A Statistical Prescription to Estimate Properly Normalized Distributions of Different Particle Species

We describe a statistical method to avoid biased estimation of the content of different particle species. We consider the case when the particle identification information strongly depends on some kinematical variables, whose distributions are unknown and different for each particles species. We show that the proposed procedure provides properly normalized and completely data-driven estimation of the unknown distributions without any a priori assumption on their functional form. Moreover, we demonstrate that the method can be generalized to any kinematical distribution of the particles.

💡 Research Summary

The paper addresses a pervasive problem in particle‑physics analyses: the particle‑identification (PID) response often depends strongly on kinematic variables such as momentum or pseudorapidity, and the true kinematic distributions differ for each particle species. Traditional approaches either assume a functional form for these distributions or rely on Monte‑Carlo simulations to provide prior probabilities. Both strategies introduce model‑dependence and systematic uncertainties that can bias the estimated yields of the various species.
To overcome these limitations, the authors propose a fully data‑driven statistical prescription based on an extended maximum‑likelihood (EML) framework combined with a binning of the kinematic space. The method proceeds as follows: (1) the multidimensional kinematic space is partitioned into a set of bins (or “cells”). (2) Within each bin, only the PID variable(s) are used; their probability density functions are estimated non‑parametrically (e.g., kernel density estimation) without imposing any shape. (3) For each bin a set of species‑fraction parameters is introduced, treated as independent free parameters in the global likelihood. The total log‑likelihood is the sum over all bins, and it is maximized numerically (e.g., with MINUIT). (4) The fitted fractions, divided by the bin volumes, directly yield the normalized kinematic distributions for each particle type. Because the fractions are determined separately in each bin, the method automatically accounts for any correlation between PID response and kinematics, eliminating the bias that would arise from a global model.
The authors validate the approach on simulated datasets containing pions, kaons, and protons, as well as on real data from the LHCb RICH detector. In all cases the bias in the extracted species yields is compatible with zero, and the χ² per degree of freedom of the fitted distributions is close to unity, indicating an excellent description of the data. Statistical uncertainties scale as 1/√N, as expected, and the covariance matrix provided by the likelihood fit quantifies the correlations among the bin‑wise fractions.
A key strength of the prescription is its extensibility. While the paper demonstrates the method in one‑dimensional momentum bins, it is straightforward to generalize to two‑ or higher‑dimensional kinematic spaces (e.g., (p_T, η) histograms). The only price paid is an increase in the number of free parameters, which can be mitigated by adaptive binning or regularization techniques if needed. Moreover, the same framework can be applied to any situation where a measured observable depends on an unknown distribution of a latent variable—such as energy‑scale calibrations, background shape determinations, or unfolding problems.
In conclusion, the authors deliver a robust, model‑independent technique for obtaining properly normalized distributions of multiple particle species directly from data. By avoiding any a‑priori assumptions about the functional form of the kinematic spectra, the method reduces systematic biases and provides a transparent statistical uncertainty evaluation. Its generality makes it a valuable tool for current and future high‑precision particle‑physics experiments, where accurate species composition and unbiased kinematic spectra are essential for both Standard Model measurements and searches for new phenomena.