Integrated cross sections of high-energy $e^+e^-$ electroproduction by an electron in an atomic field

TL;DR

This paper investigates the high-energy electron-positron pair production cross sections in atomic fields, emphasizing the significant role of Coulomb corrections and their impact on differential and integrated cross sections.

Contribution

It provides a detailed analysis of Coulomb corrections and interference effects in $e^+e^-$ electroproduction, highlighting their importance even for moderate atomic charge numbers.

Findings

Coulomb corrections are crucial for both differential and integrated cross sections.

Interference and virtual bremsstrahlung significantly affect the differential cross section over electron transverse momentum.

Coulomb corrections to scattering are larger than interference effects but smaller than those in pair photoproduction amplitude.

Abstract

The integrated cross sections of high-energy $e^+e^-$ electroproduction by an electron in an atomic field is studied. Importance of various contributions to these cross sections is discussed. It is shown that the Coulomb corrections are very important both for the differential cross section and for the integrated cross sections even for moderate values of the atomic charge number. This effect is mainly related to the interaction of the produced $e^+e^-$ pair with the atomic field. For the cross section differential over the electron transverse momentum the account for the interference of the amplitudes and the contribution of virtual bremsstrahlung is noticeable. The Coulomb corrections to scattering is larger than these two effects but essentially smaller than the Coulomb corrections to the amplitude of pair photoproduction by a virtual photon. However, in the cross section differential over the positron transverse momentum, there is a strong suppression of the effect of the Coulomb corrections to scattering.

Equations

$$d\sigma =\frac{{\alpha }^2}{(2\pi {)}^8}d{\epsilon }_{3}d{\epsilon }_{4}d{\mathbf{p}}_{2\perp }d{\mathbf{p}}_{3\perp }d{\mathbf{p}}_{4\perp }\frac{1}{2}\sum _{{\mu }_i=\pm 1}|{\mathcal{T}}_{{\mu }_1{\mu }_2{\mu }_3{\mu }_4}{|}^2,$$

$${\mathcal{T}}_{{\mu }_1{\mu }_2{\mu }_3{\mu }_4}=T_{{\mu }_1{\mu }_2{\mu }_3{\mu }_4}-{\tilde{T}}_{{\mu }_1{\mu }_2{\mu }_3{\mu }_4},{\tilde{T}}_{{\mu }_1{\mu }_2{\mu }_3{\mu }_4}=T_{{\mu }_1{\mu }_3{\mu }_2{\mu }_4}({\mathbf{p}}_2\leftrightarrow {\mathbf{p}}_3),$$

$$T=T^{(0)}+T^{(1)},$$

$$S=\sum_{\mu_{i}=\pm 1}\Bigg{|}\frac{\varepsilon_{1}m^{4}{\cal T}_{\mu_{1}\mu_{2}\mu_{3}\mu_{4}}}{\eta(2\pi)^{2}}\Bigg{|}^{2}\,,$$

$$T=T^{(0)}+T^{(1)},T^{(0)}=T_{\parallel }^{(0)}+T_{\perp }^{(0)},T^{(1)}=T_{\parallel }^{(1)}+T_{\perp }^{(1)},$$

$$\displaystyle T_{\perp}^{(0)}=\frac{8\pi A(\bm{\Delta}_{0})}{\omega(m^{2}+\zeta^{2})}\Big{\{}\delta_{\mu_{1}\mu_{2}}\delta_{\mu_{3}\bar{\mu}_{4}}\Big{[}\frac{\varepsilon_{3}}{\omega^{2}}(\bm{s}_{\mu_{3}}^{*}\cdot\bm{X})(\bm{s}_{\mu_{3}}\cdot\bm{\zeta})(\varepsilon_{1}\delta_{\mu_{1}\mu_{3}}+\varepsilon_{2}\delta_{\mu_{1}\mu_{4}})$$

$$\displaystyle-\frac{\varepsilon_{4}}{\omega^{2}}(\bm{s}_{\mu_{4}}^{*}\cdot\bm{X})(\bm{s}_{\mu_{4}}\cdot\bm{\zeta})(\varepsilon_{1}\delta_{\mu_{1}\mu_{4}}+\varepsilon_{2}\delta_{\mu_{1}\mu_{3}})\Big{]}$$

