Towards Analytic Structure of Feynman Parameter Integrals with Rational Curves

TL;DR

This paper introduces a geometric approach to analyze the analytic structure of Feynman parameter integrals with rational singularities, aiming to facilitate the construction of symbols for multiple polylogarithms and extending to higher-loop cases.

Contribution

It presents a novel geometric method to identify and compute singularities of Feynman integrals with rational components, aiding in understanding their analytic structure and symbol construction.

Findings

Method successfully applied to Aomoto polylogarithms

Effective in analyzing integrals with quadric singularities

Potential extension to higher-loop integrals discussed

Abstract

We propose a strategy to study the analytic structure of Feynman parameter integrals where singularities of the integrand consist of rational irreducible components. At the core of this strategy is the identification of a selected stratum of discontinuities induced from the integral, together with a geometric method for computing their singularities on the principal sheet. For integrals that yield multiple polylogarithms we expect the data collected in this strategy to be sufficient for the construction of their symbols. We motivate this analysis by the Aomoto polylogarithms, and further check its validity and illustrate technical details using examples with quadric integrand singularities (which the one-loop Feynman integrals belong to). Generalizations to higher-loop integrals are commented at the end.