The Generation of High-Energy Electron–Positron Pairs during the Breit–Wheeler Resonant Process in a Strong Field of an X-ray Electromagnetic Wave

Abstract

The Breit–Wheeler resonant process was theoretically studied in a strong X-ray electromagnetic wave field under conditions when the energy of one of the initial high-energy gamma quanta passes into the energy of a positron or electron. These conditions were realized when the energy of a high-energy gamma quantum significantly exceeded the characteristic Breit–Wheeler energy, which was determined using the parameters of the electromagnetic wave and the initial setup. Analytical formulas for the resonant differential cross-section were obtained. It is shown that the resonant differential cross-section significantly depends on the ratio between the energies of the initial gamma quanta and the characteristic Breit–Wheeler energy. With a decrease in the characteristic Breit–Wheeler energy, the resonant cross-section increases sharply and may exceed the corresponding non-resonant cross-section by several orders of magnitude. The results make it possible to obtain narrow beams of ultrarelativistic positrons (electrons) with energies of the order ∼${10}^2$ GeV and could also be used to explain high-energy fluxes of positrons (electrons) near neutron stars, as well as to simulate QED processes in laser fusion.

1. Introduction

Obtaining narrow beams of ultrarelativistic positrons (electrons) is an important scientific problem. The Breit–Wheeler process (BWP) is well-known and involves the production of electron–positron pairs through the interaction of two gamma quanta [1]. In this process, the energy spectrum of the generated electrons and positrons is continuous (the energy of the initial gamma quanta is distributed between the electron and the positron). With the advent of high-intensity lasers [2,3,4,5,6,7,8,9,10,11], the study of quantum electrodynamics (QED) processes in external electromagnetic fields has been widely explored (see, for example, reviews [12,13,14,15,16,17] and articles [18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48]). Moreover, these processes can occur both resonantly and non-resonantly. The resonant behavior of high-order QED processes is associated with the entering of an intermediate particle into the mass shell. These resonances were first investigated by Oleinik [18,19]. As a result, higher-order processes by the fine-structure constant effectively reduce into several consecutive lower-order processes (see reviews [5,13,14]), as well as recent articles [26,27,28,29,42,43]). It is important to note that the probability of resonant processes can significantly exceed the corresponding probabilities of non-resonant processes.

Currently, there is a considerable body of work dedicated to studying BWP in an external electromagnetic field (see, for example, [30,31,32,33,34,35,36,37,38,39,40,41,42,43]). It is important to distinguish between the external field-stimulated BWP (a first-order process with the fine-structure constant) and the external field-assisted BWP (a second-order process with the fine-structure constant). Note that the production of narrow beams of ultrarelativistic electron–positron pairs in an external electromagnetic field has been studied in the articles in [27,28,29,41,43]. At the same time, in the articles in [27,28,29], the resonant process of photogeneration of ultrarelativistic electron–positron pairs in the field of the nucleus in a weak and strong electromagnetic field was considered. The article in [41] studied the Breit–Wheeler resonant process in a weak electromagnetic field (when the classical relativistically invariant parameter $\eta$ (4) is much less than one).

In the recent article from the authors of [43], the resonant external field-assisted BWP in a strong field of a monochromatic circularly polarized wave, propagating along the z axis, was studied. However, in this paper, in contrast to previous articles devoted to the Breit–Wheeler resonant process in a strong electromagnetic field, we will study the case when the resonant energy of a positron (electron) is close to the energy of a high-energy initial gamma quantum. It is important to note that under the conditions of resonance and the absence of interference from different reaction channels, the original second-order process effectively reduces into two first-order processes: the external field-stimulated Breit–Wheeler process (EFSBWP), and the external field-stimulated Compton effect (EFSCE) [12] (see Figure 1). The resonant BWP for high-energy initial gamma quanta and ultrarelativistic final electron–positron pairs was considered

$$ħ{\omega }_{1,2}\gg m{c}^2,E_{\pm }\gg m{c}^2.$$

(1)

Here, $ħ{\omega }_{1,2}$ and $E_{\pm }$ are the energies of the initial gamma quanta and the final positron or electron. At the same time, all particles (the initial gamma quanta and the final electron–positron pair) move in a narrow cone away from the direction of wave propagation

$${\theta }_{j\pm }\equiv \angle ({\mathbf{k}}_j,{\mathbf{p}}_{\pm })\ll 1,{\theta }_i\equiv \angle ({\mathbf{k}}_1,{\mathbf{k}}_2)\ll 1,$$

(2)

$${\theta }_j\equiv \angle ({\mathbf{k}}_j,\mathbf{k})\sim 1,j=1,2;{\theta }_{\pm }\equiv \angle ({\mathbf{p}}_{\pm },\mathbf{k})\sim 1,{\theta }_{1,2}\approx {\theta }_{\pm }\equiv \theta .$$

(3)

In expression (3), ${\mathbf{p}}_{\pm }$ are momenta of the positron or the electron. It was assumed that the classical relativistic-invariant parameter satisfies the following condition [12,27,28,29]:

$$\eta =\frac{eFƛ}{m{c}^2}\ll {\eta }_{max},{\eta }_{max}=\min \left(\frac{E_{\pm }}{m{c}^2}\right).$$

(4)

Here, e and m are the charge and mass of the electron, F and $ƛ=c/\omega$ are the electric field strength and wavelength, and $\omega$ is the frequency of the wave.

