Reduction of Feynman integrals to integrals of Schläfli functions

Abstract

We show that off-shell perturbative amplitudes with arbitrary number of external lines and complex masses can be reduced to $I$-fold integrals of the generalized Schläfli functions, where $I$ is the number of lines in the corresponding vacuum diagram which is independent of the number of external lines. The Schläfli functions are obtained as analytic continuation of the volume of the spherical simplex as a function of the hyperplane parameters that define the simplex. These functions have nice and thoroughly studied analytic, geometric and number theoretic properties. They possess Gauss-Manin connection and in conformal case are expressed by iterated integrals. Our representation sheds new light on geometry of particle configuration spaces in perturbation theory.

Comments: a mistake in the appendix corrected