Quantum Entanglement of the Final Particles in the Resonant Trident Pair Production Process in a Strong Electromagnetic Wave

Abstract

The resonant trident pair production process in the collision of ultrarelativistic electrons with a strong electromagnetic wave was theoretically studied. Under resonant conditions, the intermediate virtual gamma-quantum became real. As a result, the original resonant trident pair production process effectively split into two first-order processes by the fine structure constant: the electromagnetic field-stimulated Compton effect and the electromagnetic field-stimulated Breit–Wheeler process. The kinematics of the resonant trident pair production process were studied in detail. It was shown that there are two different cases for the energies and outgoing angles of the final particles (an electron and an electron–positron pair) in which their quantum entanglement is realized. In the first case, energies and outgoing angles of the final ultrarelativistic particles are uniquely determined by the parameters of the electromagnetic field-stimulated Compton effect (the outgoing angle of the final electron and the quantum parameter of the Compton effect). In the second case, energies and outgoing angles of the final particles are uniquely determined by the electromagnetic field-stimulated Breit–Wheeler process (the electron–positron pair outgoing angle and the Breit–Wheeler quantum parameter). It was shown that in a sufficiently wide range of frequencies and intensities of a strong electromagnetic wave, and in the case of ultrarelativistic initial electrons, the differential probability of the resonant trident pair production process with simultaneous registration of the outgoing angles of the final particles can significantly (by several orders of magnitude) exceed the total probability of the electromagnetic field-stimulated Compton effect.

1. Introduction

With the advent of high-intensity lasers, as well as the source of high-energy particles and high-energy gamma-quanta [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22], the processes of quantum electrodynamics (QED) in external electromagnetic fields have become widely studied in recent decades (see, for example, reviews [23,24,25,26,27,28,29,30,31,32,33,34,35], monographs [36,37,38], and articles [39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117]). At the same time, much attention has been paid to the first-order QED processes (by the fine structure constant) that occur only in an external electromagnetic field. Such processes include the electromagnetic field-stimulated Compton effect (EFSCE) [39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54], as well as the electromagnetic field-stimulated Breit–Wheeler (EFSBW) process [55,56,57,58,59,60,61,62,63,64,65,66]. It is important to note that higher-order processes in an external electromagnetic field can proceed in both resonant and non-resonant ways. The resonant flow of higher-order QED processes is associated with intermediate virtual particles reaching a mass surface. Such resonances were first considered by Oleynik [67,68]. As a result, higher-order processes effectively break up into several consecutive lower-order processes (see the reviews [26,31], monographs [36,37], and recent studies [69,72,73,74,75]). At the same time, the probability of resonant processes can significantly exceed the corresponding probabilities of non-resonant processes.

It is important to note that when a relativistic electron beam collides with an electromagnetic wave, along with the EFSCE, a second-order process takes place, namely, electron scattering with the simultaneous generation of an electron–positron pair. This is the so-called trident pair production process. This process also attracts attention (see, for example, articles [69,70,71,76,77,78,79,80,81,82]). It should be emphasized that the trident pair production process can proceed in a resonant way, when the intermediate gamma-quantum becomes real. In this case, the original resonant trident pair production (RTPP) process effectively splits into two first-order processes: the EFSCE and EFSBW processes. Note that in the case of a weak plane electromagnetic wave, the RTPP process was considered in article [69]. It is worth mentioning that electron–positron pair creation in muon-laser-field collisions was considered in [71].In this paper, we consider the RTPP process in the collision of ultrarelativistic electrons with a strong monochromatic electromagnetic wave. We studied the conditions under which the probability of the RTPP process can significantly exceed the corresponding probability of the EFSCE.

In the considered problem, the main parameter is the classical relativistic-invariant parameter

$$\eta ={\displaystyle \frac{eFƛ}{m{c}^2}}\underset{\sim }{>}1,$$

(1)

which is numerically equal to the ratio of the field work at the wavelength to the rest energy of the electron (e and m are the charge and mass of the electron, F and $ƛ=c/\phantom{c\omega }\omega$ are the electric field strength and wavelength, and $\omega$ is the wave frequency). In addition, in this problem, the characteristic quantum parameters of the Compton effect $\left({\epsilon }_{iC}\right)$ and the Breit–Wheeler process $\left({\epsilon }_{iBW}\right)$ arise, which are equal to the ratio of the energy of the initial electron to the characteristic energies of the process:

$${\epsilon }_{iC}={\displaystyle \frac{E_i}{\hslash {\omega }_C}},{\epsilon }_{iBW}={\displaystyle \frac{E_i}{\hslash {\omega }_{BW}}}.$$

(2)

Here, $E_i$ is the energy of the initial electrons and $\hslash {\omega }_C$ and $\hslash {\omega }_{BW}$ are the characteristic quantum energies of the Compton effect and the Breit–Wheeler process [72,73,74,75]:

$$\hslash {\omega }_C={\displaystyle \frac{{\left(m_{*}{c}^2\right)}^2}{4\left(\hslash \omega \right){sin}^2\left({\theta }_i/\phantom{{\theta }_{i}2}2\right)}}={\displaystyle \frac{{\left(m{c}^2\right)}^2\left(1+{\eta }^2\right)}{4\left(\hslash \omega \right){sin}^2\left({\theta }_i/\phantom{{\theta }_{i}2}2\right)}},{\omega }_{BW}=4{\omega }_C.$$

(3)

Here, $m_{*}$ is the effective mass of an electron in the field of a circularly polarized wave (18) and ${\theta }_i$ is the angle between the momentum of the initial electron and the direction of propagation of the wave (24). Note that the characteristic energies (3) are inversely proportional to the frequency $\left(\omega \right)$, directly proportional to the intensity of the external electromagnetic wave $\left(I\sim {\eta }^2\right)$, and dependent on the angle between the wave and the initial particles’ momenta.

In this paper, we will show that the resonant energies of the final particles, as well as the resonant differential probabilities, significantly depend on the values of these quantum parameters (2), i.e., on the ratio of the initial electron energy to the corresponding characteristic energy.

Later in the article, the relativistic system of units is used: $\hslash =c=1$.

2. Process Amplitude

Let us choose the 4-potential of the external electromagnetic field in the form of a plane circularly polarized monochromatic wave propagating along the z axis:

$$A\left(\phi \right)={\displaystyle \frac{F}{\omega }}·\left(e_{x}cos\phi +\delta e_{y}sin\phi \right),\phi =\left(kx\right)=\omega (t-z).$$

(4)

Here, $\delta =\pm 1$, $e_x,e_y$—are the polarization 4-vectors of the external field, which have the following properties: $e_x=(0,{\mathit{e}}_x),e_y=(0,{\mathit{e}}_y),e_x{e}_y=0,{\left(e_x\right)}^2={\left(e_y\right)}^2=-1.$ The trident pair production process stimulated by an external field is described by two Feynman diagrams (Figure 1).

The amplitude of the process under consideration can be represented in the following form:

$$S=-i{e}^2\int d^4{x}_1{d}^4{x}_2{\overline{\psi }}_{p_f}\left(x_1\left|A\right.\right){\gamma }^{\mu }{\psi }_{p_i}\left(x_1\left|A\right.\right)D_{\mu \nu }(x_2-x_1){\overline{\psi }}_{p_{-}}\left(x_2\left|A\right.\right){\gamma }^{\nu }{\psi }_{-p_{+}}\left(x_2\left|A\right.\right)-(p_f\leftrightarrow p_{-}),$$

(5)

where $p_j=(E_j,{\mathit{p}}_j)$—are the 4-momenta of the final and initial electrons and positrons ($j=i,f,\pm$). Here, ${\psi }_{p_i}\left(x_1\left|A\right.\right)$, ${\psi }_{-p_{+}}\left(x_2\left|A\right.\right)$, ${\overline{\psi }}_{p_{-}}\left(x_2\left|A\right.\right)$, ${\overline{\psi }}_{p_f}\left(x_1\left|A\right.\right)$ are the Volkov functions in the plane electromagnetic wave field [118,119] and $D_{\mu \nu }(x_2-x_1)$ is the free photon propagator [36,37].

After performing the appropriate integrations, the amplitude (5) can be represented as

$$S=\sum _{l=-\infty }^{+\infty }{S}_l,$$

(6)

where the partial amplitude $S_l$ corresponds to the absorption or emission of $\left|l\right|$ photons of an external wave. For channel A, the partial amplitude can be represented as follows:

$$\begin{array}{c}\hfill S_l={\displaystyle \frac{16{\pi }^5{e}^2{e}^{-i{d}_0}}{\sqrt{{\tilde{E}}_i{\tilde{E}}_f{\tilde{E}}_{-}{\tilde{E}}_{+}}}}\sum _{l_{2=-\infty }}^{\infty }\left[{\overline{u}}_{p_f}{F}_{-\left(l-l_2\right)}^{\mu }\left({\tilde{p}}_f,{\tilde{p}}_i\right)u_{p_i}\right]{\displaystyle \frac{1}{q^2}}\left[{\overline{u}}_{p_{-}}{F}_{-l_2,\mu }\left({\tilde{p}}_{-},-{\tilde{p}}_{+}\right){\nu }_{p_{+}}\right]\times \\ \hfill \times {\delta }^{\left(4\right)}\left({\tilde{p}}_{+}+{\tilde{p}}_{-}-{\tilde{p}}_i+{\tilde{p}}_f-lk\right),\left(l=l_1+l_2\right),\end{array}$$

(7)

where q is the 4-momentum of the intermediate gamma-quantum, which takes the form

$$q={\tilde{p}}_{+}+{\tilde{p}}_{-}-l_{2}k.$$

(8)

In the expression (7), $u_{p_i}$ is the Dirac bispinor of the initial electron, ${\overline{u}}_{p_f},{\overline{u}}_{p_{-}},{\nu }_{p_{+}}$—are the Dirac bispinors of the final electron and electron–positron pair, and $d_0$ is the phase independent of the summation indices; the function $F_{-(l-l_2)}^{\mu }\left({\tilde{p}}_f,{\tilde{p}}_i\right)$ has the form

$$\begin{array}{c}\hfill F_{-\left(l-l_2\right)}^{\mu }\left({\tilde{p}}_f,{\tilde{p}}_i\right)=a^{\mu }\left({\tilde{p}}_f,{\tilde{p}}_i\right)L_{-\left(l-l_2\right)}+\\ \hfill +b_{-}^{\mu }\left({\tilde{p}}_f,{\tilde{p}}_i\right)L_{-\left(l-l_2\right)-1}+b_{+}^{\mu }\left({\tilde{p}}_f,{\tilde{p}}_i\right)L_{-\left(l-l_2\right)+1}.\end{array}$$

(9)

Here, the matrices $a^{\mu }$ and $b_{\pm }^{\mu }$ have the form

$$\begin{array}{c}{a}^{\mu }\left({\tilde{p}}_f,{\tilde{p}}_i\right)={\gamma }^{\mu }+{\displaystyle \frac{{\eta }^2{m}^2}{2\left(k{\tilde{p}}_f\right)\left(k{\tilde{p}}_i\right)}}\widehat{k}{k}^{\mu },\end{array}$$

(10)

$$b_{\pm }^{\mu }\left({\tilde{p}}_f,{\tilde{p}}_i\right)={\displaystyle \frac{\eta m}{4}}\left[{\displaystyle \frac{1}{\left(k{\tilde{p}}_f\right)}}{\widehat{\epsilon }}_{\pm }\widehat{k}{\gamma }^{\mu }+{\displaystyle \frac{1}{\left(k{\tilde{p}}_i\right)}}{\gamma }^{\mu }\widehat{k}{\widehat{\epsilon }}_{\pm }\right],$$

(11)

$${\epsilon }_{\pm }=e_x\pm i{e}_y.$$

(12)

In expressions (10) and (11), values with caps represent scalar products of respective 4-vectors with Dirac’s gamma matrices ($\widehat{k}={\gamma }^{\mu }{k}_{\mu }={\gamma }^0{k}^0-\gamma \mathit{k}$). In expressions (9), special functions $L_l$ [27] in the case of circular wave polarization can be represented using Bessel functions with integer indices:

$$L_{-\left(l-l_2\right)}=exp\left(i\left(l-l_2\right){\chi }_{{\tilde{p}}_f{p}_i}\right)J_{-\left(l-l_2\right)}\left({\gamma }_{{\tilde{p}}_f{p}_i}\right),$$

(13)

where the following is indicated:

$${\gamma }_{{\tilde{p}}_f{\tilde{p}}_i}=\eta m\sqrt{-Q_{{\tilde{p}}_f{\tilde{p}}_i}^2},$$

(14)

$$Q_{{\tilde{p}}_f{\tilde{p}}_i}={\displaystyle \frac{{\tilde{p}}_f}{\left(k{\tilde{p}}_f\right)}}-{\displaystyle \frac{{\tilde{p}}_i}{\left(k{\tilde{p}}_i\right)}},$$

(15)

$$tan{\chi }_{{\tilde{p}}_f{\tilde{p}}_i}=\delta {\displaystyle \frac{\left(Q_{{\tilde{p}}_f{\tilde{p}}_i}{e}_y\right)}{\left(Q_{{\tilde{p}}_f{\tilde{p}}_i}{e}_x\right)}}.$$

(16)

In the above expressions, ${\tilde{p}}_j=\left({\tilde{E}}_j,{\tilde{\mathit{p}}}_j\right)$—are the quasi-momenta of electrons and positrons and $m_{*}$- is the effective mass of the electron in the wave field (4):

$${\tilde{p}}_j=p_j+{\eta }^2{\displaystyle \frac{m^2}{2\left(k{p}_j\right)}}k,i=f,\pm .$$

(17)

$${\tilde{p}}_j^2=m_{*}^2,m_{*}=m\sqrt{1+{\eta }^2}.$$

(18)

Note that the function $F_{-l_2,\mu }\left({\tilde{p}}_{-},-{\tilde{p}}_{+}\right)$ in expression (7) is obtained from the corresponding expressions (9)–(16) by replacing ${\tilde{p}}_i\to -{\tilde{p}}_{+},{\tilde{p}}_f\to {\tilde{p}}_{-},\left(l-l_2\right)\to l_2$ and lowering the index $\mu$.

3. Resonant Kinematics

Under resonant conditions, the intermediate gamma-quantum becomes real (Oleinik resonances [67,68,69,72,73,74,75]). As a result, the initial second-order process in the wave field effectively decays into two first-order processes: the EFSCE (at the first vertex) [24,119] and the EFSBW process (at the second vertex) [24]. Diagrams of processes under resonant conditions are shown in Figure 2.

$$q^2=0.$$

(19)

Further consideration will be given to the resonant channel A. Under the conditions of (19), we write the 4-momentum conservation laws in the first and second vertices of the Feynman diagram (see Figure 2A):

$${\tilde{p}}_i+l_{1}k=q+{\tilde{p}}_f,$$

(20)

$$q+l_{2}k={\tilde{p}}_{+}+{\tilde{p}}_{-}.$$

(21)

Given that ${\tilde{p}}_{i,f}^2={\tilde{p}}_{\pm }^2=m_{*}^2$ and $q^2=k^2=0$, the expressions (20) and (21) are valid for $l_1\ge 1$ and $l_2\ge 1$.

Later in this article, we will discuss the case of ultrarelativistic energies of the initial electrons, as well as the final particles (the final electron and the electron–positron pair).

$$E_j>>m,j=i,f,\pm .$$

(22)

Due to condition (22), the final particles will mainly fly out in a narrow cone along the momentum of the initial electron. In this case, we assume that this narrow cone of particles lies far from the direction of propagation of the external electromagnetic wave (otherwise the resonances disappear [72,73,74,75], see Figure 3). Thus, the angles between the particle momenta must satisfy the following conditions:

$$\begin{array}{cc}{\theta }_f\equiv \angle \left({\mathit{p}}_i,{\mathit{p}}_f\right)\ll 1,& {\theta }_{\pm }\equiv \angle \left({\mathit{p}}_{+},{\mathit{p}}_{-}\right)\ll 1,\end{array}$$

(23)

$$\begin{array}{cc}{\theta }_i\equiv \angle \left({\mathit{p}}_i,\mathit{k}\right)\sim 1,& \angle \left({\mathit{p}}_j,\mathit{k}\right)\approx {\theta }_i\end{array},j=f,+,-.$$

(24)

We also assume that the classical parameter $\eta$ and the quantum parameter of the Compton effect are bound from above by the following conditions [72,73,74,75]:

$$\eta <<{\displaystyle \frac{E_j}{m}},j=i,f,\pm ;{\epsilon }_{iC}<<{\displaystyle \frac{E_i}{m}}.$$

(25)

Because of this, further consideration of the resonant process will be valid for sufficiently large wave intensities. However, the intensity of these fields should be less than the critical Schwinger field $F_{*}\approx 1.3·{10}^{16}\mathrm{V}/\phantom{\mathrm{V}\mathrm{cm}}\mathrm{cm}$.

