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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

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.