# Amplitudes at Infinity

## TL;DR

This paper studies the behavior of multi-loop amplitudes at infinity in supersymmetric theories, revealing that while $ ext{N}=4$ SYM has no poles at infinity up to three loops, $ ext{N}=8$ supergravity exhibits non-vanishing residues starting at two loops, indicating potential UV divergences.

## Contribution

It demonstrates the presence of non-vanishing residues at infinity in $ ext{N}=8$ supergravity from two loops onward, highlighting a tension between UV finiteness and gauge invariance.

## Key findings

- $ ext{N}=4$ SYM amplitudes have no poles at infinity up to three loops.
- $ ext{N}=8$ supergravity exhibits non-zero residues at infinity from two loops.
- Degree of poles at infinity grows with multiplicity in supergravity.

## 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.