All planar two-loop amplitudes in maximally supersymmetric Yang-Mills theory

TL;DR

This paper computes all two-loop planar scattering amplitudes in maximally supersymmetric Yang-Mills theory by deriving integral representations for key elliptic integrals, enabling a complete calculation for any particle number and helicity.

Contribution

It introduces new integral representations for elliptic integrals that facilitate the calculation of all two-loop planar amplitudes in the theory, a significant advancement over previous partial results.

Findings

Derived integral representations for elliptic integrals

Expressed all two-loop planar amplitudes in the theory

Achieved calculations for any particle number and helicity

Abstract

We calculate the general planar dual-conformally invariant double-pentagon and pentabox integrals in four dimensions. Concretely, we derive one-fold integral representations for these elliptic integrals over polylogarithms of weight three. These integral representations allow us to determine the respective symbols using consistency conditions alone. Together with the previously calculated double-box integral, these integrals suffice to express all two-loop planar scattering amplitudes in maximally supersymmetric Yang-Mills theory in four dimensions -- which we thus calculate for any number of particles and any helicity configuration!