The resonant energy of the final electron is determined by the Compton effect stimulated by an external field. Given the law of conservation of 4-momentum at the first vertex (20), the resonant condition (19), and the kinematic conditions (22)–(24), after simple manipulations, we obtain the following expression for the resonant energy of the final electron at the first vertex:

$$x_f={\displaystyle \frac{2+l_1{\epsilon }_{iC}\pm \sqrt{{\left(l_1{\epsilon }_{iC}\right)}^2-4{\delta }_f^2}}{2\left(1+{\delta }_f^2+l_1{\epsilon }_{iC}\right)}}.$$

(26)

Here, the following is indicated:

$$x_f={\displaystyle \frac{E_f}{E_i}},{\delta }_f^2={\displaystyle \frac{E_i^2{\theta }_f^2}{m_{*}^2}}.$$

(27)

In expression (26), the quantum parameter ${\epsilon }_{iC}$ is defined by the expressions (2) and (3). From expression (26), it follows that the energy of the final electron is determined by the number of absorbed photons of the wave $\left(l_1\right)$, the quantum parameter ${\epsilon }_{iC}$, and the outgoing angle of the final electron relative to the momentum of the initial electron (an ultrarelativistic parameter ${\delta }_f^2$). Expression (26) has two branches (upper $x_{f\left(u\right)}$ and lower $x_{f\left(d\right)}$) in accordance with the signs $\left(\pm \right)$ before the square root in expression (26). For fixed values $l_1$ and ${\epsilon }_{iC}$, upper and lower branches depend only on the outgoing angles of the electron $\left({\delta }_f^2\right)$. It is important to emphasize that the interval of variation of the ultrarelativistic parameter ${\delta }_f^2$ will be different for the upper and lower branches of the electron energy values. So, from expression (26), it follows that the ultrarelativistic parameter for the lower branch $\left({\delta }_{f\left(d\right)}^2\right)$ has the following interval of change:

$$0\le {\delta }_{f\left(d\right)}^2\le {\left(l_1{\epsilon }_{iC}\right)}^2/\phantom{{\left(l_1{\epsilon }_{iC}\right)}^{2}4}4.$$

(28)

In this case, the minimum and maximum value of the resonant energy of an electron on the lower branch is obtained from expression (26) when an electron is scattered at zero and maximum angles:

$$x_{f\left(d\right)}^{min}={\displaystyle \frac{1}{l_1{\epsilon }_{iC}+1}},x_{f\left(d\right)}^{max}={\displaystyle \frac{2}{l_1{\epsilon }_{iC}+2}}.$$

(29)

At the same time, for the upper branch of the resonant energy of the electron $x_{f\left(u\right)}$ (26), the parameter change interval ${\delta }_f^2$ should be different. Indeed, in this case, it is impossible to take the minimum value of the outgoing angle ${\delta }_{f\left(u\right)}^2=0$, since we obtain $x_{f\left(u\right)}=1$, something that does not make physical sense. Because of this, the desired range of change in the ultrarelativistic parameter ${\delta }_{f\left(u\right)}^2$ is obtained from matching solutions for the resonant energies of the electron (at the first vertex) and the electron–positron pair (at the second vertex) (see (51)).

The resonant energy of the electron–positron pair is determined by the Breit–Wheeler process stimulated by an external field at the second vertex. Given expressions (19) and (21)–(24), after simple manipulations, we obtain the expression for the energies of the electron $\left(x_{-}\right)$ and positron $\left(x_{+}\right)$ of a pair in units of the energy of the initial electron:

$$\left(x_{+}+x_{-}\right)\left(x_{+}+x_{-}-l_2{\epsilon }_{iBW}{x}_{+}{x}_{-}\right)+4{\delta }_{\pm }^2{x}_{+}^2{x}_{-}^2=0.$$

(30)

Here, the quantum parameter ${\epsilon }_{iBW}$ is defined by expressions (2) and (3), and the following notation is used:

$$\begin{array}{cc}{x}_{\pm }={\displaystyle \frac{E_{\pm }}{E_i}},& {\delta }_{\pm }^2={\displaystyle \frac{E_i^2{\theta }_{\pm }^2}{m_{*}^2}}.\end{array}$$

(31)

The above expression (30) implies that it is symmetric with respect to replacement $x_{+}\leftrightarrow x_{-}$. Therefore, the resonant energies of the electron and positron are equal, and, in Equation (30), it can be put that $x_{+}=x_{-}$. Given this, after simple transformations, we obtain the resonant energy of the positron (electron) of the pair:

$$x_{\pm }={\displaystyle \frac{l_2{\epsilon }_{iBW}}{{\delta }_{\pm }^2}}\left(1\pm \sqrt{1-{\displaystyle \frac{{\delta }_{\pm }^2}{l_2^2{\epsilon }_{iBW}^2}}}\right).$$

(32)

From expression (32), it follows that the energy of the positron (electron) pair is determined by the number of absorbed photons of the wave $\left(l_2\right)$, the quantum parameter ${\epsilon }_{iBW}$, and the angle between the momenta of the electron and positron of the pair (an ultrarelativistic parameter ${\delta }_{\pm }^2$). Expression (32) has two branches (upper $x_{\pm \left(u\right)}$ and lower $x_{\pm \left(d\right)}$) in accordance with the sign $\left(\pm \right)$ before the square root in the ratio (32). For fixed values $l_2$ and ${\epsilon }_{iBW}$, both branches depend only on their outgoing angle $\left({\delta }_{\pm }^2\right)$. It is important to emphasize that the ultrarelativistic parameter ${\delta }_{\pm }^2$ has a different variation interval for the upper and lower branches of the resonant energy values of the electron–positron pair. So, from expression (32), it follows that the interval of variation of the ultrarelativistic parameter for the lower branch of the resonant energy of the pair ${\delta }_{\pm \left(d\right)}^2$ has the form

$$0\le {\delta }_{\pm \left(d\right)}^2\le {\left(l_2{\epsilon }_{iBW}\right)}^2.$$

(33)

In this case, the minimum and maximum value of the resonant energy of the electron–positron pair is obtained from the ratio (32) for zero and maximum outgoing angle of the electron–positron pair:

$$2x_{\pm \left(d\right)}^{min}={\displaystyle \frac{1}{l_2{\epsilon }_{iBW}}},2x_{\pm \left(d\right)}^{max}={\displaystyle \frac{2}{l_2{\epsilon }_{iBW}}}.$$

(34)

At the same time, there is a lower limit on the number of absorbed photons of the wave at the second vertex:

$$l_2\ge l_{2\left(d\right)}^{min},l_{2\left(d\right)}^{min}=⌈2{\epsilon }_{iBW}^{-1}⌉.$$

(35)

Note that the condition of (35) is necessary but not sufficient (see the condition for (43)). For the upper branch solutions $x_{\pm \left(u\right)}$(32), you cannot take the minimum value of the outgoing angle ${\delta }_{\pm \left(u\right)}^2\to 0$ since, in this case, we obtain $x_{\pm \left(u\right)}\sim {\delta }_{\pm \left(u\right)}^{-2}\to \infty$, which does not make physical sense. The range of change in the ultrarelativistic parameter ${\delta }_{\pm \left(u\right)}^2$ is obtained from matching solutions for the resonant energies of the electron (at the first vertex) and the electron–positron pair (at the second vertex) (see (39)).

Let us write down the law of conservation of energy in general for the process under consideration:

$$x_f+2x_{\pm }\approx 1.$$

(36)

Note that in expression (36), we ignored small corrections ($\left|l\right|\omega /\phantom{\left|l\right|\omega E_i}{E}_i<<1$ and ${\eta }^2{m}^2/\phantom{m^2{E}_j^2}{E}_j^2<<1$, see (25)). The resonant energies of the electron (26) and the electron–positron pair (32) each have two possible solutions. In this case, four possible cases formally arise with the energies of the final particles: each branch of the pair energy corresponds to two branches of the final electron energy. However, due to the general law of the conservation of energy, only two cases that correspond to the upper (lower) energy branch of the electron–positron pair and the lower (upper) energy branch of the final electron make physical sense for the resonant process under study. It should also be noted that unlike the lower branches of the final particle energies, the upper branches give non-physical values and require additional investigation.

First, we find physical solutions for the upper energy branch of the electron–positron pair using the lower energy branch of the electron and the general energy conservation law (36), which, in this case, takes the form

$$2x_{\pm \left(u\right)}=1-x_{f\left(d\right)}.$$

(37)

Substituting in this expression the upper branch for the electron–positron pair (32) and the lower branch for the electron energy (26), we obtain

$$\sqrt{1-{\displaystyle \frac{{\delta }_{\pm \left(u\right)}^2}{{\left(l_2{\epsilon }_{iBW}\right)}^2}}}={\displaystyle \frac{{\delta }_{\pm \left(u\right)}^2}{2l_2{\epsilon }_{iBW}}}\left(1-x_{f\left(d\right)}\right)-1>0.$$

(38)

Using simple manipulations, we obtain a condition for matching the outgoing angles of the electron–positron pair and the final electron:

$${\delta }_{\pm \left(u\right)}^2={\displaystyle \frac{4}{\left(1-x_{f\left(d\right)}\right)}}\left[l_2{\epsilon }_{iBW}-{\displaystyle \frac{1}{\left(1-x_{f\left(d\right)}\right)}}\right]>0.$$

(39)

Expression (39) allows us to determine the interval of variation of the ultrarelativistic parameter ${\delta }_{\pm \left(u\right)}^2$ for the upper energy branch of the electron–positron pair:

$${\delta }_{\pm \left(min\right)}^2\le {\delta }_{\pm \left(u\right)}^2\le {\delta }_{\pm \left(max\right)}^2$$

(40)

where the following is indicated:

$${\delta }_{\pm \left(min\right)}^2=4\left(1+{\displaystyle \frac{1}{l_1{\epsilon }_{iC}}}\right)\left[l_2{\epsilon }_{iBW}-\left(1+{\displaystyle \frac{1}{l_1{\epsilon }_{iC}}}\right)\right]>0,$$

(41)

$${\delta }_{\pm \left(max\right)}^2=4\left(1+{\displaystyle \frac{2}{l_1{\epsilon }_{iC}}}\right)\left[l_2{\epsilon }_{iBW}-\left(1+{\displaystyle \frac{2}{l_1{\epsilon }_{iC}}}\right)\right]>0.$$

(42)

Note that the right-hand sides of expressions (38), (41), and (42) must be greater than zero. This implies different conditions for the number of absorbed photons of the wave at the second vertex. The condition under which all three equations are valid is obtained from the condition of the right-hand side of expression (38) for ${\delta }_{\pm \left(u\right)}^2={\delta }_{\pm \left(max\right)}^2,x_{f\left(d\right)}=x_{f\left(d\right)}^{max}$. As a result, we obtain the following lower restrictions on the number of absorbed photons of the wave at the second vertex for the upper energy branch of the electron–positron pair:

$$l_2\ge l_{2\left(min\right)},l_{2\left(min\right)}=⌈{\displaystyle \frac{2}{{\epsilon }_{iBW}}}\left(1+{\displaystyle \frac{2}{l_1{\epsilon }_{iC}}}\right)⌉,l_1\ge 1.$$

(43)

Note that substituting expression (39) into the expression for the upper branch of the resonant energy of the electron–positron pair (32) results in expression (37), as it should be. In this case, the maximum and minimum energies of the electron–positron pair take the form

$$x_{\pm \left(u\right)}^{max}\approx {\displaystyle \frac{1}{2}}\left[1-x_{f\left(d\right)}^{min}\right]={\displaystyle \frac{1}{2}}{\left(1+{\displaystyle \frac{1}{l_1{\epsilon }_{iC}}}\right)}^{-1},$$

(44)

$$x_{\pm \left(u\right)}^{min}\approx {\displaystyle \frac{1}{2}}\left[1-x_{f\left(d\right)}^{max}\right]={\displaystyle \frac{1}{2}}{\left(1+{\displaystyle \frac{2}{l_1{\epsilon }_{iC}}}\right)}^{-1}.$$

(45)

It is important to emphasize that the condition for the number of absorbed photons of the wave at the second vertex (43) ensures the validity of the energies of the final particles in the entire range of outgoing angles of the final electron (28) and the electron–positron pair (40). From the condition of (43), we can determine the range of the quantum parameter ${\epsilon }_{iBW}$ values with the minimum number of photons in the second vertex equal to one. After simple manipulations, we obtain

$${\epsilon }_{iBW}\ge {\epsilon }_{*},{\epsilon }_{*}=1+\sqrt{1+{\displaystyle \frac{1}{l_1}}},\left(l_1\ge 1,l_{2min}=1\right).$$

(46)

Note also that under conditions when the final particles fly out at the minimum angles ${\delta }_{f\left(d\right)}^2=0$, ${\delta }_{\pm \left(u\right)}^2={\delta }_{\pm \left(min\right)}^2$, the number of absorbed photons of the wave at the second vertex will be determined by a more “soft” condition, which follows from expression (41):

$$l_2\ge l_{2min\left(0\right)}=⌈{\displaystyle \frac{1}{{\epsilon }_{iBW}}}\left[1+{\displaystyle \frac{1}{l_1{\epsilon }_{iC}}}\right]⌉.$$

(47)

From this, we obtain a condition for the possible values of the quantum parameter ${\epsilon }_{iBW}$, under which the minimum number of photons in the second vertex is equal to one:

$${\epsilon }_{iBW}\ge {\displaystyle \frac{1}{2}}{\epsilon }_{*},\left(l_1\ge 1,l_{2min\left(0\right)}=1\right).$$

(48)

Figure 4a shows the dependences of the energies of the final particles (an electron on the lower branch and the electron–positron pair on the upper branch) on the square of the outgoing angle of the final electron for different numbers of absorbed photons of the wave at the first vertex. This figure shows that as the number of absorbed photons increases, the electron energy decreases, and the energy of the electron–positron pair increases and tends to be the energy of the initial electron. At the same time, the energies of the final particles for a different number of photons of the wave at the first vertex are in different energy regions (do not intersect). Figure 4b shows the dependences of the square of the outgoing angle of the electron–positron pair (on the upper energy branch) on the square (28) for fixed values of the number of absorbed photons of the wave at the first and second vertices. Figure 4b shows that for different numbers of absorbed photons of the wave in the first and second vertices, each outgoing angle of the final electron uniquely corresponds to a certain outgoing angle of the electron–positron pair. Expressions (37) and (39)–(45) (see also Figure 4) completely solve the problem of resonant energies of the final particles (on the lower branch of the energies of the final electron and the upper branch of the energies of the electron–positron pair). In this case, the outgoing angle of the final electron $\left({\delta }_{f\left(d\right)}^2\right)$ in the range of angles (28) and the values of the quantum parameter of the Compton effect $\left({\epsilon }_{iC}\right)$ uniquely determine the outgoing angle of the electron–positron pair $\left({\delta }_{\pm \left(u\right)}^2\right)$ (see expressions (39)–(43)), as well as the final particle energies (see expressions (26) and (37)). Because of this, there is a quantum entanglement of the final particles. Thus, the Compton effect, stimulated by an external field, which occurs at the first vertex, completely determines the resonant states of the final particles in this case.