It was shown that the resonant energy of positron and electron has a discrete spectrum. The outgoing angles of the electron and positron are uniquely related and determine their resonant energies. In addition, the resonant energies of the electron–positron pair for each reaction channel are determined by the quantum parameters ${\epsilon }_{1C\left(r^{\prime }\right)}$ (at the first vertex for the EFSCE) and ${\epsilon }_{2BW\left(r\right)}$ (at the second vertex for the EFSBWP):

$${\epsilon }_{1C\left(r^{\prime }\right)}=r^{\prime }{\epsilon }_{1C},{\epsilon }_{1C}=\frac{ħ{\omega }_1}{ħ{\omega }_C},ħ{\omega }_C=\frac{1}{4}ħ{\omega }_{BW},$$

(5)

$${\epsilon }_{2BW\left(r\right)}=r{\epsilon }_{2BW}\ge 1,{\epsilon }_{2BW}=\frac{ħ{\omega }_2}{ħ{\omega }_{BW}},$$

(6)

$$ħ{\omega }_{BW}=\frac{{\left(m{c}^2\right)}^2(1+{\eta }^2)}{ħ\omega {sin}^2(\theta /2)}.$$

(7)

In expression (5), $r^{\prime }=1,2,3\cdots$ represents the number of photons of the wave emitted at the first vertex, while $ħ{\omega }_C$ is the characteristic energy of the EFSCE [27]. In Equation (6), r represents the number of photons absorbed at the second vertex, while $ħ{\omega }_{BW}$ (7) is the characteristic energy of the EFSBWP [28,29]. Unlike the first vertex, the number of photons absorbed from the wave r at the second vertex significantly depends on the relationship between the energy of the second gamma quantum and the characteristic energy $ħ{\omega }_{BW}$. Thus, if the quantum parameter is ${\epsilon }_{2BW}<1({\omega }_2<{\omega }_{BW})$, then the number of absorbed photons of the wave at the second vertex starts from a certain minimum value: $r\ge r_{min}=\lceil {\epsilon }_{2BW}^{-1}\rceil >1$ (see condition (6) for parameter ${\epsilon }_{2BW\left(r\right)}$). However, if the quantum parameter is ${\epsilon }_{2BW}\ge 1({\omega }_2\ge {\omega }_{BW})$, then the number of absorbed photons starts from one: $r=1,2,3\cdots$.

It should be emphasized that the case where the following conditions were imposed on the energies of the initial gamma quanta within framework (1) was thoroughly investigated:

$${\omega }_2\sim {\omega }_{BW},{\omega }_1\ll {\omega }_{BW},{\omega }_1\sim {\omega }_C({\epsilon }_{2BW}\sim 1,{\epsilon }_{1BW}\ll 1,{\epsilon }_{1C}\sim 1).$$

(8)

It should be noted that under these conditions, the exchange resonant reaction channels $(k_1⟷k_2)$ were suppressed and not considered.

It is important to highlight that from a physical point of view, the most interesting case is when the energy of the second gamma quantum significantly exceeds the characteristic energy of the EFSBWP, while the energy of the first gamma quantum is on the order of, or much smaller than, the characteristic energy of the EFSCE, which had not been studied previously. In this case, instead of relationships (8), the following conditions are obtained:

$${\omega }_2\gg {\omega }_{BW},{\omega }_1\ll {\omega }_{BW},{\omega }_1\sim {\omega }_C({\epsilon }_{2BW}\gg 1,{\epsilon }_{1BW}\ll 1,{\epsilon }_{1C}\sim 1);$$

(9)

$${\omega }_2\gg {\omega }_{BW},{\omega }_1\ll {\omega }_C({\epsilon }_{2BW}\gg 1,{\epsilon }_{1C}\ll 1).$$

(10)

It is important to note that the characteristic Breit–Wheeler energy $ħ{\omega }_{BW}$ (7), at a fixed parameter $\eta$ value, is inversely proportional to the energy of the photon of the electromagnetic wave. As the photon energy of the wave increases from optical frequencies to X-ray frequencies, the characteristic Breit–Wheeler energy decreases, while the intensity of the wave increases. Let us estimate the characteristic energy of the EFSBWP $ħ{\omega }_{BW}$ (7). In deriving the estimate, we will consider frequencies of the electromagnetic wave in the X-ray range, as well as the angle $\theta =\pi$:

$$ħ{\omega }_{BW}=\left\{\begin{array}{ccc}\hfill 17.4\mathrm{G}\mathrm{e}\mathrm{V}& \mathrm{i}\mathrm{f}& \omega =30\mathrm{e}\mathrm{V},I=1.675·{10}^{21}{\mathrm{W}\mathrm{c}\mathrm{m}}^{-2}({\eta }^2=1)\hfill \\ \hfill 522\mathrm{M}\mathrm{e}\mathrm{V}& \mathrm{i}\mathrm{f}& \omega =1\mathrm{k}\mathrm{e}\mathrm{V},I=1.861·{10}^{24}{\mathrm{W}\mathrm{c}\mathrm{m}}^{-2}({\eta }^2=1)\hfill \\ \hfill 522\mathrm{M}\mathrm{e}\mathrm{V}& \mathrm{i}\mathrm{f}& \omega =10\mathrm{k}\mathrm{e}\mathrm{V},I=3.536·{10}^{27}{\mathrm{W}\mathrm{c}\mathrm{m}}^{-2}({\eta }^2=19)\hfill \end{array}\right.$$

(11)

We note that in Equation (11), for frequencies of the X-ray wave $\omega =30$ eV and $\omega =1$ keV, the wave parameters were chosen such that the characteristic Breit–Wheeler energy decreases with increasing wave intensity, from $ħ{\omega }_{BW}=17.4$ GeV to $ħ{\omega }_{BW}=522$ MeV. At the same time, for X-ray frequencies ranging from $\omega =1$ keV to $\omega =10$ keV, the wave parameters were chosen such that the characteristic energy of the EFSBWP $ħ{\omega }_{BW}=522$ MeV remains unchanged with increasing wave intensity.

In this article, we will study the resonant BWP in a strong X-ray field under the conditions of the initial gamma quanta energies specified by Equations (9) and (10). As will be shown later, under these conditions, the resonant energy of the positron (for Channel A) or electron (for Channel B) tends to be the energy of the high-energy second gamma quantum with the greatest probability.