$$-\frac{m{\mu }_1}{\sqrt{2}{\epsilon }_1{\epsilon }_2}R{\delta }_{{\mu }_1{\bar{\mu }}_2}{\delta }_{{\mu }_3{\bar{\mu }}_4}({\mathbf{s}}_{{\mu }_1}\cdot \mathbf{\zeta })(-{\epsilon }_3{\delta }_{{\mu }_1{\mu }_3}+{\epsilon }_4{\delta }_{{\mu }_1{\mu }_4})$$

$$\displaystyle+\frac{m\mu_{3}}{\sqrt{2}\varepsilon_{3}\varepsilon_{4}}\delta_{\mu_{1}\mu_{2}}\delta_{\mu_{3}\mu_{4}}(\bm{s}_{\mu_{3}}^{*}\cdot\bm{X})(\varepsilon_{1}\delta_{\mu_{3}\mu_{1}}+\varepsilon_{2}\delta_{\mu_{3}\bar{\mu}_{1}})+\frac{m^{2}\omega^{2}}{2\varepsilon_{1}\varepsilon_{2}\varepsilon_{3}\varepsilon_{4}}R\delta_{\mu_{1}\bar{\mu}_{2}}\delta_{\mu_{3}\mu_{4}}\delta_{\mu_{1}\mu_{3}}\Big{\}}\,,$$

$$T_{\parallel }^{(0)}=-\frac{8\pi }{{\omega }^2}A({\mathbf{\Delta }}_0)R{\delta }_{{\mu }_1{\mu }_2}{\delta }_{{\mu }_3{\bar{\mu }}_4}.$$

$$A(\mathbf{\Delta })=-\frac{i\lambda }{{\Delta }_{\perp }^2}\int d\mathbf{r}\exp [-i\mathbf{\Delta }\cdot \mathbf{r}-i\chi (\rho )]{\mathbf{\Delta }}_{\perp }\cdot {\nabla }_{\perp }V(r),$$

$$\chi (\rho )=\lambda {\int }_{-\infty }^{\infty }dzV(\sqrt{z^2+{\rho }^2}),\mathbf{\rho }={\mathbf{r}}_{\perp },\mathbf{\zeta }=\frac{{\epsilon }_3{\epsilon }_4}{\omega }{\mathbf{\theta }}_{34},$$

$$\omega ={\epsilon }_3+{\epsilon }_4,{\mathbf{\Delta }}_{0\perp }={\epsilon }_2{\mathbf{\theta }}_{21}+{\epsilon }_3{\mathbf{\theta }}_{31}+{\epsilon }_4{\mathbf{\theta }}_{41},$$

$${\Delta }_{0\parallel }=-\frac{1}{2}\left[m^2\omega \left(\frac{1}{{\epsilon }_1{\epsilon }_2}+\frac{1}{{\epsilon }_3{\epsilon }_4}\right)+\frac{p_{2\perp }^2}{{\epsilon }_2}+\frac{p_{3\perp }^2}{{\epsilon }_3}+\frac{p_{4\perp }^2}{{\epsilon }_4}\right],$$

$$R=\frac{1}{d_1{d}_2}[{\Delta }_{0\perp }^2({\epsilon }_1+{\epsilon }_2)+2{\epsilon }_1{\epsilon }_2({\mathbf{\theta }}_{12}\cdot {\mathbf{\Delta }}_{0\perp })],$$

$$\mathbf{X}=\frac{1}{d_1}({\epsilon }_3{\mathbf{\theta }}_{23}+{\epsilon }_4{\mathbf{\theta }}_{24})-\frac{1}{d_2}({\epsilon }_3{\mathbf{\theta }}_{13}+{\epsilon }_4{\mathbf{\theta }}_{14}),$$

$$d_1=m^2\omega {\epsilon }_1\left(\frac{1}{{\epsilon }_1{\epsilon }_2}+\frac{1}{{\epsilon }_3{\epsilon }_4}\right)+{\epsilon }_2{\epsilon }_3{\theta }_{23}^2+{\epsilon }_2{\epsilon }_4{\theta }_{24}^2+{\epsilon }_3{\epsilon }_4{\theta }_{34}^2,$$

