The Tropical Geometry of Subtraction Schemes

TL;DR

This paper introduces a tropical geometry-based approach to construct local subtraction schemes for Feynman integrals, enabling systematic Laurent expansions and analytic results for integrals with UV and IR singularities.

Contribution

It provides a necessary and sufficient condition for local finiteness of Euler integrals and develops a new subtraction scheme applicable to a broad class of Feynman integrals.

Findings

Successfully computes Laurent expansions of Feynman integrals with singularities.

Applies the scheme to effective field theories and phase-space integrals.

Provides new analytic results for generalized Feynman integrals.

Abstract

We study the construction of local subtraction schemes through the lenses of tropical geometry. We focus on individual Feynman integrals in parametric presentation, and think of them as particular instances of Euler integrals. We provide a necessary and sufficient condition for a combination of Euler integrands to be locally finite, i.e. to be expandable as a Taylor series in the exponent variables directly under sign of integration. We use this to construct a local subtraction scheme that is applicable to a class of Euler integrals that satisfy a certain geometric property. We apply this to compute the Laurent expansion in the dimensional regulator $\epsilon$ of various Feynman integrals involving both UV and IR singularities, as well as to generalizations of Feynman integrals that arise in effective field theories and in phase-space integrations, for which we provide new analytic results.