On the status of expansion by regions

TL;DR

This paper explores the expansion by regions method for multiloop Feynman integrals, using Lee-Pomeransky representation to formulate geometric prescriptions and conjecturing their broader applicability, supported by partial proofs and examples.

Contribution

It introduces a geometric formulation of the expansion by regions strategy and conjectures its validity in more general cases, with partial proofs and illustrative examples.

Findings

Formulated geometric prescriptions for expansion by regions

Conjectured broader applicability of the method

Provided partial proofs and simple examples

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.