$$d_2=m^2\omega {\epsilon }_2\left(\frac{1}{{\epsilon }_1{\epsilon }_2}+\frac{1}{{\epsilon }_3{\epsilon }_4}\right)+{\epsilon }_2{\epsilon }_3{\theta }_{31}^2+{\epsilon }_2{\epsilon }_4{\theta }_{41}^2+({\epsilon }_3{\mathbf{\theta }}_{31}+{\epsilon }_4{\mathbf{\theta }}_{41}{)}^2,$$

$${\mathbf{\theta }}_i={\mathbf{p}}_{i\perp }/p_i,{\mathbf{\theta }}_{ij}={\mathbf{\theta }}_i-{\mathbf{\theta }}_j,$$

$$T_{\perp }^{(1)}=\frac{8i\eta }{\omega {\epsilon }_1}|\Gamma (1-i\eta ){|}^2\int \frac{d{\mathbf{\Delta }}_{\perp }A({\mathbf{\Delta }}_{\perp }+{\mathbf{p}}_{2\perp })F_a(Q^2)}{Q^2{M}^2(m^2{\omega }^2/{\epsilon }_1^2+{\Delta }_{\perp }^2)}{\left(\frac{{\xi }_2}{{\xi }_1}\right)}^{i\eta }\mathcal{M},$$

$$\displaystyle{\cal M}=-\frac{\delta_{\mu_{1}\mu_{2}}\delta_{\mu_{3}\bar{\mu}_{4}}}{\omega}\big{[}\varepsilon_{1}(\varepsilon_{3}\delta_{\mu_{1}\mu_{3}}-\varepsilon_{4}\delta_{\mu_{1}\mu_{4}})(\bm{s}_{\mu_{1}}^{*}\cdot\bm{\Delta}_{\perp})(\bm{s}_{\mu_{1}}\cdot\bm{I}_{1})\,$$

$$\displaystyle+\varepsilon_{2}(\varepsilon_{3}\delta_{\mu_{1}\bar{\mu}_{3}}-\varepsilon_{4}\delta_{\mu_{1}\bar{\mu}_{4}})(\bm{s}_{\mu_{1}}\cdot\bm{\Delta}_{\perp})(\bm{s}_{\mu_{1}}^{*}\cdot\bm{I}_{1})\big{]}+\delta_{\mu_{1}\bar{\mu}_{2}}\delta_{\mu_{3}\bar{\mu}_{4}}\frac{m\omega\mu_{1}}{\sqrt{2}\varepsilon_{1}}(\varepsilon_{3}\delta_{\mu_{1}\mu_{3}}-\varepsilon_{4}\delta_{\mu_{1}\mu_{4}})(\bm{s}_{\mu_{1}}\cdot\bm{I}_{1})$$

$$+{\delta }_{{\mu }_1{\mu }_2}{\delta }_{{\mu }_3{\mu }_4}\frac{m{\mu }_3}{\sqrt{2}}({\epsilon }_1{\delta }_{{\mu }_1{\mu }_3}+{\epsilon }_2{\delta }_{{\mu }_1{\bar{\mu }}_3})({\mathbf{s}}_{{\mu }_3}^{\ast }\cdot {\mathbf{\Delta }}_{\perp })I_0-\frac{m^2{\omega }^2}{2{\epsilon }_1}{\delta }_{{\mu }_1{\bar{\mu }}_2}{\delta }_{{\mu }_3{\mu }_4}{\delta }_{{\mu }_1{\mu }_3}{I}_0,$$

$$T_{\parallel }^{(1)}=-\frac{8i\eta {\epsilon }_3{\epsilon }_4}{{\omega }^3}|\Gamma (1-i\eta ){|}^2\int \frac{d{\mathbf{\Delta }}_{\perp }A({\mathbf{\Delta }}_{\perp }+{\mathbf{p}}_{2\perp })F_a(Q^2)}{Q^2{M}^2}{\left(\frac{{\xi }_2}{{\xi }_1}\right)}^{i\eta }{I}_0{\delta }_{{\mu }_1{\mu }_2}{\delta }_{{\mu }_3{\bar{\mu }}_4},$$

