# Analytic Form of the Planar Two-Loop Five-Parton Scattering Amplitudes in QCD

## TL;DR

This paper derives the complete analytic expressions for all leading-color two-loop five-parton helicity amplitudes in QCD, facilitating precise calculations of three-jet production at hadron colliders.

## Contribution

It introduces a novel method combining finite field evaluations and variable choices to efficiently reconstruct and simplify two-loop five-parton amplitudes analytically.

## Key findings

- Analytic expressions for all leading-color two-loop five-parton amplitudes obtained.
- Amplitudes are simplified to a compact form using multivariate partial fractions.
- Results enable next-to-next-to-leading order QCD corrections for three-jet production.

## Abstract

We present the analytic form of all leading-color two-loop five-parton helicity amplitudes in QCD. The results are analytically reconstructed from exact numerical evaluations over finite fields. Combining a judicious choice of variables with a new approach to the treatment of particle states in $D$ dimensions for the numerical evaluation of amplitudes, we obtain the analytic expressions with a modest computational effort. Their systematic simplification using multivariate partial-fraction decomposition leads to a particularly compact form. Our results provide all two-loop amplitudes required for the calculation of next-to-next-to-leading order QCD corrections to the production of three jets at hadron colliders in the leading-color approximation.