Now we turn to the case of the upper energy branch of the final electron and the lower energy branch of the electron–positron pair. In this case, energy conservation law (36) takes the form

$$x_{f\left(u\right)}=1-2x_{\pm \left(d\right)}.$$

(49)

Substituting into this equation the corresponding energies of the final particles for (26) and (32), we obtain

$$\begin{array}{c}\hfill \sqrt{{\left(l_1{\epsilon }_{iC}\right)}^2-4{\delta }_{f\left(u\right)}^2}=2\left[1-2x_{\pm \left(d\right)}\right]\times \\ \hfill \times \left(1+l_1{\epsilon }_{iC}+{\delta }_{f\left(u\right)}^2\right)-\left(2+l_1{\epsilon }_{iC}\right)>0.\end{array}$$

(50)

After simple transformations from this equation, we obtain the desired dependence of the outgoing angle of the final electron on the outgoing angle of the electron–positron pair:

$${\delta }_{f\left(u\right)}^2={\left[{\displaystyle \frac{1}{2x_{\pm \left(d\right)}}}-1\right]}^{-1}\left\{l_1{\epsilon }_{iC}-{\left[{\displaystyle \frac{1}{2x_{\pm \left(d\right)}}}-1\right]}^{-1}\right\}.$$

(51)

The expression (51) allows us to determine the range of ${\delta }_{f\left(u\right)}^2$ values for the upper energy branch of the final electron:

$${\delta }_{f\left(min\right)}^2\le {\delta }_{f\left(u\right)}^2\le {\delta }_{f\left(max\right)}^2,$$

(52)

where the following is indicated:

$${\delta }_{f\left(min\right)}^2=\left({\displaystyle \frac{1}{l_2{\epsilon }_{iBW}-1}}\right)\left[l_1{\epsilon }_{iC}-\left({\displaystyle \frac{1}{l_2{\epsilon }_{iBW}-1}}\right)\right]>0,$$

(53)

$${\delta }_{f\left(max\right)}^2=\left({\displaystyle \frac{2}{l_2{\epsilon }_{iBW}-2}}\right)\left[l_1{\epsilon }_{iC}-\left({\displaystyle \frac{2}{l_2{\epsilon }_{iBW}-2}}\right)\right]>0.$$

(54)

Note that the right-hand sides of expressions (50), (53), and (54) must be greater than zero. This implies different conditions for the number of absorbed photons of the wave at the second vertex. The condition under which all three equations are valid is obtained from the condition that the right-hand side of expression (50) at ${\delta }_{f\left(u\right)}^2={\delta }_{f\left(max\right)}^2, x_{\pm \left(d\right)}=x_{\pm \left(d\right)}^{max}$ is positive. This condition matches expression (43). Note that substituting expression (51) into the expression for the upper branch of the resonant energy of the final electron (26) results in expression (49), as it should be. In this case, the maximum and minimum energies of the final electron take the form

$$x_{f\left(u\right)}^{max}=1-2x_{\pm \left(d\right)}^{min}=1-{\displaystyle \frac{1}{l_2{\epsilon }_{iBW}}},$$

(55)

$$x_{f\left(u\right)}^{min}=1-2x_{\pm \left(d\right)}^{max}=1-{\displaystyle \frac{2}{l_2{\epsilon }_{iBW}}}.$$

(56)

Figure 5a shows the dependence of the energies of the final particles (the final electron on the upper branch and the electron–positron pair on the lower branch) on the square of the angle between the momenta of the electron and positron of the pair for different numbers of absorbed photons of the wave at the second vertex when $l_{2min}=3$. It follows from this figure that with an increase in the number of absorbed photons of the wave, the energy of the electron–positron pair decreases, and the energy of the final electron increases and tends to be the energy of the initial electron. At the same time, the energies of the final particles for a different number of photons of the wave at the second vertex are in different energy regions (do not intersect). Figure 5b shows the dependences of the square of the final electron outgoing angle (on the upper energy branch) on the square of the electron–positron pair outgoing angle for fixed values of the number of absorbed photons of the wave at the first and second vertices. Figure 5b shows that for different numbers of absorbed photons of the wave at the first and second vertices, each outgoing angle of the electron–positron pair uniquely corresponds to a certain outgoing angle of the final electron. The obtained expressions (49) and (51)–(56) (see Figure 5) completely solve the problem of resonant energies of the final particles (on the lower branch of the electron–positron pair energies and the upper branch of the electron energies). In this case, the outgoing angle of the electron–positron pair $\left({\delta }_{\pm \left(d\right)}^2\right)$ in (33) and the value of the quantum parameter of the Breit–Wheeler $\left({\epsilon }_{iBW}\right)$ process uniquely determine the outgoing angle of the final electron $\left({\delta }_{f\left(u\right)}^2\right)$ (see expressions (51)–(54)), as well as the final particle energies (see expressions (32) and (49)). Because of this, there is a quantum entanglement of the final particles. Thus, the Breit–Wheeler process, stimulated by an external field, which takes place at the second vertex, completely determines the resonant states of the final particles in this case.

It is important to emphasize that the condition for the number of absorbed photons of the wave at the second vertex (43) ensures the validity of the final particles’ energies in the entire range of electron–positron pair outgoing angles (33) and the final electron outgoing angles (52). In this case, the range of the quantum parameter ${\epsilon }_{iC}$ values at which the minimum number of photons in the second vertex is equal to one is determined by expression (46). When the final particles fly out at the minimum angles ${\delta }_{\pm \left(d\right)}^2=0$, ${\delta }_{f\left(u\right)}^2={\delta }_{f\left(min\right)}^2$, the number of absorbed photons of the wave at the second vertex is determined by a “softer” condition (47), which follows from expression (53). In this case, the range of the quantum parameter ${\epsilon }_{iC}$ values at which the minimum number of photons in the second vertex is equal to one is determined by expression (48).

Let us consider the case of the initial electron energies significantly exceeding the characteristic Compton effect energy but simultaneously being of the same order as the characteristic Breit–Wheeler energy:

$$E_i>>{\omega }_C,E_i\sim {\omega }_{BW}.$$

(57)

In this case, ${\epsilon }_{iC}>>1$ and ${\epsilon }_{iBW}\sim 1$. Therefore, the resonant energy of the final electron (lower branch, see expressions (26) and (29)) will be small compared to the initial electron energy, and the resonant energy of the electron–positron pair (upper branch, see expressions (37), (44), and (45)) will be close to the energy of the initial electron:

$$x_{f\left(d\right)}\sim {\displaystyle \frac{1}{l_1{\epsilon }_{iC}}}<<1,2x_{\pm \left(u\right)}\approx 1-x_{f\left(d\right)}\approx 1.$$

(58)

In addition, as follows from expressions (40)–(42) that the ultrarelativistic parameter of the pair (39), which determines their outgoing angle, will be close to the value ${\delta }_{\pm \left(u\right)}^2\approx {\delta }_{\pm \left(*\right)}^2$:

$$\begin{array}{c}{\delta }_{\pm \left(min\right)}^2\approx {\delta }_{\pm \left(\ast \right)}^2\left(1+{\displaystyle \frac{\xi }{l_1{\epsilon }_{iC}}}\right)\approx {\delta }_{\pm \left(\ast \right)}^2,\hfill \\ {\delta }_{\pm \left(max\right)}^2\approx {\delta }_{\pm \left(\ast \right)}^2\left(1+{\displaystyle \frac{2\xi }{l_1{\epsilon }_{iC}}}\right)\approx {\delta }_{\pm \left(\ast \right)}^2,\hfill \end{array}$$

(59)

$${\delta }_{\pm \left(\ast \right)}^2=4\left(l_2{\epsilon }_{iBW}-1\right),\xi ={\displaystyle \frac{\left(l_2{\epsilon }_{iBW}-2\right)}{\left(l_2{\epsilon }_{iBW}-1\right)}}.$$

(60)

It is also important to note that the number of photons of the wave in the second vertex (43) in this case (57) weakly depends on the process parameters in the first vertex:

$$l_{2\left(min\right)}=⌈{\displaystyle \frac{8}{{\epsilon }_{iC}}}\left(1+{\displaystyle \frac{2}{l_1{\epsilon }_{iC}}}\right)⌉\approx ⌈{\displaystyle \frac{8}{{\epsilon }_{iC}}}⌉=1.$$

(61)

For the second case of the final particle energies, we have $2x_{\pm \left(d\right)}\sim x_{f\left(u\right)}$ (see expressions (32), (34), (49), (55), and (56)).

Now let us consider a more stringent condition for initial electrons’ energies than (57), when they significantly exceed the characteristic Breit–Wheeler energy:

$$E_i>>{\omega }_{BW}.$$

(62)

In this case, the characteristic parameters at the first and second vertices are significantly greater than unity: ${\epsilon }_{iC}>>1,{\epsilon }_{iBW}>>1$. Therefore, only the just described case of resonant energies of the final particles is realized here, when the energy of the initial electron mainly turns into the energy of the electron–positron pair (see expressions (58)–(60)). At the same time,

$$\xi \approx 1,{\delta }_{\pm \left(\ast \right)}^2\approx 4l_2{\epsilon }_{iBW}.$$

(63)

In addition, for the second case of the final particle energies, the energy of the electron–positron pair (lower branch, see expressions (32) and (34)) will be small compared to the energy of the initial electrons, and the resonant energy of the final electron (upper branch, see expressions (49), (55), and (56)) will be close to the energy of the initial electrons:

$$2x_{\pm \left(d\right)}^{min}\sim {\displaystyle \frac{1}{l_2{\epsilon }_{iBW}}}<<1,x_{f\left(u\right)}=1-2x_{\pm \left(d\right)}^{min}\approx 1.$$

(64)

In addition, as follows from expressions (52)–(54), the ultrarelativistic parameter of the final electron (51), which defines its outgoing angle, will be enclosed in the interval:

$${\delta }_{f\left(\ast \right)}^2\le {\delta }_{f\left(u\right)}^2\le 2{\delta }_{f\left(\ast \right)}^2,{\delta }_{f\left(\ast \right)}^2={\displaystyle \frac{l_1}{4l_2}}.$$

(65)

It is also important to note that the number of photons of the wave in the second vertex (32) in this case (62) satisfies the expression $l_2\ge 1$.

4. Resonant Differential Probability

It was shown in the previous section that, due to resonant kinematics, in the wave field, the initial second-order process effectively decays into two first-order processes: at the first vertex, we have the Compton effect stimulated by an external field and, at the second vertex, we have the Breit–Wheeler process stimulated by an external field. In this case, the energy of the final electron (on the lower branch) is determined by the first-order process at the first vertex and the energy of the electron–positron pair (on the lower branch) is given by the first-order process at the second vertex. It is important to note that the two final electrons (in the first and second vertices) have different energies since their energies are determined by different first-order processes in the first and second vertices. Because of this, the final electron and the pair electron are distinguishable (see Figure 2) and, as a result, channels A and B do not interfere and are topologically identical. Therefore, in the future, we will consider only channel A. It is important to emphasize that for the resonant channel A, the energies of the final particles for different numbers of absorbed photons of the wave are also different (see Figure 4 and Figure 5). Therefore, the amplitudes of the process with different numbers of photons of the wave $\left(l_1,l_2\right)$ also do not interfere with each other. Given the expression for (6)–(18) and the resonant conditions (19)–(21), it is easy to obtain the partial resonant differential probability of the process (per unit time) for the unpolarized initial and final particles (22)–(24):

$$\begin{array}{c}\hfill d{w}_{l_1{l}_2}={\displaystyle \frac{4\pi {\alpha }^2{m}^4}{{\tilde{E}}_i{\tilde{E}}_f{\tilde{E}}_{-}{\tilde{E}}_{+}}}{\displaystyle \frac{1}{{\left|q^2\right|}^2}}{K}_{l_1}\left(u_f,z_f\right)P_{l_2}\left(u_{\pm },z_{\pm }\right)\times \\ \hfill \times {\delta }^{\left(4\right)}\left[{\tilde{p}}_{+}+{\tilde{p}}_{-}-{\tilde{p}}_i+{\tilde{p}}_f-\left(l_1+l_2\right)k\right]d^3{\tilde{p}}_f{d}^3{\tilde{p}}_{+}{d}^3{\tilde{p}}_{-}.\end{array}$$

(66)

In expression (66), the function $K_{l_1}$ defines the probability of the Compton effect stimulated by an external field and the function $P_{l_2}$ defines the probability of the Breit–Wheeler process stimulated by an external field [24].

$$K_{l_1}\left(u_f,z_f\right)=-4J_{l_1}^2\left(z_f\right)+{\eta }^2\left(2+{\displaystyle \frac{u_f^2}{1+u_f}}\right)\left[J_{l_1-1}^2\left(z_f\right)+J_{l_1+1}^2\left(z_f\right)-2J_{l_1}^2\left(z_f\right)\right],$$

(67)

$$P_{l_2}\left(u_{\pm },z_{\pm }\right)=J_{l_2}^2\left(z_{\pm }\right)+{\eta }^2\left(2u_{\pm }-1\right)\left[\left({\displaystyle \frac{l_2^2}{z_{\pm }^2}}-1\right)J_{l_2}^2\left(z_{\pm }\right)+J_{l_2}^{\prime 2}\left(z_{\pm }\right)\right].$$

(68)

Here, the following is indicated:

$$\begin{array}{cc}{z}_j=2l_j{\displaystyle \frac{\eta }{\sqrt{1+{\eta }^2}}}\sqrt{{\displaystyle \frac{u_j}{v_j}}\left(1-{\displaystyle \frac{u_j}{v_j}}\right)},& \left(j=f,\pm \right)\end{array},$$

(69)

$$u_f={\displaystyle \frac{\left(kq\right)}{\left(k{p}_f\right)}}\approx {\displaystyle \frac{1}{x_f}}-1,v_f=2l_1{\displaystyle \frac{\left(k{p}_i\right)}{m_{*}^2}}\approx l_1{\epsilon }_{iC},$$

(70)

$$u_{\pm }={\displaystyle \frac{{\left(kq\right)}^2}{4\left(k{p}_{-}\right)\left(k{p}_{+}\right)}}\approx {\displaystyle \frac{1}{4}}\left({\displaystyle \frac{x_{+}}{x_{-}}}+{\displaystyle \frac{x_{-}}{x_{+}}}+2\right)=1,$$

(71)

$$v_{\pm }=l_2{\displaystyle \frac{\left(kq\right)}{2m_{*}^2}}\approx \left(x_{+}+x_{-}\right)l_2{\epsilon }_{iBW}=2x_{\pm }{l}_2{\epsilon }_{iBW}.$$

(72)

Substituting (70)–(72) into (69), taking into account expressions (26) and (32), we obtain

$$z_f\approx {\displaystyle \frac{2\eta }{\sqrt{1+{\eta }^2}}}{\displaystyle \frac{\sqrt{{\delta }_f^2}}{{\epsilon }_{iC}}},z_{\pm }\approx {\displaystyle \frac{\eta }{\sqrt{1+{\eta }^2}}}{\displaystyle \frac{\sqrt{{\delta }_{\pm }^2}}{{\epsilon }_{iBW}}}.$$

(73)

At the same time, if we take into account the recurrent formulas for Bessel functions, then, after simple transformations, the function (68), taking into account the value $u_2$ (71), will look as follows:

$$P_{l_2}\left(z_{\pm }\right)=J_{l_2}^2\left(z_{\pm }\right)+{\displaystyle \frac{1}{2}}{\eta }^2\left[J_{l_2+1}^2\left(z_{\pm }\right)+J_{l_2-1}^2\left(z_{\pm }\right)-2J_{l_2}^2\left(z_{\pm }\right)\right].$$

(74)

The resonant energies of the electron and positron pair are the same, i.e., there is a symmetry between the particles of the electron–positron pair. For this reason, integration over $d^3{\tilde{p}}_{+}$ or $d^3{\tilde{p}}_{-}$ can be performed with the three-dimensional delta function in resonant probability (66). In addition, it is easy to integrate along the azimuthal angles of departure of the final electron and one of the particles of the pair. Given this and the ultrarelativistic kinematics (22)–(25), the resonant differential probability (66) will look like this:

$${\displaystyle \frac{d{w}_{l_1{l}_2}}{d{\delta }_f^{2}d{\delta }_{\pm }^2}}={\displaystyle \frac{4{\pi }^3{\alpha }^2{m}^4{m}_{*}^4}{E_i^3{\left|q^2\right|}^2}}{K}_{l_1}\left(u_f,z_f\right)P_{l_2}\left(z_{\pm }\right)\delta \left(1-x_f-x_{-}-x_{+}\right){\displaystyle \frac{x_f{x}_{\pm }}{x_{\mp }}}d{x}_{f}d{x}_{\pm }.$$

(75)

The delta function in expression (75) makes it easy to integrate over the energy of the final electron or one of the particles of the pair. At the same time, two possible variants of the final particle energy should be considered. When the energy of the final electron lies on the lower branch $x_{f\left(d\right)}$, in expression (75), integration should be performed with respect to the energy of the particle pair. If the energy of the pair particle lies on the lower branch, then, in expression (75), integration should be performed with respect to the energy of the final electron. Given this, expression (75) looks like

$${\displaystyle \frac{d{w}_{l_1{l}_2}^{\left(f\right)}}{d{\delta }_{\pm \left(u\right)}^{2}d{x}_{f\left(d\right)}}}={\displaystyle \frac{4{\pi }^3{\alpha }^2{m}^4}{E_i^3}}{\displaystyle \frac{m_{*}^4{x}_{f\left(d\right)}}{{\left|q^2\left(x_{f\left(d\right)}\right)\right|}^2}}{K}_{l_1}\left(u_{f\left(d\right)},z_{f\left(d\right)}\right)P_{l_2}\left(z_{\pm \left(u\right)}\right)d{\delta }_f^2,$$

(76)

$${\displaystyle \frac{d{w}_{l_1{l}_2}^{\left(\pm \right)}}{d{\delta }_{f\left(u\right)}^{2}d{x}_{\pm \left(d\right)}}}={\displaystyle \frac{4{\pi }^3{\alpha }^2{m}^4}{E_i^3}}{\displaystyle \frac{m_{*}^4\left(1-2x_{\pm \left(d\right)}\right)}{{\left|q^2\left(x_{\pm \left(d\right)}\right)\right|}^2}}{K}_{l_1}\left(u_{f\left(u\right)},z_{f\left(u\right)}\right)P_{l_2}\left(z_{\pm \left(d\right)}\right)d{\delta }_{\pm }^2.$$

(77)

Here, functions $K_{l_1}$ and $P_{l_2}$ are defined by expressions (67) and (74), in which the parameters $u_f$ (70) and $z_f,z_{\pm }$ (73) are taken on the lower and upper energy branches of the final electron and electron–positron pair (see Section 3).

