Local Integrand Representations of All Two-Loop Amplitudes in Planar SYM

TL;DR

The paper delivers a complete, local integrand-level construction of all two-loop amplitudes in planar $\mathcal{N}=4$ SYM, built from a minimal set of on-shell cuts and decorated with chiral, dual-conformally invariant integrands. It separates divergent and finite pieces to expose IR-divergence exponentiation at the integrand level and demonstrates consistency with all-loop BCFW recursion, including a closed-form two-loop solution. A merge framework is introduced to make infrared-safe observables like ratio functions manifestly finite, and a publicly available Mathematica package implements these results. The approach preserves dual conformal symmetry, avoids basis-dependent linear algebra, and provides a pathway toward higher-loop generalizations in a broad class of theories.

Abstract

We use generalized unitarity at the integrand-level to directly construct local, manifestly dual-conformally invariant formulae for all two-loop scattering amplitudes in planar, maximally supersymmetric Yang-Mills theory (SYM). This representation separates contributions into manifestly finite and divergent terms---in a way that makes manifest the exponentiation of infrared divergences at the integrand-level. These results perfectly match the all-loop BCFW recursion relations, to which we provide a closed-form solution valid through two-loop-order. Finally, we describe and document a Mathematica package which implements these results, available as part of this work's source files on the arXiv.