On the status of expansion by regions

TL;DR

This work recasts expansion by regions for two-scale, dimensionally regularized Feynman integrals in the Lee–Pomeransky parametric framework, treating the problem geometrically via the Newton polytope of $P=U+F$. The main conjecture identifies essential facets of $\mathcal N_P$ as the sources of asymptotic regions, with leading terms governed by operators $M_{\gamma}$ acting on $P$ and yielding a sum over facet contributions, while convergence is established through analytic regularization and facet-based criteria. The authors prove equivalence with the conventional Feynman-parametric prescriptions and derive leading-order results (and, in the simple one-facet case, general-order expansions) using sector-decomposition ideas, providing a rigorous foundation for region expansions and a practical, geometry-driven algorithm implemented in FIESTA/asy.m. These results offer a robust mathematical underpinning for region-based asymptotics and facilitate reliable, automated calculations of multi-loop integrals in disparate scaling regimes.

Abstract

We discuss the status of expansion by regions, i.e. a well-known strategy to obtain an expansion of a given multiloop Feynman integral in a given limit where some kinematic invariants and/or masses have certain scaling measured in powers of a given small parameter. Using the Lee-Pomeransky parametric representation, we formulate the corresponding prescriptions in a simple geometrical language and make a conjecture that they hold even in a much more general case. We prove this conjecture in some partial cases and illustrate them in a simple example.