Subsequently, the relativistic system of units is used: $c=ħ=1$.

2. Resonant Energies of the Positron (Electron)

Oleinik resonances occur when an intermediate electron or positron in an external field enters the mass shell. As a result, for Channels A and B, we obtain (see Figure 1):

$${\tilde{q}}_{-}^2=m_{*}^2,{\tilde{q}}_{-}=k_2-{\tilde{p}}_{+}+rk={\tilde{p}}_{-}-k_1+r^{\prime }k,$$

(12)

$${\tilde{q}}_{+}^2=m_{*}^2,{\tilde{q}}_{+}=k_2-{\tilde{p}}_{-}+rk={\tilde{p}}_{+}-k_1+r^{\prime }k.$$

(13)

In Expressions (12) and (13), $k_{1,2}=({\omega }_{1,2},{\mathbf{k}}_{\mathbf{1},\mathbf{2}})$ and $k=(\omega ,\mathbf{k})$ are 4-momenta of the first and second gamma quanta and the external wave photon, ${\tilde{p}}_{\pm }=({\tilde{E}}_{\pm },{\tilde{\mathbf{p}}}_{\pm })$ and ${\tilde{q}}_{\mp }$ are 4-quasimomenta of the final positron (electron) and intermediate electron (positron), and $m_{*}$ is the effective mass of the electron in the field of a circularly polarized wave [12]:

$${\tilde{p}}_{\pm }=p_{\pm }+{\eta }^2\frac{m^2}{2\left(k{p}_{\pm }\right)}k,{\tilde{q}}_{\mp }=q_{\mp }+{\eta }^2\frac{m^2}{2\left(k{q}_{\mp }\right)}k,$$

(14)

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

(15)

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

(16)

In expression (14), $p_{\pm }=(E_{\pm },{\mathbf{p}}_{\pm })$—4-momenta of the positron (electron).

In this article, within the framework of conditions (9) and (10), we will consider the energies of the second gamma quantum ${\omega }_2\lesssim {10}^3$ GeV, as well as the range of X-ray frequencies $10\mathrm{e}\mathrm{V}\lesssim \omega \lesssim {10}^5\mathrm{e}\mathrm{V}$. At the same time, we will consider the intensities of the electromagnetic wave to be significantly smaller than the critical intensities of the Schwinger [17,46] ($I\ll I_{*}\sim {10}^{29}$ W/cm${}^2$).

Considering Equations (1)–(4), (12) and (13), it is straightforward to obtain the resonant energies of positron $E_{+\left(r\right)}$ (for Channel A) or electron $E_{-\left(r\right)}$ (for Channel B) at the second vertex, where the EFSBWP occurs (see Figure 1):

$$E_{\pm \left(r\right)}=\frac{{\omega }_2}{2({\epsilon }_{2BW\left(r\right)}+{\delta }_{2\pm }^2)}\left[{\epsilon }_{2BW\left(r\right)}+\sqrt{{\epsilon }_{2BW\left(r\right)}({\epsilon }_{2BW\left(r\right)}-1)-{\delta }_{2\pm }^2}\right],$$

(17)

where the ultrarelativistic parameter

$${\delta }_{2\pm }^2=\frac{{\omega }_2^2}{{\left(2m_{*}\right)}^2}{\theta }_{2\pm }^2,$$

(18)

associated with the outgoing angle of the positron (electron) relative to the momentum of the second gamma quantum, can vary in the interval:

$$0\le {\delta }_{2\pm }^2\le {\delta }_{2max}^2,{\delta }_{2max}^2={\epsilon }_{2BW\left(r\right)}({\epsilon }_{2BW\left(r\right)}-1).$$

(19)

It should be noted that the quantity (17) determines the maximum energy of the positron (electron). There is also a solution with a minus sign in front of the square root, which determines the minimum value of the resonant energy. However, the minimum resonant energy is unlikely and not used in the text below. Under the conditions (9) and (10), the quantum parameter ${\epsilon }_{2BW\left(r\right)}=r{\epsilon }_{2BW}\gg 1(r\ge 1)$. As a result, the resonant energy of the positron or electron (17) will be close to the energy of the high-energy second gamma quantum:

$$E_{\pm \left(r\right)}\approx {\omega }_2\left[1-\frac{(1+4{\delta }_{2\pm }^2)}{4r{\epsilon }_{2BW}}\right]⟶{\omega }_2({\delta }_{2\pm }^2\ll r{\epsilon }_{2BW}).$$

(20)

Figure 2 illustrates the resonant energy of the positron (for Channel A) or electron (for Channel B) as a function of the corresponding ultrarelativistic parameter ${\delta }_{2\pm }^2$ for the energies of the second gamma quantum ${\omega }_2=600$ MeV and ${\omega }_2=100$ GeV. This graph presents two solutions for the resonant energy, corresponding to the “plus” (solid curves) and “minus” (dashed curves) signs before the square root in Equation (17). As mentioned earlier, only the maximum energy of the positron (electron) will be used in future (solid curves in Figure 2). From Figure 2, it can be observed that there is a significant difference in the resonant energies of the positron (electron) for these second gamma quantum energies. Unlike the case with ${\omega }_2=600$ MeV, the resonant energy of the positron (electron) for the case with ${\omega }_2=100$ GeV tends to converge to the energy of the second gamma quantum at small values of the parameter ${\delta }_{2\pm }^2$ (see solid curve 2 in Figure 2, as well as Formula (20)).

