Interpolating amplitudes

TL;DR

This paper reviews and evaluates various interpolation methods for scattering amplitudes in high-dimensional phase spaces, aiming to enable faster and more reliable amplitude evaluations in complex particle physics calculations.

Contribution

It compares polynomial, spline, sparse grid, and neural network interpolation techniques for multi-variable amplitudes, highlighting their performance and potential in high-dimensional phase space applications.

Findings

Neural networks and sparse grids show promise for high-dimensional interpolation.

Interpolation methods can significantly reduce computational costs.

Performance varies depending on data availability and complexity.

Abstract

The calculation of scattering amplitudes at higher orders in perturbation theory has reached a high degree of maturity. However, their usage to produce physical predictions within Monte Carlo programs is often precluded by the slow evaluation of two- and higher-loop virtual amplitudes, particularly those calculated numerically. As a remedy, interpolation frameworks have been successfully used for amplitudes depending on up to two kinematic invariants. For amplitude interpolation with more variables, such as the five dimensions of a 2 -> 3 phase space, efficient and reliable solutions are sparse. This work aims to pave the way for using amplitude interpolation in higher-dimensional phase spaces by reviewing state-of-the-art interpolation methods, and assessing their performance on a selection of 2 -> 3 scattering amplitudes. Specifically, we investigate interpolation methods based on polynomials, splines, spatially adaptive sparse grids, and neural networks (multilayer perceptron and Lorentz-Equivariant Geometric Algebra Transformer), all under the constraint of limited obtainable data. Our additional aim is to motivate further studies of the interpolation of scattering amplitudes among both physicists and mathematicians.