Direct Integration for Multi-leg Amplitudes: Tips, Tricks, and When They Fail

TL;DR

This paper reviews the use of direct hyperlogarithmic integration for multi-particle Feynman diagrams, highlighting its successes, limitations, and connection to Calabi-Yau manifolds in complex cases.

Contribution

It provides a comprehensive overview of the method's capabilities, including practical tips, and discusses the algebraic obstructions that cause failures.

Findings

Successfully computed diagrams with up to eight particles and four loops.

Identified algebraic obstructions related to Calabi-Yau manifolds.

Outlined scenarios where the method fails due to algebraic complexities.

Abstract

Direct hyperlogarithmic integration offers a strong alternative to differential equation methods for Feynman integration, particularly for multi-particle diagrams. We review a variety of results by the authors in which this method, employed with some care, can compute diagrams of up to eight particles and four loops. We also highlight situations in which this method fails due to an algebraic obstruction. In a large number of cases the obstruction can be associated with a Calabi-Yau manifold.