Analytic evaluation of the three-loop three-point form factor of $\operatorname{tr}\phi^3$ in $\mathcal{N}=4$ sYM

TL;DR

This paper analytically computes the three-loop three-point form factor of the operator tr φ^3 in N=4 sYM, expressing results with Chen iterated integrals and polylogarithms, confirming previous numerical findings.

Contribution

It provides the first analytical three-loop calculation of the tr φ^3 form factor in N=4 sYM, revealing simplified singularity structure and symbol relations.

Findings

Results agree with previous numerical computations.

Fewer kinematic singularities than individual Feynman integrals.

Finite part satisfies symbol adjacency relations similar to the tr φ^2 case.

Abstract

We compute analytically the three-loop correlation function of the local operator $\text{tr} \, \phi^3$ inserted into three on-shell states, in maximally supersymmetric Yang-Mills theory. The result is expressed in terms of Chen iterated integrals. We also present our result using generalised polylogarithms, and evaluate them numerically, finding agreement with a previous numerical result in the literature. We observe that the result depends on fewer kinematic singularities compared to individual Feynman integrals. Furthermore, upon choosing a suitable definition of the finite part, we find that the latter satisfies powerful symbol adjacency relations similar to those previously observed for the $\text{tr} \, \phi^2$ case.