Multivariate cumulants in flow analyses: The Next Generation

TL;DR

This paper formalizes multivariate cumulants in flow analyses, introduces new estimators and cumulants for azimuthal angles and flow amplitudes, and demonstrates their utility in constraining flow fluctuation distributions in high-energy nuclear collisions.

Contribution

It provides a rigorous mathematical foundation for multivariate cumulants in flow analyses, introduces novel cumulants and estimators, and applies them to improve understanding of flow fluctuations.

Findings

New multivariate cumulants of azimuthal angles defined event-by-event.

Introduction of Asymmetric Cumulants for flow amplitudes.

First realisation of Cumulants of Symmetry Plane Correlations.

Abstract

We reconcile for the first time the strict mathematical formalism of multivariate cumulants with the usage of cumulants in anisotropic flow analyses in high-energy nuclear collisions. This reconciliation yields to the next generation of estimators to be used in flow analyses. We review all fundamental properties of multivariate cumulants and use them as a foundation to establish two simple necessary conditions to determine whether some multivariate random variable is a multivariate cumulant in the basis they are expressed in. We argue that properties of cumulants are preserved only for the stochastic variables on which the cumulant expansion has been performed directly, and if there are no underlying symmetries due to which some terms in the cumulant expansion are identically zero. We illustrate one possibility of new multivariate cumulants of azimuthal angles by defining them event-by-event and by keeping all non-isotropic terms in the cumulant expansion. Further, we introduce new cumulants of flow amplitudes named Asymmetric Cumulants, which generalize recently introduced Symmetric Cumulants for the case when flow amplitudes are raised to different powers. Finally, we present the new concept of Cumulants of Symmetry Plane Correlations and provide the first realisation for the lowest orders. The new estimators can be used directly to constrain the multivariate probability density function of flow fluctuations, since its functional form can be reconstructed only from its true moments or cumulants. The new definition for cumulants of azimuthal angles enables separation of nonflow and flow contributions, and offers first insights into how the combinatorial background contributes for small multiplicities to flow measurements with correlation techniques. All the presented results are supported by Monte Carlo studies using state-of-the-art models.