IBP reduction coefficients made simple

TL;DR

This paper introduces an improved method for simplifying IBP reduction coefficients in multi-loop Feynman integrals, leveraging a modern partial fraction algorithm and properties of uniform transcendental bases to significantly reduce complexity.

Contribution

The authors develop an enhanced partial fraction algorithm and demonstrate its effectiveness in simplifying IBP coefficients, especially when using a UT basis, with substantial size reductions.

Findings

IBP reduction coefficients can be significantly simplified using the new algorithm.

The method is particularly effective with a UT basis, reducing coefficient size by up to 100 times.

The approach also performs well without a UT basis, showing broad applicability.

Abstract

We present an efficient method to shorten the analytic integration-by-parts (IBP) reduction coefficients of multi-loop Feynman integrals. For our approach, we develop an improved version of Leinartas' multivariate partial fraction algorithm, and provide a modern implementation based on the computer algebra system Singular. Furthermore, We observe that for an integral basis with uniform transcendental (UT) weights, the denominators of IBP reduction coefficients with respect to the UT basis are either symbol letters or polynomials purely in the spacetime dimension $D$. With a UT basis, the partial fraction algorithm is more efficient both with respect to its performance and the size reduction. We show that in complicated examples with existence of a UT basis, the IBP reduction coefficients size can be reduced by a factor of as large as $\sim 100$. We observe that our algorithm also works well for settings without a UT basis.