Elliptic flow in the Gaussian model of eccentricity fluctuations

TL;DR

This paper presents a Gaussian model of initial eccentricity fluctuations in heavy-ion collisions, showing that elliptic flow from 4-particle cumulants equals the average reaction plane flow, with implications for flow measurement methods.

Contribution

It demonstrates that in the Gaussian fluctuation model, the 4-particle cumulant flow equals the average reaction plane flow, linking flow measurements to initial eccentricity fluctuations.

Findings

$ u_4$ equals the average flow in the reaction plane.

All higher order cumulants are equal to $ u_4$.

The $Q$-distribution is fully characterized by $ u_2$ and $ u_4$.

Abstract

We discuss a specific model of elliptic flow fluctuations due to Gaussian fluctuations in the initial spatial $x$ and $y$ eccentricity components $\left\{\mean{(\sigma_y^2-\sigma_x^2)/(\sigma_x^2+\sigma_y^2)}, \mean{2\sigma_{xy}/(\sigma_x^2+\sigma_y^2)} \right\}$. We find that in this model $\vfour$, elliptic flow determined from 4-particle cumulants, exactly equals the average flow value in the reaction plane coordinate system, $\mean{v_{RP}}$, the relation which, in an approximate form, was found earlier by Bhalerao and Ollitrault in a more general analysis, but under the same assumption that $v_2$ is proportional to the initial system eccentricity. We further show that in the Gaussian model all higher order cumulants are equal to $\vfour$. Analysis of the distribution in the magnitude of the flow vector, the $Q-$distribution, reveals that it is totally defined by two parameters, $\vtwo$, the flow from 2-particle cumulants, and $\vfour$, thus providing equivalent information compared to the method of cumulants. The flow obtained from the $Q-$distribution is again $\vfour=\mean{v_{RP}}$.