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

摘要

The long-standing problem of representing the general massive one-loop Feynman integral as a meromorphic function of the space-time dimensiondhas 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 functionsF12, vertices need additionally Kampé de Fériet-Appell functionsF1, and box integrals also Lauricella-Saran functionsFS. In this study, alternative recursive Mellin-Barnes representations are used for the representation ofn-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 functionsF1and Saran functionsFSat arbitrary kinematical arguments.