Loop Amplitudes in the Coulomb Branch of $\mathcal{N}=4$ Super-Yang-Mills Theory

TL;DR

This paper investigates four-point planar loop amplitudes in the Coulomb branch of $ ext{N}=4$ super-Yang-Mills theory, confirming the absence of certain diagrams at multiple loops and deriving an all-loop recursion relation for specific graphs.

Contribution

It provides explicit verification of diagram contributions at various loop levels and introduces an all-loop recursion relation for two-particle reducible graphs.

Findings

Bubble and triangle graphs do not contribute at one loop.

Results extend to higher loop levels.

An all-loop recursion relation for reducible graphs is derived.

Abstract

We study four point planar loop amplitudes at an arbitrary point in the Coulomb branch of $\mathcal{N}=4$ super-Yang-Mills theory. We study two particle unitary cuts up to four loop order. We explicitly verify that bubble and triangle graphs do not contribute at one loop level and show that the results hold at higher loop level as well. We also write down an all loop recursion relation for two particle reducible graphs for four point amplitudes.