Statistical uncertainties of the $v_{n}\{2k\}$ harmonics from Q-cumulants

TL;DR

This paper derives analytic formulas for calculating statistical uncertainties of $v_{n} ext{ }\{2k\}$ harmonics from Q-cumulants, validated through toy models, sub-set analysis, and bootstrap methods, enhancing precision in flow measurements.

Contribution

It introduces new analytic formulas for uncertainties of $v_{n} ext{ }\{2k\}$ harmonics from Q-cumulants, including variances and covariances, and presents a recurrence equation for Q-cumulant calculation.

Findings

Analytic uncertainties agree well with bootstrap and sub-set methods.

Formulas incorporate variances and covariances of azimuthal correlations.

Recurrence equation simplifies Q-cumulant calculations.

Abstract

Analytic formulas to calculate statistical uncertainties of $v_{n}\{2k\}$ harmonics extracted from the Q-cumulants are presented. The Q-cumulants are multivariate polynomial functions of the weighted means of $2m$-particle azimuthal correlations, $\llangle 2m \rrangle$. Variances and covariances of the $\llangle 2m \rrangle$ are included in the analytic formulas of the uncertainties that can be calculated simultaneously with the calculations of the $v_{n}\{2k\}$ harmonics. The calculations are performed using a simple toy model which roughly simulates elliptic flow azimuthal anisotropy with magnitudes around 0.05. The results are compared with the results obtained by the many data sub-sets, and by the bootstrapping method. The first one is a common way of estimation of the statistical uncertainties of the $v_{n}\{2k\}$ harmonics in a real experiment. In order to increase precision in the measurement of the $v_{n}\{2k\}$ harmonics, a large number of 15000 sub-sets and the same number of the re-sampling in the bootstrap method is used. A very good agreement in the comparison with the proposed way of the analytic calculation of the $v_{n}\{2k\}$ statistical uncertainties is found. Additionally, a recurrence equation for Q-cumulants calculation is also presented.