Study of effects caused by second pulsed laser wave presence is of interest due to the fact that probability of stimulated processes in the interference kinematics is generally greater than in other one.
Scattering of an electron by an electron in the field of a laser wave as a process of the second order in the fine structure constant can be resonant (Oleinik,[39,40]). Oleinik resonances are one of fundamental effects of QED in a laser field. The processes of the first order, which are forbidden by conservation laws in the external field absence, can occur in an electromagnetic wave. The resonant character relates to the fact that within a certain range of energy and momentum values a particle in an intermediate state may fall within the mass shell. Then a higher-order process is effectively reduced into two consecutive lower-order processes. Resonant cross sections of particles scattering can exceed the corresponding ones in the external field absence by several orders of the magnitude [5][6][7]17,[39][40][41][42][43][44][45][46][47][48][49]. Thus, resonant scattering of an electron by an electron in the external light field was studied in [5][6][7]17,[41][42][43][44][45].
It important to emphasize that when resonant conditions are satisfied the amplitude of particles scattering in the field of a plane monochromatic wave formally becomes infinite. Elimination of resonant infinity is usually performed by including of radiative corrections into the Green function of the intermediate particle by the Breit-Wigner procedure [39][40][41][42][43][44]. Elimination of the resonant infinity strictly mathematically becomes possible in the frame of the quasimonochromatic wave model, which can describe the field of the powerful pulsed laser up to the femtosecond range [6][7]17,[45][46][47][48][49]. General processes (both resonant and nonresonant) of the both first and second order in the field of a single pulsed wave were studied in detail in [45][46][47][48][49][50][51][52][53][54][55], and results have been summarized in monographs [6,17] and reviews [7,8]. Resonant scattering of a lepton by a lepton was studied in [45]. It is found that within a broad range of particle relative velocity resonant differential cross section of scattering of an electron by an electron (positron, muon) in the field of a single pulsed wave exceeds the corresponding cross section in the external field absence by several orders of the magnitude within the whole range of entrance polar angles.
The present paper is devoted to studying of the influence of two pulsed waves on resonant scattering of an electron by an electron in the interference region. The model of two pulsed plane waves allows us to describe the field of the two-mode pulsed laser, when laser radiation is characterized by presence of two frequency allotments in the field spectral distribution. At that the process is considered in the first order of the perturbation theory with respect to the interaction between electrons (the Born approximation), i.e. for sufficiently fast particles ( j c υ α  , j υ is the electron velocity, c is the light velocity in a vacuum, α is the fine structure constant). Thereby during the interaction time electrons do not have time to derivate significantly from the initial direction of movement at the expense of ponderomotive effect, consequently, spatial inhomogeneity in the transverse direction can be neglected [14]. The external light field is taken into account exactly.
The amplitude of the field strength of powerful ultra short laser pulses changes greatly in both space and time. Thus, we consider the field of two pulsed laser waves as two plane nonmonochromatic waves, propagating in the same direction. At that their planes of polarization belong to the plane xy , and the waves propagate along the z -axis. The four-potential of such a field has the form ( ) The index n labels the number of the laser wave. Each term in (1) corresponds to the field of the first and second wave, respectively. Here, n ϕ is the wave phase; 0n F is the strength amplitude of the wave electric field in the pulse peak; n ω is the characteristic frequency of wave field oscillation; ( ) in (1) are the envelope functions of the four-potential of external wave that allows to take into account the pulsed character of the laser field. Generally the envelope function is chosen to be equal to unity in the center of a pulse and to decrease exponentially when n n n ϕ ω τ  . Thereby the quantity n τ can be considered as the characteristic pulse duration [14]. It is assumed that pulse duration considerably exceeds the characteristic period of wave oscillation (see, (5)). The process is studied in the frame of the quasimonochromatic approximation when each of the waves performs a lot of amplitude oscillation, i.e. the fallowing condition is met: The quantity 2 n n ω τ π is the characteristic number of oscillation of the external field amplitude in the electromagnetic pulse. The given condition (5) allows to apply the derived differential cross-section for the real experiment description. It is shown in [14], that in addition to fulfillment of the condition of ponderomotive scattering neglecting, which is caused by spatial inhomogeneity of the laser field amplitude, the pulse duration must to be small enough that spreading of the initial particles packet can be neglected. Within the optical frequency range n τ can be even tens of femtoseconds. Consequently, the condition (5) is rather weak and is satisfied for the majority of modern lasers. Such fields, for which the condition (5) is met, are named the quasimonochromatic [6][7][8]14]. We emphasize here that field description by the potential (1)-(3) doesn't take into account the possible shift between laser waves and stipulates that laser pulses' maxima coincide. We do not study the effects introduced by the phase shift and dispersion in the pulses' peaks in the presented paper.
