Reduction of Feynman Integrals in the Parametric Representation II: Reduction of Tensor Integrals

TL;DR

This paper presents new methods for reducing tensor Feynman integrals in the parametric representation, utilizing polynomial operator equations, auxiliary parameters, and Gr"obner basis techniques to improve reduction efficiency.

Contribution

It introduces two novel approaches for tensor integral reduction, one avoiding dimension shifts with auxiliary parameters and another combining Gr"obner basis with symbolic rules.

Findings

Polynomial equations for tensor integral operators are derived.

Two reduction methods are developed: one using auxiliary parameters, another using Gr"obner basis.

Unreduced integrals can be further simplified via parametric IBP identities.

Abstract

In a recent paper by the author (Chen in JHEP 02:115, 2020), the reduction of Feynman integrals in the parametric representation was considered. Tensor integrals were directly parametrized by using a generator method. The resulting parametric integrals were reduced by constructing and solving parametric integration-by-parts (IBP) identities. In this paper, we furthermore show that polynomial equations for the operators that generate tensor integrals can be derived. Based on these equations, two methods to reduce tensor integrals are developed. In the first method, by introducing some auxiliary parameters, tensor integrals are parametrized without shifting the spacetime dimension. The resulting parametric integrals can be reduced by using the standard IBP method. In the second method, tensor integrals are (partially) reduced by using the technique of Gr\"obner basis combined with the application of symbolic rules. The unreduced integrals can further be reduced by solving parametric IBP identities.