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Towards Collinear Evolution Equations in Electroweak Theory

M. Ciafaloni, P. Ciafaloni, D. Comelli

TL;DR

The paper investigates how collinear radiative corrections factorize in a spontaneously broken electroweak theory, where initial-state isospin charges and incomplete Ward identities alter the standard DGLAP structure. It derives one-loop electroweak splitting functions, including infrared-sensitive pieces for nonzero t-channel isospin, and proposes a DGLAP-like evolution framework for the electroweak sector. A key contribution is showing how collinear Ward identities enable a factorization formula and identifying new T≠0 splitting structures tied to the symmetry-breaking scale, providing a path to extending to the full Standard Model. The work clarifies the high-energy behavior of electroweak logarithms and sets up evolution equations for the overlap matrix that encode the broken-gauge dynamics.

Abstract

We consider electroweak radiative corrections to hard inclusive processes at the TeV scale, and we investigate how collinear logarithms factorize in a spontaneously broken gauge theory, similarly to the DGLAP analysis in QCD. Due to the uncancelled double logs noticed previously, we find a factorization pattern which is qualitatively different from the analogous one in QCD. New types of splitting functions emerge which are needed to describe the initial beam charges and are infrared-sensitive, that is dependent on an infrared cutoff provided, ultimately, by the symmetry breaking scale. We derive such splitting functions at one-loop level in the example of SU(2) gauge theory, and we also discuss the structure functions' evolution equations, under the assumption that isospin breaking terms present in the Ward identities of the theory are sufficiently subleading at higher orders.

Towards Collinear Evolution Equations in Electroweak Theory

TL;DR

The paper investigates how collinear radiative corrections factorize in a spontaneously broken electroweak theory, where initial-state isospin charges and incomplete Ward identities alter the standard DGLAP structure. It derives one-loop electroweak splitting functions, including infrared-sensitive pieces for nonzero t-channel isospin, and proposes a DGLAP-like evolution framework for the electroweak sector. A key contribution is showing how collinear Ward identities enable a factorization formula and identifying new T≠0 splitting structures tied to the symmetry-breaking scale, providing a path to extending to the full Standard Model. The work clarifies the high-energy behavior of electroweak logarithms and sets up evolution equations for the overlap matrix that encode the broken-gauge dynamics.

Abstract

We consider electroweak radiative corrections to hard inclusive processes at the TeV scale, and we investigate how collinear logarithms factorize in a spontaneously broken gauge theory, similarly to the DGLAP analysis in QCD. Due to the uncancelled double logs noticed previously, we find a factorization pattern which is qualitatively different from the analogous one in QCD. New types of splitting functions emerge which are needed to describe the initial beam charges and are infrared-sensitive, that is dependent on an infrared cutoff provided, ultimately, by the symmetry breaking scale. We derive such splitting functions at one-loop level in the example of SU(2) gauge theory, and we also discuss the structure functions' evolution equations, under the assumption that isospin breaking terms present in the Ward identities of the theory are sufficiently subleading at higher orders.

Paper Structure

This paper contains 3 sections, 11 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Collinear Ward Identities
  3. Evolution Equations

Figures (3)

  • Figure 1: Unitarity diagrams for one-loop real emission corrections to the inclusive hard overlap matrix ${\cal O}^{Hf}$. The symmetrical counterpart of diagram $b)$ is not shown.
  • Figure 2: Examples of one-loop Goldstone boson contributions to CWI for (a) fermion-initiated and (b) boson-initiated processes. The wavy (dashed) lines label vector boson (Goldstone boson) exchanges, the small circles denote the $M$ couplings arising from the use of CWI's.
  • Figure 3: Picture of the collinear evolution equations of eq. (\ref{['sys']}) in the light-fermion, vector-boson sector.