Electron-positron annihilation to photons at $O(\alpha^3)$ revisited

TL;DR

This paper recalculates the electron-positron annihilation cross sections to photons at third order in alpha using modern multiloop methods, correcting previous inaccuracies and confirming some earlier results.

Contribution

It applies advanced multiloop techniques to precisely compute cross sections, identifies and corrects errors in prior differential cross section calculations, and clarifies the asymptotic behavior.

Findings

Agreement with some previous results after correction

Identification of omitted terms in earlier differential cross sections

Correction of errors in prior theoretical calculations

Abstract

We apply the modern multiloop methods to the calculation of the total cross sections of electron-positron annihilation to 2 and 3 photons exactly in $s/m^2$ with the accuracy $O(\alpha^3)$. Examining the asymptotics of our results, we find agreement with Ref. [Andreassi_et_al_1962] and discover mistakes in the results of Refs. [Eidelman&Kuraev_1978,Berends&Kleiss_1981]. This mistake is due to the terms, omitted in differential cross section in Refs. \[Eidelman&Kuraev_1978,Berends&Kleiss_1981], which are peaked in the kinematic region with all three photons being quasi-parallel to the collision axis. After restoring these terms, we find an agreement of the corrected result of Ref. [Berends&Kleiss_1981] with our result.