# The $O(\alpha^2)$ Initial State QED Corrections to $e^+e^-$ Annihilation to a Neutral Vector Boson Revisited

## TL;DR

This paper provides a precise calculation of second-order QED initial state corrections for electron-positron annihilation into a neutral vector boson, highlighting discrepancies with previous results and confirming the factorization approach.

## Contribution

It offers an exact analytic computation of $O( ext{alpha}^2)$ corrections, including non-logarithmic terms, and clarifies the validity of factorization in high-energy $e^+e^-$ processes.

## Key findings

- Discrepancies with earlier $O( ext{alpha}^2)$ results at high energies.
- Confirmation of the factorization of massive partons in Drell-Yan processes.
- Addition of non-logarithmic terms previously omitted.

## Abstract

We calculate the non-singlet, the pure singlet contribution, and their interference term, at $O(\alpha^2)$ due to electron-pair initial state radiation to $e^+ e^-$ annihilation into a neutral vector boson in a direct analytic computation without any approximation. The correction is represented in terms of iterated incomplete elliptic integrals. Performing the limit $s \gg m_e^2$ we find discrepancies with the earlier results of Ref.~\cite{Berends:1987ab} and confirm results obtained in Ref.~\cite{Blumlein:2011mi} where the effective method of massive operator matrix elements has been used, which works for all but the power corrections in $m^2/s$. In this way, we also confirm the validity of the factorization of massive partons in the Drell-Yan process. We also add non-logarithmic terms at $O(\alpha^2)$ which have not been considered in \cite{Berends:1987ab}. The corrections are of central importance for precision analyzes in $e^+e^-$ annihilation into $\gamma^*/Z^*$ at high luminosity.