# Evaluating Feynman integrals by the hypergeometry

## TL;DR

This paper introduces a hypergeometric function approach to analytically evaluate Feynman integrals, deriving differential equations and enabling numerical continuation across kinematic domains, applicable to various Feynman diagrams.

## Contribution

The paper develops a hypergeometric function method that provides explicit analytic expressions and differential equations for scalar Feynman integrals, facilitating their numerical evaluation.

## Key findings

- Verified expressions for one-loop and two-loop scalar integrals match known results.
- Established systems of differential equations satisfied by these integrals.
- Proposed a numerical continuation method using finite element techniques.

## Abstract

The hypergeometric function method naturally provides the analytic expressions of scalar integrals from concerned Feynman diagrams in some connected regions of independent kinematic variables, also presents the systems of homogeneous linear partial differential equations satisfied by the corresponding scalar integrals. Taking examples of the one-loop $B_{_0}$ and massless $C_{_0}$ functions, as well as the scalar integrals of two-loop vacuum and sunset diagrams, we verify our expressions coinciding with the well-known results of literatures. Based on the multiple hypergeometric functions of independent kinematic variables, the systems of homogeneous linear partial differential equations satisfied by the mentioned scalar integrals are established. Using the calculus of variations, one recognizes the system of linear partial differential equations as stationary conditions of a functional under some given restrictions, which is the cornerstone to perform the continuation of the scalar integrals to whole kinematic domains numerically with the finite element methods. In principle this method can be used to evaluate the scalar integrals of any Feynman diagrams.