Nontrivial One-loop Recursive Reduction Relation

TL;DR

This paper introduces a new recursion relation for one-loop integrals that simplifies the reduction process by explicitly expressing lower terms, improving efficiency over previous methods.

Contribution

It derives a non-trivial recursion relation for one-loop integrals, explicitly expressing lower terms to enhance reduction speed and simplicity.

Findings

Derived explicit expressions for lower terms in the recursion relation

Simplified the reduction process for one-loop integrals

Improved computational efficiency in tensor integral reduction

Abstract

In arXiv:2204.03190, we proposed a universal method to reduce one-loop integrals with both tensor structure and higher-power propagators. But the method is quite redundant as it does not utilize the results of lower rank cases when addressing certain tensor integrals. Recently, we found a remarkable recursion relation arXiv:2203.16881,2205.03000, where a tensor integral is reduced to lower-rank integrals and \textit{lower terms} corresponding to integrals with one or more propagators being canceled. However, the expression of the lower terms is unknown. In this paper, we derive this non-trivial recursion relation for non-degenerate and degenerate cases and provides an explicit expression for the lower terms, thus simplifying and speeding up the reduction process.