Five legs @ three loops: slightly off-shell dual conformal integrals

TL;DR

This paper computes three-loop five-point integrals in N=4 SYM theory on the Coulomb branch, introducing a new method to factor out dual conformal cross ratios and simplifying most integrals.

Contribution

It presents a novel approach to factor out dual conformal cross ratios in three-loop integrals, simplifying the calculation of five-point amplitudes in N=4 SYM.

Findings

Successfully computed most of the three-loop five-point integrals.

Developed a method to factor out dependence on dual conformal cross ratios.

Performed IBP reduction and evaluated integrals with HyperInt for complex regions.

Abstract

We calculate the three-loop master integrals contributing to the three-loop five-point amplitude on the special Coulomb branch of $\mathcal{N}=4$ SYM theory. For the genuine pentagon integrals, we follow the approach of Ref. [JHEP 12 (2025) 107], which includes a regularization preserving dual conformal invariance (DCI). As a new ingredient, we introduce a simple method, allowing to factor out the dependence on the DCI cross ratios from the contribution of each region. The remaining integrals are then essentially simplified by taking successive limits of vanishing external invariants. For 3 out of 82 regions contributing to the most complicated integral $\mathcal{I}_5^{(3)}$ we were not able to perform the integration even after these simplifications. For these three regions, we perform the integration-by-parts (IBP) reduction in parametric representation and evaluate the resulting locally finite integrals using HyperInt.