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

TL;DR

The paper delivers the first analytic expressions for the complete set of planar two-loop five-gluon amplitudes in QCD in the leading-color limit, including the MHV configuration, by reconstructing results from numerical unitarity samples on finite fields. It employs a momentum-twistor parametrization and a pentagon-function basis, exploiting the 26-letter pentagon alphabet to constrain denominators and using basis changes to minimize polynomial degree in the coefficients. The amplitudes are expressed as finite remainders decomposed into pentagon functions, with ancillary files providing remainders, one-loop inputs, and evaluation scripts, enabling efficient and stable phase-space integration for NNLO applications. The approach demonstrates substantial efficiency and resilience against intermediate expression growth and is expected to inform future precision QCD studies and formal developments in scattering amplitudes.

Abstract

We present the analytic form of the two-loop five-gluon scattering amplitudes in QCD for a complete set of independent helicity configurations of external gluons. These include the first analytic results for five-point two-loop amplitudes relevant for the computation of next-to-next-to-leading-order QCD corrections at hadron colliders. The results were obtained by reconstructing analytic expressions from numerical evaluations. The complexity of the computation is reduced by exploiting physical and analytical properties of the amplitudes, employing a minimal basis of so-called pentagon functions that have recently been classified.