Local integrands for two-loop all-plus Yang-Mills amplitudes

TL;DR

This work develops and applies $D$-dimensional local integrands to the planar five- and six-gluon all-plus amplitudes at two loops, fixing the integrands via six-dimensional unitarity cuts. The approach yields IR-safe, locality-preserving representations with reduced basis size and no spurious poles, and it makes the infrared structure explicit at the integrand level. Rational terms are obtained from one-loop squared topologies, and the results for five and six points agree with existing literature, including augmented BCFW methods. The framework leverages connections to $\mathcal{N}=4$ SYM and momentum-twistor parameterizations, offering a path toward generalizing to nonplanar and more complex two-loop amplitudes.

Abstract

We express the planar five- and six-gluon two-loop Yang-Mills amplitudes with all positive helicities in compact analytic form using D-dimensional local integrands that are free of spurious singularities. The integrand is fixed from on-shell tree amplitudes in six dimensions using D-dimensional generalised unitarity cuts. The resulting expressions are shown to have manifest infrared behaviour at the integrand level. We also find simple representations of the rational terms obtained after integration in 4-2epsilon dimensions.