Statistical uncertainties of the vn{2k} harmonics from Q cumulants

Nadderd, L.; Milosevic, J.; Wang, F.

doi:10.1103/physrevc.104.034906

Title: Statistical uncertainties of the $v_{n} {2 k}$ harmonics from $Q$ cumulants

Journal Article · 31 August 2021 · Physical Review. C

DOI: https://doi.org/10.1103/physrevc.104.034906 · OSTI ID:1851773

Nadderd, L. ^[1];

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Univ. of Belgrade (Serbia). Institute of Nuclear Science, National Institute of the Republic of Serbia; Huzhou Univ., Zhejiang (China). Zhejiang - Serbia Strong Coupling Physics Research Laboratory
Huzhou Univ., Zhejiang (China). Zhejiang - Serbia Strong Coupling Physics Research Laboratory; Huzhou Univ., Zhejiang (China). College of Science; Purdue Univ., West Lafayette, IN (United States)

Analytic formulas to calculate statistical uncertainties of $v_{n} {2 k}$ harmonics extracted from the Q cumulants are presented. The Q cumulants are multivariate polynomial functions of the weighted means of 2m-particle azimuthal correlations, $$\langle\langle 2 m \rangle\rangle$$. Variances and covariances of the $$\langle\langle 2 m \rangle\rangle$$ are included in the analytic formulas of the uncertainties that can be calculated simultaneously with the calculations of the $v_{n} {2 k}$ 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 subsets, and by the bootstrapping method. The first one is a common way of estimation of the statistical uncertainties of the $v_{n} {2 k}$ harmonics in a real experiment. In order to increase precision in the measurement of the $v_{n} {2 k}$ harmonics, a large number of 15 000 subsets and the same number of the resampling in the bootstrap method is used. Unlike the other ways of the analytic calculation of the $v_{n} {2 k}$ statistical uncertainties, our proposal that includes the use of squared weights in the calculation of both the variances and covariances, gives the best agreement with the results obtained from the subsets and bootstrap method. Additionally, a recurrence equation between Q cumulants of any order is also presented.

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Research Organization:: Purdue Univ., West Lafayette, IN (United States)

Sponsoring Organization:: Ministry of Education Science and Technological Development, Republic of Serbia; National Natural Science Foundation of China; USDOE Office of Science (SC)

Grant/Contract Number:: SC0012910

OSTI ID:: 1851773

Journal Information:: Physical Review. C, Journal Name: Physical Review. C Journal Issue: 3 Vol. 104; ISSN PRVCAN; ISSN 2469-9985

Publisher:: American Physical Society (APS)Copyright Statement

Country of Publication:: United States

Language:: English

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