Reduction of Feynman integrals to integrals of Schl\"afli functions

TL;DR

This paper demonstrates how off-shell perturbative amplitudes with complex masses and many external lines can be expressed as integrals of Schl"afli functions, linking particle physics calculations to geometric and number theoretic properties.

Contribution

It introduces a novel reduction of Feynman integrals to Schl"afli functions, revealing geometric and analytic structures in perturbation theory.

Findings

Feynman amplitudes reduced to integrals of Schl"afli functions

Schl"afli functions linked to spherical simplex volumes and their properties

New geometric insights into particle configuration spaces

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 Schl\"{a}fli functions, where $I$ is the number of lines in the corresponding vacuum diagram which is independent of the number of external lines. The Schl\"{a}fli 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.