Reduction to master integrals and transverse integration identities

TL;DR

This paper explores how transverse integration identities can simplify the reduction of complex Feynman integrals to master integrals, leading to more efficient calculations in loop diagram evaluations.

Contribution

It introduces a novel approach using transverse tensor decompositions to transform integral families into simpler forms, improving reduction efficiency.

Findings

Significant reduction in computational complexity for integral reduction.

Successful application to advanced integral families with improved performance.

Demonstrated potential for integrating into existing reduction algorithms.

Abstract

The reduction of Feynman integrals to a basis of linearly independent master integrals is a pivotal step in loop calculations, but also one of the main bottlenecks. In this paper, we assess the impact of using transverse integration identities for the reduction to master integrals. Given an integral family, some of its sectors correspond to diagrams with fewer external legs or to diagrams that can be factorized as products of lower-loop integrals. Using transverse integration identities, i.e. a tensor decomposition in the subspace that is transverse to the external momenta of the diagrams, one can map integrals belonging to such sectors and their subsectors to (products of) integrals belonging to new and simpler integral families, characterized by either fewer generalized denominators, fewer external invariants, fewer loops or combinations thereof. Integral reduction is thus drastically simpler for these new families. We describe a proof-of-concept implementation of the application of transverse integration identities in the context of integral reduction. We include some applications to cutting-edge integral families, showing significant improvements over traditional algorithms.