Specializations of partial differential equations for Feynman integrals

TL;DR

This paper develops an algorithm to derive systems of partial differential equations for Feynman integrals using Mellin-Barnes representations, enabling advanced analysis and reduction of hypergeometric functions and integrals.

Contribution

The authors introduce a novel algorithm that derives PDE systems for Feynman integrals with arbitrary propagator powers, extending existing methods.

Findings

Derived a fourth-order differential equation for a one-loop two-point diagram.

Algorithm handles arbitrary propagator powers and multiple hypergeometric sums.

Demonstrated application to a specific Feynman diagram with different masses.

Abstract

Starting from the Mellin-Barnes integral representation of a Feynman integral depending on set of kinematic variables $z_i$, we derive a system of partial differential equations w.r.t.\ new variables $x_j$, which parameterize the differentiable constraints $z_i=y_i(x_j)$. In our algorithm, the powers of propagators can be considered as arbitrary parameters. Our algorithm can also be used for the reduction of multiple hypergeometric sums to sums of lower dimension, finding special values and reduction equations of hypergeometric functions in a singular locus of continuous variables, or finding systems of partial differential equations for master integrals with arbitrary powers of propagators. As an illustration, we produce a differential equation of fourth order in one variable for the one-loop two-point Feynman diagram with two different masses and arbitrary propagator powers.