Event-Plane Correlations in Heavy-Ion Collisions

• Event-plane correlations are defined as statistical averages of cosine functions that quantify the mutual orientation of symmetry plane angles in heavy-ion collisions.
• They are measured using experimental techniques like the EP, SP, and MLE methods, with resolution corrections to account for finite multiplicity and nonflow effects.
• These observables reveal nonlinear hydrodynamic responses and initial geometry fluctuations in the quark-gluon plasma, informing strategies for flow analysis and jet-background subtraction.

Event-plane correlations are observables in heavy-ion collisions that quantify the mutual orientation of the symmetry planes (event planes) defined by the azimuthal anisotropies of final-state particle distributions. They probe not only the collective response of the produced medium to the initial-state geometry but also encapsulate information about underlying nonlinear hydrodynamic dynamics and event-by-event fluctuations. Modern analyses employ these multi-plane correlators to unravel the interplay between initial geometry, its fluctuations, and the nonlinear evolution of the quark-gluon plasma (QGP).

1. Definitions and Mathematical Formalism

The azimuthal distribution of outgoing particles in a single collision event is typically described via a Fourier expansion: $$\frac{dN}{d\phi} \propto 1 + 2\sum_{n=1}^\infty v_n \cos(n[\phi - \Psi_n]),$$ where $v_n$ is the magnitude of the $n$th-order flow harmonic (collective anisotropy), and $\Psi_n$ is the corresponding event-plane angle. The flow vectors can be written compactly as

$V_n = v_n e^{i n \Psi_n}.$

Event-plane correlations refer to the statistical expectation values of cosines of linear combinations of these angles across events: $$\left\langle \cos\left(k_1 n_1 \Psi_{n_1} + k_2 n_2 \Psi_{n_2} + \cdots + k_m n_m \Psi_{n_m} \right) \right\rangle,$$ subject to the rotational invariance constraint $\sum_{j=1}^m k_j n_j = 0$ (Collaboration, 2014, Bhalerao et al., 2013, Jia et al., 2012). Two-plane correlators, such as $\langle\cos[k(\Psi_n - \Psi_m)]\rangle$, probe mutual alignment (or anti-alignment) between harmonic planes of order $n$ and $m$. Higher-order (three- or four-plane) correlators extend this concept.

The experimentally accessible event-plane angles are estimated via flow vectors built from measured final-state particles: $$Q_n \equiv \sum_j w_j e^{i n \phi_j} = |Q_n| e^{i n \Psi_n},$$ with $w_j$ typically unity or $p_T$. Finite-multiplicity requires resolution corrections to recover unbiased physical correlators (Collaboration, 2014, Jia, 2012).

2. Measurement Methodologies and Experimental Systematics

Event-plane correlations are extracted using several complementary experimental techniques:

• Standard Event-Plane (EP) Method: Event-plane angles $\Psi_n$ are estimated in multiple detector sub-events or pseudorapidity windows to suppress autocorrelations. Correlators are formed using these independently reconstructed planes and divided by the relevant product of single-plane resolution factors, typically obtained via two- or three-subevent methods (Collaboration, 2014, Jia, 2012).
• Scalar-Product (SP) Method: Correlators are constructed from the full complex flow vectors $Q_n$, with correction for resolution effects in the denominator. SP correlators tend to be systematically larger (up to 30%) than the EP method due to their stronger weighting of high-flow events, and are free from certain projection ambiguities (Collaboration, 2014, Bhalerao et al., 2013).
• Maximum-Likelihood Estimators (MLE): Recent theoretical work demonstrates that MLE-based extraction of flow harmonics and event planes reproduces standard methodologies and extends naturally to arbitrarily differential and higher-order correlators with robust statistical properties (Ye et al., 2024).

Event-plane correlations are measured differentially in centrality, transverse momentum, and pseudorapidity. Correction for finite event-plane resolution, as well as systematic uncertainties from nonflow correlations and detector effects, are standard (Jia, 2012, Collaboration, 2014).

