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

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.