How to choose master integrals
A. V. Smirnov, V. A. Smirnov
TL;DR
The paper tackles the problem that master integral bases produced by IBP-reduction codes can contain large, unwieldy denominators. It proposes a Sabbah-theorem–inspired algorithm to systematically improve a given basis by replacing problematic master integrals and analyzing reductions sector-by-sector and level-by-level, with an implementation embedded in FIRE. A detailed three-loop vertex example demonstrates the method's effectiveness, yielding a good basis where reductions become more tractable. The work enhances the reliability and efficiency of multiloop IBP reductions and supports downstream techniques such as modular arithmetic and differential equations.
Abstract
The standard procedure when evaluating integrals of a given family of Feynman integrals, corresponding to some Feynman graph, is to construct an algorithm which provides the possibility to write any particular integral as a linear combination of so-called master integrals. To do this, public (AIR, FIRE, REDUZE, LiteRed, KIRA) and private codes based on solving integration by parts relations are used. However, the choice of the master integrals provided by these codes is not always optimal. We present an algorithm to improve a given basis of the master integrals, as well as its computer implementation; see also a competitive variant [1].