Evaluating Feynman integrals by the hypergeometry

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.

39 pages, including 2 ps figures

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Cited by in corpus (11)

• GKZ-hypergeometric systems for Feynman integrals
• The system of partial differential equations for the $C_{{}_0}$ function
• $\epsilon$-Expansion of Multivariable Hypergeometric Functions Appearing in Feynman Integral Calculus
• Analytic continuations and numerical evaluation of the Appell $F_1$, $F_3$, Lauricella $F_D^{(3)}$ and Lauricella-Saran $F_S^{(3)}$ and their Application to Feynman Integrals
• Analytic continuations of the Horn $H_1$ and $H_5$ functions
• Feynman Integrals of Grassmannians
• GKZ-system of the 2-loop self energy with 4 propagators
• GKZ hypergeometric systems of the three-loop vacuum Feynman integrals
• The four-propagator three-loop vacuum integral by the hypergeometry
• Electric potentials and field lines for uniformly-charged tube and cylinder expressed by Appell's hypergeometric function and integration of $Z(u|m)\mathrm{sc}(u|m)$
• GKZ hypergeometric systems of the four-loop vacuum Feynman integrals