Considering Equations (1)–(4), (12) and (13), it is straightforward to obtain the resonant energies of electron $E_{-\left(r^{\prime }\right)}$ (for Channel A) or positron $E_{+\left(r^{\prime }\right)}$ (for Channel B) at the first vertex, where the EFSCE occurs (see Figure 1):

$$E_{\mp \left(r^{\prime }\right)}=\frac{{\omega }_1}{2({\epsilon }_{1C\left(r^{\prime }\right)}-{\delta }_{1\mp }^2)}\left[{\epsilon }_{1C\left(r^{\prime }\right)}+\sqrt{{\epsilon }_{1C\left(r^{\prime }\right)}^2+4({\epsilon }_{1C\left(r^{\prime }\right)}-{\delta }_{1\mp }^2)}\right].$$

(21)

Here, the ultrarelativistic parameter, associated with the outgoing angle of the electron (positron) relative to the momentum of the first gamma quantum, is defined:

$${\delta }_{1\mp }^2=\frac{{\omega }_1^2}{m_{*}^2}{\theta }_{1\mp }^2.$$

(22)

It should be noted that there are no constraints on the parameter ${\epsilon }_{1C\left(r^{\prime }\right)}$ for the EFSCE. At the same time, ${\delta }_{1\mp }^2<{\epsilon }_{1C\left(r^{\prime }\right)}$.

It is important to note that in the conditions of resonance (12) and (13) for each of the reaction channels, the resonant energies of the positron and electron are determined by different physical processes: the EFSBWP (17) and the EFSCE (21). At the same time, the energies of the electron–positron pair are related to each other by the general law of conservation of energy

$$E_{+}+E_{-}\approx {\omega }_1+{\omega }_2.$$

(23)

It should be noted that in Equation (23) we have neglected a small correction term $\sim |r-r^{\prime }|\omega /{\omega }_2\ll 1$. Taking into account relations (20) and (21), as well as the law of energy conservation (23) for Channels A and B, we obtain the following equation relating the outgoing angles of the positron and electron within the conditions (9) and (10):

$${\delta }_{1\mp }^2={\epsilon }_{1C}\left\{r^{\prime }-\frac{r{\epsilon }_{1C}}{1+4{\delta }_{2\pm }^2+r{\epsilon }_{1C}}\left[r^{\prime }+\frac{r}{1+4{\delta }_{2\pm }^2+r{\epsilon }_{1C}}\right]\right\}.$$

(24)

Here, the upper (lower) sign corresponds to Channel A (B). In relation (24), to the left of the equality sign is the ultrarelativistic parameter responsible for the outgoing angle of electron (positron) relative to the momentum of the first gamma quantum, and to the right of the equality sign is the function of the ultrarelativistic parameter ${\delta }_{2\pm }^2$, which corresponds to the outgoing angle of the positron (electron) relative to the momentum of the second gamma quantum. Under the given parameters ${\epsilon }_{1C}$ and $(r,r^{\prime })$, the relation (24) uniquely determines the outgoing angles of the electron and positron, and therefore their resonant energies. If we substitute expression (24) into relation (21), we find that, under conditions (9), the energy of the electron (for Channel A) or positron (for Channel B) is determined by the expression (also see relations (20), (23)):

$$E_{\mp }\approx {\omega }_1+\frac{1+4{\delta }_{2\pm }^2}{r}{\omega }_C.$$

(25)

Moreover, under conditions (10), the expression (25) simplifies:

$$E_{\mp }\approx \frac{1+4{\delta }_{2\pm }^2}{r}{\omega }_C.$$

(26)

Thus, the resonant energy of the electron–positron pair in conditions (9) is determined by Equations (20) and (25), and in conditions (10) by Equations (20) and (26). It is important to emphasize that in Conditions (9) and (10), the energy of the positron (for Channel A) or electron (for Channel B) is determined by the energy of the second high-energy gamma quantum, while the energy of the electron (for Channel A) or positron (for Channel B) in conditions (10) is primarily determined, not by the energy of the first gamma quantum, but by the characteristic energy of the EFSCE.

3. Maximum Breit–Wheeler Resonant Cross-Section

In conditions (9) and (10), in Channels A and B the EFSBWP proceeds with the number of absorbed photons of the wave $r\ge 1$, and for exchange resonant diagrams A’ and B’ the number of absorbed photons is $r\ge r_{1min}=\lceil {\epsilon }_{1BW}^{-1}\rceil \gg 1$. Therefore, the resonant Channels A’ and B’ will be suppressed, and it is sufficient to consider only two resonant Channels, A and B. It should also be noted that for Channel A, the resonant energy of the electron–positron pair is determined by the outgoing angle of the positron relative to the momentum of the second gamma quantum, while for Channel B—by the outgoing angle of the electron relative to the momentum of the second gamma quantum. Therefore, Channels A and B are distinguishable and do not interfere with each other.

In the article [43], a general relativistic expression for the resonant BWP modified by a strong electromagnetic field was obtained. The resonance infinity, which occurs in the field of a plane monochromatic wave, was eliminated using the Breit–Wigner procedure [27,29,48]. As a result, the maximum resonant differential cross-section (at the point of maximum Breit–Wigner distribution) for Channel A (with a “plus” sign) and Channel B (with a “minus” sign) has the following form:

$$R_{\pm \left(r{r}^{\prime }\right)}=\frac{d{\sigma }_{r{r}^{\prime }}^{max}}{d{\delta }_{2\pm }^2}=r_e^2{c}_{\eta i}{\mathsf{\psi }}_{\pm \left(r{r}^{\prime }\right)}.$$

(27)

Here, $r_e=e^2/m$ is the classical electron radius, and the function $c_{\eta i}$ is determined by the initial setup parameters:

$$c_{\eta i}=\frac{2{\left(4\pi \right)}^3}{{\alpha }^2{K}^2\left({\epsilon }_{1C}\right)}{\left(\frac{m}{{\delta }_{\eta i}{\omega }_{BW}}\right)}^2\approx 0.745·{10}^8{\left(\frac{m}{{\delta }_{\eta i}{\omega }_{BW}K\left({\epsilon }_{1C}\right)}\right)}^2,$$