The elimination of resonant infinity in expressions (76) and (77) is performed by the Breit–Wigner procedure [24,120]:

$$E_i\to E_i-i{\Gamma }_{BW}/\phantom{{\Gamma }_{BW}2}2,E_j\to E_j+i{\Gamma }_{BW}/\phantom{{\Gamma }_{BW}2}2,$$

(78)

$$\begin{array}{c}\hfill j=f,\pm ;{\Gamma }_{BW}={\displaystyle \frac{1}{2}}{W}_{BW}.\end{array}$$

where $W_{BW}$ is the total probability (per unit time) of the Breit–Wheeler process stimulated by an external field [24]:

$$W_{BW}\left(\eta ,{\epsilon }_{iC}\right)={\displaystyle \frac{\alpha m^2}{8\pi q_0}}\mathrm{P}\left(\eta ,{\epsilon }_{iC}\right),$$

(79)

$$\mathrm{P}\left(\eta ,{\epsilon }_{iC}\right)=\sum _{k=k_{min}=⌈{{\epsilon }_{iC}}^{-1}⌉}^{\infty }{\mathrm{P}}_k\left(\eta ,{\epsilon }_{iC}\right),$$

(80)

$$\begin{array}{c}\hfill {\mathrm{P}}_k\left({\epsilon }_{iC}\right)={\int }_1^{k{\epsilon }_{iC}}{\displaystyle \frac{du}{u\sqrt{u\left(u-1\right)}}}P\left(u,\eta ,k{\epsilon }_{iC}\right)\end{array}$$

$$P\left(\eta ,u,k{\epsilon }_{iC}\right)=J_r^2\left(z\right)+{\eta }^2\left(u-{\displaystyle \frac{1}{2}}\right)[J_{l_2+1}^2\left(z\right)+J_{l_2-1}^2\left(z\right)-2J_{l_2}^2\left(z\right)],$$

(81)

$$z=2k{\displaystyle \frac{\eta }{\sqrt{1+{\eta }^2}}}\sqrt{{\displaystyle \frac{u}{k{\epsilon }_{iC}}}\left(1-{\displaystyle \frac{u}{k{\epsilon }_{iC}}}\right)}.$$

(82)

Due to expression (78), the square of the 4-momentum of the intermediate gamma-quantum obtains an imaginary additive:

$$q_0\to q_0-i{\Gamma }_{BW},q^2\to q^2-2i{q}_0{\Gamma }_{BW}.$$

(83)

Taking this into account, the resonant differential probabilities (76) and (77) will take the following form:

$${\displaystyle \frac{d{w}_{l_1{l}_2}^{\left(f\right)}}{d{\delta }_{\pm \left(u\right)}^{2}d{x}_{f\left(d\right)}}}={\displaystyle \frac{4{\pi }^3{\alpha }^2{m}^4}{E_i^3{x}_{f\left(d\right)}}}{K}_{l_1}\left(u_{f\left(d\right)},z_{f\left(d\right)}\right)P_{l_2}\left(z_{\pm \left(u\right)}\right){\int }_0^{\infty }{\displaystyle \frac{d{\delta }_f^2}{\left[{\left({\delta }_f^2-{\delta }_{f\left(d\right)}^2\right)}^2+{{\rm Y}}_f^2\right]}},$$

(84)

$${\displaystyle \frac{d{w}_{l_1{l}_2}^{\left(\pm \right)}}{d{\delta }_{f\left(u\right)}^{2}d{x}_{\pm \left(d\right)}}}={\displaystyle \frac{4{\pi }^3{\alpha }^2{m}^4}{E_i^3}}{K}_{l_1}\left(u_{f\left(u\right)},z_{f\left(u\right)}\right)P_{l_2}\left(z_{\pm \left(d\right)}\right){\displaystyle \frac{\left(1-2x_{\pm \left(d\right)}\right)}{x_{\pm \left(d\right)}^4}}{\int }_0^{\infty }{\displaystyle \frac{d{\delta }_{\pm }^2}{\left[{\left({\delta }_{\pm }^2-{\delta }_{\pm \left(d\right)}^2\right)}^2+{{\rm Y}}_{\pm }^2\right]}}.$$

(85)

Here, the ultrarelativistic parameters ${\delta }_{f\left(d\right)}^2$ and ${\delta }_{\pm \left(d\right)}^2$ determine the resonant energies of the final electron and electron–positron pair (see expressions (26) and (32)), and the parameters ${\delta }_f^2$ and ${\delta }_{\pm }^2$ can take arbitrary values that do not depend on the energy of the final particles and can, therefore, be integrated according to these parameters. In this case, the angular resonant widths are determined by the following expressions:

$${{\rm Y}}_f={\displaystyle \frac{\alpha m^2}{8\pi m_{*}^2{x}_{f\left(d\right)}}}P\left(\eta ,{\epsilon }_{iBW}\right),$$

(86)

$${{\rm Y}}_{\pm }={\displaystyle \frac{\alpha m^2}{8\pi m_{*}^2{x}_{\pm \left(d\right)}^2}}P\left(\eta ,{\epsilon }_{iBW}\right).$$

(87)

Note that expressions (84) and (85) have a characteristic Breit–Wigner resonant structure and take the maximum value at ${\delta }_f^2={\delta }_{f\left(d\right)}^2$ and ${\delta }_{\pm }^2={\delta }_{\pm \left(d\right)}^2$, respectively. Let us integrate expressions (84) and (85) with respect to ultrarelativistic parameters ${\delta }_f^2$ and ${\delta }_{\pm }^2$ near their respective resonant values ${\delta }_{f\left(d\right)}^2$ and ${\delta }_{\pm \left(d\right)}^2$. It should be taken into account that the resonant width in these expressions has a dominant value only in the resonant denominators (in all other expressions, this width can be ignored). Therefore, this integration is reduced to an integral of only the resonant denominator. For example, for expression (84), we obtain

$${\int }_{-\infty }^{\infty }{\displaystyle \frac{dy}{\left[y^2+{{\rm Y}}_f^2\right]}}={\displaystyle \frac{\pi }{{{\rm Y}}_f}}\left(y={\delta }_f^2-{\delta }_{f\left(d\right)}^2\right).$$

(88)

Here, the limits of integration are extended to $\pm \infty$ due to the fast convergence of the integral. Taking this into account, the expressions for the resonant probabilities (84) and (85) will take the form

$${\displaystyle \frac{d{w}_{l_1{l}_2}^{\left(f\right)}}{d{\delta }_{\pm \left(u\right)}^{2}d{x}_{f\left(d\right)}}}=a_i{\Psi }_{l_1{l}_2}^{\left(f\right)}\left(x_{f\left(d\right)},{\delta }_{\pm \left(u\right)}^2\right),$$

(89)

$${\displaystyle \frac{d{w}_{l_1{l}_2}^{\left(\pm \right)}}{d{\delta }_{f\left(u\right)}^{2}d{x}_{\pm \left(d\right)}}}=a_i{\Psi }_{l_1{l}_2}^{\left(\pm \right)}\left(x_{\pm \left(d\right)},{\delta }_{f\left(u\right)}^2\right).$$

(90)

Here, the function $a_i$ is defined by the initial installation parameters:

$$a_i={\displaystyle \frac{32{\pi }^5\alpha m^2{m}_{*}^2}{E_i^{3}P\left(\eta ,{\epsilon }_{iBW}\right)}},$$

(91)

and the functions ${\Psi }_{l_1{l}_2}^{\left(f\right)}$ and ${\Psi }_{l_1{l}_2}^{\left(\pm \right)}$ determine the spectral–angular distribution of the final particles:

$${\Psi }_{l_1{l}_2}^{\left(f\right)}\left(x_{f\left(d\right)},{\delta }_{\pm \left(u\right)}^2\right)=K_{l_1}\left(u_{f\left(d\right)},z_{f\left(d\right)}\right)P_{l_2}\left(z_{\pm \left(u\right)}\right),$$

(92)

$${\Psi }_{l_1{l}_2}^{\left(\pm \right)}\left(x_{\pm \left(d\right)},{\delta }_{f\left(u\right)}^2\right)={\displaystyle \frac{\left(1-2x_{\pm \left(d\right)}\right)}{x_{\pm \left(d\right)}^2}}{K}_{l_1}\left(u_{f\left(u\right)},z_{f\left(u\right)}\right)P_{l_2}\left(z_{\pm \left(d\right)}\right).$$

(93)

Note that expressions (89), (91), and (92) are determined by the resonant differential probability of the RTPP process with the simultaneous registration of the energy of the final electron and the angle between momenta of the pair. In this case, the energy of the final electron uniquely determines the outgoing angle of the pair (see expression (39)). Expressions (90), (91), and (93) determine the differential probability of the RTPP process with the simultaneous registration of the pair energy and the final electron outgoing angle. In this case, the energy of the pair uniquely determines the outgoing angle of the final electron (see expression (51)). Thus, the differential probabilities (89) and (90) together with expressions (39) and (51) reflect the spectral–angular entanglement of the final particles.

Given the expressions for the energies of the electron (26) and the electron–positron pair (32) on their lower branches, we can obtain the expression of differential $d{x}_{f\left(d\right)}$ with $d{\delta }_{f\left(d\right)}^2$ as well as the expression of differential $d{x}_{\pm \left(d\right)}$ with $d{\delta }_{\pm \left(d\right)}^2$. Thus, we obtain the corresponding differential probabilities of the RTPP process with the angular entanglement of the final particles:

$${\displaystyle \frac{d{w}_{l_1{l}_2}^{\left(f\right)}}{d{\delta }_{\pm \left(u\right)}^{2}d{\delta }_{f\left(d\right)}^2}}=a_i{\Phi }_{l_1{l}_2}^{\left(f\right)}\left({\delta }_{f\left(d\right)}^2,{\delta }_{\pm \left(u\right)}^2\right),$$

(94)

$${\displaystyle \frac{d{w}_{l_1{l}_2}^{\left(\pm \right)}}{d{\delta }_{f\left(u\right)}^{2}d{\delta }_{\pm \left(d\right)}^2}}=a_i{\Phi }_{l_1{l}_2}^{\left(\pm \right)}\left({\delta }_{\pm \left(d\right)}^2,{\delta }_{f\left(u\right)}^2\right).$$

(95)

Here, the function ${\Phi }_{l_1{l}_2}^{\left(f\right)}$ determines the outgoing probability of the final electron by an angle ${\delta }_{f\left(d\right)}^2$ (on the lower branch of the electron energy) and the electron–positron pair by an angle ${\delta }_{\pm \left(u\right)}^2$ (on the upper branch of the pair energy). At the same time, these angles are uniquely related by expression (39).

$${\Phi }_{l_1{l}_2}^{\left(f\right)}\left({\delta }_{f\left(d\right)}^2,{\delta }_{\pm \left(u\right)}^2\right)={\displaystyle \frac{2\sqrt{{\left(l_1{\epsilon }_{iC}\right)}^2-4{\delta }_{f\left(d\right)}^2}{\left(1+{\delta }_{f\left(d\right)}^2+l_1{\epsilon }_{iC}\right)}^2{K}_{l_1}\left(u_{f\left(d\right)},z_{f\left(d\right)}\right)P_{l_2}\left(z_{\pm \left(u\right)}\right)}{2\left(1-{\delta }_{f\left(d\right)}^2\right)+\left(2+l_1{\epsilon }_{iC}\right)\left(l_1{\epsilon }_{iC}-\sqrt{{\left(l_1{\epsilon }_{iC}\right)}^2-4{\delta }_{f\left(d\right)}^2}\right)}}.$$

(96)

The function ${\Phi }_{l_1{l}_2}^{\left(\pm \right)}$ determines the outgoing probability of the electron–positron pair by an angle ${\delta }_{\pm \left(d\right)}^2$ (on the lower branch of the pair energy) and the final electron by an angle ${\delta }_{f\left(u\right)}^2$ (on the upper branch of the electron energy). At the same time, these angles are uniquely related by expression (51).

$${\Phi }_{l_1{l}_2}^{\left(\pm \right)}\left({\delta }_{f\left(u\right)}^2,{\delta }_{\pm \left(d\right)}^2\right)={\displaystyle \frac{{\delta }_{\pm \left(d\right)}^2\left(1-2x_{\pm \left(d\right)}\right)\sqrt{1-{\delta }_{\pm \left(d\right)}^2/\phantom{{\delta }_{\pm \left(d\right)}^2\left(l_2^2{\epsilon }_{iBW}^2\right)}\left(l_2^2{\epsilon }_{iBW}^2\right)}}{x_{\pm \left(d\right)}^2\left[x_{\pm \left(d\right)}-1/\phantom{1\left(2l_2{\epsilon }_{iBW}\right)}\left(2l_2{\epsilon }_{iBW}\right)\right]}}{K}_{l_1}\left(u_{f\left(u\right)},z_{f\left(u\right)}\right)P_{l_2}\left(z_{\pm \left(d\right)}\right)$$

(97)

We would like to emphasize that when ultrarelativistic electrons interact with a strong electromagnetic wave, there are two first-order processes that occur sequentially (the effective trident effect). The first process is the Compton effect stimulated by an external field. The second process is the Breit–Wheeler process stimulated by an external field, which takes place with the final gamma-quantum of the nonlinear Compton scattering [70]. At the same time, the energy and kinematics of the final electron and gamma-quantum are determined only by the nonlinear Compton effect and are not related to the nonlinear Breit–Wheeler process. The probability of the effective trident effect is significantly less than the probability of each of the two components of the process. At the same time, when resonant conditions are met, resonant trident pair production process also takes place, which proceeds in a special resonant kinematics. At the same time, two sequential first-order processes (EFSCE and EFSBW) are interconnected by an intermediate real gamma-quantum and the laws of the conservation of energy and momentum common to the process. This, as well as the small resonant width, makes the resonant trident pair production process, which is studied in this article, dominant in certain cases. Further, we will consider the resonant differential probability in units of the total probability (per unit time) of the Compton effect stimulated by an external field.

$$W_C\left(\eta ,{\epsilon }_{iC}\right)={\displaystyle \frac{\alpha m^2}{4\pi E_i}}\mathrm{K}\left(\eta ,{\epsilon }_{iC}\right),$$

(98)

where

$$\mathrm{K}\left(\eta ,{\epsilon }_{iC}\right)=\sum _{n=1}^{\infty }{\int }_0^{n{\epsilon }_{iC}}{\displaystyle \frac{du}{{\left(1+u\right)}^2}}{K}_n\left(\eta ,u,n{\epsilon }_{iC}\right).$$

(99)

Here, functions $K_n\left(\eta ,u,n{\epsilon }_{iC}\right)$ are defined by the following expression:

$$\begin{array}{c}\hfill K_n\left(\eta ,u,n{\epsilon }_{iC}\right)=-4J_n^2\left(z^{\prime }\right)+{\eta }^2\left(2+{\displaystyle \frac{u^2}{1+u}}\right)\left(J_{n+1}^2+J_{n-1}^2-2J_n^2\right),\end{array}$$

(100)

$$z^{\prime }=2n{\displaystyle \frac{\eta }{\sqrt{1+{\eta }^2}}}\sqrt{{\displaystyle \frac{u}{n{\epsilon }_{iC}}}\left(1-{\displaystyle \frac{u}{n{\epsilon }_{iC}}}\right)}.$$

(101)

Dividing expressions (89), (90) and (94), (95) by the expression (98), we obtain the relative resonant differential probabilities (in units of the total probability of having a Compton effect stimulated by an external field):

• If the final electron energy on the lower branch $\left(x_{f\left(d\right)}\right)$ and the angle between momenta of the pair on the upper branch are simultaneously recorded $\left({\delta }_{\pm \left(u\right)}^2\right)$:

  $$R_{l_1{l}_2}^{\left(f\right)}\left(x_{f\left(d\right)},{\delta }_{\pm \left(u\right)}^2\right)={\displaystyle \frac{1}{W_C\left(\eta ,{\epsilon }_{iC}\right)}}{\displaystyle \frac{d{w}_{l_1{l}_2}^{\left(f\right)}}{d{\delta }_{\pm \left(u\right)}^{2}d{x}_{f\left(d\right)}}}=b_i{\Psi }_{l_1{l}_2}^{\left(f\right)}\left(x_{f\left(d\right)},{\delta }_{\pm \left(u\right)}^2\right).$$

  (102)
• If the outgoing angle of the final electron on the lower branch $\left({\delta }_{f\left(d\right)}^2\right)$ and the outgoing angle of the electron–positron pair on the upper branch are simultaneously recorded $\left({\delta }_{\pm \left(u\right)}^2\right)$:

  $${\mathrm{H}}_{l_1{l}_2}^{\left(f\right)}\left({\delta }_{f\left(d\right)}^2,{\delta }_{\pm \left(u\right)}^2\right)={\displaystyle \frac{1}{W_C\left(\eta ,{\epsilon }_{iC}\right)}}{\displaystyle \frac{d{w}_{l_1{l}_2}^{\left(f\right)}}{d{\delta }_{\pm \left(u\right)}^{2}d{\delta }_{f\left(d\right)}^2}}=b_i{\Phi }_{l_1{l}_2}^{\left(f\right)}\left({\delta }_{f\left(d\right)}^2,{\delta }_{\pm \left(u\right)}^2\right).$$

