Two-loop electroweak next-to-leading logarithmic corrections to massless fermionic processes

TL;DR

This work derives the complete one‑ and two‑loop electroweak corrections to high‑energy processes with external massless fermions, focusing on leading and next‑to‑leading logarithms in the regime $Q\gg M_W$. By employing sector decomposition and expansion by regions, the authors extract mass singularities as $L=\ln(Q^2/M_W^2)$ plus $1/\epsilon$ poles in $D=4-2\epsilon$, within the full spontaneously broken electroweak theory. A central result is that mass‑singular terms factorize from the Born amplitude and that non‑factorizable contributions cancel via collinear Ward identities, leaving a universal, process‑independent structure built from one‑loop building blocks; the two‑loop corrections can be written in terms of these one‑loop operators plus beta‑function coefficients, and can be expressed as a product of three exponentials corresponding to symmetric EW, $M_Z$–$M_W$ mixing, and QED mass‑gap effects. The EW two‑loop results closely mirror the singular structure of massless QCD and are consistent with Catani’s formula and EW resummation prescriptions, while also yielding new terms in $\ln(M_Z^2/M_W^2)$. The formalism is then specialized to neutral‑current and charged‑current four‑fermion processes, providing explicit, gauge‑invariant corrections that enhance precision for high‑energy collider phenomenology and validate resummation frameworks in the electroweak sector.

Abstract

We consider two-loop leading and next-to-leading logarithmic virtual corrections to arbitrary processes with external massless fermions in the electroweak Standard Model at energies well above the electroweak scale. Using the sector-decomposition method and alternatively the strategy of regions we calculate the mass singularities that arise as logarithms of Q^2/MW^2, where Q is the energy scale of the considered process, and 1/εpoles in D=4-2εdimensions, to one- and two-loop next-to-leading logarithmic accuracy. The derivations are performed within the complete electroweak theory with spontaneous symmetry breaking. Our results indicate a close analogy between the form of two-loop electroweak logarithmic corrections and the singular structure of scattering amplitudes in massless QCD. We find agreement with the resummation prescriptions that have been proposed in the literature based on a symmetric SU(2) \times U(1) theory matched with QED at the electroweak scale and provide new next-to-leading contributions proportional to ln(MZ^2/MW^2).