Nontrivial one-loop recursive reduction relation

Abstract

In [1], 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 [2, 3], where a tensor integral is reduced to lower-rank integrals and 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.

An explicit expression of generating function for one-loop tensor reduction

“…After introducing the above notations, we begin to derive the recursive relations for the generating function of the reduction coefficients. As pointed out in [100] and [90], there exists a non-trivial recursion relation for one-loop tensor integrals for 𝑟 ≥ 1, 1…”

“…However, in both studies, authors provide only an iterative approach to compute the generating functions, and it remains strenuous to directly write generating functions explicitly, even at the one-loop level. Fortunately, we discovered a new recursive relation of generating functions based on investigations into Feynman parametrization in the projective space [90,99,100]. This new relation consists of a single ordinary differential equation on 𝓉 instead of a complex set of partial differential equations, allowing us to directly write an explicit expression of the generating function for the reduction of 𝑛-gon to (𝑛 − 𝑘)-gon for universal 𝑘 without recursion.…”