High-energy electron-positron photoproduction cross section close to the end of the spectrum

We consider the cross section of electron-positron pair production by a high-energy photon in a strong Coulomb field close to the end of the electron or positron spectrum. We show that the cross section essentially differs from the result obtained in the Born approximation as well as from the result which takes into account the Coulomb corrections under the assumption that both electron and positron are ultrarelativistic. The cross section of bremsstrahlung in a strong Coulomb field by a high-energy electron is also obtained in the region where the final electron is not ultrarelativistic.

Figure 1

Contour of integration in the complex plane of the variable $u$ used to perform the integral in Eq. (13).

Figure 2

The cross section $\omega {\mathrm{\sigma ̃}}^{-1}d\sigma /d{\epsilon }_{-}$ of $e^{+}{e}^{-}$ pair production at zero electron velocity in units $\mathrm{\sigma ̃}=\alpha (Z\alpha ){}^3/m^2$. Solid curve, our results via Eq. (14); dashed curve, the results of Ref. [22] [see also Eq. (16)]; dotted curve, the results of Ref. [7] obtained at $\omega =40\hspace{0.5em}$MeV and ${\beta }_{-}=0.1265$.

Figure 3

The dependence of the Coulomb corrections $\omega {\sigma }_0^{-1}d{\sigma }_C/d{\epsilon }_{-}$ (continuous line) with ${\sigma }_0=\alpha (Z\alpha ){}^2/m^2$ on the scaled electron energy ${\epsilon }_{-}/m$ at different values of $Z$. The dashed line represents the leading-order Coulomb corrections in the limit ${\epsilon }_{-}\gg m$, while the dotted line also includes corrections proportional to $m/{\epsilon }_{-}$. The dashed and the dotted lines start at ${\epsilon }_{-}=5\hspace{0.5em}m$ because the corresponding asymptotics are valid at ${\epsilon }_{-}\gg m$.

Figure 4

The dependence of the Coulomb corrections $\omega {\sigma }_0^{-1}d{\sigma }_C/d{\epsilon }_{+}$ (continuous line) with ${\sigma }_0=\alpha (Z\alpha ){}^2/m^2$ on the scaled positron energy ${\epsilon }_{+}/m$ at different values of $Z$. The dashed line represents the leading-order Coulomb corrections in the limit ${\epsilon }_{+}\gg m$, while the dotted line also includes corrections proportional to $m/{\epsilon }_{+}$. The dashed and the dotted lines start at ${\epsilon }_{+}=5\hspace{0.5em}m$ because the corresponding asymptotics are valid at ${\epsilon }_{+}\gg m$.