Nonlinear effects in processes of interaction of electrons with the wave field and with each other are specified by the following parameters. The first one is the relativistically invariant classical parameter [2] 276 Resonant Interference Effect in Scattering of an Electron by an Electron in the Field of Two Pulsed Laser Waves η ω = 1, 2, j = (6) which numerically equals to the ratio of the work done by the field within the wavelength to the electron rest energy (here, e is the electron charge and m is the electron mass, the index j labels the particle). The parameter (6) is one the external-field most important characteristics, and if ( ) 0 1 j n η  it has the meaning of the velocity of particle oscillation in the external laser field. The multiplicity of multiphoton processes occurring in Coulomb interaction of particles in the laser field in nonresonant scattering are determined by the quantum Bunkin-Fedorov multiphoton parameter [56] which numerically equals to the ratio of the work done by the field within the space, passing by an electron during the characteristic duration of wave oscillation ( 1 n ω − ), to the wave photon energy (here,  is the Planck constant) (the exact expression for the parameter ( ) 0 j n γ in electron-electron scattering is given by (33)). In the relativistic case and for laser wave elliptical polarization the quantum parameter ( )  As previously mentioned the special quite narrow region named the interference one appears in the field of two laser waves, where the Bunkin-Fedorov quantum parameter doesn't appear. When processes in the laser field of two unidirectional waves of different frequency are studied the quantum interference parameter may be relevant [5,24]: Parameters ( ) δ δ = = + . Also we emphasize that in our case the frequency of waves should not be close for waves' different polarization, because it corresponds to the single wave case (see, [24]).
For a certain intensity of the external field the parameters (6), (7) and for the interference region we have In (10) the values of the classical parameter ( ) 0 j n η 0.3  are mentioned [10].
It is important to underline that for nonresonant scattering of an electron by an electron in the field of the both single wave and two waves the Bunkin-Fedorov quantum parameter ( ) 0 j n γ (7) is the main multiphoton parameter [5][6][7][8].
In resonant scattering of an electron by an electron the influence of the parameter In this case the classical parameter ( ) 0 j n η is the main multiphoton parameter [45]. At that in the field of two waves in the interference resonant region the quantum parameter ( ) 0 j α ± (9) also becomes classical (see, (54)). Therefore hereinafter we consider resonant scattering of an electron by an electron in the intensity range (10), i.e. in the first order of the perturbation theory with respect to the laser field.
The relativistic system of units, where 1 c = =  , and the standard metric for four-vectors: ( ) 0 0 ab a b = − ab will be used throughout this paper.

Process General Amplitude
We limit to the Feynman direct diagram only when calculating the amplitude ( Figure 1) [19,41,45]. Such a problem statement is possible due to fact that the resonances for the direct diagrams of scattering type and the ones for the exchange diagrams within the intensity range (10) occur within the considerably different nonoverlapping kinematical regions [41,45]. Note also that for the direct scattering amplitude within the fields range (10) resonant scattering of an electron by an electron occurs when electrons scatter forwards into the small angles in the frame of the reference related to the center of inertia of initial particles [41,45] and is effectively divided into two processes of the first order similar to the Compton scattering of a wave by an electron (laser-induced Compton-effect). The general expression for the process scattering amplitude (for the direct diagram) has the form [18]: Here, ν γ and ν γ ′  ( , 0,1, 2,3 ν ν ′ = After computations analogous to the single pulsed wave case (see, [22,41,45]) the direct amplitude of scattering of an electron with the four-momentum 1 p by an electron with the four-momentum 2 p in the field of two pulsed moderately strong waves (5), (10) with the four-potential (1)-(3) can be represented as a sum over partial components with emission ( ( ) where ( ) j n l is the number of n -wave's photons emitted or absorbed by an j -particle.
