$\texttt{AlgRel.wl}$: Algebraic Relations for the Product of Propagators in Feynman integrals

TL;DR

This paper introduces the $ exttt{AlgRel.wl}$ Mathematica package that automates the derivation of algebraic relations for products of propagators in Feynman integrals, enabling their reduction to simpler forms for computational efficiency.

Contribution

The paper presents a modified, automatable method and a software package for deriving algebraic relations that reduce the complexity of Feynman integrals, applicable to both one-loop and higher-loop cases.

Findings

Successfully derived reduction formulae for various one-loop and higher-loop integrals.

Demonstrated the package's capability to handle arbitrary kinematic parameters.

Provided a practical tool to facilitate algebraic reduction of Feynman integrals.

Abstract

Motivated by the foundational work of Tarasov, who pointed out that the algebraic relations of the type considered here can lead to functional reduction of Feynman integrals, we suitably modify the original method to be able to implement and automatize it and present a $\textit{Mathematica}$ package $\texttt{AlgRel.wl}$. The purpose of this package is to help derive the algebraic relations with arbitrary kinematic quantities, for the product of propagators. Under specific choices of the arbitrary parameters that appear in these relations, we can write the original integral with all massive propagators in general, as a sum of integrals which have fewer massive propagators. The resulting integrals are of reduced complexity for computational purposes. For the one-loop cases, with all different and non-zero masses, this would result in integrals with one massive propagator. We also devise a strategy so that the method can also be applied to higher-loop integrals. We demonstrate the procedure and the results obtained using the package for various one-loop and higher-loop examples. Due to the fact that the Feynman integrals are intimately related to the hypergeometric functions, a useful consequence of these algebraic relations is in deriving the sets of non-trivial reduction formulae. We present various such reduction formulae and further discuss how, more such formulae can be obtained than described here. The $\texttt{AlgRel.wl}$ package and an example notebook $\texttt{Examples.nb}$ can be found at https://github.com/TanayPathak-17/Algebraic-relation-for-the-product-of-propagators