On the reconstruction of kinematic distributions computed with Monte Carlo methods using orthogonal basis functions

TL;DR

This paper introduces an orthogonal basis function approach for reconstructing kinematic distributions from Monte Carlo data, offering smoother results and reduced fluctuations compared to traditional histograms.

Contribution

It proposes a novel method using orthogonal basis functions for distribution reconstruction, improving smoothness and stability over conventional histogram methods.

Findings

The method produces smooth approximations to distributions.

It eliminates bin-to-bin fluctuations in high-order calculations.

Applied to Higgs boson production, it outperforms traditional histograms.

Abstract

Reconstruction of one-dimensional kinematic distributions from calculations based on high-dimensional Monte-Carlo integration is a standard problem in high-energy physics. Traditionally, this is done by collecting randomly-generated events in histograms. In this article, we explore an alternative approach, whose main idea is to approximate the target distribution by a weighted sum of orthogonal basis functions whose coefficients are calculated using the Monte-Carlo integration. This method has the advantage of directly yielding smooth approximations to target distributions. Furthermore, in the context of high-order perturbative calculations with local subtractions, it eliminates the so-called bin-to-bin fluctuations, which often severely affect the quality of conventional histograms. We also demonstrate that the availability of a high-quality approximation to the target distribution, for example the leading-order result in the perturbative expansion, can be exploited to construct an optimized orthonormal basis. We compare the performance of this method to conventional histograms in both toy-model and real Monte-Carlo settings, applying it to Higgs boson production in weak boson fusion as an example.