An algorithm for the high-energy expansion of multi-loop diagrams to next-to-leading logarithmic accuracy

TL;DR

The paper develops an automated algorithm to compute multi‑loop massive Feynman diagrams in the high‑energy regime where $s\gg M^2$, by combining asymptotic expansion with sector decomposition to isolate UV and mass singularities. It formulates a systematic FP representation, identifies the origin of singularities, and implements primary and iterated sector decompositions to factorize all dangerous regions, enabling analytic extraction of leading poles and logarithms and numerical evaluation of finite parts. The method explicitly yields LL and NLL contributions at arbitrary loop order and scales, with applications to electroweak two‑loop corrections and cross‑checks of resummation prescriptions, and it can be extended to higher logarithmic accuracy. Overall, this approach provides a robust, automated toolkit for precise high‑energy predictions in theories with multiple mass and energy scales.

Abstract

We present an algorithm to compute arbitrary multi-loop massive Feynman diagrams in the region where the typical energy scale \sqrt{s} is much larger than the typical mass scale M, i.e. s>>M^2, while various different energy and mass parameters may be present. In this region we perform an asymptotic expansion and, using sector decomposition, we extract the leading contributions resulting from ultraviolet and mass singularities, which consist of large logarithms log(s/M^2) and 1/εpoles in D=4-2εdimensions. To next-to-leading accuracy, at L loops all terms of the form α^L ε^{-k} log^j(s/M^2) with j+k=2L and j+k=2L-1 are taken into account. This algorithm permits, in particular, to compute higher-order next-to-leading logarithmic electroweak corrections for processes involving various kinematical invariants of the order of hundreds of GeV and masses M_W \sim M_Z \sim M_H \sim M_t of the order of the electroweak scale, in the approximation where the masses of the light fermions are neglected.