Here, the arguments of delta-functions are defined by the components of the four-vector x y z z q q In the expression (21) q′ is the four-momentum of an intermediate virtual photon: The integration variables n φ have the form , n n n n (the expression for n ϕ is given by (2)). .
Through j κ and j κ ′ the following relations are designated where z e is the unit vector along the direction of external waves propagating z n n = e k k .

In expressions for
with the arguments ,  For the interference region (see, (46)) and wave circular polarization ( 2 1 n δ = ) the expression for the special function(42) simplifies considerably [5,57] emitted or absorbed by electrons.
One can see from (44) that in the interference kinematical region for circular polarization of both waves the scattering process formally looks like the field of a single wave, but with emission and absorption of photons of combination frequency ± Ω (45). Hereinafter we consider 1 2 ω ω > . Note also that if waves' frequency and polarization are close to each other, then it is easy to demonstrate that the obtained amplitude (17)-(28) is transformed into the amplitude for the single laser wave case [24].

Interference Resonant Amplitude
It is of interest to define and study the interference kinematical region, where parameters ( ) 0 j n γ are small due to the special geometry of the considered process in the field of two waves. This region is determined by the following condition: where j   g is the projection of the vector j  g onto the plane, perpendicular to z e . From the expression (47) we can conclude that it is possible for both field arbitrary intensity and polarization in the kinematical region if the vector j   g is parallel or antiparallel to the z -axis (i.e., is perpendicular to the polarization plane): These conditions obviously can be met only if electrons scatter in the plane formed by their initial momenta and the wave vector [24].
It was mentioned above that scattering of an electron by an electron in the laser-wave field can proceed resonantly. We consider the case when an intermediate photon falls within the mass shell. It means that the squared four-momentum of intermediate photon belongs to a very narrow region near zero. Consider the field pulsed character in the frame of quasimonochromatic wave (5) the condition of Oleinik resonance has the form: This region depends on initial momenta of both scattered particles and their scattering angles. Four-momentum conservation laws for the vertices of the resonant diagram ( Figure 2) can be represented in the form of two equalities: Here, the following condition was used (see, (22) and [24]): The equality (50)   It is easy to demonstrate that for waves' circular polarization under the resonance condition (49) within the interference region the Bessel function arguments ( )  (54) and also use the frame of reference related to the center of inertia of initial particles, where 1 2 0 + = p p . At that initial particle energy have the form Final particle momenta and energy are determined as where ′ P is the momentum of the inertia center after scattering. The scattering angle of electrons in the frame of reference related to the center of inertia can be determined as wave propagation and initial relative momentum. Note that for the process direct amplitude there two angles which correspond to the first and second wave. Thereby the resonant region of electron scattering angles in the frame of reference related to the center of inertia is the following: 1 2 E E E ′ ′ ′ = + are the particle total energy before and after scattering, respectively; The following designations are input in the function (66): The function "erf" in (67), (68) is the error function. The parameters 1 β (67) and 2 β (68) in (66) specify how the squared four-momentum of intermediate photon is close to the mass shell. These parameters are defined by both the process kinematics and characteristics of pulsed waves: The delta-functions presence in the resonant amplitude (64) indicates fulfillment of the following conservation laws: where ( ) , , is the momentum of the inertia center after scattering.

