Event Topological Separation Method

• Event Topological Separation Method is a framework that classifies collision events by their particle emission patterns to isolate signature charge separation effects indicative of topological QCD phenomena.
• It employs Fourier decomposition and event shape variables like q₂² to control and subtract backgrounds such as collective elliptic flow and non-topological two-particle correlations.
• Experimental applications across varied collision energies demonstrate significant background suppression and residual signals consistent with the chiral magnetic effect.

The Event Topological Separation Method encompasses a set of theoretical and experimental techniques that enable the differentiation and statistical isolation of phenomena or observables linked to underlying topological effects, particularly within the context of complex quantum many-body systems and high-energy nuclear collisions. In heavy-ion physics, this framework is essential for distinguishing signatures of topological quantum chromodynamics (QCD) phenomena such as the chiral magnetic effect (CME), which manifests as charge separation along an intense magnetic field generated in relativistic nucleus–nucleus collisions. A central challenge is disentangling the genuine topological signal from substantial backgrounds originating from conventional (non-topological) processes, such as collective flow and conventional two-particle correlations. The Event Shape Selection (ESS) method (Collaboration, 30 May 2025) represents a key methodological advance in this domain, providing a rigorous statistical procedure for event-wise separation and background control.

1. Event Shape Selection (ESS): Principles and Implementation

The ESS method is predicated on the classification of collision events according to their final-state particle emission pattern, thereby constructing an "event-shape variable" that is sensitive to the anisotropic flow (e.g., elliptic flow, $v_2$) characterizing the event geometry and collective expansion. In contrast to event shape engineering (ESE), which employs an independent pseudorapidity or momentum window for the event-shape variable, ESS constructs this variable using either the particles of interest (POI) or pairs of POIs (PPOI).

A prototypical ESS variable is formed from the squared second-harmonic flow vector:

$$q_2^2 = \frac{\left(\sum_{i=1}^N \sin 2\phi_i\right)^2 + \left(\sum_{i=1}^N \cos 2\phi_i\right)^2}{N(1 + N v_2^2)},$$

where $\phi_i$ are the azimuthal angles of the POIs, $N$ is their multiplicity, and $v_2$ is the elliptic flow strength. Binning the data by $q_2^2$ enables the extraction of observables at fixed event shapes.

To isolate the topological signal (e.g., CME-driven charge separation), the observable $\Delta\gamma^{112}$—constructed from the three-particle correlator

$$\gamma^{112} = \langle \cos(\phi_\alpha + \phi_\beta - 2\Psi_{\rm RP}) \rangle$$

and measured as the difference between opposite-sign ($\alpha\ne\beta$) and same-sign ($\alpha=\beta$) pairs—is evaluated as a function of $v_2$. A linear fit of $\Delta\gamma^{112}$ vs. $v_2$ allows for an extrapolation to $v_2 = 0$, yielding the ESS-background-subtracted value:

$$\Delta\gamma^{112}_{\rm ESS} = \text{intercept}\left[ \Delta\gamma^{112}(v_2 \rightarrow 0) \right],$$

with an additional correction factor $(1 - \langle v_2 \rangle)^2$ ensuring insensitivity to non-interdependent collectivity.

This procedure operationalizes the statistical isolation of the topological component by nullifying the leading background sources, which scale linearly (or otherwise predictably) with $v_2$.

2. Charge Separation Observables and Magnetic Field Alignment

A primary experimental signature of topological QCD effects such as the CME is the electric charge separation of produced hadrons along the direction of the strong magnetic field $\vec{B}$ generated by the colliding heavy ions. This effect is quantified by correlation observables such as $\Delta\gamma^{112}$, which are maximized when measured relative to the true magnetic field direction.

In contemporary analyses, the event plane angle $\Psi_1$ is reconstructed using spectator nucleons registered in forward detectors (e.g., ZDC-SMD or EPD at RHIC), providing an unbiased estimate of $\vec{B}$ orientation:

$\Psi_1 \simeq \Psi_{\text{B-field}},$

thus minimizing backgrounds unrelated to the underlying QCD topology. Background contributions from elliptic flow and other collective phenomena remain sensitive to $v_2$ and are systematically suppressed via the ESS approach.

