Some recent results on evaluating Feynman integrals

TL;DR

The paper surveys recent progress in evaluating Feynman integrals, contrasting Mellin-Barnes (MB) representations for single integrals with a Groebner-bases (GB) based framework for IBP reduction to master integrals. MB methods provide a systematic loop-by-loop construction, pole-resolution strategies (MB1/MB2), and epsilon expansions, enabling cross-checks in partial limits and in supersymmetric contexts. The Groebner-bases approach is developed into a sector-based 2S strategy that builds sector-specific bases to recursively reduce to master integrals across index sectors, with successful demonstrations on multi-index and HQET problems. The perspectives highlight complementary numerical algorithms that can bypass analytic reduction when necessary, advocating a hybrid workflow that combines MB, GB, and numerical techniques for robust evaluation of complex Feynman integrals.

Abstract

Some recent results on evaluating Feynman integrals are reviewed. The status of the method based on Mellin-Barnes representation as a powerful tool to evaluate individual Feynman integrals is characterized. A new method based on Groebner bases to solve integration by parts relations in an automatic way is described.