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Electroweak Evolution Equations

Paolo Ciafaloni, Denis Comelli

TL;DR

This work develops infrared–collinear electroweak evolution equations for the full Standard Model, extending prior analyses that included only fermions and transverse gauge bosons. It introduces infrared-singular splitting functions to handle Bloch–Nordsieck-violating double logs and incorporates third-family Yukawa effects, while CKM mixing and ultraviolet running are neglected. The authors formulate matrix evolution equations in isospin space and then project onto states with definite total t-channel quantum numbers (T,Y,CP), yielding a tractable set of scalar evolution equations with a clear block-diagonal structure. The resulting framework provides a DGLAP-like resummation for electroweak logarithms at high energies and specifies how to rotate from gauge to mass eigenstates for physical observables.

Abstract

Enlarging a previous analysis, where only fermions and transverse gauge bosons were taken into account, we write down infrared-collinear evolution equations for the Standard Model of electroweak interactions computing the full set of splitting functions. Due to the presence of double logs which are characteristic of electroweak interactions (Bloch-Nordsieck violation), new infrared singular splitting functions have to be introduced. We also include corrections related to the third generation Yukawa couplings.

Electroweak Evolution Equations

TL;DR

This work develops infrared–collinear electroweak evolution equations for the full Standard Model, extending prior analyses that included only fermions and transverse gauge bosons. It introduces infrared-singular splitting functions to handle Bloch–Nordsieck-violating double logs and incorporates third-family Yukawa effects, while CKM mixing and ultraviolet running are neglected. The authors formulate matrix evolution equations in isospin space and then project onto states with definite total t-channel quantum numbers (T,Y,CP), yielding a tractable set of scalar evolution equations with a clear block-diagonal structure. The resulting framework provides a DGLAP-like resummation for electroweak logarithms at high energies and specifies how to rotate from gauge to mass eigenstates for physical observables.

Abstract

Enlarging a previous analysis, where only fermions and transverse gauge bosons were taken into account, we write down infrared-collinear evolution equations for the Standard Model of electroweak interactions computing the full set of splitting functions. Due to the presence of double logs which are characteristic of electroweak interactions (Bloch-Nordsieck violation), new infrared singular splitting functions have to be introduced. We also include corrections related to the third generation Yukawa couplings.

Paper Structure

This paper contains 4 sections, 39 equations, 3 figures.

Table of Contents

  1. Introduction
  2. A working example: left fermions in the $g'\to 0$ limit
  3. Full EW evolution equations
  4. Appendix:

Figures (3)

  • Figure 1: Graphical picture of the factorization formula \ref{['fact']}.)
  • Figure 2: Graphical picture showing the evolution of the single electron component, see eqs. (\ref{['system1']}).
  • Figure 3: Leading real emission Feynman diagrams in axial gauge: A) Feynman diagrams contributing to the evolution of the fermionic structure functions; B) Feynman diagrams contributing to the evolution of the transverse gauge boson structure functions; C) Feynman diagrams contributing to the evolution of the scalar structure functions. The wavy lines are transverse gauge bosons, dashed lines stay for Higgs sector particles and straight lines for fermions.