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

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.