3. Central Physical Findings: Experimental Results and Model Interpretation

Systematic measurements by ATLAS, STAR, PHENIX, and ALICE at RHIC and the LHC have established a suite of robust experimental facts:

• Two-plane correlators: The correlation $\langle\cos 4(\Psi_2 - \Psi_4)\rangle$ is large and positive (up to $\sim$0.6), growing in peripheral collisions as elliptic flow $v_2$ strengthens. Analogously, $\langle\cos 6(\Psi_2 - \Psi_3)\rangle$ is negligible, indicating near-random mutual orientation of elliptic and triangular flow (Collaboration, 2014, Jia, 2012).
• Three-plane and higher correlators: Combinations such as $\langle\cos(2\Psi_2 + 3\Psi_3 - 5\Psi_5)\rangle$ are positive and increase in non-central events, while e.g. $\langle\cos(2\Psi_2 - 6\Psi_3 + 4\Psi_4)\rangle$ are negative, reflecting anti-alignment driven by non-trivial mode-coupling (Collaboration, 2014, Teaney et al., 2013, Jia, 2012, Jia et al., 2012).
• p$_T$ and $\eta$ decorrelation: Event planes reconstructed in separated pseudorapidity intervals or $p_T$ bins exhibit nontrivial decorrelation, decreasing steadily with increasing separation, and even becoming anticorrelated for certain harmonics (notably triangular flow at large $\Delta \eta$) (Xiao et al., 2012, Xiao et al., 2015).
• Jet and hadron correlations: Event-plane–differential dihadron and jet-hadron correlations reveal path-length–dependent modification of the away-side structure (from single to double peak), and suppression of yields out-of-plane versus in-plane, consistent with path-length–dependent jet quenching in the QGP (Collaboration, 2014, Adare et al., 2018, Zhang, 2019). However, high-$p_T$ jet–hadron correlations at top LHC energies show vanishing event-plane dependence within current uncertainties (Collaboration, 2019, Collaboration et al., 2023).

Correlator	Exp. Value (Mid-central)	Physical Interpretation
$\langle\cos 4(\Psi_2 - \Psi_4)\rangle$	0.4–0.6	Nonlinear $v_4 \sim v_2^2$ mode mixing
$\langle\cos 6(\Psi_2 - \Psi_3)\rangle$	$<0.02$	Weak/no $v_2$–$v_3$ geometric coupling
$\langle\cos(2\Psi_2 + 3\Psi_3 - 5\Psi_5)\rangle$	$\sim$0.05–0.15	$v_5$ nonlinear from $v_2 v_3$ coupling
$C_n(\Delta\eta)$ for $n=3$	$0 \to -0.2$ (anticorr.)	Twisted triangular geometry

4. Theoretical Mechanisms and Hydrodynamic Response

The physical origin of event-plane correlations is established as a competition between three key effects:

• Initial-state participant-plane (eccentricity) correlations: In the initial nuclear overlap, participant-plane angles $\Phi_n$ are correlated via geometry and density fluctuations; both geometry-dominated (peripheral) and fluctuation-dominated (central) patterns are accessible (Jia et al., 2012, Yan, 2015, Yang et al., 2019).
• Hydrodynamic evolution and nonlinear mode mixing: The final anisotropic flow vectors can be written as

$$V_n = V_{n,L} + \sum_{k} \chi_{n;k} \prod_{i} V_{n_i},$$

where $V_{n,L}$ is the linear response to $\epsilon_n$, and $\chi_{n;k}$ terms encode quadratic or higher nonlinear mixing (e.g., $V_4 \propto V_2^2$, $V_5 \propto V_2 V_3$). The relative contribution of linear vs. nonlinear response determines the magnitude and centrality dependence of event-plane correlations (Collaboration, 2014, Teaney et al., 2013, Qian et al., 2017, Teaney et al., 2012).

