Reduction with Degenerate Gram matrix for One-loop Integrals

TL;DR

This paper introduces a systematic algorithm to handle Gram determinant divergences in one-loop integral reductions, improving the stability and applicability of existing reduction methods.

Contribution

It presents a novel decomposition approach that cancels divergences by expressing master integrals as Taylor series in the Gram determinant.

Findings

Successfully cancels Gram determinant divergences in reduction coefficients

Decomposes highest topology master integrals into lower topologies

Enhances the robustness of one-loop integral reduction methods

Abstract

An improved PV-reduction method for one-loop integrals with auxiliary vector $R$ has been proposed in \cite{Feng:2021enk,Hu:2021nia}. It has also been shown that the new method is a self-completed method in \cite{Feng:2022uqp}. Analytic reduction coefficients can be easily produced by recursion relations in this method, where the Gram determinant appears in denominators. The singularity caused by Gram determinant is a well-known fact and it is important to address these divergences in a given frame. In this paper, we propose a systematical algorithm to deal with this problem in our method. The key idea is that now the master integral of the highest topology will be decomposed into combinations of master integrals of lower topologies. By demanding the cancellation of divergence for obtained general reduction coefficients, we solve decomposition coefficients as a Taylor series of the Gram determinant. Moreover, the same idea can be applied to other kinds of divergences.