(28)

where $\alpha$ is the fine-structure constant, and ${\delta }_{\eta i}^2$ is the ultrarelativistic parameter, which determines the angle between the momenta of the initial gamma quanta (see Equation (2));

$${\delta }_{\eta i}^2=\frac{{\omega }_1{\omega }_2}{m_{*}^2}{\theta }_i^2.$$

(29)

The function $K\left({\epsilon }_{1C}\right)$ is defined by the total probability (per unit of time) of the EFSCE on the intermediate electron (for Channel A) or positron (for Channel B):

$$K\left({\epsilon }_{1C}\right)=\sum _{s=1}^{\infty }{\int }_0^{s{\epsilon }_{1C}}\frac{du}{{(1+u)}^2}K(u,s{\epsilon }_{1C}).$$

(30)

Here, the function $K(u,s{\epsilon }_{1C})$ is determined by the expressions:

$$K(u,s{\epsilon }_{1C})=-4J_s^2\left(y_{1\left(s\right)}\right)+{\eta }^2\left[2+\frac{u^2}{1+u}\right](J_{s-1}^2+J_{s+1}^2-2J_s^2),$$

(31)

$$y_{1\left(s\right)}=2s\frac{\eta }{\sqrt{1+{\eta }^2}}\sqrt{\frac{u}{s{\epsilon }_{1C}}\left(1-\frac{u}{s{\epsilon }_{1C}}\right)}.$$

(32)

In Equation (27), functions ${\mathsf{\Psi }}_{\pm \left(r{r}^{\prime }\right)}$ determine the spectral-angular distribution of the generated electron–positron pair:

$${\mathsf{\psi }}_{\pm \left(r{r}^{\prime }\right)}=\frac{x_{\pm \left(r\right)}}{{\epsilon }_{2BW}^2(1-x_{\pm \left(r\right)})}{K}_{1\mp \left(r^{\prime }\right)}{P}_{2\pm \left(r\right)}.$$

(33)

In expression (33), the function $P_{2\pm \left(r\right)}$ determines the probability of the EFSBWP, and the function $K_{1\mp \left(r^{\prime }\right)}$ determines the probability of the EFSCE [12]:

$$P_{2\pm \left(r\right)}=J_r^2\left({\gamma }_{2\pm \left(r\right)}\right)+{\eta }^2(2u_{2\pm \left(r\right)}-1)\left[\left(\frac{r^2}{{\gamma }_{2\pm \left(r\right)}^2}-1\right)J_r^2+J_r^{\prime 2}\right],$$

(34)

$$K_{1\mp \left(r^{\prime }\right)}=-4J_{r^{\prime }}^2\left({\gamma }_{1\mp \left(r^{\prime }\right)}\right)+{\eta }^2\left[2+\frac{u_{1\mp \left(r^{\prime }\right)}^2}{1+u_{1\mp \left(r^{\prime }\right)}}\right](J_{r^{\prime }-1}^2+J_{r^{\prime }+1}^2-2J_{r^{\prime }}^2).$$

(35)

The arguments of the Bessel functions have the following form:

$${\gamma }_{2\pm \left(r\right)}=2r\frac{\eta }{\sqrt{1+{\eta }^2}}\sqrt{\frac{u_{2\pm \left(r\right)}}{v_{2\pm \left(r\right)}}\left(1-\frac{u_{2\pm \left(r\right)}}{v_{2\pm \left(r\right)}}\right)},$$

(36)

$${\gamma }_{1\mp \left(r^{\prime }\right)}=2r^{\prime }\frac{\eta }{\sqrt{1+{\eta }^2}}\sqrt{\frac{u_{1\mp \left(r^{\prime }\right)}}{v_{1\mp \left(r^{\prime }\right)}}\left(1-\frac{u_{1\mp \left(r^{\prime }\right)}}{v_{1\mp \left(r^{\prime }\right)}}\right)}.$$

(37)

Here, the relativistic-invariant parameters are equal to

$$u_{1\mp \left(r^{\prime }\right)}=\frac{\left(k_{1}k\right)}{\left(p_{\mp }k\right)}\approx \frac{{\omega }_1}{E_{\mp \left(r^{\prime }\right)}}\approx \frac{{\omega }_1}{{\omega }_1+{\omega }_2-E_{\pm \left(r\right)}},$$

(38)

$$v_{1\mp \left(r^{\prime }\right)}=\frac{2r^{\prime }\left(q_{\mp }k\right)}{m_{*}^2}\approx {\epsilon }_{1C\left(r^{\prime }\right)}\left(\frac{E_{\mp \left(r^{\prime }\right)}}{{\omega }_1}-1\right)\approx {\epsilon }_{1C\left(r^{\prime }\right)}\frac{{\omega }_2-E_{\pm \left(r\right)}}{{\omega }_1},$$

(39)

$$u_{2\pm \left(r\right)}=\frac{{\left(k_{2}k\right)}^2}{4\left(p_{\pm }k\right)\left(q_{\mp }k\right)}\approx \frac{{\omega }_2^2}{4E_{\pm \left(r\right)}\left({\omega }_2-E_{\pm \left(r\right)}\right)},v_{2\pm \left(r\right)}=r\frac{\left(k_{2}k\right)}{2m_{*}^2}\approx {\epsilon }_{2BW\left(r\right)}.$$

(40)

It should be noted that in the right-hand sides of Equations (38) and (39), the general law of conservation of energy (23) has been taken into account. Therefore, the relativistic-invariant parameters for the EFSCE $u_{1\mp \left(r^{\prime }\right)}$ and $v_{1\mp \left(r^{\prime }\right)}$ can be expressed in terms of the energy of the positron (electron) for the EFSBWP.

