Resummation of angular dependent corrections in spontaneously broken gauge theories

TL;DR

The paper develops an all-orders resummation framework for angular dependent subleading electroweak logarithms in spontaneously broken gauge theories. By employing an infrared evolution equation approach in a symmetric basis with matching to soft QED below the weak scale, it incorporates CKM mixing, mass gaps, and the scalar sector, yielding an exponentiated operator description in a CKM-extended isospin space. The method reproduces known one-loop results and aligns with higher-order findings in massless four-fermion cases, establishing a robust tool for precision electroweak corrections at future colliders. This work enables controlled high-energy predictions by separating universal and non-universal angular SL effects and leveraging real soft emissions for consistency.

Abstract

Recent investigations of electroweak radiative corrections have revealed the importance of higher order contributions in high energy processes, where the size of typical corrections can exceed those associated with QCD considerably. Beyond one loop, only universal (angular independent) corrections are known to all orders except for massless $e^+ e^- \longrightarrow f {\overline f}$ processes where also angular dependent corrections exist in the literature. In this paper we present general arguments for the consistent resummation of angular dependent subleading (SL) logarithmic corrections to all orders in the regime where all invariants are still large compared to the gauge boson masses. We discuss soft isospin correlations, fermion mass and gauge boson mass gap effects, the longitudinal and Higgs boson sector as well as mixing contributions including CKM effects for massive quarks. Two loop arguments are interpreted in the context of the effective high energy effective theory based on the Standard Model Lagrangian in the symmetric basis with the appropriate matching conditions to include the soft QED regime. The result is expressed in exponentiated operator form in a CKM-extended isospin space in the symmetric basis. Thus, a full electroweak SL treatment based on the infrared evolution equation method is formulated for arbitrary high energy processes at future colliders. Comparisons with known results are presented.

Figures (2)

• Figure 1: Angular dependent two loop on-shell QED diagrams. The sum of II$^a_\theta$ and II$^b_\theta$ factorizes in massive QED into the product of the two one loop corrections (each with a different invariant and mass terms) in leading order.
• Figure 2: Electroweak angular dependent two loop on-shell diagrams involving CKM matrix elements (denoted by $V_{ij}$). The wavy line denotes a photon, the zigzag line a $W^\pm$. The sum of the virtual diagrams II$^a_\theta$, II$^b_\theta$ and II$^c_\theta$ must factorize for the leading mass singular terms into the product of the two one loop corrections (each with a different invariant and mass terms). The formal reason for this factorization is provided by the fact that the real emission diagrams II$^a_r$ and II$^b_r$ are free of fermion mass singular terms due to the off-shellness generated by the $W^\pm$ exchange. Thus only the factorized real corrections of diagrams II$^c_r$ and II$^d_r$ contribute fermion mass singular terms and the KLN theorem then yields the analogous factorization for the above virtual corrections. For the full SM corrections, however, one must include all diagrams since the photon cannot be separated from the other gauge bosons in a gauge invariant way.