Cross-Order Integral Relations from Maximal Cuts

TL;DR

This paper explores the ABDK relation using maximal cuts of multi-loop integrals, revealing how to identify integral combinations that satisfy the relation and reconstruct terms with one-loop squared integrals, advancing understanding of integral identities.

Contribution

It introduces a method to find special integral combinations for ABDK relations and demonstrates how maximal cuts can reveal integral identities across loop orders.

Findings

Identified specific integral combinations satisfying ABDK relations.

Showed integrals across different loop orders can share unitarity cuts.

Highlighted the role of nonhomologous contours in integral analysis.

Abstract

We study the ABDK relation using maximal cuts of one- and two-loop integrals with up to five external legs. We show how to find a special combination of integrals that allows the relation to exist, and how to reconstruct the terms with one-loop integrals squared. The reconstruction relies on the observation that integrals across different loop orders can have support on the same generalized unitarity cuts and can share global poles. We discuss the appearance of nonhomologous integration contours in multivariate residues. Their origin can be understood in simple terms, and their existence enables us to distinguish contributions from different integrals. Our analysis suggests that maximal and near-maximal cuts can be used to infer the existence of integral identities more generally.