However, when considering the correlation of processes at the first and second vertices of the resonant Feynman diagrams (see Figure 1), it is necessary to require the fulfillment of the condition $u_{1\mp \left(r^{\prime }\right)}\le v_{1\mp \left(r^{\prime }\right)}$, to ensure that the argument of the Bessel functions ${\gamma }_{1\mp \left(r^{\prime }\right)}$ (37) is a real quantity. This condition can be expressed as a requirement for the physically admissible number of emitted photons at the first vertex:

$$r^{\prime }\ge \lceil r_{*}^{\prime }\rceil ,r_{*}^{\prime }=\frac{{\omega }_1^2}{{\epsilon }_{1C}}{\left[{\omega }_1+{\omega }_2-E_{\pm \left(r\right)}^{max}\right]}^{-1}{\left({\omega }_2-E_{\pm \left(r\right)}^{max}\right)}^{-1}.$$

(41)

Here, $E_{\pm \left(r\right)}^{max}$ is the maximum value of the resonant energy of the positron (electron), which is obtained from expression (17) at ${\delta }_{2\pm }^2=0$. Thus, the conditions (24) and (41) uniquely determine the outgoing angle of the electron (positron).

From Equations (27), (28), and (33), it can be seen that the magnitude of the resonant differential cross-section is mainly determined by function $c_{\eta i}$ (28). With a fixed value of the initial ultrarelativistic parameter ${\delta }_{\eta i}^2$ (29), the function $c_{\eta i}$ is significantly dependent on the characteristic energy of the EFSBWP ${\omega }_{BW}$ (7), as well as on function $K\left({\epsilon }_{1C}\right)$ (30), which determines the resonance width. As the characteristic energy of EFSBWP decreases, the resonant cross-section increases quite rapidly (see Figure 3, Figure 4 and Figure 5).

Within the scope of the case of the very high energies of the second gamma quantum (9) studied in this article, the resonant energy of the positron (electron) takes the form (20). Taking this into account, expressions (36)–(40) will take the following form:

$${\gamma }_{2\pm }\approx 4r\frac{\eta }{\sqrt{1+{\eta }^2}}\frac{{\delta }_{2\pm }}{1+4{\delta }_{2\pm }^2},$$

(42)

$$u_{2\pm }\approx \frac{r{\epsilon }_{2BW}}{1+4{\delta }_{2\pm }^2}\gg 1,$$

(43)

$${\gamma }_{1\mp }\approx 2r\frac{\eta }{\sqrt{1+{\eta }^2}}\frac{\sqrt{(r^{\prime }/r){\beta }_{\pm }-1}}{{\beta }_{\pm }},$$

(44)

$${\beta }_{\pm }=(1+4{\delta }_{2\pm }^2)\left[1+\frac{1+4{\delta }_{2\pm }^2}{r{\epsilon }_{1C}}\right],$$

(45)

$$u_{1\mp }\approx {\left[1+\frac{1+4{\delta }_{2\pm }^2}{r{\epsilon }_{1C}}\right]}^{-1}.$$

(46)

Due to this, the functions ${\mathsf{\psi }}_{\pm \left(r{r}^{\prime }\right)}$ (33) in the expression for the resonant differential cross-section (27) is simplified and takes the following form:

$${\mathsf{\psi }}_{\pm \left(r{r}^{\prime }\right)}=\frac{8{\eta }^2{r}^2{K}_{1\mp \left(r^{\prime }\right)}}{(1+4{\delta }_{2\pm }^2+r{\epsilon }_{1C})(1+4{\delta }_{2\pm }^2)}\left[\left(\frac{r^2}{{\gamma }_{2\pm }^2}-1\right)J_r^2\left({\gamma }_{2\pm }\right)+J_r^{\prime 2}\right].$$

(47)

Here, the argument of the Bessel functions ${\gamma }_{2\pm }$ takes the form (42). It should be noted that the obtained resonant cross-section (27), (28), (47) depends on the frequency of the high-energy second gamma quantum (9) only through the ultrarelativistic parameters (29) and (18).

Let us consider the case when the energies of the initial gamma quanta satisfy the conditions (10). Then, the parameter $u_{1\mp }$ (46) takes the following form:

$$u_{1\mp }\approx \frac{r{\epsilon }_{1C}}{1+4{\delta }_{2\pm }^2}\ll 1.$$

(48)

As a result, expressions (35) and (44) are simplified:

$$K_{1\mp \left(r^{\prime }\right)}=-4J_{r^{\prime }}^2\left({\gamma }_{1\mp }\right)+2{\eta }^2(J_{r^{\prime }-1}^2+J_{r^{\prime }+1}^2-2J_{r^{\prime }}^2).$$

(49)

$${\gamma }_{1\mp }\approx 2r\frac{\eta }{\sqrt{1+{\eta }^2}}\frac{\sqrt{r^{\prime }{\epsilon }_{1C}}}{1+4{\delta }_{2\pm }^2}.$$

(50)

Thus, the resonant cross-section for the initial gamma quanta energies (10) is given by Equations (27), (28) and (47) in which the functions $K_{1\mp \left(r^{\prime }\right)}$, ${\gamma }_{1\mp }$ and ${\gamma }_{2\pm }$ are determined by relationships (49), (50), and (42), respectively.

Figure 3, Figure 4 and Figure 5 show the distributions of the resonant cross-section in the strong X-ray field with varying numbers of absorbed and emitted photons as a function of the square of the positron (for Channel A) or electron (for Channel B) outgoing angle relative to the momentum of the second gamma quantum under the conditions (8), (9), and (10) for various characteristic energies of the EFSBWP. The initial ultrarelativistic parameter ${\delta }_{\eta i}^2={10}^{-2}$.

Table 1. The values of the ultrarelativistic parameter ${\delta }_{2\pm }^{2(*)}$ that correspond to maximum values $R_{\pm (r,r^{\prime })}^{(*)}$ of the spectral-angular distribution for the resonant differential cross-sections (27) under the conditions (8) (see Figure 3 and expression (11)).