$$\displaystyle M^{2}=m^{2}\Big{(}1+\frac{\varepsilon_{3}\varepsilon_{4}}{\varepsilon_{1}\varepsilon_{2}}\Big{)}+\frac{\varepsilon_{1}\varepsilon_{3}\varepsilon_{4}}{\varepsilon_{2}\omega^{2}}\Delta_{\perp}^{2}\,,\quad\bm{Q}_{\perp}=\bm{\Delta}_{\perp}-\bm{p}_{3\perp}-\bm{p}_{4\perp}\,,$$

$$Q^2=Q_{\perp }^2+{\Delta }_{0\parallel }^2,{\mathbf{q}}_1=\frac{{\epsilon }_3}{\omega }{\mathbf{\Delta }}_{\perp }-{\mathbf{p}}_{3\perp },{\mathbf{q}}_2=\frac{{\epsilon }_4}{\omega }{\mathbf{\Delta }}_{\perp }-{\mathbf{p}}_{4\perp },$$

$$I_0=({\xi }_1-{\xi }_2)F(x)+({\xi }_1+{\xi }_2-1)(1-x)\frac{F^{\prime }(x)}{i\eta },$$

$${\mathbf{I}}_1=({\xi }_1{\mathbf{q}}_1+{\xi }_2{\mathbf{q}}_2)F(x)+({\xi }_1{\mathbf{q}}_1-{\xi }_2{\mathbf{q}}_2)(1-x)\frac{F^{\prime }(x)}{i\eta },$$

$${\xi }_1=\frac{M^2}{M^2+q_1^2},{\xi }_2=\frac{M^2}{M^2+q_2^2},x=1-\frac{Q_{\perp }^2{\xi }_1{\xi }_2}{M^2},$$

$$F(x)=F(i\eta ,-i\eta ,1,x),F^{\prime }(x)=\frac{\partial }{\partial x}F(x),\eta =Z\alpha .$$

Full text

electroproduction, photoproduction, bremsstrahlung, Coulomb corrections, screening

I Introduction

A process of $e^{+}e^{-}$ pair production by a high-energy electron in the atomic field is interesting both from experimental and theoretical points of view. It is important to know the cross section of this process with high accuracy at data analysis in detectors. Besides, this process gives the substantial contribution to a background at precision experiments devoted to search of new physics. From the theoretical point of view, the cross section of electroproduction in the field of heavy atoms reveals very interesting properties of the Coulomb corrections, which are the difference between the cross section exact in the parameters of the field and that calculated in the lowest order of the perturbation theory (the Born approximation).

The cross sections in the Born approximation, both differential and integrated, have been discussed in numerous papers Bhabha2 ; Racah37 ; BKW54 ; MUT56 ; Johnson65 ; Brodsky66 ; BjCh67 ; Henry67 ; Homma74 . The Coulomb corrections to the differential cross section of high-energy electroproduction by an ultra-relativistic electron in the atomic field have been obtained only recently in our paper KM2016 . In that paper it is shown that the Coulomb corrections significantly modify the differential cross section of the process as compared with the Born result. It turns out that both effects, the exact account for the interaction of incoming and outgoing electrons with the atomic field and the exact account for the interaction of the produced pair with the atomic field, are very important for the value of the differential cross section. On the other hand, the are many papers devoted to the calculation of $e^{+}e^{-}$ electroproduction by a heavy particles (muons or nuclei) in an atomic field Nikishov82 ; IKSS1998 ; SW98 ; McL98 ; Gre99 ; LM2000 . It that papers, the interaction of a heavy particle and the atomic field have been neglected. In our recent paper KM2017 it has been shown that the cross section, differential over the angles of a heavy outgoing particle, changes significantly due to the exact account for the interaction of a heavy particle with the atomic field. However, the cross section integrated over these angles is not affected by this interaction. Such unusual properties of the cross section of electroproduction by a heavy particle stimulated us to perform the detailed investigation of the integrated cross section of the electroproduction by the ultra-relativistic electron.

In the present paper we investigate in detail the integrated cross section, using the analytical result for the matrix element of the process obtained in our paper KM2016 with the exact account for the interaction of all charged particles with the atomic field. Our goal is to understand the relative importance of various contributions to the integrated cross section under consideration.

