Collinear and Soft Limits of Multi-Loop Integrands in N=4 Yang-Mills

TL;DR

This paper explores how collinear and soft limits can uniquely determine multi-loop integrands in N=4 super Yang-Mills theory, extending previous dual conformal invariance constraints and analyzing their effectiveness at various particle numbers.

Contribution

It reformulates the collinear constraint in terms of the amplitude itself and demonstrates its effectiveness in determining integrands for multiple loops and particle numbers.

Findings

Collinear and soft constraints can uniquely determine integrands for all n at two and three loops.

The reformulation simplifies the application of collinear constraints.

Constraints become more effective at larger n for soft behavior.

Abstract

It has been argued in arXiv:1112.6432 that the planar four-point integrand in N=4 super Yang-Mills theory is uniquely determined by dual conformal invariance together with the absence of a double pole in the integrand of the logarithm in the limit as a loop integration variable becomes collinear with an external momentum. In this paper we reformulate this condition in a simple way in terms of the amplitude itself, rather than its logarithm, and verify that it holds for two- and three-loop MHV integrands for n>4. We investigate the extent to which this collinear constraint and a constraint on the soft behavior of integrands can be used to determine integrands. We find an interesting complementarity whereby the soft constraint becomes stronger while the collinear constraint becomes weaker at larger n. For certain reasonable choices of basis at two and three loops the two constraints in unison appear strong enough to determine MHV integrands uniquely for all n.