Singularity-Free Feynman Integral Bases

TL;DR

This paper introduces two algorithms to construct Feynman integral bases free of singular coefficients in dimensional regularization, simplifying amplitude calculations and revealing clearer physical structures.

Contribution

It presents novel algorithms for creating singularity-free Feynman integral bases, improving the clarity and simplicity of scattering amplitude evaluations.

Findings

Algorithms successfully applied to two-loop double-box integrals

Bases reduce singularities in reduction coefficients

Simpler amplitude representations achieved

Abstract

Standard integration-by-parts (IBP) reduction methods typically yield Feynman integral bases where the reduction of some integrals gives rise to coefficients singular as the dimensional regulator $\epsilon\rightarrow 0$. These singular coefficients can also appear in scattering amplitudes, obscuring their structure, and rendering their evaluation more complicated. We investigate the use of bases in which the reduction of any integral is free of singular coefficients. We present two general algorithms for constructing such bases. The first is based on sequential $D=4$ IBP reduction. It constructs a basis iteratively by projecting onto the finite part of the set of IBP relations. The second algorithm performs Gaussian elimination within a local ring forbidding division by $\epsilon$ while permitting division by polynomials in $\epsilon$ finite at $\epsilon=0$. We study the application of both algorithms to a pair of two-loop examples, the planar and nonplanar double-box families of integrals. We also explore the incorporation of finite Feynman integrals into these bases. In one example, the resulting basis provides a simpler and more compact representation of a scattering amplitude.