Resonant Cross Section
We obtain the differential cross-section of resonant scattering of nonpolarized electron by an electron in the field of two pulsed laser wave into the element of solid angle within the interference region by a standard mode [18] ( ) 4 , ,  Resonant Interference Effect in Scattering of an Electron by an Electron in the Field of Two Pulsed Laser Waves scattering into the angles close to the resonant ones. The distance between the peaks is determined by the difference of frequency of the external laser waves. If waves' frequency differs too little, then peaks are blended and we get the single wave case [24]. Moreover, the difference of peak altitude also is determined by the ratio of frequency of the field of two pulsed laser waves.
We may integrate differential cross-section (73) within the narrow range of scattering angles near the resonance (61).
Integration over the scattering angle θ is substituted by integration over parameters n β (71). For the case of the opposite directions of rotation of the field strength ( 1 2 1 δ δ = − = ± ) we obtain the expression for the differential cross section of resonant scattering of an electron by an electron in the field of two pulsed laser waves within the interference region in the form: The quantities n Γ (85) are the resonance transit widths [49]. It is easy to notice that 2 6 1 1 1 10 , and to estimate the ratio of the widths 1 Γ and 2 Γ : Broadening of the resonant peak in particle scattering in the pulsed-wave field is specified by both the wave finite duration and observation time. It was noticed in works by Ritus and Nikishov [2] that observation of a process during N wave periods results in fulfillment of the energy conservation law with an accuracy 1 N − [14]. Consequently, broadening of resonance peaks at the expense of wave finite duration in scattering of an electron by an electron in the field of two pulsed laser waves appears for an arbitrary values of the parameter ρ .
For the case of coincident directions of rotation of the field strength ( 1 2 1 δ δ = + = ± ) we obtain the expression for the differential cross section of resonant scattering of an electron by an electron in the field of two pulsed laser waves within the interference region in the form: Transit widths n Γ in (91) are determined by (85). The significant issue is the influence of finite duration of laser wave on resonant behavior of scattered electrons. The assertion that the pulsewidth must exceed the time required for realization of the Compton scattering of an external field photons by each of the electrons is to be taken into consideration. If this condition is not satisfied than particles do not have time to interact with the wave under the resonant conditions [6][7]. Consequently the following correlation is valid for the pulse duration 1 , is the radiation width, which is determined by the total probability (per time unit) of the Compton scattering of an external field photon on each of electrons with the relative momentum p . Inconsequence of the expression (94) we derive that the value of the parameter n n ω τ must satisfy the condition Due to correlation (96) one can define that for the frequency optical range and the external pulsed wave intensity range (10) the pulsewidth meets the condition (the cases of particle ultrarelativistic energy when their movement within the narrow cone with an external field photon are excluded from consideration) 12 10 .
Using expressions (73)-(78) one can estimate the ratio of the resonant cross section of scattering of an electron by an electron in the field of two pulsed laser waves (within the interference region) to the cross section in the absence of the external field. Thereby, for the nonrelativistic case, the waves' optical frequency and the number of oscillation of the field amplitude at the characteristic pulse duration

Conclusion
Study of resonant scattering of an electron by an electron in the field of two pulsed laser waves within the interference region results in following conclusion: 1. Resonant scattering of an electron by an electron in the field of two pulsed laser waves within the interference region appears under process certain geometry. Thus, for electron nonrelativistic energy it corresponds to the case when the particle ingoing angle is close to in the frame of the reference related to the center of inertia of initial particles. 2. The form of resonant cross section of the studied process considerably depends on the direction of rotation of vectors of the field strength of the waves. Thus, for opposite directions of rotation of the strength ( ) the cross section consists of two summands corresponding to emission and absorption of photons of the same wave. If directions of rotation of vectors of the field strength of laser waves coincide ( ), then processes with absorption of a photon from one of the waves and emission of a photon of another wave are allowed. 3.The resonant differential cross section of scattering of nonrelativistic electrons by each other in the field of two pulsed laser waves of the optical frequency (10) within the interference region is approximately ten times greater in the magnitude than the corresponding cross section in the external field absence.