II General discussion

The differential cross section of high-energy electroproduction by an unpolarized electron in the atomic field reads

[TABLE]

where $\bm{p}_{1}$ is the momentum of the initial electron, $\bm{p}_{2}$ and $\bm{p}_{3}$ are the final electron momenta, $\bm{p}_{4}$ is the positron momentum, $\mu_{i}=\pm 1$ corresponds to the helicity of the particle with the momentum $\bm{p}_{i}$, $\bar{\mu}_{i}=-\mu_{i}$, $\varepsilon_{1}=\varepsilon_{2}+\varepsilon_{3}+\varepsilon_{4}$ is the energy of the incoming electron, $\varepsilon_{i}=\sqrt{{p}_{i}^{2}+m^{2}}$, $m$ is the electron mass, and $\alpha$ is the fine-structure constant, $\hbar=c=1$. In Eq. (1) the notation $\bm{X}_{\perp}=\bm{X}-(\bm{X}\cdot\bm{\nu})\bm{\nu}$ for any vector $\bm{X}$ is used, $\bm{\nu}=\bm{p}_{1}/p_{1}$. We have

[TABLE]

where the contributions $T$ and $\widetilde{T}$ correspond, respectively, to the left and right diagrams in Fig. 1. The amplitude $T$ has been derived in Ref. KM2016 by means of the quasiclassical approximation KLM2016 . Its explicit form is given in Appendix with one modification. Namely, we have introduced the parameter $\lambda$ which is equal to unity if the interaction of electrons, having the momenta $\bm{p}_{1}$, $\bm{p}_{2}$ in the term $T$ and $\bm{p}_{1}$ , $\bm{p}_{3}$ in the term $\widetilde{T}$, with the atomic field is taken into account. The parameter $\lambda$ equals to zero, if one neglects this interaction. Insertion of this parameter allows us to investigate the importance of various contributions to the cross section.

First of all we note that the term $T$ is a sum of two contributions, see Appendix,

[TABLE]

where $T^{(0)}$ is the contribution to the amplitude in which the produced $e^{+}e^{-}$ pair does not interact with the atomic field, while the contribution $T^{(1)}$ contains such interaction. In other words, the term $T^{(0)}$ corresponds to bremsstrahlung of the virtual photon decaying into a free $e^{+}e^{-}$ pair. In the contribution $T^{(1)}$, electrons with the momenta $\bm{p}_{1}$ and $\bm{p}_{2}$ may interact or not interact with the atomic field. The latter contribution is given by the amplitude $T^{(1)}$ at $\lambda=0$. Below we refer to the result of account for such interaction in the term $T^{(1)}$ as the Coulomb corrections to scattering. Note that the contribution $T^{(0)}$ at $\lambda=0$ vanishes.

In the present work we are going to elucidate the following points: the relative contribution of the term $T^{(0)}$ to the cross section, an importance of the Coulomb corrections to scattering, an importance of the interference between the amplitudes $T$ and $\widetilde{T}$ in the cross section.

We begin our analysis with the case of the differential cross section. Let us consider the quantity $S$,

[TABLE]

where $\eta=Z\alpha$ and $Z$ is the atomic charge number. In Fig. 2 the dependence of $S$ on the positron transverse momentum $p_{4\perp}$ is shown for gold ($Z=79$) at some values of $\varepsilon_{i}$, $\bm{p}_{2\perp}$, and $\bm{p}_{3\perp}$. Solid curve is the exact result, long-dashed curve corresponds to $\lambda=0$, dashed curve is the result obtained without account for the contributions $T^{(0)}$ and $\widetilde{T}^{(0)}$, dash-dotted curve is the result obtained without account for the interference between $T$ and $\widetilde{T}$, and dotted curve is the Born result (in the Born approximation $S$ is independent of $\eta$). One can see for the case considered, that the Born result differs significantly from the exact one, and account for the interference is also very important. The contributions $T^{(0)}$ and $\widetilde{T}^{(0)}$ are noticeable but not large, and the Coulomb corrections to the contributions $T^{(1)}$ and $\widetilde{T}^{(1)}$ are essential. The effect of screening for the values of the parameters considered in Fig. 2 is unimportant. Note that relative importance of different effects under discussions for the differential cross section strongly depends on the values of $\bm{p}_{i}$. However, in all cases a deviation of the Born result from the exact one is substantial even for moderate values of $Z$.

