p-adic reconstruction of rational functions in multi-loop amplitudes

TL;DR

This paper introduces a p-adic based method for directly reconstructing rational functions in partial-fractioned form, significantly reducing computational effort in multi-loop amplitude calculations in QCD.

Contribution

The authors develop a novel p-adic reconstruction technique that simplifies multi-loop rational functions directly in partial-fractioned form, improving efficiency over traditional methods.

Findings

Requires about 25 times fewer numerical probes per prime field.

Achieves a 130-fold reduction in the size of the reconstructed rational function.

Reveals additional structure in the rational functions that could further optimize calculations.

Abstract

Numerical reconstruction techniques are widely employed in the calculation of multi-loop scattering amplitudes. In recent years, it has been observed that the rational functions in multi-loop calculations greatly simplify under partial fractioning. In this article, we present a technique to reconstruct rational functions directly in partial-fractioned form, by evaluating the functions at special integer points chosen for their properties under a p-adic metric. As an application, we apply this technique to reconstruct the largest rational function in the integration-by-parts reduction of one of the rank-5 integrals appearing in 2-loop 5-point full-colour massless amplitude calculations in Quantum Chromodynamics (QCD). The number of required numerical probes (per prime field) is found to be around 25 times smaller than in conventional techniques, and the obtained result is 130 times smaller. The reconstructed result displays signs of additional structure that could be used to further reduce its size and the number of required probes.