Four-Particle Subevent Cumulants

• Four-particle subevent cumulants are multiparticle correlation observables that isolate genuine collective behavior by dividing event data into distinct pseudorapidity subevents.
• They quantify flow fluctuations by distinguishing collective flow from nonflow contributions like jets and resonance decays in high-energy nuclear collisions.
• Their multi-subevent approach enables robust characterization of the quark–gluon plasma by disentangling effects of global conservation laws from intrinsic flow dynamics.

Four-particle subevent cumulants are a class of multiparticle correlation observables designed to quantify genuine collective multiparticle correlations in high-energy nuclear collisions, while efficiently suppressing nonflow effects such as jets, resonance decays, and global conservation constraints. These cumulants are constructed by forming azimuthal multiparticle correlators from particles selected in distinct pseudorapidity intervals ("subevents"), and play a central role in the measurement of flow fluctuations, the disentanglement of collective effects from trivial correlations, and the characterization of the quark–gluon plasma's dynamical properties.

1. Mathematical Formulation and Construction

The construction of four-particle subevent cumulants begins with the definition of multi-particle azimuthal correlators. For a given harmonic $n$, the standard four-particle cumulant is: $$c_n\{4\} = \langle e^{in(\phi_1 + \phi_2 - \phi_3 - \phi_4)} \rangle - 2 \langle e^{in(\phi_1 - \phi_2)} \rangle^2$$ where angle brackets denote averaging over all quadruplets in an event and then over events.

In the subevent cumulant method, the particle sample is divided in pseudorapidity into disjoint regions (subevents), and only quadruplets where the particles are drawn from separate subevents are used. For example, the two- and three-subevent four-particle correlators are

$$\langle 4 \rangle_{\textrm{2-sub}} = \langle e^{in(\phi_1^a + \phi_2^a - \phi_3^b - \phi_4^b)} \rangle$$

$$\langle 4 \rangle_{\textrm{3-sub}} = \langle e^{in(\phi_1^a + \phi_2^a - \phi_3^b - \phi_4^c)} \rangle$$

with particles indexed by subevent assignment. The corresponding four-particle subevent cumulant is: $$c_n^{\textrm{sub}}\{4\} = \langle 4 \rangle_{\textrm{sub}} - 2 \langle 2 \rangle_{ab} \langle 2 \rangle_{ac}$$ where $\langle 2 \rangle_{ab}$ denotes two-particle Q-vector correlations between subevents $a$ and $b$.

The symmetric cumulant generalizes to mixed harmonics as

$$sc_{n,m}\{4\} = \langle v_n^2 v_m^2 \rangle - \langle v_n^2 \rangle \langle v_m^2 \rangle$$

which, for subevent implementation, uses the aforementioned restrictions in pseudorapidity.

2. Relation to Flow Fluctuations and Collectivity

Four-particle subevent cumulants are robust probes of genuine collective flow. By correlating four particles from separated regions, the method systematically suppresses nonflow contributions—such as dijets or resonance decays—that are typically localized in rapidity. For instance, in proton–proton and proton–lead collisions, the standard cumulant method yields $c_2\{4\}$ with strong dependence on event class definition (e.g., $p_T$ or multiplicity selection), and may even return positive or ambiguous values when nonflow dominates. The subevent method, particularly with three or more subevents, yields a negative $c_2\{4\}$ throughout a broad multiplicity range—signaling the presence of long-range collectivity and allowing a robust extraction of flow coefficients, $v_n\{4\} = \left(-c_n\{4\}\right)^{1/4}$ (Jia et al., 2017, Collaboration, 2017).

By comparing two- and four-particle cumulants ($v_n\{2\}$ and $v_n\{4\}$), event-by-event flow fluctuations can be quantified. The observed hierarchy, $v_n\{4\} < v_n\{2\}$, is a manifestation of these fluctuations. The quantity

$$F(v_n) = \sqrt{\frac{v_n\{2\}^2 - v_n\{4\}^2}{v_n\{2\}^2 + v_n\{4\}^2}}$$

serves as a relative measure of flow fluctuations, with larger $F$ in more central or small systems (Collaboration, 2014).

3. Suppression of Nonflow and Centrality/Multiplicity Dependence

Subevent cumulant methods (employing two, three, or four subevents) exhibit strong suppression of short-range nonflow effects, as demonstrated by simulation in PYTHIA and AMPT models (Jia et al., 2017, Nie et al., 2018, Huo et al., 2017). The residual difference in the cumulants between different subevent configurations characterizes the magnitude of the remaining nonflow.

• At high event multiplicity, standard and subevent methods yield similar results, indicating nonflow is negligible.
• At low multiplicity, the divergence of cumulants extracted with different numbers of subevents signals the increasing role of few-particle short-range correlations, which are eliminated in the four-subevent approach (Collaboration, 2019, Collaboration, 2018).

These features establish the subevent method as a robust baseline for the identification of collective signatures at the onset of collectivity, and for mapping system-size or energy-dependence of the phenomenon (Collaboration, 2019, Collaboration, 2017, Bhatta et al., 2021).