  (103)
• If the energy of the electron–positron pair on the lower branch $\left(x_{\pm \left(d\right)}\right)$ and the outgoing angle of the elecron on the upper branch are simultaneously recorded $\left({\delta }_{f\left(u\right)}^2\right)$:

  $$R_{l_1{l}_2}^{\left(\pm \right)}\left(x_{\pm \left(d\right)},{\delta }_{f\left(u\right)}^2\right)={\displaystyle \frac{1}{W_C\left(\eta ,{\epsilon }_{iC}\right)}}{\displaystyle \frac{d{w}_{l_1{l}_2}^{\left(\pm \right)}}{d{\delta }_{f\left(u\right)}^{2}d{x}_{\pm \left(d\right)}}}=b_i{\Psi }_{l_1{l}_2}^{\left(\pm \right)}\left(x_{\pm \left(d\right)},{\delta }_{f\left(u\right)}^2\right).$$

  (104)
• If the outgoing angle of the electron–positron pair on the lower branch $\left({\delta }_{\pm \left(d\right)}^2\right)$ and the outgoing angle of the final electron on the upper branch are simultaneously recorded $\left({\delta }_{f\left(u\right)}^2\right)$:

  $${\mathrm{H}}_{l_1{l}_2}^{\left(\pm \right)}\left({\delta }_{\pm \left(d\right)}^2,{\delta }_{f\left(u\right)}^2\right)={\displaystyle \frac{1}{W_C\left(\eta ,{\epsilon }_{iC}\right)}}{\displaystyle \frac{d{w}_{l_1{l}_2}^{\left(\pm \right)}}{d{\delta }_{f\left(u\right)}^{2}d{\delta }_{\pm \left(d\right)}^2}}=b_i{\Phi }_{l_1{l}_2}^{\left(\pm \right)}\left({\delta }_{\pm \left(d\right)}^2,{\delta }_{f\left(u\right)}^2\right).$$

  (105)

Here, the dimensionless function $b_i$ is determined by the initial setup parameters:

$$b_i={\displaystyle \frac{2{\left(4{\pi }^2\right)}^3\left(1+{\eta }^2\right)}{P\left(\eta ,{\epsilon }_{iBW}\right)\mathrm{K}\left(\eta ,{\epsilon }_{iC}\right){\epsilon }_{iC}^2}}{\left({\displaystyle \frac{m}{{\omega }_C}}\right)}^2.$$

(106)

It is important to emphasize that the initial installation $\eta$ (1), ${\omega }_C$ (3), ${\epsilon }_{iC}$ (2), as well as the number of absorbed photons of the wave at the first vertex $l_1\ge 1$ and the outgoing angle of the final electron ${\delta }_{f\left(d\right)}^2$ (27) and (28), uniquely determine the resonant relative probabilities (102) and (103). At the same time, the parameters that define the Breit–Wheeler process stimulated by an external field in the second vertex (see the functions $P_{l_2}\left(z_{\pm \left(u\right)}\right)$ (74)) are determined by the following expressions: ${\epsilon }_{iBW}={\epsilon }_{iC}/\phantom{{\epsilon }_{iC}4}4$, $l_{2\left(min\right)}$ (43), ${\delta }_{\pm \left(u\right)}^2$ (39). On the other hand, the initial setup $\eta$ (1), ${\omega }_{BW}$ (3), ${\epsilon }_{iBW}$ (2), as well as the number of absorbed photons of the wave in the second vertex $l_2\ge l_{2\left(min\right)}$ (43) and the outgoing angle of the electron–positron pair ${\delta }_{\pm \left(d\right)}^2$ (31) and (33), uniquely determine the resonant relative probabilities (104) and (105). At the same time, the parameters that determine the Compton effect stimulated by an external field at the first vertex (see the functions $K_{l_1}\left(u_{f\left(u\right)},z_{f\left(u\right)}\right)$ (67)) are determined by the following expressions: ${\epsilon }_{iC}=4{\epsilon }_{iBW}$, $l_1\ge 1$ (43), ${\delta }_{f\left(u\right)}^2$ (51). Taking this into account, we can plot the corresponding relative probabilities multiplied by the parameter ${\left({\omega }_C/\phantom{{\omega }_{C}m}m\right)}^2$ as a function of the outgoing angles of the final particles at different values of the initial parameters $\eta$ and ${\epsilon }_{iC}\left({\epsilon }_{iBW}\right)$, which determine the parameters of the plane electromagnetic wave and the initial electron energy. In order for the resonant probability to take maximum values, we assume that the characteristic quantum parameter of the Breit–Wheeler process is chosen from the condition ${\epsilon }_{iBW}\ge {\epsilon }_{*}$$\left({\epsilon }_{iC}\ge 4{\epsilon }_{*}\right)$. This condition allows us to select the minimum number of absorbed photons of the wave at the second vertex equal to one (see expression (46)). As was shown earlier (see expressions (57) and (62)), for large values of quantum parameters ${\epsilon }_{iC}\left({\epsilon }_{iBW}\right)$, the energy of the initial electron mainly goes into the electron–positron pair energy or the final electron energy. It is important to emphasize that for a fixed energy of the initial electron, the quantum parameter ${\epsilon }_{iC}\left({\epsilon }_{iBW}\right)$ increases as the corresponding characteristic energy decreases ${\omega }_C\left({\omega }_{BW}\right)$. If we fix the characteristic energy of the process, then the quantum parameter ${\epsilon }_{iC}\left({\epsilon }_{iBW}\right)$ increases with the increasing energy of the initial electron.

Figure 6 and Figure 7 show graphs of the relative probabilities $R_{l_1{l}_2}^{\left(f\right)}$ (102) and ${\mathrm{H}}_{l_1{l}_2}^{\left(f\right)}$ (103) (multiplied by a factor ${\left({\omega }_C/\phantom{{\omega }_{C}m}m\right)}^2$) depending on the outgoing angle of the final electron (on the lower branch of its energies) for fixed initial setup parameters and different numbers of absorbed photons of the wave.

Table 1. The maximum values of the resonant relative probability $R_{l_1{l}_2\left(max\right)}^{\left(f\right)}\times {\left({\omega }_C/\phantom{{\omega }_{C}m}m\right)}^2$ (102) and the corresponding values of the energies and outgoing angles of the final particles for different numbers of absorbed photons of the wave at the first and second vertices and fixed quantum parameters of the Breit–Wheeler process and the Compton effect.

	${\mathit{l}}_1,{\mathit{l}}_2$	${\mathit{\delta }}_{\mathit{f}\left(\mathit{d}\right)}^{2*}$	${\mathit{\delta }}_{\pm \left(\mathit{u}\right)}^{2*}$	${\mathit{x}}_{\mathit{f}\left(\mathit{d}\right)}$	$2{\mathit{x}}_{\pm \left(\mathit{u}\right)}$	${\mathit{R}}_{{\mathit{l}}_1{\mathit{l}}_2\left(max\right)}^{\left(\mathit{f}\right)}\times {\left(\frac{{\mathit{\omega }}_{\mathit{C}}}{\mathit{m}}\right)}^2$
$\eta =1$	1, 1	0	5.79	0.09	0.91	$1.41\times {10}^3$
${\epsilon }_{iBW}\approx 2.41$	1, 2	0	16.44	0.09	0.91	$4.45\times {10}^2$
${\epsilon }_{iC}\approx 9.66$	2, 1	62.40	5.75	0.06	0.94	$4.0\times {10}^2$
	2, 2	62.84	16.04	0.06	0.94	125
$\eta =1$	1, 1	0	36.80	0.02	0.98	167
${\epsilon }_{iBW}\approx 10$	1, 2	0	77.80	0.02	0.98	$16.15$
${\epsilon }_{iC}\approx 40$	2, 1	1083	36.51	0.02	0.98	50
	2, 2	1086	77.15	0.02	0.98	$4.81$

and

Table 2. The maximum values of the resonant relative probability $H_{l_1{l}_2\left(max\right)}^{\left(f\right)}\times {\left({\omega }_C/\phantom{{\omega }_{C}m}m\right)}^2$ (103) and the corresponding values of the energies and outgoing angles of the final particles for different numbers of absorbed photons of the wave at the first and second vertices and fixed quantum parameters of the Breit–Wheeler process and the Compton effect.

	${\mathit{l}}_1,{\mathit{l}}_2$	${\mathit{\delta }}_{\mathit{f}\left(\mathit{d}\right)}^{2*}$	${\mathit{\delta }}_{\pm \left(\mathit{u}\right)}^{2*}$	${\mathit{x}}_{\mathit{f}\left(\mathit{d}\right)}$	$2{\mathit{x}}_{\pm \left(\mathit{u}\right)}$	${\mathit{H}}_{{\mathit{l}}_1{\mathit{l}}_2\left(max\right)}^{\left(\mathit{f}\right)}\times {\left(\frac{{\mathit{\omega }}_{\mathit{C}}}{\mathit{m}}\right)}^2$
$\eta =1$	1, 1	0	5.79	0.09	0.91	$1.545\times {10}^6$
${\epsilon }_{iBW}\approx 2.41$	1, 2	0	16.44	0.09	0.91	$4.90\times {10}^5$
${\epsilon }_{iC}\approx 9.66$	2, 1	34.95	5.74	0.05	0.95	$1.67\times {10}^6$
	2, 2	35.05	15.95	0.05	0.95	$5.25\times {10}^5$
$\eta =1$	1, 1	0	36.80	0.02	0.98	$1.125\times {10}^7$
${\epsilon }_{iBW}\approx 10$	1, 2	0	77.80	0.02	0.98	$1.09\times {10}^6$
${\epsilon }_{iC}\approx 40$	2, 1	597.61	36.45	0.01	0.99	$1.38\times {10}^7$
	2, 2	598.34	77.00	0.01	0.99	$1.32\times {10}^6$

show the corresponding maximum probabilities for different numbers of absorbed photons of the wave (at points ${\delta }_{f\left(d\right)}^{2*}$ corresponding to the maxima of the relative probability distributions in Figure 6 and Figure 7), as well as the energies of the final electron and electron–positron pair (in units of the energy of the initial electron). It can be seen from the figures and tables that for $l_1=1$ and $l_2=1,2,\dots$, the maximum of relative probabilities occurs when an electron is scattered at a zero angle $\left({\delta }_{f\left(d\right)}^{2*}=0\right)$. If $l_1=2,3,\dots$ (for any values $l_2$), then the maximum of resonant probabilities is shifted to the right, into the region of non-zero values of the electron outgoing angles. Figure 6 and Table 1 show that the maximum relative probability $R_{l_1{l}_2\left(max\right)}^{\left(f\right)}$ holds for the minimum number of absorbed photons of the wave $l_1=l_2=1$. As the number of absorbed photons increases, the relative probability decreases. In addition, when the quantum parameter of the Compton effect increases by a factor of 4, the maximum relative probability decreases by one order of magnitude: $\left(R_{11\left(max\right)}^{\left(f\right)}{|}_{{\epsilon }_{iC=40}}\right)/\phantom{\left(R_{11\left(max\right)}^{\left(f\right)}{|}_{{\epsilon }_{iC=40}}\right)\left(R_{11\left(max\right)}^{\left(f\right)}{|}_{{\epsilon }_{iC=9.66}}\right)}\left(R_{11\left(max\right)}^{\left(f\right)}{|}_{{\epsilon }_{iC=9.66}}\right)\approx 0.09$. Figure 7 and Table 2 show that the maximum relative probability $H_{l_1{l}_2\left(max\right)}^{\left(f\right)}$ holds for the number of absorbed photons of the wave $l_1=2,l_2=1$. At the same time, $H_{11\left(max\right)}^{\left(f\right)}\underset{\sim }{<}{H}_{21\left(max\right)}^{\left(f\right)}$. As the number of absorbed photons increases, the relative probability decreases. In addition, when the quantum parameter of the Compton effect is increased by a factor of 4, the maximum relative probability $H_{21\left(max\right)}^{\left(f\right)}$ is reduced by one order of magnitude: $\left(H_{21\left(max\right)}^{\left(f\right)}{|}_{{\epsilon }_{iC=40}}\right)/\phantom{\left(H_{21\left(max\right)}^{\left(f\right)}{|}_{{\epsilon }_{iC=40}}\right)\left(H_{21\left(max\right)}^{\left(f\right)}{|}_{{\epsilon }_{iC=9.66}}\right)}\left(H_{21\left(max\right)}^{\left(f\right)}{|}_{{\epsilon }_{iC=9.66}}\right)\approx 8.33$.

Figure 8 shows the dependence of the maximum relative probabilities $R_{11\left(max\right)}^{\left(f\right)}$ (102) and $H_{21\left(max\right)}^{\left(f\right)}$ (103) on the final electron outgoing angle and the quantum parameter of the Compton effect at the optimal number of absorbed photons in the first and second vertices. Figure 8a shows that the maximum relative probability with the simultaneous registration of the electron outgoing angle and the electron–positron pair $R_{11\left(max\right)}^{\left(f\right)}$ energy has a maximum value at zero outgoing angle of the final electron and when the quantum parameter ${\epsilon }_{iC}\approx 4{\epsilon }_{*}$. As the value of this quantum parameter increases, the relative probability $R_{11\left(max\right)}^{\left(f\right)}$ decreases rather rapidly. On the other hand, Figure 8b shows that the maximum relative probability with the simultaneous registration of the final electron and electron–positron pair outgoing angles $H_{21\left(max\right)}^{\left(f\right)}$ increases quite rapidly with increasing quantum parameter ${\epsilon }_{iC}$, taking maximum values for sufficiently large outgoing angles of the final electron. It is important to note that the final electron outgoing angle uniquely determines the outgoing angle of the electron–positron pair (see expressions (39) and (59)) and the energies of the final particles (see expressions (26) and (37)), i.e., quantum entanglement of the final particles takes place. At the same time, as the quantum parameter of the Compton effect increases ${\epsilon }_{iC}$, the energy of the electron–positron pair tends to be the energy of the initial electrons (58).

Figure 9 and Figure 10 show graphs of the relative probabilities $R_{l_1{l}_2}^{\left(\pm \right)}$ (104) and ${\mathrm{H}}_{l_1{l}_2}^{\left(\pm \right)}$ (105) (multiplied by a factor ${\left({\omega }_C/\phantom{{\omega }_{C}m}m\right)}^2$) as a function of the angle between momenta of electron and positron pairs on the lower branch of their energies for fixed initial setup parameters and different numbers of absorbed photons of the wave.

Table 3. The maximum values of the resonant relative probability $R_{l_1{l}_2\left(max\right)}^{\left(\pm \right)}\times {\left({\omega }_C/\phantom{{\omega }_{C}m}m\right)}^2$(104) and the corresponding values of the energies and outgoing angles of the final particles for different numbers of absorbed photons of the wave at the first and second vertices and fixed quantum parameters of the Breit–Wheeler process and the Compton effect.

	${\mathit{l}}_1,{\mathit{l}}_2$	${\mathit{\delta }}_{\mathit{f}\left(\mathit{u}\right)}^{2*}$	${\mathit{\delta }}_{\pm \left(\mathit{d}\right)}^{2*}$	${\mathit{x}}_{\mathit{f}\left(\mathit{u}\right)}$	$2{\mathit{x}}_{\pm \left(\mathit{d}\right)}$	${\mathit{R}}_{{\mathit{l}}_1{\mathit{l}}_2\left(max\right)}^{\left(\pm \right)}\times {\left(\frac{{\mathit{\omega }}_{\mathit{C}}}{\mathit{m}}\right)}^2$
$\eta =1$	1, 1	6.33	0	0.59	0.41	$4.28\times {10}^3$
${\epsilon }_{iBW}\approx 2.41$	1, 2	3.17	13.98	0.75	0.25	$3.23\times {10}^3$
${\epsilon }_{iC}\approx 9.66$	2, 1	13.16	0	0.59	0.41	$3.25\times {10}^2$
	2, 2	7.48	18.34	0.72	0.28	$123.75$
$\eta =1$	1, 1	4.43	0	0.90	0.10	$3.21\times {10}^3$
${\epsilon }_{iBW}\approx 10$	1, 2	2.65	253.18	0.94	0.06	$1.98\times {10}^3$
${\epsilon }_{iC}\approx 40$	2, 1	8.88	0	0.90	0.10	$8.925$
	2, 2	5.88	315.62	0.93	0,07	$3.430$

and

Table 4. The maximum values of the resonant relative probability $H_{l_1{l}_2\left(max\right)}^{\left(\pm \right)}\times {\left({\omega }_C/\phantom{{\omega }_{C}m}m\right)}^2$ (105) and the corresponding values of the energies and outgoing angles of the final particles for different numbers of absorbed photons of the wave at the first and second vertices and fixed quantum parameters of the Breit–Wheeler process and the Compton effect.

