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We calculate the complete $O(\alpha^2)$ initial state radiation correctionsto $e^+ e^-$ annihilation into a neutral vector boson in a direct analyticcomputation without any approximation. The corrections are represented in termsof iterated incomplete (elliptic) integrals over alphabets of square--rootvalued letters. Performing the limit $s \gg m_e^2$, we find discrepancies withthe earlier results of Ref.~\cite{Berends:1987ab} and confirm results obtainedin Ref.~\cite{Blumlein:2011mi} where the effective method of massive operatormatrix elements has been used, which works for all but the power corrections in$m^2_e/s$. In this way, we also confirm the validity of the factorization ofmassive partons in the Drell--Yan process. We add non--logarithmic terms at$O(\alpha^2)$ which have not been considered in previous calculations. Thefinal results in the limit $s \gg m_e^2$ can be given in terms of Nielsenintegrals.

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

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