Two-loop electroweak angular-dependent logarithms at high energies

TL;DR

This work computes the two-loop leading and angular-dependent next-to-leading electroweak logarithms for arbitrary high-energy processes in the spontaneously broken Standard Model. It employs an eikonal framework with the Sudakov technique and sector decomposition to extract mass-singular logs across a multi-scale hierarchy, revealing that the total corrections factorize into a product of two exponentials: one for the SU(2)×U(1) sector and one for the mass-gap QED-like sector, i.e. $\mathcal{M}_2 = \mathcal{M}_0 \exp(\delta_{\rm sew}) \exp(\delta_{\rm sem})$. A nontrivial commutator between $\delta_{\rm sew}$ and $\delta_{\rm sem}$ fixes the ordering of these exponentials at angular-dependent NL levels, aligning with resummation prescriptions based on SU(2)×U(1) symmetry matched to QED. The analysis, corroborated by two independent integral methods, supports the exponentiation picture for high-energy EW corrections and extends its applicability to extensions of the electroweak sector that preserve the gauge content and mass-scale hierarchy.

Abstract

We present results on the two-loop leading and angular-dependent next-to-leading logarithmic virtual corrections to arbitrary processes at energies above the electroweak scale. In the `t Hooft-Feynman gauge the relevant Feynman diagrams involving soft and collinear gauge bosons γ, Z, W^\pm coupling to external legs are evaluated in the eikonal approximation in the region where all kinematical invariants are much larger than the electroweak scale. The logarithmic mass singularities are extracted from massive multi-scale loop integrals using the Sudakov method and alternatively the sector-decomposition method in the Feynman-parameter representation. The derivations are performed within the spontaneously broken phase of the electroweak theory, and the two-loop results are in agreement with the exponentiation prescriptions that have been proposed in the literature based on a symmetric SU(2) x U(1) theory matched with QED at the electroweak scale.