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

Understand

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.