Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Electroweak splitting functions in the Standard Model and beyond

Stefan Dittmaier, Max Reyer

TL;DR

This work derives polarized leading-order electroweak splitting functions in generic spontaneously broken gauge theories, addressing the challenge of mass-singular terms in longitudinal polarizations. It introduces two gauge-invariant strategies to ensure consistent factorization: a Ward-identity–guided approach that keeps standard polarization vectors, and an alternative cancelation-based method. The authors provide a comprehensive, frame-independent set of bosonic and fermionic splitting functions, including detailed massless and heavy-vector limits, and establish symmetry and crossing relations, azimuthal averages, and IS/FS correspondences. The results lay a solid foundation for electroweak parton showers and subtraction schemes at high energies, with explicit comparisons to existing literature showing agreement where applicable and clarifying methodological differences. Overall, the paper advances practical tools for precision predictions in the TeV-scale and future colliders by delivering a complete, symmetry-consistent framework for EW quasi-collinear radiation.

Abstract

We derive quasi-collinear factorization formulas in generic spontaneously broken gauge theories with scalars, fermions, and vector bosons. Specifically, we obtain polarized leading-order splitting functions for all possible final-state and initial-state 1->2 processes in the considered gauge theory. The main complication lies in the presence of mass-singular terms in longitudinal polarization vectors, prohibiting the direct application of the usual factorization procedure known from Quantum Electrodynamics and Quantum Chromodynamics. We overcome this issue with two different strategies, using gauge invariance and Ward identities as guiding principle. Our derivations do not use any explicit component-wise parametrizations of momenta and wave functions and bear no reference to a particular Lorentz frame. Furthermore, our results are valid for completely general definitions of the spin reference axes of the individual external particles. The various massless limits, the special case of the Electroweak Standard Model, the reproduction of existing literature results, and symmetry relations among our splitting functions are discussed in detail.

Electroweak splitting functions in the Standard Model and beyond

TL;DR

This work derives polarized leading-order electroweak splitting functions in generic spontaneously broken gauge theories, addressing the challenge of mass-singular terms in longitudinal polarizations. It introduces two gauge-invariant strategies to ensure consistent factorization: a Ward-identity–guided approach that keeps standard polarization vectors, and an alternative cancelation-based method. The authors provide a comprehensive, frame-independent set of bosonic and fermionic splitting functions, including detailed massless and heavy-vector limits, and establish symmetry and crossing relations, azimuthal averages, and IS/FS correspondences. The results lay a solid foundation for electroweak parton showers and subtraction schemes at high energies, with explicit comparisons to existing literature showing agreement where applicable and clarifying methodological differences. Overall, the paper advances practical tools for precision predictions in the TeV-scale and future colliders by delivering a complete, symmetry-consistent framework for EW quasi-collinear radiation.

Abstract

We derive quasi-collinear factorization formulas in generic spontaneously broken gauge theories with scalars, fermions, and vector bosons. Specifically, we obtain polarized leading-order splitting functions for all possible final-state and initial-state 1->2 processes in the considered gauge theory. The main complication lies in the presence of mass-singular terms in longitudinal polarization vectors, prohibiting the direct application of the usual factorization procedure known from Quantum Electrodynamics and Quantum Chromodynamics. We overcome this issue with two different strategies, using gauge invariance and Ward identities as guiding principle. Our derivations do not use any explicit component-wise parametrizations of momenta and wave functions and bear no reference to a particular Lorentz frame. Furthermore, our results are valid for completely general definitions of the spin reference axes of the individual external particles. The various massless limits, the special case of the Electroweak Standard Model, the reproduction of existing literature results, and symmetry relations among our splitting functions are discussed in detail.

Paper Structure

This paper contains 135 sections, 441 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Generic treatment of spontaneously broken gauge theories
  3. Generic gauge couplings
  4. Quantization and gauge fixing
  5. Spontaneous symmetry breaking
  6. Feynman rules
  7. Ward and Slavnov--Taylor identities
  8. Generic would-be Goldstone couplings
  9. $VGH$ couplings.
  10. $GVV$ couplings.
  11. $VGG$ couplings.
  12. $GGH$ couplings.
  13. $GHH$ couplings.
  14. $GGG$ couplings.
  15. $G\bar{f}f$ couplings.
  16. ...and 120 more sections

Figures (1)

  • Figure 1: Some representative tree-level diagrams contributing to the partonic process $u\bar{d}\to {\mathrm{W}}\xspace^+ {\mathrm{W}}\xspace^- {\mathrm{W}}\xspace^+$ (adapted from Ref. Dittmaier:2017bnh). In the limit in which ${\mathrm{W}}\xspace^+_{(i)}$ and ${\mathrm{W}}\xspace^-_{(j)}$ become collinear, graphs (b), (c), (d) are of type I, while graphs (a), (e) are of type II.