	(r,r’)	${\mathit{\delta }}_{2\pm }^{2(*)}$	${\mathit{R}}_{\pm (\mathit{r},{\mathit{r}}^{\prime })}^{(*)}$
$I=1.675·{10}^{21}{\mathrm{W}\mathrm{c}\mathrm{m}}^{-2}$, ${\omega }_1=1\mathrm{G}\mathrm{e}\mathrm{V}$, ${\omega }_2=35\mathrm{G}\mathrm{e}\mathrm{V}$	(1,1)	0	$1.13·{10}^2$
	(2,1)	0.095	$3.90·{10}^1$
	(1,2)	0	$1.69·{10}^1$
	(2,2)	0.060	$1.53·{10}^1$
$I=1.861·{10}^{24}{\mathrm{W}\mathrm{c}\mathrm{m}}^{-2}$, ${\omega }_1=50\mathrm{M}\mathrm{e}\mathrm{V}$, ${\omega }_2=1\mathrm{G}\mathrm{e}\mathrm{V}$	(1,1)	0	$4.36·{10}^4$
	(2,1)	0.047	$1.66·{10}^4$
	(1,2)	0	$9.85·{10}^3$
	(2,2)	0.069	$6.83·{10}^3$
$I=3.536·{10}^{27}{\mathrm{W}\mathrm{c}\mathrm{m}}^{-2}$, ${\omega }_1=50\mathrm{M}\mathrm{e}\mathrm{V}$, ${\omega }_2=1\mathrm{G}\mathrm{e}\mathrm{V}$	(1,1)	0	$1.41·{10}^4$
	(2,1)	0.025	$6.41·{10}^3$
	(1,2)	0	$6.13·{10}^3$
	(2,2)	0.048	$3.49·{10}^3$

,

Table 2. The values of the ultrarelativistic parameter ${\delta }_{2\pm }^{2(*)}$ that correspond to maximum values $R_{\pm (r,r^{\prime })}^{(*)}$ of the spectral-angular distribution for the resonant differential cross-sections (27) under the conditions (9) (see Figure 4 and expression (11)).

	(r,r’)	${\mathit{\delta }}_{2\pm }^{2(*)}$	${\mathit{R}}_{\pm (\mathit{r},{\mathit{r}}^{\prime })}^{(*)}$
$I=1.675·{10}^{21}{\mathrm{W}\mathrm{c}\mathrm{m}}^{-2}$, ${\omega }_1=1\mathrm{G}\mathrm{e}\mathrm{V}$	(1,1)	0	$1.68·{10}^2$
	(2,1)	0.070	$4.66·{10}^1$
	(1,2)	0	$3.40·{10}^1$
	(2,2)	0.055	$1.99·{10}^1$
$I=1.861·{10}^{24}{\mathrm{W}\mathrm{c}\mathrm{m}}^{-2}$, ${\omega }_1=50\mathrm{M}\mathrm{e}\mathrm{V}$	(1,1)	0	$6.65·{10}^4$
	(2,1)	0.033	$2.59·{10}^4$
	(1,2)	0	$2.02·{10}^4$
	(2,2)	0.063	$8.84·{10}^3$
$I=3.536·{10}^{27}{\mathrm{W}\mathrm{c}\mathrm{m}}^{-2}$, ${\omega }_1=50\mathrm{M}\mathrm{e}\mathrm{V}$	(1,1)	0	$2.00·{10}^4$
	(2,1)	0.020	$1.15·{10}^4$
	(1,2)	0	$1.18·{10}^4$
	(2,2)	0.043	$4.19·{10}^3$

and

Table 3. The values of the ultrarelativistic parameter ${\delta }_{2\pm }^{2(*)}$ that correspond to the maximum values $R_{\pm (r,r^{\prime })}^{(*)}$ of the spectral-angular distribution for the resonant differential cross-sections (27) under the conditions (10) (see Figure 5 and expression (11)).

	(r,r’)	${\mathit{\delta }}_{2\pm }^{2(*)}$	${\mathit{R}}_{\pm (\mathit{r},{\mathit{r}}^{\prime })}^{(*)}$
$I=1.675·{10}^{21}{\mathrm{W}\mathrm{c}\mathrm{m}}^{-2}$, ${\omega }_1=0.35\mathrm{G}\mathrm{e}\mathrm{V}$	(1,1)	0	$1.43·{10}^3$
	(2,1)	0.070	$5.01·{10}^2$
	(1,2)	0	$1.10·{10}^2$
	(2,2)	0.043	$1.05·{10}^2$
$I=1.861·{10}^{24}{\mathrm{W}\mathrm{c}\mathrm{m}}^{-2}$, ${\omega }_1=12\mathrm{M}\mathrm{e}\mathrm{V}$	(1,1)	0	$1.27·{10}^6$
	(2,1)	0.073	$2.33·{10}^5$
	(1,2)	0	$1.12·{10}^5$
	(2,2)	0.044	$9.88·{10}^4$
$I=3.536·{10}^{27}{\mathrm{W}\mathrm{c}\mathrm{m}}^{-2}$, ${\omega }_1=12\mathrm{M}\mathrm{e}\mathrm{V}$	(1,1)	0	$3.39·{10}^5$
	(2,1)	0.073	$1.08·{10}^5$
	(1,2)	0	$6.11·{10}^4$
	(2,2)	0.044	$5.92·{10}^4$

