Vanishing cycles and analysis of singularities of Feynman diagrams

TL;DR

This paper studies the geometric nature of singularities in Feynman loop integrals using vanishing cycles and spectral sequences, providing a classification and explicit asymptotic expansions relevant for high-precision physics calculations.

Contribution

It introduces a novel geometric framework for analyzing Feynman diagram singularities and derives explicit formulas for asymptotic expansions near these singularities.

Findings

Complete classification of vanishing geometries for Feynman integrals

Explicit formulas for asymptotic expansion coefficients

Potential for exact calculations of complex Feynman diagrams

Abstract

In this work, we analyze vanishing cycles of Feynman loop integrals by means of the Mayer-Vietoris spectral sequence. A complete classification of possible vanishing geometries are obtained. We employ this result for establishing an asymptotic expansion for the loop integrals near their singularity locus, then give explicit formulas for the coefficients of such an expansion. The further development of this framework may potentially lead to exact calculations of one- and two-loop Feynman diagrams, as well as other next-to-leading and higher-order diagrams, in studies of radiative corrections for upcoming lepton-hadron scattering experiments.