Understanding the correlation between elliptic and triangular flow

TL;DR

This paper addresses the centrality-dependent correlation between elliptic and triangular flow in ultrarelativistic heavy-ion collisions, focusing on the observable ${\rm nsc}_{2,3}\{4\}$. It shows that non-Gaussian fluctuations in the initial state—especially a mixed skewness between the elliptic deformation in the reaction plane and the triangularity—drive the correlation, and introduces an intensive parameter $\Gamma_S$ to characterize this effect. Centrality (impact-parameter) fluctuations further shape the signal, notably in ultracentral collisions, and a data-driven treatment using ATLAS centrality estimators yields $\Gamma_S \approx -2$, in line with Monte Carlo initial-condition models like TRÉNTo. Together, these results explain the sign-change and centrality evolution of ${\rm nsc}_{2,3}\{4\}$ and provide a framework to connect initial geometry fluctuations to final-state flow across collision systems. The methodology, centered on non-Gaussian cumulants and intensive skewness measures, offers a practical tool for interpreting flow correlations in future O+O, p+Pb, or other system studies, with potential refinements from centrality resolution and hydrodynamic response variations.

Abstract

The relative correlation between the magnitudes of elliptic flow ($v_2$) and triangular flow ($v_3$) has been accurately measured in nucleus-nucleus collisions at the LHC collider. As a function of the centrality of the collision, it changes sign and varies non-monotonically. We show that this is naturally explained by two combined effects. The first effect is a skewness in initial-state fluctuations, which is quantified by the correlation between the geometry-driven elliptic deformation in the reaction plane and the fluctuation-driven triangularity $\varepsilon_3$. We introduce an intensive measure of this skewness, which is generically of order unity and depends weakly on the system size and centrality. We evaluate its magnitude using Monte Carlo simulations of the initial state, which show that it is sensitive to the nucleon width. The second effect is the fluctuation of impact parameter relative to centrality classifiers used by experiment. The ATLAS collaboration uses two different centrality classifiers, the multiplicity $N_{ch}$ and the transverse energy $E_T$. We fit both sets of results for Pb+Pb collisions up to $\approx 40\%$ centrality with a single parameter, the intensive mixed skewness. Its value inferred from experiment agrees with theoretical expectations.

Figures (2)

• Figure 1: (a) and (b): Histograms of the two centrality classifiers used by ATLAS ATLAS:2019peb: (a) Multiplicity $N_{ch}$ of charged particles with $p_T>0.5$ GeV/$c$ and $|\eta|<2.5$, where $p_T$ is the transverse momentum and $\eta$ the pseudorapidity; (b) Transverse energy $E_T$ in $3.2<|\eta|<4.5$. Arrows indicate specific values of the centrality fraction for reference. Solid lines are fits assuming Gaussian fluctuations at fixed impact parameter $b$Das:2017ned, and the vertical dashed line is the position of the knee, defined as the expectation value of the centrality classifier at $b=0$. The region around the knee corresponds to ultracentral collisions. (c), (d): Symbols: $v_2\{2\}$, $v_2\{4\}$ and $v_3\{2\}$ data ATLAS:2019peb. Lines: fits taking into account impact parameter ($b$) fluctuations Roubertie:2025qps. (e), (f): ATLAS data on ${\rm nsc}_{2,3}\{4\}$ data ATLAS:2019peb. Dashed lines: One-parameter fits with skewness and without $b$ fluctuations (Eq. (\ref{['nscng']})). Thin solid lines: Calculated value (not a fit) without skewness and kurtosis, but taking into account $b$ fluctuations (Sec. \ref{['s:centrality']}). Dot-dashed lines: One-parameter fits with skewness and$b$ fluctuations. Full lines: Two-parameter fits with skewness, kurtosis and $b$ fluctuations.
• Figure 2: Intensive mixed skewness $\Gamma_S$ (a) and mixed kurtosis $\Gamma_K$ (b), defined by Eqs. (\ref{['v2v3cumul']}) and (\ref{['defmixedskewness']}), as a function of the centrality fraction $c$ in T$\mathrel{\raisebox{-2.1pt}{R}}$ENTo simulations. We define $c\equiv \pi b^2/\sigma_{\rm PbPb}$, where $\sigma_{\rm PbPb}\approx 785$ fm$^2$ is the total inelastic Pb+Pb cross section (see Sec. \ref{['s:centrality']}) The band in panel (a) is the range of values inferred by fitting ATLAS data on ${\rm nsc}_{2,3}\{4\}$ with impact parameter fluctuations taken into account (dot-dashed and full lines in Fig. \ref{['fig:atlasfits']} (e) and (f), see Sec. \ref{['s:centrality']}).