The diamond rule for multi-loop Feynman diagrams

TL;DR

This paper introduces the diamond rule, an extension of the triangle rule, for simplifying multi-loop Feynman integrals in high energy physics, enabling more efficient calculations in higher-loop diagrams.

Contribution

The paper presents the diamond rule, a new recursive reduction technique for multi-loop Feynman integrals, with explicit solutions that avoid spurious poles.

Findings

Applicable to three, four, and five-loop diagrams

Prevents spurious poles in intermediate steps

Enhances efficiency of high-loop calculations

Abstract

An important aspect of improving perturbative predictions in high energy physics is efficiently reducing dimensionally regularised Feynman integrals through integration by parts (IBP) relations. The well-known triangle rule has been used to achieve simple reduction schemes. In this work we introduce an extensible, multi-loop version of the triangle rule, which we refer to as the diamond rule. Such a structure appears frequently in higher-loop calculations. We derive an explicit solution for the recursion, which prevents spurious poles in intermediate steps of the computations. Applications for massless propagator type diagrams at three, four, and five loops are discussed.