Spectral Decomposition of Euler-Mellin Integrals

TL;DR

This paper develops a spectral decomposition framework for Euler-Mellin integrals, including Feynman integrals, by explicitly constructing characteristic varieties and cycles, and analyzing singularities and Landau loci.

Contribution

It introduces explicit methods to compute characteristic varieties and cycles for Euler-Mellin integrals and Feynman integrals, advancing singularity analysis and algebraic geometry tools.

Findings

Explicit constructions of characteristic varieties and cycles.

Method to compute Landau singularities of integrals.

Procedure for calculating Euler obstructions and complex links.

Abstract

We consider the spectral decomposition of singularities of integrals and their integrands. Our results apply to any integral of Euler-Mellin type, and thus especially to every scalar Feynman integral. Specifically we provide for both the integrand and integral respectively; two explicit constructions of the characteristic variety and characteristic cycle of the constructible function and $D$-module they are associated with. From this we also obtain the singular locus or Landau singularities of the integral. En route we give a simple procedure to compute the local Euler obstruction function of a variety, and using this, to compute the Euler characteristic of the complex link of a Whitney stratum.