Analytic Form of the Three-loop Four-gluon Scattering Amplitudes in Yang-Mills Theory

TL;DR

This paper derives explicit analytic three-loop four-gluon scattering amplitudes in pure Yang-Mills theory, crucial for high-precision collider predictions, using advanced unitarity and reduction techniques.

Contribution

It provides the first explicit analytic form of three-loop four-gluon amplitudes in Yang-Mills theory, including gauge-invariant bases and master integrals.

Findings

Amplitudes are explicitly invariant under cyclic permutations.

Remaining divergences match known infrared structures.

Results are essential for N^3LO jet production calculations.

Abstract

We present analytic forms of three-loop four-gluon planar amplitudes in pure Yang-Mills theory in this letter. Gauge invariant bases and a set of proper master integrals are chosen such that the amplitudes are explicitly invariant under cyclic permutations of external particles. The $D$-dimensional unitarity method and integration-by-parts reductions with cut conditions are performed to determine the coefficients of the master integrals. Helicity amplitudes in the conventional dimension regularization scheme are obtained by setting gauge invariant bases to helicity configurations. After renormalizing the ultraviolet divergences, remaining divergences of the amplitudes agree with known infrared divergence structures exactly. Our results provide an essential piece of three-loop amplitudes required for the ${\rm N^3LO}$ order corrections to the production of two jets at hadron colliders.