	${\mathit{l}}_1,{\mathit{l}}_2$	${\mathit{\delta }}_{\mathit{f}\left(\mathit{u}\right)}^{2*}$	${\mathit{\delta }}_{\pm \left(\mathit{d}\right)}^{2*}$	${\mathit{x}}_{\mathit{f}\left(\mathit{u}\right)}$	$2{\mathit{x}}_{\pm \left(\mathit{d}\right)}$	${\mathit{H}}_{{\mathit{l}}_1{\mathit{l}}_2\left(max\right)}^{\left(\pm \right)}\times {\left(\frac{{\mathit{\omega }}_{\mathit{C}}}{\mathit{m}}\right)}^2$
$\eta =1$	1, 1	6.33	0	0.59	0.41	$5.00\times {10}^5$
${\epsilon }_{iBW}\approx 2.41$	1, 2	2.79	8.35	0.77	0.23	$1.62\times {10}^6$
${\epsilon }_{iC}\approx 9.66$	2, 1	13.16	0	0.59	0.41	$3.80\times {10}^4$
	2, 2	5.80	9.47	0.77	0.23	$5.0\times {10}^4$
$\eta =1$	1, 1	4.43	0	0.90	0.10	$2.60\times {10}^7$
${\epsilon }_{iBW}\approx 10$	1, 2	2.36	146.94	0.94	0.06	$6.80\times {10}^7$
${\epsilon }_{iC}\approx 40$	2, 1	8.88	0	0.90	0.10	$7.0\times {10}^4$
	2, 2	4.79	162.89	0.94	0.06	$1.0\times {10}^5$

show the corresponding maximum probabilities for different numbers of absorbed photons of the wave (at points ${\delta }_{\pm \left(d\right)}^{2*}$ corresponding to the maxima of the relative probability distributions in Figure 9 and Figure 10), as well as the energy of the final electron and the electron–positron pair (in units of the energy of the initial electron). It can be seen from the figures and tables that for $l_2=1$ and $l_1=1,2,\dots$, the maximum of relative probabilities occurs at zero outgoing angle of the pair $\left({\delta }_{\pm \left(d\right)}^{2*}=0\right)$. If $l_2=2,3,\dots$ (for any values $l_1$), then the maximum of resonant probabilities is shifted to the right, into the region of non-zero values of the ultrarelativistic parameter ${\delta }_{\pm \left(d\right)}^{2*}$. Figure 9 and Table 3 show that the maximum relative probability $R_{l_1{l}_2\left(max\right)}^{\left(\pm \right)}$ holds for the minimum number of absorbed photons of the wave $l_1=l_2=1$. As the number of absorbed photons increases, the relative probability decreases. In addition, when the quantum parameter of the Breit–Wheeler process increases by about 4 times, the maximum relative probability $R_{l_1{l}_2\left(max\right)}^{\left(\pm \right)}$ decreases slightly: $\left(R_{11\left(max\right)}^{\left(\pm \right)}{|}_{{\epsilon }_{iBW=10}}\right)/\phantom{\left(R_{11\left(max\right)}^{\left(\pm \right)}{|}_{{\epsilon }_{iBW=10}}\right)\left(R_{11\left(max\right)}^{\left(\pm \right)}{|}_{{\epsilon }_{iBW=2.41}}\right)}\left(R_{11\left(max\right)}^{\left(\pm \right)}{|}_{{\epsilon }_{iBW=2.41}}\right)\approx 0.75$. Figure 10 and Table 4 show that the maximum relative probability $H_{l_1{l}_2\left(max\right)}^{\left(\pm \right)}$ holds for the optimal number of absorbed photons of the wave $l_1=1,l_2=2$. At the same time, $H_{11\left(max\right)}^{\left(\pm \right)}\underset{\sim }{<}{H}_{12\left(max\right)}^{\left(\pm \right)}$. As the number of absorbed photons increases, the relative probability decreases. In addition, when the quantum parameter of the Breit–Wheeler process increases by about 4 times, the maximum relative probability $H_{12\left(max\right)}^{\left(f\right)}$ increases by 42 times: $\left(H_{12\left(max\right)}^{\left(\pm \right)}{|}_{{\epsilon }_{iBW=10}}\right)/\phantom{\left(H_{12\left(max\right)}^{\left(\pm \right)}{|}_{{\epsilon }_{iBW=10}}\right)\left(H_{12\left(max\right)}^{\left(\pm \right)}{|}_{{\epsilon }_{iBW=2.41}}\right)}\left(H_{12\left(max\right)}^{\left(\pm \right)}{|}_{{\epsilon }_{iBW=2.41}}\right)\approx 42$.

Figure 11 shows the dependence of the maximum relative probabilities $R_{11\left(max\right)}^{\left(\pm \right)}$ (104) and $H_{12\left(max\right)}^{\left(\pm \right)}$ (105) on the electron–positron pair outgoing angle and the quantum parameter of the Breit–Wheeler process at the optimal number of absorbed photons at the first and second vertices. Figure 11a shows that the maximum relative probability with the simultaneous registration of the electron–positron pair outgoing angle and the energy of the final electron $R_{11\left(max\right)}^{\left(\pm \right)}$ has a maximum value when the electron–positron pair outgoing angle is zero and the quantum parameter ${\epsilon }_{iBW}\approx {\epsilon }_{*}$. As the value of this quantum parameter increases, the relative probability $R_{11\left(max\right)}^{\left(\pm \right)}$ decreases. On the other hand, Figure 11b shows that the maximum relative probability with the simultaneous registration of the electron and electron–positron pair outgoing angles $H_{12\left(max\right)}^{\left(\pm \right)}$ increases significantly with an increase in the quantum parameter ${\epsilon }_{iBW}$. So, if the quantum parameter ${\epsilon }_{iBW}$ increases by four times, then the corresponding probability increases by two orders of magnitude. In this case, the outgoing angles of the electron–positron pair increase as ${\delta }_{\pm \left(d\right)}^{2*}\sim {\epsilon }_{iBW}^2$. It is important to note that the outgoing angle of the electron–positron pair uniquely determines the outgoing angle of the final electron (see expression (51)), as well as the energies of the final particles (see expressions (32) and (49)), i.e., quantum entanglement of the final particles occurs. At the same time, as the quantum parameter of the Breit–Wheeler process increases ${\epsilon }_{iBW}$, the energy of the final electron tends to be the energy of the initial electrons (64).

The expressions for relative probabilities (102)–(105) are significantly simplified under conditions when final particles fly out at minimal angles. Moreover, if ${\delta }_{f\left(d\right)}^2=0$, ${\delta }_{\pm \left(u\right)}^2={\delta }_{\pm \left(min\right)}^2$, then, for the functions $R_{l_1{l}_2}^{\left(f\right)}\left(x_{f\left(d\right)}^{min},{\delta }_{\pm \left(min\right)}^2\right)$ (102) and ${\mathrm{H}}_{l_1{l}_2}^{\left(f\right)}\left(0,{\delta }_{\pm \left(min\right)}^2\right)$ (103), the main contribution to the probability of the process is made by the process with the absorption of one photon of the wave in the first vertex (for the Compton effect, (67) and (73) with the Bessel functions $z_f=0$ and $l_1=1$). In this case, we obtain

$$R_{1l_2}^{\left(f\right)}\left(x_{f\left(d\right)}^{min},{\delta }_{\pm \left(min\right)}^2\right)=b_i^{\prime }{P}_{l_2}\left(z_{\pm \left(min\right)}\right),$$

(107)

$${\mathrm{H}}_{1l_2}^{\left(f\right)}\left(0,{\delta }_{\pm \left(min\right)}^2\right)={\epsilon }_{iC}{\left(1+{\epsilon }_{iC}\right)}^2{b_i}^{\prime }{P}_{l_2}\left(z_{\pm \left(min\right)}\right).$$

(108)

Here, the following is indicated:

$$b_i^{\prime }={\eta }^2\left[\left(1+{\epsilon }_{iC}\right)+{\displaystyle \frac{1}{\left(1+{\epsilon }_{iC}\right)}}\right]b_i.$$

(109)

In expressions (107) and (108), the functions $P_{l_2}\left(z_{\pm \left(min\right)}\right)$ have the form (74) with the argument of the Bessel functions equal to

$$z_{\pm \left(min\right)}\approx {\displaystyle \frac{4\eta }{{\epsilon }_{iC}\sqrt{1+{\eta }^2}}}\sqrt{\left(1+{\epsilon }_{iC}\right)\left[l_2-{\displaystyle \frac{4}{{\epsilon }_{iC}}}\left(1+{\displaystyle \frac{1}{{\epsilon }_{iC}}}\right)\right]}.$$

(110)

We emphasize that in expressions (107), (108), and (110), the number of photons absorbed at the second vertex $l_2$ is determined by expression (47). Expressions (107) and (108) show that these relative probabilities have the same dependence on the classical parameter $\eta$, but different dependences on the quantum parameter: the Compton effect. Given this, we can obtain the ratio of these relative probabilities, which depends only on the quantum parameter ${\epsilon }_{iC}$:

$${\gamma }^{\left(f\right)}\left({\epsilon }_{iC}\right)={\displaystyle \frac{{\mathrm{H}}_{11}^{\left(f\right)}\left(0,{\delta }_{\pm \left(min\right)}^2\right)}{R_{11}^{\left(f\right)}\left(x_{f\left(d\right)}^{min},{\delta }_{\pm \left(min\right)}^2\right)}}={\epsilon }_{iC}{\left(1+{\epsilon }_{iC}\right)}^2.$$

(111)

At the same time, the relative probability of the RTPP process with the simultaneous registration of the electron and pair outgoing angles at minimum angles exceeds the corresponding probability with the simultaneous registration of the electron outgoing angle and the pair energy. Moreover, this excess will be significant ${\gamma }^{\left(f\right)}\approx {\epsilon }_{iC}^3>>1$ for large quantum parameters of the Compton effect.

Now, consider the case where ${\delta }_{\pm \left(d\right)}^2=0$, ${\delta }_{f\left(u\right)}^2={\delta }_{f\left(min\right)}^2$. In this case, for the functions $R_{l_1{l}_2}^{\left(\pm \right)}\left(x_{\pm \left(d\right)}^{min},{\delta }_{f\left(min\right)}^2\right)$ (104) and ${\mathrm{H}}_{l_1{l}_2}^{\left(\pm \right)}\left(0,{\delta }_{f\left(min\right)}^2\right)$ (105), the main contribution to the relative probability is made by the process with the absorption of one photon of the wave at the second vertex, provided condition (48) is met (for the external field-stimulated Breit–Wheeler (68) and (73) argument of the Bessel functions $z_2=0$ and $l_2=1$). In this case, we obtain

$$R_{l_{1}1}^{\left(\pm \right)}\left(x_{\pm \left(d\right)}^{min},{\delta }_{f\left(min\right)}^2\right)=2b_i{\eta }^2{\epsilon }_{iBW}\left({\epsilon }_{iBW}-1\right)K_{l_1}\left(u_{f\left(min\right)},z_{f\left(min\right)}\right),$$

(112)

$${\mathrm{H}}_{l_{1}1}^{\left(\pm \right)}\left(0,{\delta }_{f\left(min\right)}^2\right)=32b_i{\eta }^2{\epsilon }_{iBW}^4\left({\epsilon }_{iBW}-1\right)K_{l_1}\left(u_{f\left(min\right)},z_{f\left(min\right)}\right).$$

(113)

Note that when obtaining expression (113) at ${\delta }_{\pm \left(d\right)}^2\to 0$ energy of the electron–positron pair on the lower branch, (32) is put into Taylor series up to second-order terms. In expression (113), the function $K_{l_1}\left(u_{f\left(min\right)},z_{f\left(min\right)}\right)$ is defined by expressions (67) and (70), where $u_{f\left(min\right)}$ and $z_{f\left(min\right)}$ are defined as

$$\begin{array}{c}{u}_{f\left(min\right)}={\displaystyle \frac{1}{\left({\epsilon }_{iBW}-1\right)}},\hfill \\ z_{f\left(min\right)}\approx {\displaystyle \frac{\eta }{\sqrt{1+{\eta }^2}}}{\displaystyle \frac{\sqrt{4l_1{\epsilon }_{iBW}\left({\epsilon }_{iW}-1\right)-1}}{4{\epsilon }_{iBW}\left({\epsilon }_{iBW}-1\right)}},\hfill \\ l_1\ge 1.\hfill \end{array}$$

(114)

Note that in expressions (112)–(114), the value of the quantum parameter ${\epsilon }_{iBW}$ satisfies the condition of (48). Expressions (112) and (113) show that these relative probabilities have the same dependence on the classical parameter $\eta$ but different dependences on the quantum parameter of the Breit–Wheeler process. Given this, we can obtain the ratio of these relative probabilities, which depends only on the quantum parameter ${\epsilon }_{iBW}$:

$${\gamma }^{\left(\pm \right)}\left({\epsilon }_{iBW}\right)={\displaystyle \frac{{\mathrm{H}}_{l_{1}1}^{\left(\pm \right)}\left(0,{\delta }_{f\left(min\right)}^2\right)}{R_{l_{1}1}^{\left(\pm \right)}\left(x_{\pm \left(d\right)}^{min},{\delta }_{f\left(min\right)}^2\right)}}=16{\epsilon }_{iBW}^3.$$

(115)

At the same time, the relative probability of the RTPP process with the simultaneous registration of the outgoing angles of the pair and the electron by the minimum angles exceeds the corresponding probability with the simultaneous registration of the electron outgoing angle and the energy of the pair. Moreover, this excess will be significant for large quantum parameters of the Breit–Wheeler process.

Expressions for resonant relative probabilities (107), (108) and (112), (113) are significantly simplified under the condition of (62), when the energy of the initial electrons significantly exceeds the characteristic Breit–Wheeler energy:

$${\epsilon }_{iBW}>>1\left({\epsilon }_{iC}>>1\right).$$

(116)

In this case, the resonant relative probabilities (107) and (108) will take the form

$$R_{1l_2}^{\left(f\right)}\left(x_{f\left(d\right)}^{min},{\delta }_{\pm \left(min\right)}^2\right)\approx {\displaystyle \frac{g_i}{{\epsilon }_{iC}}}{\left({\displaystyle \frac{m}{{\omega }_C}}\right)}^2{P}_{l_2}\left(z_{\pm \left(min\right)}\right),$$

(117)

$${\mathrm{H}}_{1l_2}^{\left(f\right)}\left(0,{\delta }_{\pm \left(min\right)}^2\right)\approx g_i{\epsilon }_{iC}^2{\left({\displaystyle \frac{m}{{\omega }_C}}\right)}^2{P}_{l_2}\left(z_{\pm \left(min\right)}\right).$$

(118)

Here, the following is indicated:

$$g_i={\displaystyle \frac{2{\left(4{\pi }^2\right)}^3{\eta }^2\left(1+{\eta }^2\right)}{P\left(\eta ,{\epsilon }_{iBW}\right)\mathrm{K}\left(\eta ,{\epsilon }_{iC}\right)}}\approx {\displaystyle \frac{1.23\times {10}^5{\eta }^2\left(1+{\eta }^2\right)}{P\left(\eta ,{\epsilon }_{iBW}\right)\mathrm{K}\left(\eta ,{\epsilon }_{iC}\right)}}.$$

(119)

The argument of the functions $P_{l_2}\left(z_{\pm \left(min\right)}\right)$ and the minimum angle between the electron and positron momenta of the pair take the form

$$z_{\pm \left(min\right)}\approx {\displaystyle \frac{2\eta }{\sqrt{1+{\eta }^2}}}\sqrt{{\displaystyle \frac{l_2}{{\epsilon }_{iBW}}}},{\delta }_{\pm \left(min\right)}^2\approx 4l_2{\epsilon }_{iBW}.$$

(120)

In this case, the energy of the initial electron is mainly converted into the energy of the electron–positron pair:

$$x_{f\left(d\right)}^{min}\approx {\displaystyle \frac{1}{{\epsilon }_{iC}}}<<1,2x_{\pm \left(u\right)}^{max}\approx 1-{\displaystyle \frac{1}{{\epsilon }_{iC}}}\approx 1.$$

(121)

It is important to note that if the Breit–Wheeler quantum parameter satisfies a more stringent condition

$$\sqrt{{\epsilon }_{iBW}}>>1,$$

(122)

then the argument of the functions $P_{l_2}\left(z_{\pm \left(min\right)}\right)$ in expressions (117) and (118) becomes small $\left(z_{\pm \left(min\right)}<<1\right)$. Because of this, the process with the absorption of one photon of the wave at the second vertex becomes most likely. Given this, the relative resonant probabilities (117) and (118) within the (122) condition takes the following form:

$$R_{11}^{\left(f\right)}\left(x_{f\left(d\right)}^{min},{\delta }_{\pm \left(min\right)}^2\right)\approx {\displaystyle \frac{1}{2}}{g}_i{\displaystyle \frac{{\eta }^2}{{\epsilon }_{iC}}}{\left({\displaystyle \frac{m}{{\omega }_C}}\right)}^2,$$

(123)

$$\begin{array}{c}\hfill {\mathrm{H}}_{11}^{\left(f\right)}\left(0,{\delta }_{\pm \left(min\right)}^2\right)\approx {\displaystyle \frac{1}{2}}{g}_i{\eta }^2{\epsilon }_{iC}^2{\left({\displaystyle \frac{m}{{\omega }_C}}\right)}^2\\ \hfill \left({\delta }_{\pm \left(min\right)}^2\approx {\epsilon }_{iC}>>1\right).\end{array}$$

(124)

This shows that the ratio of these probabilities in accordance with expression (111) has the form

$${\gamma }^{\left(f\right)}\approx {\epsilon }_{iC}^3>>1.$$

(125)

Now, let us consider the resonant relative probabilities $R_{l_{1}1}^{\left(\pm \right)}$ (112) and $H_{l_{1}1}^{\left(\pm \right)}$ (113) under the condition on the Breit–Wheeler quantum parameter (116). In this case, the main contribution to the function $K_{l_1}\left(u_{f\left(min\right)},z_{f\left(min\right)}\right)\approx 2{\eta }^2$ is made by the term $l_1=1$ ($u_{f\left(min\right)}\approx {\epsilon }_{iBW}^{-1}<<1$, $z_{f\left(min\right)}\sim {\epsilon }_{iBW}^{-1}<<1$). Therefore, the required relative resonant probabilities take the form

$$R_{11}^{\left(\pm \right)}\left(x_{\pm \left(d\right)}^{min},{\delta }_{f\left(min\right)}^2\right)={\displaystyle \frac{1}{4}}{g}_i{\eta }^2{\left({\displaystyle \frac{m}{{\omega }_C}}\right)}^2,$$

(126)

$${\mathrm{H}}_{11}^{\left(\pm \right)}\left(0,{\delta }_{f\left(min\right)}^2\right)=4g_i{\eta }^2{\epsilon }_{iBW}^3{\left({\displaystyle \frac{m}{{\omega }_C}}\right)}^2.$$

(127)

In this case, the outgoing angles of the final particles are given by the following expression:

$${\delta }_{\pm \left(d\right)}^2=0,{\delta }_{f\left(min\right)}^2\approx 4,$$

(128)

and the energy of the initial electron basically transforms into the energy of the final electron:

$$2x_{\pm \left(d\right)}^{min}={\displaystyle \frac{1}{{\epsilon }_{iBW}}}<<1,x_{f\left(u\right)}^{max}\approx 1-{\displaystyle \frac{1}{{\epsilon }_{iBW}}}\approx 1.$$

(129)

Note that the probability ratio of (112) to (126) is (115).

