Asymptotics of Feynman Diagrams and The Mellin-Barnes Representation
Samuel Friot, David Greynat, Eduardo de Rafael
TL;DR
The paper introduces a streamlined framework for analytic asymptotics of Feynman diagrams with multiple mass scales by marrying Feynman parameterization with Mellin–Barnes representations and the Converse Mapping Theorem. This approach decouples mass-ratio dependence from parameter integrals, turning the problem into a pole-analysis in the complex plane to read off large- and small-ρ expansions while preserving ε-dependence. Through concrete examples—the muon g-2 vacuum-polarization contributions, a two-loop master integral, and a three-loop calculation—the authors demonstrate that the method yields known results with significantly reduced complexity and also provides higher-order terms. The technique offers a flexible, general toolkit for analytic asymptotics in quantum field theory with multiple scales, with clear connections to traditional region-based methods but with greater systematic efficiency.
Abstract
It is shown that the integral representation of Feynman diagrams in terms of the traditional Feynman parameters, when combined with properties of the Mellin--Barnes representation and the so called {\it converse mapping theorem}, provide a very simple and efficient way to obtain the analytic asymptotic behaviours in both the large and small ratios of mass scales.