A related correlation, $\Delta\gamma^{132}$—built analogously but with a different harmonic structure—serves as a background-dominated control observable. The ESS methodology predicts $\Delta\gamma^{132}_{\rm ESS} \simeq 0$ in the absence of residual background contamination.

3. Results: Suppression of Backgrounds and Energy Dependence

Application of the ESS method to Au+Au collisions across a broad range of center-of-mass energies (7.7–200 GeV) in the RHIC Beam Energy Scan (BES-II) revealed the following:

• $\Delta\gamma^{112}_{\rm ESS}$ is strongly suppressed compared to the uncorrected $\Delta\gamma^{112}$, reduced to at most 20% of its raw value in all considered centrality bins and energies.
• In the 20–50% centrality range, nonzero residual charge separation at the $2.6\sigma$, $3.1\sigma$, and $3.3\sigma$ significance level was observed at $\sqrt{s_{NN}} = 11.5, 14.6, 19.6$ GeV, respectively.
• For 17.3 and 27 GeV, $\Delta\gamma^{112}_{\rm ESS}$ is positive but with lower significance ($1.3\sigma$, $1.1\sigma$); at the lowest and highest energies, $\Delta\gamma^{112}_{\rm ESS}$ is statistically consistent with zero.

A plausible implication is that the topological charge separation signal attributable to the CME is prominent only in a specific intermediate energy regime, potentially linked to optimal conditions for chiral symmetry restoration and sufficiently strong, long-lived magnetic fields.

4. Mathematical Formalism of Event Separation

The Fourier decomposition of the final-state particle distribution with respect to the reaction plane $\Psi_{\rm RP}$ is given by:

$$\frac{dN_\pm}{d\phi} \propto 1 + 2a_1^\pm \sin(\phi - \Psi_{\rm RP}) + \sum_{n} 2v_n^\pm \cos[n(\phi - \Psi_{\rm RP})],$$

where $a_1^\pm$ represents charge-dependent modulation (CME), and $v_n^\pm$ are the harmonic flow coefficients. The ESS workflow systematically varies $v_2$ and extrapolates correlators such as $\Delta\gamma^{112}$ to the point where elliptic flow backgrounds vanish, thereby isolating $a_1^\pm$.

For background validation, the control observable $\Delta\gamma^{132}$, expected to scale with $v_2$ via local charge conservation effects, satisfies the relation:

$\Delta\gamma^{132} \approx v_2 \Delta\delta,$

where $\Delta\delta$ is the difference in two-particle correlations between opposite- and same-sign pairs.

5. Significance, Limitations, and Broader Context

The ESS method constitutes a statistically rigorous protocol for topological event separation in the presence of dominating collective backgrounds. Its reliance on event classification by the POI or PPOI, tight association with the reconstructed magnetic field direction, and extrapolation to the zero-flow limit collectively ensure that the signal extracted (residual nonzero $\Delta\gamma^{112}_{\rm ESS}$) is robustly associated with nontrivial topological QCD domains.

Significance levels approaching or exceeding $3\sigma$ at intermediate energies provide evidence compatible with CME-induced effects, though the persistence of any residual background or possible nonflow bias remains a subject of experimental and theoretical scrutiny.

A plausible implication is that such topological event separation methodologies—potentially generalizable to other systems or observables where topological order coexists with strong background episodes—offer a template for precision measurements of topological quantum phenomena amidst overwhelming conventional signals.

6. Perspective and Future Directions

ESS provides a controlled approach to experimental topological separation in relativistic heavy-ion collisions and sets a methodological standard for future searches for CME and related QCD topological effects. Ongoing efforts include refining the extraction and interpretation of background-free signals at new collision energies, further statistical validation against alternative background models, extension to other observables sensitive to topological effects, and broader adoption in other physical systems exhibiting event-level topological coexistence or separation (Collaboration, 30 May 2025).