Amplitudes at Infinity

TL;DR

This work probes the asymptotic behavior of multi-loop amplitudes in four dimensions by testing residue-theorem identities among on-shell, color-dressed leading singularities. Using generalized unitarity and compact color-dressed MHV cuts, the authors show that ${ m N}=4$ SYM amplitudes have no poles at infinity through three loops, strengthening the view of a hidden symmetry structure beyond planar limits. In contrast, ${ m N}=8$ supergravity exhibits poles at infinity starting at two loops with pole degree increasing with multiplicity, highlighting a tension between manifest UV finiteness via color-kinematics duality and gauge-invariance constraints from unitarity-based methods. These results suggest the need for a gauge-invariant, possibly geometric description of gravity integrands and point to fundamental differences between gauge and gravity theories at high loop order. The ancillary data and methods provided enable reproducibility of the residue-test findings and illuminate the landscape of non-planar amplitude structures.

Abstract

We investigate the asymptotically large loop-momentum behavior of multi-loop amplitudes in maximally supersymmetric quantum field theories in four dimensions. We check residue-theorem identities among color-dressed leading singularities in $\mathcal{N}=4$ supersymmetric Yang-Mills theory to demonstrate the absence of poles at infinity of all MHV amplitudes through three loops. Considering the same test for $\mathcal{N}=8$ supergravity leads us to discover that this theory does support non-vanishing residues at infinity starting at two loops, and the degree of these poles grow arbitrarily with multiplicity. This causes a tension between simultaneously manifesting ultraviolet finiteness---which would be automatic in a representation obtained by color-kinematic duality---and gauge invariance---which would follow from unitarity-based methods.