4. Impact of Global Conservation Laws and Interplay with Flow

Transverse momentum conservation (TMC) induces kinematic correlations among final-state particles that manifest in four-particle cumulants, especially at small multiplicity. Analytical calculations show that TMC contributions to $c_2\{4\}$ and mixed symmetric cumulants $sc_{n,m}\{4\}$ are positive and suppress the negative (flow-driven) values at low $N$, leading to a possible sign change as observed in ATLAS data (Bzdak et al., 2018, Pei et al., 9 Mar 2024, Pei et al., 17 Mar 2025). The general form of the cumulant under TMC and flow is: $$c_n\{4\} \approx (v_2(p))^4 - \text{TMC corrections}\, - 2\,(c_2\{2\})^2$$ where the TMC corrections scale as $1/N^k$ for $k$-particle cumulants. At high multiplicity, flow dominates and $c_2\{4\}$ is negative, while at low multiplicity the TMC term dominates and $c_2\{4\}$ becomes positive (Bzdak et al., 2018). Subevent cumulants maintain this feature; the interplay between flow and TMC must be carefully modeled for the correct interpretation of low-multiplicity data (Pei et al., 9 Mar 2024, Pei et al., 17 Mar 2025).

5. Multiparticle and Mixed-Harmonic Cumulants

Beyond single-harmonic cumulants, four-particle subevent methods facilitate the extraction of symmetric cumulants $sc_{n,m}\{4\}$ (correlations between $v_n$ and $v_m$), and their normalized variants (e.g., $$nsc_{n,m}\{4\} = sc_{n,m}\{4\}/(v_n\{2\}^2 v_m\{2\}^2)$$). These observables probe the joint probability distribution $P(v_n,v_m,\Psi_n,\Psi_m)$ of the flow coefficients and symmetry-plane angles. For instance,

• $sc_{2,3}\{4\}$ is negative, indicating anti-correlation between $v_2$ and $v_3$;
• $sc_{2,4}\{4\}$ is positive, indicating positive correlation between $v_2$ and $v_4$ (Pei et al., 17 Mar 2025, Collaboration, 2018);
• Higher-order cumulants (e.g., six-particle $sc_{2,3,4}\{6\}$) probe three-mode (tri-harmonic) correlations that cannot be decomposed into two-mode contributions (Pei et al., 17 Mar 2025, Taghavi, 2020).

Subevent implementation extends directly: four- (and higher-) particle correlators are built from particles in three or four η-separated intervals, providing maximal rejection of nonflow even for complex correlation structures (Collaboration, 2018, Huo et al., 2017).

6. Extraction of Source and Fluctuation Properties

Four-particle cumulants are also employed for emission source characterization in femtoscopy, where measured momentum-space cumulants ($\kappa_2,\kappa_4$ etc.) are related to the corresponding spatial cumulants of the emission source via Fourier transforms. The extraction of, e.g., the fourth-order source cumulant (or x-kurtosis) from measured momentum cumulants is achieved using ratios of series expansions (Gram–Charlier and, preferably, Edgeworth). The Edgeworth expansion, properly ordered per the Central Limit Theorem, provides an accurate and systematically converging estimate, crucial for model-independent source imaging (Eggers et al., 2010).

Additionally, in critical point searches, decomposition of particle-number cumulants in terms of genuine multiparticle (factorial) correlation functions, via the relations

$K_4 = \langle N \rangle + 7C_2 + 6C_3 + C_4$

allows identification of genuine four-particle (critical) fluctuations, as opposed to accidental contributions from lower-order correlations (Bzdak et al., 2016, Pandav, 2020).

7. Applications, Model Comparisons, and Future Perspectives

Extensive comparisons to data from ATLAS, CMS, ALICE, and STAR demonstrate that four-particle subevent cumulants provide an essential discriminator:

• Subevent cumulants reproduce collective flow signals and are consistent across small and large collision systems for a given event-multiplicity, while standard cumulants are contaminated by nonflow (Collaboration, 2017, Collaboration, 2019, Collaboration, 2018, Collaboration, 2019).
• Deviations between subevent cumulants and hydrodynamic model predictions at low multiplicity or in small systems motivate further improvements in modeling initial fluctuations, non-Gaussian dynamics, and conservation laws (Taghavi, 2020, Abbasi et al., 2017).
• The framework is being extended to pT-fluctuations, event-shape engineering (ESC), and centrality fluctuations, where subevent cumulants help in quantifying long-range, collective, initial-state geometry fluctuations with reduced sensitivity to finite acceptance or autocorrelation biases (Wei et al., 2020, Bhatta et al., 2021, Zhou et al., 2018).

A plausible implication is that the continued development of subevent cumulant observables (including higher-order and mixed-harmonic variants) will provide deeper insight into the collective evolution, nonlinear response, and initial-state fluctuations of the quark–gluon plasma, and will clarify the relative role of collectivity vs. global conservation effects in small collision systems.

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Summary Table: Key Four-Particle Subevent Cumulant Observables

Observable	Formulation (Subevent)	Physics Probed
$c_n\{4\}$	$$⟨ e^{in(\phi_1^a + \phi_2^a - \phi_3^b - \phi_4^b)} ⟩$$ etc.	Flow fluctuations, collectivity
$sc_{n,m}\{4\}$	$⟨ v_n^2 v_m^2 ⟩ - ⟨ v_n^2 ⟩⟨ v_m^2 ⟩$ (subevents)	Harmonic correlations
$nsc_{n,m}\{4\}$	$sc_{n,m}\{4\}/(v_n\{2\}^2 v_m\{2\}^2)$	Normalized harmonic correl.
Edgeworth moments	Expansion of measured cumulants	Source kurtosis/skewness

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Four-particle subevent cumulants constitute a rigorous and experimentally validated framework for isolating true multiparticle collectivity in high-energy nuclear collisions, enabling the systematic paper of flow fluctuations, initial-state geometry, and the emergence of collective behavior even in small systems. Their extension to higher-order, mixed-harmonic, and pT fluctuations continues to deepen the understanding of the QCD medium created in these collisions.