Two-loop QED Operator Matrix Elements with Massive External Fermion Lines
Johannes Blümlein, Abilio De Freitas, Wilhelmus van Neerven
TL;DR
The paper computes the two-loop massive operator matrix elements for fermionic twist-2 operators with external massive fermion lines in QED, including constant $\varepsilon$ terms. It implements a Renormalization Group framework to factorize the $O(\alpha^2)$ initial-state corrections into massive OMEs and massless Drell–Yan Wilson coefficients, separating universal mass effects from process-dependent pieces. It provides the full renormalization program and explicit results for the OMEs up to $O(a^2)$, including non-singlet and pure-singlet contributions and the one-loop OMEs $A_{e\gamma}^{(1)}$, $A_{\gamma e}^{(1)}$. It assesses the validity and limitations of the factorization approach and supplies the machinery to improve precision in predictions for $e^+e^-$ annihilation and related processes.
Abstract
The two-loop massive operator matrix elements for the fermionic local twist--2 operators with external massive fermion lines in Quantum Electrodynamics (QED) are calculated up to the constant terms in the dimensional parameter $ε= D - 4$. We investigate the hypothesis of Ref. \cite{BBN} that the 2--loop QED initial state corrections to $e^+e^-$ annihilation into a virtual neutral gauge boson, except power corrections of $O((m_f^2/s)^k), k \geq 1$, can be represented in terms of these matrix elements and the massless 2-loop Wilson coefficients of the Drell-Yan process.