• Mapping from participant to flow angles: Hydrodynamic simulations have shown that strong final-state event-plane correlations are mainly determined by large initial eccentricities rather than pre-existing plane correlations, except in very peripheral events (Yang et al., 2019).

This response structure explains the centrality trends and sign patterns of experimentally measured correlators, as well as the dependence on viscosity ($\eta/s$). Quantitative agreement between models and data is achieved for low-order harmonics and principal correlators (Teaney et al., 2013).

5. Differential and Higher-Order Correlators

Recent research explores the differential structure of event-plane correlations:

• $p_T$–differential correlations: Event-plane angles resolved in $p_T$-bins exhibit decorrelation, but the inter-bin correlators factorize to high accuracy through a single "global" event-plane (the $p_T$-integrated plane), not the initial participant plane (Xiao et al., 2015, Ye et al., 2024). Decorrelations may reach 10–20% at large $p_T$ separations.
• Higher- and mixed-harmonic correlators: Scalar-product and MLE-based analyses can extract correlators involving arbitrary numbers of harmonics and test new hypotheses—for example, predicted four-plane correlators (Bhalerao et al., 2013). The magnitude and structure of these observables are sensitive to nonlinear hydrodynamic response and initial geometry models.
• Small systems: Plane correlations (especially anti-correlations between elliptic and triangular flow) have been proposed and quantified as litmus tests of collectivity in p–A, d–A, and $^3$He–A collisions, with model predictions for their sign and scaling (Yan, 2015).

6. Practical Implications for Flow Analysis and Background Subtraction

Event-plane correlations undergird a range of experimental strategies:

• Flow-modulated background modeling in jet and dihadron correlations requires full account of event-plane dependence, including the correct inclusion of harmonic mixing, phase shifts for asymmetric trigger sectors, inter-plane correlations, and finite event-plane resolution (Nattrass et al., 2018, Collaboration, 2014). Neglecting these effects can bias interpretation at the few-percent level.
• Pseudorapidity-gap methods for nonflow suppression must consider decorrelation of event planes over $\eta$, as large $\eta$ gaps can systematically underestimate $v_n$ and impact extracted QGP properties (e.g., $\eta/s$) (Xiao et al., 2012).
• Unified Fourier methods allow for efficient, unbiased extraction of all possible plane correlators within an event, facilitate direct model–data comparison, and are robust to acceptance limitations (Jia et al., 2012, Bhalerao et al., 2013).
• MLE approaches yield statistically optimal (asymptotically normal and unbiased) event-plane and correlator estimates, enable new mixed-harmonic and non-canceling sum correlators, and provide diagnostic decompositions into flow-magnitude vs. event-plane decorrelation sources (Ye et al., 2024).

7. Current Limitations and Outlook

Event-plane correlations have established themselves as critical multi-particle observables for QGP tomography and medium property extraction. Limitations persist in only moderate sensitivity to initial participant-plane couplings in central collisions, statistical uncertainties for high-order/multi-plane observables, and model ambiguities at high $n$. Ongoing data from LHC Run 3, high-luminosity RHIC campaigns, and continuous development of hydrodynamic and transport theory promise further progress in using event-plane correlators to constrain the fluctuating dynamics and transport coefficients of hot QCD matter (Collaboration, 2014, Teaney et al., 2013, Ye et al., 2024).

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References:

(Collaboration, 2014, Jia, 2012, Bhalerao et al., 2013, Jia et al., 2012, Collaboration, 2014, Nattrass et al., 2018, Xiao et al., 2015, Xiao et al., 2012, Yan, 2015, Qian et al., 2017, Teaney et al., 2013, Teaney et al., 2012, Adare et al., 2018, Yang et al., 2019, Ye et al., 2024, Collaboration, 2019, Collaboration et al., 2023, Zhang, 2019, Castilho et al., 2017)