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Five-Parton Scattering in QCD at Two Loops

Bakul Agarwal, Federico Buccioni, Federica Devoto, Giulio Gambuti, Andreas von Manteuffel, Lorenzo Tancredi

TL;DR

This work delivers the complete two-loop QCD corrections for five-parton scattering with full color dependence, covering gg→gggg, q q̄→ggg, and related channels. It combines tensor projection in the t Hooft-Veltman scheme with a color-decomposed framework, reducing vast sets of loop integrals to a basis of master integrals expressed through massless pentagon functions. The calculation employs state-of-the-art techniques—IBP and differential equations, finite-field methods, and multivariate partial fractioning—to manage algebraic complexity and yield compact analytic finite remainders. The results provide essential full-color NNLO inputs for three-jet observables at the LHC and offer new insights into high-energy QCD regimes such as multi-Regge kinematics and collinear factorization breaking.

Abstract

We compute all helicity amplitudes for the scattering of five partons in two-loop QCD in all the relevant flavor configurations, retaining all contributing color structures. We employ tensor projection to obtain helicity amplitudes in the 't Hooft-Veltman scheme starting from a set of primitive amplitudes. Our analytic results are expressed in terms of massless pentagon functions, and are easy to evaluate numerically. These amplitudes provide important input to investigations of collinear-factorization breaking and to studies of the multi-Regge kinematics regime.

Five-Parton Scattering in QCD at Two Loops

TL;DR

This work delivers the complete two-loop QCD corrections for five-parton scattering with full color dependence, covering gg→gggg, q q̄→ggg, and related channels. It combines tensor projection in the t Hooft-Veltman scheme with a color-decomposed framework, reducing vast sets of loop integrals to a basis of master integrals expressed through massless pentagon functions. The calculation employs state-of-the-art techniques—IBP and differential equations, finite-field methods, and multivariate partial fractioning—to manage algebraic complexity and yield compact analytic finite remainders. The results provide essential full-color NNLO inputs for three-jet observables at the LHC and offer new insights into high-energy QCD regimes such as multi-Regge kinematics and collinear factorization breaking.

Abstract

We compute all helicity amplitudes for the scattering of five partons in two-loop QCD in all the relevant flavor configurations, retaining all contributing color structures. We employ tensor projection to obtain helicity amplitudes in the 't Hooft-Veltman scheme starting from a set of primitive amplitudes. Our analytic results are expressed in terms of massless pentagon functions, and are easy to evaluate numerically. These amplitudes provide important input to investigations of collinear-factorization breaking and to studies of the multi-Regge kinematics regime.
Paper Structure (5 sections, 30 equations, 1 figure, 2 tables)

This paper contains 5 sections, 30 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. Kinematics and color decomposition
  3. Helicity Amplitudes
  4. Details of the calculation
  5. Final Results and Conclusions

Figures (1)

  • Figure 1: Definition of the minimal set of helicity and color configurations for the different processes. The list of helicity configuration is given in the left-most column of each table. The middle columns contain our definition of the spinor factors $\Phi_c({\boldsymbol\lambda})$ for the different color factors. The right columns give the list of partial amplitudes we computed analytically for the corresponding helicity configuration. Spinor factors are chosen to set the relative tree-level (when non-vanishing) to 1. Because of this, two spinor factor choices are needed for the $\bar{q} q \bar{q} ' q' g$ channel.