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

Abstract

We calculate the non-singlet, the pure singlet contribution, and their interference term, at $O(α^2)$ 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 $O(α^2)$ which have not been considered in \cite{Berends:1987ab}. The corrections are of central importance for precision analyzes in $e^+e^-$ annihilation into $γ^*/Z^*$ at high luminosity.

Figures (2)

• Figure 1: Relative deviations of the results of Ref. Berends:1987ab from the exact result in % for the $O(\alpha^2)$ corrections. The non--singlet contribution (process II): dash-dotted line; the pure singlet contribution (process III): dashed; the interference term between both contributions (process IV): dots; for $s = M_Z^2$, $M_Z = 91.1879$ GeV.
• Figure 2: The initial state $O(\alpha^2)$ corrections to $\gamma^*/Z^*$ production due to $e^+ e^-$ pair production multiplied by $z(1-z)$. The non--singlet contribution (process II): dash-dotted line; the pure singlet contribution (process III): dashes; the interference term between both contributions (process IV) $\times 10$: dotted; the vector contributions implied by process B, Ref. Hamberg:1990np, and interferences $\times 100$: long dash-dotted; all contributions: full line for $s = M_Z^2$.