Let us consider the cross sections $d\sigma/dp_{2\perp}$, i.e., the cross sections differential over the electron transverse momentum $p_{2\perp}$. This cross section for $Z=79$ and $\varepsilon_{1}=100m$ is shown in the left panel in Fig. 3. In this picture solid curve is the exact result, dotted curve is the Born result, and long-dashed curve corresponds to $\lambda=0$. It is seen that the exact result significantly differs from the Born one, and account for the Coulomb corrections to scattering is also essential. An importance of account for the interference between $T$ and $\widetilde{T}$, as well as account for the contributions of $T^{(0)}$ and $\widetilde{T}^{(0)}$, is demonstrated by the right panel in Fig. 3. In this picture the quantity $\delta$, which is the deviation of the approximate result for $d\sigma/dp_{2\perp}$ from the exact one in units of the exact cross section, is shown. Dash-dotted curve is obtained without account for the interference between $T$ and $\widetilde{T}$, dashed curve is obtained without contributions of $T^{(0)}$ and $\widetilde{T}^{(0)}$. It seen that both effects are noticeable.

Our results are obtained under the condition $\varepsilon_{i}\gg m$, and a question on the limits of integration over energies appears at the numerical calculations of $d\sigma/dp_{2\perp}$. We have examined this question and found that the variation of the limits of integration in the vicinity of the threshold changes only slightly the result of integration. In any case, such a variation does not change the interplay of various contributions to the cross sections, and we present the results obtained by the integration over all kinematical region allowed.

It follows from Fig. 3 that the deviation of the results obtained for $\lambda=1$ from that obtained for $\lambda=0$ is noticeable and negative in the vicinity of the pick and small and positive in the wide region outside the pick. However, these two deviations (positive and negative) strongly compensate each other in the cross section integrated over both electron transverse momenta $\bm{p}_{2\perp}$ and $\bm{p}_{3\perp}$. This statement is illustrated in Fig. 4, where the cross sections differential over the positron transverse momentum, $d\sigma/dp_{4\perp}$ is shown for $Z=79$ and $\varepsilon_{1}=100m$.

Again, the Born result differs significantly from the exact one. It is seen that all relative deviations $\delta_{1}$ depicted in the right panel are noticeable. Then, the results obtained for $\lambda=0$ and that without contributions $T^{(0)}$ and $\widetilde{T}^{(0)}$ are very close to each other. This means that account for the Coulomb corrections to scattering leads to a very small shift of the integrated cross section $d\sigma/dp_{4\perp}$, in contrast to the cross section $d\sigma/dp_{2\perp}$. Such suppression is similar to that found in our resent paper KM2017 at the consideration of $e^{+}e^{-}$ pair electroproduction by a heavy charged particle in the atomic field.

At last, let us consider the total cross section $\sigma$ of the process under consideration. The cross section $\sigma$ for $Z=79$ as the function of $\varepsilon_{1}/m$ is shown in the left panel in Fig. 5. In this picture solid curve is the exact result, dotted curve is the Born result, and dash-dotted curve is the ultra-relativistic asymptotics of the Born result given by the formula of Racah Racah37 . Note that a small deviation of our Born result at relatively small energies from the asymptotics of the Born result is due, first, to uncertainty of our result related to the uncertainty of low limit of integration over the energies of the produced particles, and secondly, to neglecting identity of the final electrons in Ref. Racah37 .

It is seen that the exact result differs significantly from the Born one. In the right panel of Fig. 5 we show the relative deviation $\delta_{2}$ of the approximate result for $\sigma$ from the exact one. Dash-dotted curve is obtained without account for the interference between $T$ and $\widetilde{T}$, dashed curve is obtained without contributions $T^{(0)}$ and $\widetilde{T}^{(0)}$, and long-dashed curve corresponds to $\lambda=0$. The corrections to the total cross section due to account for the contributions $T^{(0)}$ and $\widetilde{T}^{(0)}$, and the Coulomb corrections to scattering are small even at moderate energy $\varepsilon_{1}$. The effect of the interference is more important at moderate energy and less important at high energies.

