Factorization of denominators in integration-by-parts reductions

TL;DR

The paper introduces findFactorizedBasis.m, a Mathematica tool that selects a master-integral basis for IBP reductions so that the $d$-dependence in denominators factorizes into polynomials independent of kinematics. The approach uses a two-stage reduction with differing scales to identify denominator-building blocks and iteratively optimize the basis until all blocks depend only on $d$, enabling efficient analytic reconstruction and reduced memory usage. Applied to a three-loop non-planar form-factor and to the double-pentagon topology, the method achieves fully factorized $d$-dependence across all denominators, with the number of master integrals per topology highlighted (159 and 108, respectively). The work emphasizes practical gains when combining with finite-field reductions and algebraic reconstruction, offering a path to faster, more scalable multi-loop IBP reductions.

Abstract

We present a Mathematica package which finds a basis of master integrals for the Feynman integral reduction. In this basis the dependence on the dimensional regularization in the denominators factorizes in kinematic independent polynomials.

Figures (2)

• Figure 1: Three-loop Feynman diagram for the top level sector of an integral in Eq. \ref{['eq:threeLoop']}.
• Figure 2: This is a Feynman diagram for one top level sector integral for the double pentagon.