Interpolation of rational functions in loop calculations by using p-adic numbers

TL;DR

This paper introduces a p-adic based interpolation method for large rational functions in multi-loop calculations, significantly reducing computational complexity in quantum field theory amplitude evaluations.

Contribution

The paper presents a novel p-adic interpolation technique for rational functions, enabling more efficient multi-loop amplitude calculations in QCD.

Findings

Number of probes reduced by a factor of 25

Result size reduced by a factor of 130

Effective in 2-loop 5-point amplitude calculations

Abstract

The calculation and manipulation of large multi-variable rational functions is a key bottleneck in multi-loop calculations. In these conference proceedings, based on my article [Chawdhry (2023) arXiv:2312.03672], I present a technique to interpolate such rational functions in a compact form by evaluating them at special integer points chosen for their properties under a $p$-adic absolute value. I apply the technique to examples of large rational functions appearing in 2-loop 5-point massless non-planar amplitude calculations in Quantum Chromodynamics (QCD). The number of required numerical probes (per field) is found to be around 25 times smaller than in conventional techniques, and the obtained result is 130 times smaller.