The $O({\alpha }^2)$ Initial State QED Corrections to $e^{+}{e}^{-}\to {\gamma }^{\ast }/Z_0^{\ast }$

Abstract

We calculate the complete $O({\alpha }^2)$ initial state radiation corrections to $e^{+}{e}^{-}$ annihilation into a neutral vector boson in a direct analytic computation without any approximation. The corrections are represented in terms of iterated incomplete (elliptic) integrals over alphabets of square--root valued letters. 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 $m_e^2/s$. In this way, we also confirm the validity of the factorization of massive partons in the Drell--Yan process. We add non--logarithmic terms at $O({\alpha }^2)$ which have not been considered in previous calculations. The final results in the limit $s\gg m_e^2$ can be given in terms of Nielsen integrals.

32 pages Latex, several figures

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