High-energy $e^{+}{e}^{-}$ photoproduction in the field of a heavy atom accompanied by bremsstrahlung

Helicity amplitudes and the differential cross section of high-energy $e^{+}{e}^{-}$ photoproduction accompanied by bremsstrahlung in the electric field of a heavy atom (i.e., the amplitudes of the process ${\gamma }_{1}Z\to e^{+}{e}^{-}{\gamma }_{2}Z)$ are derived. The results are exact in the nuclear charge number and are obtained in the leading quasiclassical approximation. They correspond to the leading high-energy small-angle asymptotics of the amplitude. It is shown that, in general, accounting for the Coulomb corrections essentially modify the differential cross section, which is different from the Born result. When the initial photon is circularly polarized, the Coulomb corrections lead to the asymmetry in the distribution over the azimuth angles ${\phi }_i$ of produced particles with respect to the replacement ${\phi }_i\to -{\phi }_i$.

Figure 1

Diagrams of the process ${\gamma }_{1}Z\to e^{+}{e}^{-}{\gamma }_{2}Z$. Thick solid lines denote exact propagators in the nuclear field.

Figure 2

The quantity $S$ [the differential cross section in units of ${\sigma }_0$, averaged over the polarization of the initial photon and summed over polarizations of the final particles; see Eq. (29)] as a function of $k_{2x}/m$ for ${\varepsilon }_p=0.4{\omega }_1,{\varepsilon }_q=0.25{\omega }_1,p_x=4.7m,q_x=-0.8m,p_y=q_y=k_{2y}=0$: Born result (dotted curve), $Z=47$ (Ag, dash-dotted curve), $Z=82$ (Pb, dashed curve), and $Z=92$ (U, solid curve). The quantity $S$, calculated for the Coulomb field, is independent of ${\omega }_1$. Accounting for screening, where the dependence on ${\omega }_1$ still exists due to the atomic form factor, modifies $S$ only in the narrow vicinity of the point ${\Delta }_{\perp }=0$ $(k_{2x}=-3.9m$ here), where the Coulomb corrections are unimportant.

Figure 3

Same as Fig. 2, but for $p_x=0.7m,q_x=2.33m$.

Figure 4

Asymmetry $A$, Eq. (30), as a function of the angle $\phi$ between ${\mathit{k}}_{2\perp }$ and ${\mathit{p}}_{\perp }$ for ${\varepsilon }_p=0.4{\omega }_1,{\varepsilon }_q=0.25{\omega }_1,{\mathit{p}}_{\perp }\parallel -{\mathit{q}}_{\perp },p_{\perp }=4.7m,q_{\perp }=0.8m,k_{2\perp }=m$: Born result (dotted curve), $Z=47$ (Ag, dash-dotted curve), $Z=82$ (Pb, dashed curve), and $Z=92$ (U, solid curve).

Figure 5

Same as Fig. 4, but for ${\mathit{p}}_{\perp }\parallel {\mathit{q}}_{\perp },p_{\perp }=0.7m,q_{\perp }=2.33m$.