represent the maximum values of the resonant cross-sections $R_{\pm (r,r^{\prime })}^{*}$ in the corresponding peaks of the distributions as functions of ${\delta }_{2\pm }^2$. From the data in the figures and corresponding tables, it is evident how the resonant cross-sections change when the characteristic Breit–Wheeler energy varies from ${\omega }_{BW}=17.4$ GeV to ${\omega }_{BW}=522$ MeV. In this case, the magnitude of ${\omega }_{BW}$ changes due to both the variation of the wave frequency and the parameter $\eta$ (see Equation (11)). Thus, if the wave intensity is confined to the interval $I=1.675·{10}^{21}÷1.861·{10}^{24}(3.536·{10}^{27})$ Wcm${}^{-2}$, the maximum resonant cross-sections (in units $r_e^2$) for $r=r^{\prime }=1$ change as follows: under the conditions of the initial gamma quanta energies (8) in the interval $R_{\pm (1,1)}^{*}\approx 1.13·{10}^2÷4.36·{10}^4(1.41·{10}^4)$; under the conditions of the initial gamma quanta energies (9) in the interval $R_{\pm (1,1)}^{*}\approx 1.68·{10}^2÷6.65·{10}^4(2.00·{10}^4)$; under the conditions of the initial gamma quanta energies (10) in the interval $R_{\pm (1,1)}^{*}\approx 1.43·{10}^3÷1.27·{10}^6(3.39·{10}^5)$. Therefore, when the characteristic energy of the EFSBWP is reduced by a factor of 33, the maximum resonant cross-section increases by a factor of $3.86·{10}^2(1.25·{10}^2)$ under the conditions of the initial gamma quanta energies (8), by a factor of $3.96·{10}^2(1.19·{10}^2)$ under the conditions of the initial gamma quanta energies (9), and by a factor of $0.89·{10}^3(2.37·{10}^2)$ under the conditions of the initial gamma quanta energies (10).

Consequently, the maximum value of the resonant cross-section occurs under the conditions of the initial gamma quanta energies (10) $({\omega }_1\ll {\omega }_C,{\omega }_2\gg {\omega }_{BW})$ and, depending on the value of the characteristic Breit–Wheeler energy in the interval ${\omega }_{BW}\sim 10÷1$ GeV, can reach a magnitude of $R_{\pm (1,1)}^{*}\sim {10}^3÷{10}^6{r}_e^2$. At the same time, for Channel A, narrow beams of high-energy positrons $(E_{+}\to {\omega }_2)$ are obtained, while for Channel B, narrow beams of high-energy electrons $(E_{-}\to {\omega }_2)$ are obtained (see relation (20)).

However, if the characteristic Breit–Wheeler energy remains unchanged with changes in the frequency and intensity of the wave, then the resonant cross-section changes less significantly (compare the values of the resonant cross-section without parentheses and within parentheses in the above text).

It should be emphasized that in strong electromagnetic fields, various QED processes can take place, both stimulated and modified by an external field (see, for example, [5,12,13,14,15,16,17]). At the same time, the resonant QED processes modified by an external field, such as the Breit–Wheeler process, photogeneration of electron–positron pairs on the nucleus, spontaneous bremshtralung radiation during electron scattering on nuclei, the Compton effect, the generation of electron–positron pairs when an electron beam collides with an electromagnetic wave, and others, will have the greatest probability (see, for example, [13,14]). These resonant processes, depending on specific conditions, can occur in parallel with different or comparable probabilities.

We would like to note that the experimental implementation of the Breit–Wheeler resonant process in strong X-ray fields is currently very problematic. This is due to the need to have a powerful source of gamma radiation with energies, as well as strong X-ray fields with intensities. However, this process can take place near neutron stars, where appropriate resonant conditions can be created. It should also be noted that the Breit–Wheeler resonant process, as one of the resonant QED processes, may influence the implementation of laser fusion, the course of which is accompanied by various QED processes in the electromagnetic wave field.

4. Conclusions

We considered the resonant BWP modified by a strong X-ray field for high-energy initial gamma quanta. The following results were obtained:

• It is shown that the resonant cross-section significantly depends on the magnitude of the characteristic Breit–Wheeler energy ${\omega }_{BW}$ (7) and the characteristic energy of the Compton effect ${\omega }_C$ (5). The ratios of the initial energies of gamma quanta with these characteristic energies significantly affect the magnitude of the resonant cross-section.
• Under the conditions when the energy of the second gamma quantum significantly exceeds the characteristic Breit–Wheeler energy $({\omega }_2\gg {\omega }_{BW})$, the resonant energy of the positron (for Channel A) or electron (for Channel B) tends toward the energy of the high-energy second gamma quantum (20) $(E_{\pm }\to {\omega }_2)$;
• The magnitude of the resonant cross-section significantly depends on the characteristic Breit–Wheeler energy, as well as the width of the resonance $(R_{\pm \left(r{r}^{\prime }\right)}\sim {\omega }_{BW}^{-2}{K}^{-2}\left({\epsilon }_{1C}\right))$ (see Equations (27) and (28)). Consequently, by decreasing the characteristic Breit–Wheeler energy by one order of magnitude, the resonant cross-section increases by two orders of magnitude. Moreover, the highest resonant differential cross-section is achieved when the energy of the second gamma quantum significantly exceeds the characteristic Breit–Wheeler energy $({\omega }_2\gg {\omega }_{BW})$, while the energy of the first gamma quantum is significantly lower than the characteristic Compton effect energy $({\omega }_1\ll {\omega }_C)$. In this case, when reducing the characteristic Breit–Wheeler energy within the range ${\omega }_{BW}\sim 10÷1$ GeV, the maximum resonant cross-section can reach values of $R_{\pm (1,1)}^{*}\sim {10}^3÷{10}^6{r}_e^2$.

The obtained results could be used to explain the narrow fluxes of high-energy ultrarelativistic positrons (electrons) in the vicinity of neutron stars, such as accretion-powered X-ray binaries [49,50], rotation-powered X/Gamma pulsars [51,52] and magnetic-field-powered magnetars [53,54], as well as to simulate physical processes in laser thermonuclear fusion [55].