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Two-loop leading colour helicity amplitudes for $W^\pmγ+j$ production at the LHC

Simon Badger, Heribertus Bayu Hartanto, Jakub Kryś, Simone Zoia

TL;DR

This paper computes the first two-loop leading-color QCD helicity amplitudes for pp → Wγj with leptonic W decay, employing finite-field reduction and analytic reconstruction in a basis of special functions to produce compact finite remainders. It detaches the W decay, analyzes five- and four-point W-production amplitudes via a projector method, and reconstructs the results using IBP reduction to canonical master integrals and a specialized function basis. Extensive validation includes cross-checks with full six-point computations, gauge invariance tests, scale-dependence checks, and comparisons to known tree-level and one-loop results, confirming robustness. The analytic expressions, together with efficient numerical evaluation strategies and permutation handling, pave the way for NNLO QCD predictions in Wγj and inform potential SMEFT global analyses, while highlighting areas for future improvement such as non-planar and sub-leading color contributions.

Abstract

We present the two-loop leading colour QCD helicity amplitudes for the process $pp\to W(\to lν)γ+j$. We implement a complete reduction of the amplitudes, including the leptonic decay of the $W$-boson, using finite field arithmetic, and extract the analytic finite remainders using a recently identified basis of special functions. Simplified analytic expressions are obtained after considering permutations of a rational kinematic parametrisation and multivariate partial fractioning. We demonstrate efficient numerical evaluation of the two-loop colour and helicity summed finite remainders for physical kinematics, and hence the suitability for phenomenological applications.

Two-loop leading colour helicity amplitudes for $W^\pmγ+j$ production at the LHC

TL;DR

This paper computes the first two-loop leading-color QCD helicity amplitudes for pp → Wγj with leptonic W decay, employing finite-field reduction and analytic reconstruction in a basis of special functions to produce compact finite remainders. It detaches the W decay, analyzes five- and four-point W-production amplitudes via a projector method, and reconstructs the results using IBP reduction to canonical master integrals and a specialized function basis. Extensive validation includes cross-checks with full six-point computations, gauge invariance tests, scale-dependence checks, and comparisons to known tree-level and one-loop results, confirming robustness. The analytic expressions, together with efficient numerical evaluation strategies and permutation handling, pave the way for NNLO QCD predictions in Wγj and inform potential SMEFT global analyses, while highlighting areas for future improvement such as non-planar and sub-leading color contributions.

Abstract

We present the two-loop leading colour QCD helicity amplitudes for the process . We implement a complete reduction of the amplitudes, including the leptonic decay of the -boson, using finite field arithmetic, and extract the analytic finite remainders using a recently identified basis of special functions. Simplified analytic expressions are obtained after considering permutations of a rational kinematic parametrisation and multivariate partial fractioning. We demonstrate efficient numerical evaluation of the two-loop colour and helicity summed finite remainders for physical kinematics, and hence the suitability for phenomenological applications.
Paper Structure (19 sections, 81 equations, 3 figures, 6 tables)

This paper contains 19 sections, 81 equations, 3 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Structure of the Amplitudes
  3. Amplitude Computation
  4. Kinematics
  5. Structure of the five-particle $W$-production amplitudes
  6. Structure of the four-particle $W$-production amplitudes
  7. Amplitude reduction and analytic reconstruction
  8. Simplification of the rational coefficients
  9. Numerical evaluation and permutations of the amplitudes
  10. Validation
  11. Comparison against full six-point computation
  12. Gauge invariance
  13. Renormalisation scale dependence
  14. Tree-level and one-loop checks
  15. Four-point amplitude comparison
  16. ...and 4 more sections

Figures (3)

  • Figure 1: Sample two-loop Feynman diagrams for $W^+\gamma j$ production.
  • Figure 2: Sample two-loop Feynman diagrams for $W^+\gamma j$ production containing closed fermion loop. $A_{6,q}^{(2),n_f}$ vanishes due to Furry's theorem.
  • Figure 3: Reduced squared finite remainders $\mathcal{H}^{(L)}$ at tree level, one and two loops evaluated on the univariate phase-space slice defined by Eqs. \ref{['eq:unislice1']}, \ref{['eq:unislice2']} and \ref{['eq:unislice3']}, with the parameters given in Eq. \ref{['eq:unisliceparams']}, for all channels of $W^+\gamma j$ production defined in Eq. \ref{['eq:wplus_channel_definition']}.