We obtain the ratio of the relative probabilities ${\mathrm{H}}_{11}^{\left(\pm \right)}$ (127) and ${\mathrm{H}}_{11}^{\left(f\right)}$ (124) under conditions of large values of the Breit–Wheeler quantum parameters (122):

$$\beta ={\displaystyle \frac{{\mathrm{H}}_{11}^{\left(\pm \right)}\left(0,{\delta }_{f\left(min\right)}^2\right)}{{\mathrm{H}}_{11}^{\left(f\right)}\left(0,{\delta }_{\pm \left(min\right)}^2\right)}}={\displaystyle \frac{1}{2}}{\epsilon }_{iBW}>>1.$$

(130)

Hence, it can be seen that for large values of the Breit–Wheeler quantum parameters, the probability of the RTPP process with the simultaneous registration of the outgoing angles of the final particles at minimum angles $\left({\delta }_{\pm }^2=0,{\delta }_{f\left(min\right)}^2\right)$ and the transition of the main energy of the initial electrons to the energy of the final electrons $\left(E_{f\left(max\right)}=E_i-{\omega }_{BW},E_{\pm \left(min\right)}={\omega }_{BW}/\phantom{{\omega }_{BW}2}2\right)$ is several orders of magnitude higher than the corresponding probability with the simultaneous registration of the outgoing angles of the final particles at minimum angles $\left({\delta }_f^2=0,{\delta }_{\pm \left(min\right)}^2\right)$ and the transition of the main energy of the initial electrons to the energy of the electron–positron pair $\left(2E_{\pm \left(max\right)}=E_i-E_{f\left(min\right)},E_{f\left(min\right)}\approx E_i/\phantom{E_i{\epsilon }_{iC}}{\epsilon }_{iC}\right)$.

Figure 12a shows graphs of the maximum relative probabilities $R_{11\left(max\right)}^{\left(f\right)}\times {\left({\omega }_C/\phantom{{\omega }_{C}m}m\right)}^2$ (107) and $H_{11\left(max\right)}^{\left(f\right)}\times {\left({\omega }_C/\phantom{{\omega }_{C}m}m\right)}^2$ (108) on the quantum parameter of the Compton effect ${\epsilon }_{iC}\ge 2{\epsilon }_{*}$ (see expression (48)) for minimum outgoing angles of the final electron $\left({\delta }_{f\left(d\right)}^2=0\right)$ and electron–positron pair $\left({\delta }_{\pm \left(min\right)}^2\right)$. It can be seen from this figure that the relative probability $R_{11\left(max\right)}^{\left(f\right)}$(with the simultaneous registration of the electron outgoing angle and pair energy) decreases quite rapidly with an increase in the quantum parameter of the Compton effect (see expressions (107), (117), and (123)). On the other hand, the relative probability $H_{11\left(max\right)}^{\left(f\right)}$ (with the simultaneous registration of the electron and electron–positron pair outgoing angles) increases rapidly with the growth of the quantum parameter of the Compton effect and then proceeds to a smooth increase (see expressions (108), (118), and (124)). Note that in this case, the energy of the initial electrons is mainly converted into the energy of the electron–positron pair $\left(E_{+}=E_{-}\approx E_i/\phantom{E_{i}2}2\right)$.

Figure 12b shows graphs of the maximum relative probabilities $R_{11\left(max\right)}^{\left(\pm \right)}\times {\left({\omega }_C/\phantom{{\omega }_{C}m}m\right)}^2$ (112) and $H_{11\left(max\right)}^{\left(\pm \right)}\times {\left({\omega }_C/\phantom{{\omega }_{C}m}m\right)}^2$ (113) on the quantum parameter of the Breit–Wheeler ${\epsilon }_{iBW}\ge {\epsilon }_{*}/\phantom{{\epsilon }_{*}2}2$ process (see expression (48)) at the minimum outgoing angles of the electron–positron pair $\left({\delta }_{\pm \left(d\right)}^2=0\right)$ of the final electron $\left({\delta }_{f\left(min\right)}^2\right)$. This figure shows that the relative probability $R_{11\left(max\right)}^{\left(\pm \right)}$(with the simultaneous registration of the pair outgoing angle and electron energy) gradually decreases with increasing quantum parameter of the Breit–Wheeler process (see expressions (112) and (126)). On the other hand, the relative probability $H_{11\left(max\right)}^{\left(\pm \right)}$(with the simultaneous registration of the outgoing angles of the electron–positron pair and electron) increases quite rapidly with the growth of the quantum parameter of the Breit–Wheeler process (see expressions (113) and (127)). Note that in this case, the energy of the initial electrons is mainly converted into the energy of the final electron $\left(E_f\approx E_i\right)$.

Figure 13 shows the dependence of the maximum relative probabilities $H_{11\left(max\right)}^{\left(f\right)}\times {\left({\omega }_C/\phantom{{\omega }_{C}m}m\right)}^2$ (108) (case a) and $H_{11\left(max\right)}^{\left(\pm \right)}\times {\left({\omega }_C/\phantom{{\omega }_{C}m}m\right)}^2$ (113) (case b) on the classical parameter $\eta$ (1) and quantum parameters of the Compton effect and the Breit–Wheeler process at minimum outgoing angles of the final particles. It can be seen from these graphs that as the parameters grow $\eta$, the ${\epsilon }_{iC}\left({\epsilon }_{iBW}\right)$ relative resonant probabilities also increase significantly, and the function increases most strongly $H_{11\left(max\right)}^{\left(\pm \right)}$. Indeed, we take the ratio of these functions at two points for $\eta =10,{\epsilon }_{iC}=40\left({\epsilon }_{iBW}=10\right)$, and $\eta =1,{\epsilon }_{iC}=4.84\left({\epsilon }_{iBW}=1.21\right)$. Then, from Figure 13, we obtain

$$H_{11\left(max\right)}^{\left(f\right)}{|}_{\begin{array}{c}\eta =10\hfill \\ {\epsilon }_{iC}=40\hfill \end{array}}/\phantom{H_{11\left(max\right)}^{\left(f\right)}{|}_{\begin{array}{c}\eta =10\hfill \\ {\epsilon }_{iC}=40\hfill \end{array}}{H}_{11\left(max\right)}^{\left(f\right)}{|}_{\begin{array}{c}\eta =1\hfill \\ {\epsilon }_{iC}=4.84\hfill \end{array}}}{H}_{11\left(max\right)}^{\left(f\right)}{|}_{\begin{array}{c}\eta =1\hfill \\ {\epsilon }_{iC}=4.84\hfill \end{array}}\approx 2.07\times {10}^4,$$

(131)

$$H_{11\left(max\right)}^{\left(\pm \right)}{|}_{\begin{array}{c}\eta =10\hfill \\ {\epsilon }_{iBW}=10\hfill \end{array}}/\phantom{H_{11\left(max\right)}^{\left(\pm \right)}{|}_{\begin{array}{c}\eta =10\hfill \\ {\epsilon }_{iBW}=10\hfill \end{array}}{H}_{11\left(max\right)}^{\left(\pm \right)}{|}_{\begin{array}{c}\eta =1\hfill \\ {\epsilon }_{iBW}=1.21\hfill \end{array}}}{H}_{11\left(max\right)}^{\left(\pm \right)}{|}_{\begin{array}{c}\eta =1\hfill \\ {\epsilon }_{iBW}=1.21\hfill \end{array}}\approx 0.76\times {10}^6.$$

(132)

5. Discussion

In the previous section, we demonstrated that the excess resonant probability of the studied process, compared to the total probability of the Compton effect stimulated by an external field, significantly depends on the quantum parameter ${\epsilon }_{iBW}\left({\epsilon }_{iC}\right)$, the number of absorbed photons of the wave at the first and second vertices, and the characteristic energy of the Compton effect (multiplier ${\left(m/\phantom{m{\omega }_C}{\omega }_C\right)}^2$).

Let us determine the characteristic energy of the Compton effect (3) for various frequencies and intensities in case of head-on collision with an external wave (${\theta }_i=\pi$, see (24)). For the characteristic energy of the Compton effect to decrease as the frequency and intensity of a wave increases, we assume that the classical parameter $\eta$ is constant.

$${\omega }_C\approx \left\{\begin{array}{c}43.5\mathrm{GeV},\mathrm{if}\omega =3\mathrm{eV},I=1.675·{10}^{19}{\mathrm{Wcm}}^{-2}\left(\eta =1\right),\\ 130.56\mathrm{MeV},\mathrm{if}\omega =1\mathrm{keV},I=1.86·{10}^{24}{\mathrm{Wcm}}^{-2}\left(\eta =1\right),\\ 6.528\mathrm{MeV},\mathrm{if}\omega =20\mathrm{keV},I=7.44·{10}^{26}{\mathrm{Wcm}}^{-2}\left(\eta =1\right).\end{array}\right.$$

(133)

Note that ${\omega }_{BW}=4{\omega }_C$ (see expression (3)). From expression (133), it follows that in the optical range of frequencies, the characteristic energy of the Compton effect is quite high (tens of GeV). Moving into the region of X-ray waves and high field strengths, the value ${\omega }_C$ decreases quite quickly.

Given expression (133), as well as Figure 12, we can obtain the data presented in

Table 5. The maximum values of the resonant relative probability $H_{11\left(max\right)}^{\left(f\right)}$ (108) and the corresponding values of the energies and outgoing angles of the final particles for different characteristic energies of the Compton effect and energies of the initial electrons.

${\mathit{\omega }}_{\mathit{C}},\mathit{MeV}$	${\mathit{E}}_{\mathit{i}},\mathit{GeV}$	${\mathit{\delta }}_{\pm \left(min\right)}^2$	${\mathit{E}}_{\pm },\mathit{GeV}$	${\mathit{E}}_{\mathit{f}},\mathit{GeV}$	${\mathit{H}}_{11\left(max\right)}^{\left(\mathit{f}\right)}$
$130.56$	$2.175\times {10}^4$	$496.98$	$1.085\times {10}^4$	$43.41$	$9.55\times {10}^{-2}$
	$8.70\times {10}^4$	$1997.00$	$4.348\times {10}^4$	$43.48$	$1.050$
	$0.63$	$0.0022$	$0.261$	$0.108$	$16.571$
	$65.28$	$496.98$	$32.58$	$0.1303$	$1.061\times {10}^4$
$6.528$	$261.12$	$1997.00$	$130.50$	$0.1305$	$1.165\times {10}^5$
	$3.153\times {10}^{-2}$	$0.0022$	$1.306\times {10}^{-2}$	$5.408\times {10}^{-3}$	$6.630\times {10}^3$
	$3.264$	$496.98$	$1.629$	$6.515\times {10}^{-3}$	$4.245\times {10}^6$
	$13.06$	$1997.00$	$6.525$	$6.525\times {10}^{-3}$	$4.660\times {10}^7$

and

Table 6. The maximum values of the resonant relative probability $H_{11\left(max\right)}^{\left(\pm \right)}$ (113) and the corresponding values of the energies and outgoing angles of the final particles for different characteristic energies of the Compton effect and energies of the initial electrons.

${\mathit{\omega }}_{\mathit{C}},\mathit{MeV}$	${\mathit{E}}_{\mathit{i}},\mathit{GeV}$	${\mathit{\delta }}_{\mathit{f}\left(min\right)}^2$	${\mathit{E}}_{\mathit{f}},\mathit{GeV}$	${\mathit{E}}_{\pm },\mathit{GeV}$	${\mathit{H}}_{11\left(max\right)}^{\left(\pm \right)}$
$43.5\times {10}^3$	$2.101\times {10}^2$	$0.0517$	$36.11$	$87.0$	$4.098\times {10}^{-4}$
	$2.480\times {10}^2$	$7.8754$	$73.95$	$87.0$	$1.001$
	$2.175\times {10}^4$	$4.032$	$2.158\times {10}^4$	$87.0$	$65.012$
	$8.70\times {10}^4$	$4.008$	$8.68\times {10}^4$	$87.0$	$4.191\times {10}^3$
$130.56$	$0.63$	$0.0517$	$0.108$	$0.261$	$45.490$
	$65.28$	$4.032$	$64.76$	$0.261$	$1.055\times {10}^7$
	$261.12$	$4.008$	$260.60$	$0.261$	$4.653\times {10}^8$
$6.528$	$3.153\times {10}^{-2}$	$0.0517$	$5.418\times {10}^{-3}$	$1.306\times {10}^{-2}$	$1.820\times {10}^4$
	$3.264$	$4.032$	$3.238$	$1.306\times {10}^{-2}$	$4.219\times {10}^9$
	$13.06$	$4.008$	$13.03$	$1.306\times {10}^{-2}$	$1.861\times {10}^{11}$

for the case where the characteristic energy of the Compton effect is fixed, and the energy of the initial electron is chosen from the condition $E_i\ge 2{\epsilon }_{*}{\omega }_C\approx 4.83{\omega }_C$ (see expression (48)). At the same time, Table 5 (see Figure 12a) presents the results for the relative probability $H_{11\left(max\right)}^{\left(f\right)}$ (at the minimum outgoing angles of the final electron $\left({\delta }_{f\left(d\right)}^2=0\right)$ and the electron–positron pair $\left({\delta }_{\pm \left(min\right)}^2\right)$). Table 6 (see Figure 12b) shows the results for the relative probability $H_{11\left(max\right)}^{\left(\pm \right)}$ (at the minimum outgoing angles of the electron–positron pair $\left({\delta }_{\pm \left(d\right)}^2=0\right)$ and the final electron $\left({\delta }_{f\left(min\right)}^2\right)$).

In the region of optical frequencies $\left(\omega =3\mathrm{eV}\right)$ and wave intensities ($I=1.675·{10}^{19}{\mathrm{Wcm}}^{-2}$), the characteristic energy of the Compton effect is quite large $\left({\omega }_C\approx 43.5\mathrm{GeV}\right)$. Because of this, in a sufficiently wide range of very high energies of the initial electrons $\left(E_i\le 22\mathrm{TeV}\right)$, the relative probability $H_{11\left(max\right)}^{\left(f\right)}$ is small $\left(H_{11\left(max\right)}^{\left(f\right)}\le 0.1\right)$. It is only for very large energies of the initial electrons $E_i=87\mathrm{TeV}$ (which are evenly distributed between the electron and positron of the pair) that the probability of the RTPP process becomes of the same order with the total probability of the Compton effect stimulated by an external field $\left(H_{11\left(max\right)}^{\left(f\right)}\approx 1\right)$. On the other hand, in the region of optical wave frequencies, the relative probability $H_{11\left(max\right)}^{\left(\pm \right)}$ becomes of the same order as the total probability of the Compton effect stimulated by an external field for the initial electron energies $E_i\underset{\sim }{>}0.25\mathrm{TeV}$. As the initial electron energy increases, this probability increases rapidly and $E_i=87\mathrm{TeV}$ exceeds the total probability of the Compton effect stimulated by an external field by three orders of magnitude $\left(H_{11\left(max\right)}^{\left(\pm \right)}\approx 4.2\times {10}^3\right)$. Thus, in the region of optical frequencies and wave intensities $I\underset{\sim }{<}{10}^{19}{\mathrm{Wcm}}^{-2}$ with initial energies $E_i\underset{\sim }{>}1\mathrm{TeV}$, the RTPP process with the generation of the electron–positron pair at zero angle and the scattering of an electron at an angle ${\delta }_{f\left(min\right)}^2$ will be dominant. In this case, the energy of the initial electrons will mainly convert into the energy of the final electron and the remaining energy will be evenly distributed between the electron and the positron of the pair.

