The four-propagator three-loop vacuum integral by the hypergeometry

TL;DR

This paper introduces a hypergeometric function approach to compute three-loop vacuum integrals with four propagators, establishing their PDE systems for numerical continuation across kinematic regions.

Contribution

It proposes a hypergeometric method for three-loop vacuum integrals and derives PDEs for their numerical evaluation, advancing computational techniques in quantum field theory.

Findings

Scalar integrals expressed as generalized hypergeometric functions

Derived PDE systems for scalar integrals

Numerical continuation enabled by PDEs and element method

Abstract

Hypergeometric function method is proposed to calculate the scalar integrals of Feynman diagrams. For the scalar integral of three-loop vacuum diagram with four-propagator, we verify the equivalency of Feynman parametrization and the hypergeometric technique. The result can be described as generalized hypergeometric functions of triple variables. Based on the triple hypergeometric functions, we also establish the systems of homogeneous linear partial differential equations(PDEs) satisfied by the mentioned scalar integral. The continuation of the scalar integral from its convergent regions to whole kinematic domains can be made numerically through the system of homogeneous linear PDEs with the help of the element method.