Numerical Scattering Amplitudes with pySecDec

TL;DR

The paper addresses the challenge of computing high-order scattering amplitudes by numerical evaluation of multi-loop Feynman integrals with multiple scales and regulators. It introduces pySecDec version 1.6, featuring a distributed Quasi-Monte-Carlo evaluator (Disteval), median lattice rules, automatic extra regulators for expansion by regions, and flexible treatment of master-integral coefficients. Key contributions include substantial speedups (up to an order of magnitude), the ability to compute amplitudes as weighted sums with user-defined precision, and GPU-accelerated evaluation. The results are demonstrated across diverse multi-loop examples, illustrating improved robustness and applicability to precision collider phenomenology.

Abstract

We present a major update of the program pySecDec, a toolbox for the evaluation of dimensionally regulated parameter integrals. The new version enables the evaluation of multi-loop integrals as well as amplitudes in a highly distributed and flexible way, optionally on GPUs. The program has been optimised and runs up to an order of magnitude faster than the previous release. A new integration procedure that utilises construction-free median Quasi-Monte Carlo rules is implemented. The median lattice rules can outperform our previous component-by-component rules by a factor of 5 and remove the limitation on the maximum number of sampling points. The expansion by regions procedures have been extended to support Feynman integrals with numerators, and functions for automatically determining when and how analytic regulators should be introduced are now available. The new features and performance are illustrated with several examples.

Figures (13)

• Figure 1: The RQMC integration error (i.e. $\sqrt{\textrm{var}(I_i)/m}$) after $m=32$ repetitions for lattices of different sizes. The integrals are sectors of the elliptic2L_physical example from \ref{['sec:timings']}. The lattices are taken from the Qmc library, and are the same for both integrals. The result of one particularly unlucky lattice is marked with a star; note that this lattice is only unlucky for one of the sectors and performs normally for the others.
• Figure 2: Integration time of the elliptic2L_physical example from \ref{['sec:timings']} using the median QMC rules compared to the integration using CBC construction of the generating vectors. This plot uses the same benchmarking setup as \ref{['tab:elliptic2L_physical-median-timings']}.
• Figure 3: A 2-loop three point integral with three mass scales.
• Figure 4: Generation script for the two-loop muon decay example.
• Figure 5: Integration script for the two-loop muon decay example.