Scalar 1-loop Feynman integrals as meromorphic functions in space-time dimension d

Abstract

The long-standing problem of representing the general massive one-loop Feynman integral as a meromorphic function of the space-time dimension $d$ has been solved for the basis of scalar one- to four-point functions with indices one. In 2003 the solution of difference equations in the space-time dimension allowed to determine the necessary classes of special functions: self-energies need ordinary logarithms and Gauss hypergeometric functions ${}_2{F}_1$, vertices need additionally Kampé de Fériet-Appell functions $F_1$, and box integrals also Lauricella-Saran functions $F_S$. In this study, alternative recursive Mellin-Barnes representations are used for the representation of $n$-point functions in terms of $(n-1)$-point functions. The approach enabled the first derivation of explicit solutions for the Feynman integrals at arbitrary kinematics. In this article, we scetch our new representations for the general massive vertex and box Feynman integrals and derive a numerical approach for the necessary Appell functions $F_1$ and Saran functions $F_S$ at arbitrary kinematical arguments.

9 pages, 3 tables

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Cited by in corpus (10)

• From positive geometries to a coaction on hypergeometric functions
• Functional reduction of one-loop Feynman integrals with arbitrary masses
• $\epsilon$-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
• New Techniques for Worldline Integration
• Analytic continuations of the Horn $H_1$ and $H_5$ functions
• One-loop three-point Feynman integrals with Appell $F_1$ hypergeometric functions
• Scalar 1-loop Feynman integrals as meromorphic functions in space-time dimension d, II: Special kinematics
• One-loop contributions to the decay $H\to {\nu }_l{\bar{\nu }}_l\gamma$ in standard model revisited
• Hypergeometric presentation for one-loop contributing to $H\to Z\gamma$