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Double-Virtual NNLO QCD Corrections for Five-Parton Scattering: The Quark Channels

Giuseppe De Laurentis, Harald Ita, Vasily Sotnikov

TL;DR

This work completes the analytic two-loop NNLO QCD description for five-parton scattering in quark channels, going beyond leading color through non-planar, subleading-color contributions. It combines numerical unitarity with finite-field reconstruction to obtain compact analytic expressions for the finite remainders, aided by cross-channel relations with gluon amplitudes. A generating set of finite remainders is systematically constructed, with new color-identity results in the four-quark sector and an efficient reconstruction workflow that leverages rescaled gluon coefficients. The results are implemented in a public C++ library (FivePointAmplitudes) and validated extensively, enabling robust NNLO predictions for three-jet production and informing future higher-order QCD studies.

Abstract

We complete the computation of two-loop helicity amplitudes required to obtain next-to-next-to-leading order QCD corrections for three-jet production at hadron colliders, including all contributions beyond the leading-color approximation. The analytic expressions are reconstructed from finite-field samples obtained with the numerical unitarity method. We find that the reconstruction is significantly facilitated by exploiting the overlaps between rational coefficient functions of quark and gluon processes, and we display their compact generating sets in the appendix of the paper. We implement our results in a public code, and demonstrate its suitability for phenomenological applications.

Double-Virtual NNLO QCD Corrections for Five-Parton Scattering: The Quark Channels

TL;DR

This work completes the analytic two-loop NNLO QCD description for five-parton scattering in quark channels, going beyond leading color through non-planar, subleading-color contributions. It combines numerical unitarity with finite-field reconstruction to obtain compact analytic expressions for the finite remainders, aided by cross-channel relations with gluon amplitudes. A generating set of finite remainders is systematically constructed, with new color-identity results in the four-quark sector and an efficient reconstruction workflow that leverages rescaled gluon coefficients. The results are implemented in a public C++ library (FivePointAmplitudes) and validated extensively, enabling robust NNLO predictions for three-jet production and informing future higher-order QCD studies.

Abstract

We complete the computation of two-loop helicity amplitudes required to obtain next-to-next-to-leading order QCD corrections for three-jet production at hadron colliders, including all contributions beyond the leading-color approximation. The analytic expressions are reconstructed from finite-field samples obtained with the numerical unitarity method. We find that the reconstruction is significantly facilitated by exploiting the overlaps between rational coefficient functions of quark and gluon processes, and we display their compact generating sets in the appendix of the paper. We implement our results in a public code, and demonstrate its suitability for phenomenological applications.
Paper Structure (25 sections, 52 equations, 1 figure, 5 tables)

This paper contains 25 sections, 52 equations, 1 figure, 5 tables.

Table of Contents

  1. Introduction
  2. Notation and Conventions
  3. Helicity amplitudes
  4. Kinematics and permutation groups
  5. Color space
  6. Renormalization
  7. Generating set of finite remainders
  8. NNLO hard function
  9. IR scheme change
  10. Numerical sampling of remainders
  11. Identities between partial amplitudes
  12. Two-quark channel
  13. Four-quark channel
  14. Analytic reconstruction
  15. Rescaled coefficient functions
  16. ...and 10 more sections

Figures (1)

  • Figure 1: Distributions of the base 10 logarithm of the relative error (correct digits) of the NNLO hard functions defined in \ref{['eq:hard-function']} for representative physical channels contributing to three-jet production at hadron colliders. The dashed curves represent respective cumulative distributions. Here $N_f$ is set to $5$, and the renormalization scale is set dynamically to (half of) the sum of the transverse momentum of the final-state partons. The phase space definition is taken from ref. ATLAS:2014qmg.