Feynman integral reduction with intersection theory made simple

TL;DR

This paper introduces a simplified method for Feynman integral reduction using intersection theory and branch representation, significantly reducing the number of variables needed for multi-loop, multi-leg integrals.

Contribution

The authors demonstrate that employing branch representation allows reduction of L-loop Feynman integrals with arbitrary external legs via at most (3L-3) variables, simplifying previous approaches.

Findings

Achieved reduction of two-loop diagrams efficiently.

Reduced variable count from traditional methods for multi-leg integrals.

Validated method shows substantial computational improvements.

Abstract

Feynman integral reduction based on intersection theory provides an alternative to the traditional integration-by-parts method, yet its practical application has been constrained by the large number of variables required in the computation. In this Letter, we demonstrate that by employing the recently introduced branch representation, the reduction of $L$-loop Feynman integrals with an arbitrary number of external legs can be achieved through the computation of at most $(3L-3)$-variable intersection numbers. This constitutes a significant simplification compared to existing approaches, particularly for multi-leg integrals where the number of variables in conventional methods scales with the total number of propagators. We validate the proposed method through explicit calculations of two-loop diagrams, demonstrating substantial improvements in computational efficiency relative to both traditional intersection-theory approaches and standard integration-by-parts reduction techniques.