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

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 $m^2\mathrm{/}s$. 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 ${\gamma }^{\ast }\mathrm{/}{Z}^{\ast }$ at high luminosity.