In the region of X-ray frequencies $\left(\omega =1\mathrm{keV}\right)$ and sufficiently high wave intensities $\left(I=1.86·{10}^{24}{\mathrm{Wcm}}^{-2}\right)$, the characteristic energy of the Compton effect becomes significantly lower $\left({\omega }_C\approx 130.56\mathrm{MeV}\right)$. Therefore, for the initial electron energies in the interval $0.63\mathrm{GeV}\le E_i\le 261\mathrm{GeV}$, the probability of the RTPP process significantly (by several orders of magnitude) exceeds the total probability of the Compton effect stimulated by an external field. At the same time, Table 5 and Table 6 show that $16.6\le H_{11\left(max\right)}^{\left(f\right)}\le 1.2\times {10}^5$ and $45.49\le H_{11\left(max\right)}^{\left(\pm \right)}\le 4.65\times {10}^8$. Thus, for the energy of the initial electrons $E_i=0.63\mathrm{GeV}$, the relative probabilities $H_{11\left(max\right)}^{\left(f\right)}$ and $H_{11\left(max\right)}^{\left(\pm \right)}$ differ by several times $H_{11\left(max\right)}^{\left(\pm \right)}/\phantom{H_{11\left(max\right)}^{\left(\pm \right)}{H}_{11\left(max\right)}^{\left(f\right)}\approx 2.7}{H}_{11\left(max\right)}^{\left(f\right)}\approx 2.7$. However, as the energy of the initial electrons increases, this difference quickly increases to three orders of magnitude.

For higher X-ray frequencies $\left(\omega =20\mathrm{keV}\right)$ and wave intensities ($I=7.44·{10}^{26}\mathrm{W}/{\mathrm{cm}}^2$), the characteristic energy of the Compton effect becomes small $\left({\omega }_C\approx 6.528\mathrm{MeV}\right)$. Therefore, in this case, even for small initial electron energies $31\mathrm{MeV}\le E_i\le 13\mathrm{GeV}$, the probability of the RTPP process is much higher than the total probability of the Compton effect stimulated by an external field. So, Table 5 and Table 6 show that $6.6\times {10}^3\le H_{11\left(max\right)}^{\left(f\right)}\le 4.7\times {10}^7$ and $1.82\times {10}^4\le H_{11\left(max\right)}^{\left(\pm \right)}\le 1.86\times {10}^{11}$.

Thus, the RTPP process with the generation of the electron–positron pair at zero angle and the scattering of the electron by angle ${\delta }_{f\left(min\right)}^2$ will be dominant. At the same time, when the Breit–Wheeler quantum parameter is much larger than unity, the energy of the initial electrons will mainly pass into the energy of the final electron and the remaining part of the ultrarelativistic energy will be evenly distributed between the electron and the positron of the pair.

Figure 14 shows the dependences of the relative probabilities $H_{1l_{2min}\left(max\right)}^{\left(f\right)}$ (108) (case a) and $H_{11\left(max\right)}^{\left(\pm \right)}$ (113) (case b) on the characteristic energy of the Compton effect ${\omega }_C$ (3) for three values of the initial electron energy. Note that for a constant value of the parameter $\eta =1$, the characteristic energy of the Compton effect ${\omega }_C$ increases with a decrease in the frequency and intensity of the electromagnetic wave. It was assumed that ${\omega }_C>>m$ (see the inequality (25) for the quantum parameter ${\epsilon }_{iC}$). The interval of change in the characteristic energy of the Compton effect was chosen from ${\omega }_C=5\mathrm{MeV}$$\left(\omega \approx 26\mathrm{keV},I\approx 1.3\times {10}^{27}{\mathrm{Wcm}}^{-2}\right)$ to ${\omega }_C=7.4\mathrm{GeV}$$\left(\omega \approx 18\mathrm{eV},I\approx 5.85\times {10}^{20}{\mathrm{Wcm}}^{-2}\right)$. It can be seen from these figures that for the case when $l_1=l_2=1$, these dependences are approximately linear on a logarithmic scale. Moreover, for the minimum values of the characteristic energy of the Compton effect (maximum frequencies and intensities of the electromagnetic wave), the relative probabilities $H_{11\left(max\right)}^{\left(f\right)}$ and $H_{11\left(max\right)}^{\left(\pm \right)}$ take maximum values. With an increase in the characteristic energy of the Compton effect (a decrease in the frequency and intensity of the wave), the corresponding relative probabilities decrease approximately linearly, except for the energy case $E_i=50\mathrm{MeV}$ (see Figure 14a), where there are five characteristic regions of relative probability variation $H_{1l_{2min}\left(max\right)}^{\left(f\right)}$ that correspond to the following values: $l_{2min}=1,2,3,4,5$. At the same time, with an increase in the number of absorbed photons of the wave, the relative probability $H_{1l_{2min}\left(max\right)}^{\left(f\right)}$ decreases sharply. For energies $E_i=5\mathrm{GeV},500\mathrm{GeV}$, the number of photons of the wave $l_{2min}=1$. Figure 14 shows that for each energy of the initial electrons, there is a region of change in the characteristic energy of the Compton effect, in which the corresponding probabilities of the RTPP process exceed the total probability of the Compton effect stimulated by an external field. Moreover, with an increase in the energy of the initial electrons, this region of ${\omega }_C$, as well as the relative probabilities $H_{1l_{2min}\left(max\right)}^{\left(f\right)}$ and $H_{11\left(max\right)}^{\left(\pm \right)}$, increase. So, for ${\omega }_C=5\mathrm{MeV}$ and the energies of the initial electrons $E_i=50\mathrm{MeV},5\mathrm{GeV},500\mathrm{GeV}$, the relative probabilities take the following values: $H_{11\left(max\right)}^{\left(f\right)}\approx 1.7\times {10}^4;2.4\times {10}^7;9.0\times {10}^{10}$ and $H_{11\left(max\right)}^{\left(\pm \right)}\approx 1.3\times {10}^4;3.0\times {10}^9;1.1\times {10}^{15}$. From this, it can be seen that for small values of the Breit–Wheeler parameter $\left({\epsilon }_{iBW}=2.5;E_i=50\mathrm{MeV}\right)$, the corresponding relative probabilities are of the same order of magnitude $\left(H_{11\left(max\right)}^{\left(f\right)}\underset{\sim }{>}{H}_{11\left(max\right)}^{\left(\pm \right)}\sim {10}^4\right)$. However, with an increase in the Breit–Wheeler quantum parameter $\left({\epsilon }_{iBW}>>1\right)$ (with an increase in the energy of the initial electrons), the relative probability $H_{11\left(max\right)}^{\left(\pm \right)}>>H_{11\left(max\right)}^{\left(f\right)}$ (in accordance with expression (130)).

It should also be noted that along with high-probability processes of the outgoing of the final particles at minimum angles, there are also high-probability processes with the outgoing of the final particles at angles far from the minimum (see the peaks of relative probability distributions $H_{21}^{\left(f\right)}$ and $H_{12}^{\left(\pm \right)}$ in Figure 7 and Figure 10). Indeed, from Figure 7 and Figure 10, as well as Table 2 and Table 4, it follows that for ${\epsilon }_{iBW}=2.41$ and $E_i=50\mathrm{MeV}$, the corresponding relative probabilities take the following values: $H_{21\left(max\right)}^{\left(f\right)}\approx 1.62\times {10}^4$ (at the outgoing angles of the final particles ${\delta }_{f\left(d\right)}^{2*}=34.95$ and ${\delta }_{\pm \left(u\right)}^{2*}=5.74$) and $H_{12\left(max\right)}^{\left(\pm \right)}\approx 1.57\times {10}^4$ (at the outgoing angles of the final particles ${\delta }_{f\left(u\right)}^{2*}=2.79$ and ${\delta }_{\pm \left(d\right)}^{2*}=8.35$). Thus, for small electron energies and values of the Breit–Wheeler quantum parameter $\left(E_i=50\mathrm{MeV},{\epsilon }_{iBW}=2.41\right)$ in strong X-ray fields, relative probabilities $\left(H_{12\left(max\right)}^{\left(\pm \right)}\sim H_{21\left(max\right)}^{\left(f\right)}\right)\sim \left(H_{11\left(max\right)}^{\left(\pm \right)}\sim H_{11\left(max\right)}^{\left(f\right)}\right)\sim {10}^4$ are obtained. At the same time, for large energies and values of the Breit–Wheeler quantum parameter $\left(E_i=500\mathrm{GeV},{\epsilon }_{iBW}={10}^5\right)$ in strong X-ray fields, the relative probabilities are $H_{21\left(max\right)}^{\left(f\right)}\approx 2.3\times {10}^{11}$ (at the outgoing angles of the final particles ${\delta }_{f\left(d\right)}^{2*}\approx 3.7\times {10}^9$ and ${\delta }_{\pm \left(u\right)}^{2*}\approx {10}^5$) and $H_{12\left(max\right)}^{\left(\pm \right)}\approx 2.8\times {10}^{15}$ (at the outgoing angles of the final particles ${\delta }_{f\left(u\right)}^{2*}\approx 2.23$ and ${\delta }_{\pm \left(d\right)}^{2*}\approx 9.2\times {10}^8$). Thus, $H_{21\left(max\right)}^{\left(f\right)}\sim H_{11\left(max\right)}^{\left(f\right)}\sim {10}^{11}$ and $H_{12\left(max\right)}^{\left(\pm \right)}\sim H_{11\left(max\right)}^{\left(\pm \right)}\sim {10}^{15}$.

It is important to note that Oleinik resonances occur not only in the field of a plane monochromatic wave but also in the field of a plane pulsed wave, provided that the pulse time $\tau$ significantly exceeds the wave oscillation period $\tau >>{\omega }^{-1}$ [31]. However, for very short pulses $\tau \sim {\omega }^{-1}$, Oleinik resonances may not occur. In this paper, an idealized case of a plane monochromatic electromagnetic wave was considered. In a real experiment, as well as in the vicinity of pulsars and magnetars, the electromagnetic wave is inhomogeneous in space and time. The study of Oleinik resonances in such fields is a rather complex task and can be performed only by numerically solving the corresponding mathematical problem. The solution of the resonant problem in the field of a plane monochromatic wave, however, allows us to solve several important problems. First, we can determine the basic physical parameters of the problem (the characteristic energy of the process, the corresponding quantum parameters), which determine the resonant energy of the final particles, as well as the value of the resonant probability of the process. Secondly, we can obtain analytical expressions for the resonant differential probability of the process. Note that all this is very important for the subsequent numerical analysis of this resonant process in an inhomogeneous electromagnetic field.

The present study of the resonant process was carried out for the circular polarization of an external electromagnetic wave. At the same time, the cases $\delta =\pm 1$ in these resonant kinematics give the same result for the resonant energies of final particles and the corresponding differential cross-sections. The case of linear wave polarization will not qualitatively change the results obtained for resonant differential cross-sections. However, mathematically, this is a more complicated case. We plan to consider the case of linear polarization in future publications.

We also note that observations of the RTPP process in modern experimental facilities require fluxes of ultrarelativistic electrons of sufficiently high energies $\left(E_i\underset{\sim }{>}250\mathrm{GeV}\right)$. However, in the universe, in particular, near pulsars and magnetars, high-energy electron fluxes colliding with a strong X-ray wave are possible. At the same time, near such objects in strong X-ray fields, cascades of resonant QED processes are possible, such as the resonant spontaneous bremsstrahlung radiation during the scattering of ultrarelativistic electrons by nuclei [72], the Bethe–Heitler resonant process [73], the resonant Breit–Wheeler process [74], the resonant Compton effect [75], etc. These processes are interconnected and can generate fluxes of high-energy, gamma-quanta, ultrarelativistic electrons and positrons. Thus, the obtained results can be used to explain narrow fluxes of high-energy gamma-quanta near neutron stars, such as double X-ray systems operating on accretion [121,122], X-ray/gamma pulsars operating on rotation [123,124], and magnetars operating on a magnetic field [125,126].

It is important to note that the laser-stimulated (nonlinear) Compton scattering and Breit–Wheeler mechanism represent very important elementary processes within numerical simulations of laser–plasma interactions via particle-in-cell codes. At the same time, the trident effect is taken into account in the form of a sequential course of these two processes [70]. We emphasize that this numerical scheme may not always be fair. If the resonant conditions described above are implemented in laser–plasma interactions, then the resonant trident pair production process may be dominant.

In conclusion, we formulate the main results:

• Under resonant conditions, the intermediate virtual gamma-quantum becomes real. As a result, the initial second-order process by the fine structure constant in the wave field effectively splits into two first-order processes. In the first vertex, the Compton effect stimulated by an external field occurs, and, in the second vertex, the Breit–Wheeler process stimulated by an external field takes place.
• Due to the laws of the conservation of energy and momentum, the RTPP process can take place in two cases. In the first case, the energies and outgoing angles of the final particles are determined by the Compton effect stimulated by an external field. In the second case, the energies and outgoing angles of the final particles are determined by the Breit–Wheeler process stimulated by an external field. It is important to note that in the RTPP process, electrons and positrons of the pair are generated with equal energies.
• In the first case, the outgoing angle of the final electron (ultrarelativistic parameter ${\delta }_{f\left(d\right)}^2$), the quantum parameter of the Compton effect $\left({\epsilon }_{iC}\right)$, and the number of absorbed photons of the wave at the first vertex $\left(l_1\right)$ determine the outgoing angle of the electron–positron pair (ultrarelativistic parameter ${\delta }_{\pm \left(u\right)}^2$), as well as the energies of the electron (on the lower branch), electron–positron pair (on the upper branch), and the number of absorbed photons of the wave at the second vertex $\left(l_2\right)$. In this case, if the quantum parameter ${\epsilon }_{iC}>>1$, then the energy of the initial electron basically transforms into the energy of the electron–positron pair.
• In the second case, the outgoing angle of the electron–positron pair (an ultrarelativistic parameter ${\delta }_{\pm \left(d\right)}^2$), as well as the quantum parameter of the Breit–Wheeler process $\left({\epsilon }_{iBW}\right)$ and the number of absorbed photons of the wave at the second vertex $\left(l_2\right)$, determine the outgoing angle of the electron (an ultrarelativistic parameter ${\delta }_{f\left(u\right)}^2$), as well as the energies of the electron–positron pair (on the lower branch) and the electron (on the upper branches). In this case, if the quantum parameter ${\epsilon }_{iBW}>>1$, then the energy of the initial electron transforms to the energy of the final electron.
• The value of the differential probability of the RTPP process essentially depends on the quantum parameters of the Compton effect $\left({\epsilon }_{iC}\right)$ and the Breit–Wheeler process $\left({\epsilon }_{iBW}\right)$, the characteristic energy of the Compton effect $\left({\omega }_C\right)$, and the number of photons absorbed at the first and second vertices of the wave.
• In the region of optical frequencies and small electromagnetic wave intensities ($\omega \sim 1\mathrm{eV},\mathrm{I}{\underset{\sim }{<}10}^{20}{\mathrm{Wcm}}^{-2}$), the relative differential probability of the RTPP process (in units of the total probability of the Compton effect stimulated by an external field) exceeds unity for sufficiently large initial electron energies $\left(E_i\underset{\sim }{>}250\mathrm{GeV}\right)$. For very high initial electron energies $\left(E_i=87\mathrm{TeV}\right)$, the RTPP process becomes dominant $\left(H_{11\left(max\right)}^{\left(\pm \right)}\approx 4.2\times {10}^3\right)$ (see Figure 12 and Figure 13, as well as Table 6).
• In a sufficiently wide range of frequencies and intensities of a strong X-ray wave from $\omega \approx 18\mathrm{eV},I\approx 5.85\times {10}^{20}{\mathrm{Wcm}}^{-2}$ to $\omega \approx 26\mathrm{keV},I\approx 1.3\times {10}^{27}{\mathrm{Wcm}}^{-2}$, as well as in a wide range of the initial electron energies $\left(E_i\underset{\sim }{>}50\mathrm{MeV}\right)$, the RTPP process becomes dominant and can exceed the total probability of the Compton effect stimulated by an external field by many orders of magnitude (see Figure 14).
• In the RTPP process, quantum entanglement of the final particles occurs when the measurement of the electron (electron–positron pair) outgoing angle uniquely determines the energies of the final particles and the electron–positron pair (electron) outgoing angle. Moreover, this effect can significantly (by many orders of magnitude) exceed the corresponding Compton effect stimulated by an external field.