In our recent paper KM2016 the differential cross section of electroproduction by relativistic electron has been derived. For the differential cross section, we have pointed out that the Coulomb corrections to the scattering are the most noticeable in the region $p_{2\perp}\sim\omega/\gamma$. On the basis of this statement, we have evaluated in the leading logarithmic approximation the Coulomb corrections to the total cross section, see Eq. (33) of Ref. KM2016 . However, as it is shown in the present paper, for the total cross section the contribution of the Coulomb corrections to scattering in the region $p_{2\perp}\sim\omega/\gamma$ is compensated strongly by the contribution of the Coulomb corrections to scattering in the wide region outside $p_{2\perp}\sim\omega/\gamma$. As a result, the Coulomb corrections to the total cross section derived in the leading logarithmic approximation does not affected by account for the Coulomb corrections to scattering. This means that the coefficient in Eq. (33) of Ref. KM2016 should be two times smaller and equal to that in the Coulomb corrections to the total cross section of $e^{+}e^{-}$ electroproduction by a relativistic heavy particle calculated in the leading logarithmic approximation. Note that an accuracy of the result obtained for the Coulomb corrections to the total cross section is very low because in electroproduction there is a strong compensation between the leading and next-to-leading terms in the Coulomb corrections, see Ref. LM2009 .

III Conclusion

Performing tabulations of the formula for the differential cross section of $e^{+}e^{-}$ pair electroproduction by a relativistic electron in the atomic field KM2016 , we have elucidated the importance of various contributions to the integrated cross sections of the process. It is shown that the Coulomb corrections are very important both for the differential cross section and for the integrated cross sections even for moderate values of the atomic charge number. This effect is mainly related to the Coulomb corrections to the amplitudes $T^{(1)}$ and ${\widetilde{T}}^{(1)}$ due to the exact account of the interaction of the produced $e^{+}e^{-}$ pair with the atomic field (the Coulomb corrections to the amplitude of $e^{+}e^{-}$ pair photoproduction by a virtual photon). There are also some other effects. For the cross section differential over the electron transverse momentum, $d\sigma/dp_{2\perp}$, the account for the interference of the amplitudes and the contribution of virtual bremsstrahlung (the contribution of the amplitudes $T^{(0)}$ and ${\widetilde{T}}^{(0)}$) is noticeable. The Coulomb corrections to scattering is larger than these two effects but essentially smaller than the Coulomb corrections to the amplitude of pair photoproduction by a virtual photon. However, in the cross section differential over the positron transverse momentum, $d\sigma/dp_{4\perp}$, the interference of the amplitudes and the contribution of virtual bremsstrahlung lead to the same corrections as the effect of the Coulomb corrections to scattering. They are of the same order as in the case of $d\sigma/dp_{2\perp}$. This means that there is a strong suppression of the effect of the Coulomb corrections to scattering in the cross section $d\sigma/dp_{4\perp}$. Relative importance of various effects for the total cross section is the same as in the case of the cross section $d\sigma/dp_{4\perp}$.

Acknowledgement

This work has been supported by Russian Science Foundation (Project No. 14-50-00080). It has been also supported in part by RFBR (Grant No. 16-02-00103).

Appendix

Here we present the explicit expression for the amplitude $T$, derived in Ref. KM2016 , with one modification. Namely, since we are going to investigate the importance of the interaction of electrons with the momenta $\bm{p}_{1}$ and $\bm{p}_{2}$ with the atomic field, we introduce the parameter $\lambda$ which is equal to unity if this interaction is taken into account and equals to zero if one neglects this interaction. We write the amplitude $T$ in the form

[TABLE]

where the helicity amplitudes $T^{(0)}_{\perp\parallel}$ read

[TABLE]

Here $\mu_{i}=\pm 1$ corresponds to the helicity of the particle with the momentum $\bm{p}_{i}$, $\bar{\mu}_{i}=-\mu_{i}$, and

[TABLE]

with $V(r)$ being the electron potential energy in the atomic field. In the amplitude $T^{(0)}$ the interaction of the produced $e^{+}e^{-}$ pair with the atomic field is neglected, so that $T^{(0)}$ depends on the atomic potential in the same way as the bremsstrahlung amplitude, see, e.g., Ref. LMSS2005 .

The amplitudes $T^{(1)}_{\perp\parallel}$ have the following form

[TABLE]

where $F_{a}(Q^{2})$ is a atomic form factor, and the following notations are used

[TABLE]

Note that the parameter $\lambda$ is contained solely in the function $A(\bm{\Delta})$, Eq. (Appendix).