A Tree-Loop Duality Relation at Two Loops and Beyond

TL;DR

This paper extends the duality relation between one-loop integrals and phase-space integrals to higher loops, providing a new framework for calculating multi-loop scalar integrals with modified propagator prescriptions.

Contribution

It introduces a generalized duality relation applicable to two- and three-loop integrals, improving the understanding of loop integral structures and cuts.

Findings

Derived duality relations for two- and three-loop integrals

Reformulated the duality theorem for iterative higher-loop application

Analyzed the structure of cuts in multi-loop integrals

Abstract

The duality relation between one-loop integrals and phase-space integrals, developed in a previous work, is extended to higher-order loops. The duality relation is realized by a modification of the customary +i0 prescription of the Feynman propagators, which compensates for the absence of the multiple-cut contributions that appear in the Feynman tree theorem. We rederive the duality theorem at one-loop order in a form that is more suitable for its iterative extension to higher-loop orders. We explicitly show its application to two- and three-loop scalar master integrals, and we discuss the structure of the occurring cuts and the ensuing results in detail.