Prescriptive Unitarity with Elliptic Leading Singularities

TL;DR

This paper explores how elliptic leading singularities influence the structure of two-loop amplitudes in supersymmetric Yang-Mills theory, proposing methods to achieve pure integrand bases and analyzing their properties.

Contribution

It introduces a novel approach to incorporate elliptic singularities into unitarity-based amplitude representations, enhancing the understanding of their mathematical structure.

Findings

Diagonalizing with respect to elliptic leading singularities yields a term-wise pure integrand basis.

An alternative basis based on differential forms offers reduced complexity despite lacking certain invariances.

The methods improve the mathematical understanding of two-loop amplitude structures in supersymmetric theories.

Abstract

We investigate the consequences of elliptic leading singularities for the unitarity-based representations of two-loop amplitudes in planar, maximally supersymmetric Yang-Mills theory. We show that diagonalizing with respect to these leading singularities ensures that the integrand basis is term-wise pure (suitably generalized, to the elliptic multiple polylogarithms, as necessary). We also investigate an alternative strategy based on diagonalizing a basis of integrands on differential forms; this strategy, while neither term-wise Yangian-invariant nor pure, offers several advantages in terms of complexity.