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Helicity Amplitudes for ${\cal O}(α_s)$ Production of $W^+ W^-, W^\pm Z, Z Z, W^\pm γ,$ or $Z γ$ Pairs at Hadron Colliders

L. Dixon, Z. Kunszt, A. Signer

TL;DR

This work delivers a complete set of helicity amplitudes at ${\rm O}(\alpha_s)$ for the hadronic production of vector-boson pairs ($W^+W^-$, $ZZ$, $WZ$, $W\gamma$, $Z\gamma$) with full spin correlations and leptonic decays in the narrow-width limit. The authors introduce primitive amplitudes $A^{a}$ and $A^{b}$ to organize one-loop and real-emission contributions, and they show how these building blocks can be dressed with electroweak couplings to obtain the full physical amplitudes for all target processes. They provide explicit tree-level and one-loop expressions for the various production channels, including real-gluon and real-photon emissions, and demonstrate how the results reproduce known virtual corrections while enabling accurate, spin-aware NLO predictions. The framework supports flexible Monte Carlo implementations and facilitates investigations of decay-angle correlations and potential non-standard triple-gauge-boson couplings at hadron colliders.

Abstract

We present the one-loop QCD corrections to the helicity amplitudes for the processes $q\qb \to W^+ W^-, Z Z, W^\pm Z, W^\pm γ$, or $Z γ$, including the subsequent decay of each massive vector boson into a pair of leptons. We also give the corresponding tree-level amplitudes with an additional gluon radiated off the quark line. Together, these amplitudes provide all the necessary input for the calculation of the next-to-leading order QCD corrections to the production of any electroweak vector boson pair at hadron colliders, including the full spin and decay angle correlations.

Helicity Amplitudes for ${\cal O}(α_s)$ Production of $W^+ W^-, W^\pm Z, Z Z, W^\pm γ,$ or $Z γ$ Pairs at Hadron Colliders

TL;DR

This work delivers a complete set of helicity amplitudes at for the hadronic production of vector-boson pairs (, , , , ) with full spin correlations and leptonic decays in the narrow-width limit. The authors introduce primitive amplitudes and to organize one-loop and real-emission contributions, and they show how these building blocks can be dressed with electroweak couplings to obtain the full physical amplitudes for all target processes. They provide explicit tree-level and one-loop expressions for the various production channels, including real-gluon and real-photon emissions, and demonstrate how the results reproduce known virtual corrections while enabling accurate, spin-aware NLO predictions. The framework supports flexible Monte Carlo implementations and facilitates investigations of decay-angle correlations and potential non-standard triple-gauge-boson couplings at hadron colliders.

Abstract

We present the one-loop QCD corrections to the helicity amplitudes for the processes , or , including the subsequent decay of each massive vector boson into a pair of leptons. We also give the corresponding tree-level amplitudes with an additional gluon radiated off the quark line. Together, these amplitudes provide all the necessary input for the calculation of the next-to-leading order QCD corrections to the production of any electroweak vector boson pair at hadron colliders, including the full spin and decay angle correlations.
Paper Structure (18 sections, 55 equations, 2 figures)

This paper contains 18 sections, 55 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Primitive amplitudes for $W^+ W^-,\ Z \, Z,\ W^\pm Z$
  3. Preliminaries
  4. Relation to subleading-color $e^+e^- \rightarrow q{\bar{q}} Q{\bar{Q}}$ amplitude
  5. Virtual primitive amplitudes
  6. Real (Bremsstrahlung) primitive amplitudes
  7. Dressing with electroweak couplings for $W^+ W^-,\ Z \, Z,\ W^\pm Z$
  8. ${\bar{q}} q \rightarrow W^- W^+ \rightarrow \ell {\bar{\nu}}_\ell \, {\bar{\ell}}' \nu_{\ell'}$
  9. ${\bar{q}} q \rightarrow Z Z \rightarrow \ell {\bar{\ell}} \, {\bar{\ell}}' \ell'$
  10. ${\bar{u}} d \rightarrow W^- Z \rightarrow \ell {\bar{\nu}}_\ell \, {\bar{\ell}}' \ell'$
  11. Loop and Bremsstrahlung amplitudes
  12. Real photons in the final state
  13. Virtual primitive amplitudes for $q{\bar{q}} \rightarrow V_1 \gamma$
  14. Real (Bremsstrahlung) primitive amplitudes
  15. Dressing with electroweak couplings
  16. ...and 3 more sections

Figures (2)

  • Figure 1: (a$_0$) box-parent tree graph; (a$_1$) box-parent one-loop graph; (a$_{1,1}$) -- (a$_{1,3}$) additional one-loop graphs obtained from the box-parent graph; (b$_0$) triangle-parent tree graph; (b$_1$) triangle-parent one-loop graph. Solid (dashed) lines represent quarks (leptons). In order to distinguish the possibly different vector bosons we used zigzag and wavy lines. For the graphs b$_{0}$ and b$_{1}$ we show the diagrams for the $W^+ W^-$ intermediate state.
  • Figure 2: Parent diagrams for $A_6^{\rm sl}(1,2,3,4)$, i.e. the subleading-in-color part of $e^+ e^- \rightarrow q {\bar{q}} Q {\bar{Q}}$. Solid (dotted) lines represent quarks (electrons). Note that the labeling used in this figure and in section \ref{['relation']} corresponds to the one used in ref. BDKW4q.