Third-Order Strongly Intensive Observables

• Third-order strongly intensive observables are dimensionless statistical measures constructed from third-order cumulants that remain insensitive to volume fluctuations.
• They isolate genuine three-particle correlations and non-Gaussian features, providing robust probes for phenomena like the QCD critical point and multi-string dynamics.
• Their formulation using scaled moments and joint cumulants enables experimental strategies that differentiate critical behavior from trivial multiplicity variations.

A third-order strongly intensive observable is a statistically constructed function of third-order cumulants or moments that remains strictly insensitive to fluctuations in system size or the number of sources (volume fluctuations) in multiparticle production processes. These observables uniquely enable the isolation of genuine three-particle correlations and higher-order non-Gaussian features of event-by-event distributions, providing robust probes of nontrivial dynamics such as the QCD critical point, string fusion effects, or non-Poissonian source structure, independent of trivial variations in event multiplicity or geometric overlap.

1. Mathematical Definitions and Formulations

Strongly intensive observables of third order can be constructed for both inclusive single-particle quantities (such as transverse momentum) and multi-particle event properties (such as forward-backward multiplicities).

For transverse momentum, consider the $n$-th moment of $p_T$ in the ensemble:

$m_n \equiv \langle p_T^n \rangle$

The $n$-th scaled moment is

$$C_n = \frac{m_n}{m_1^n} = \frac{\langle p_T^n \rangle}{\langle p_T \rangle^n}\,,\qquad D_n = \frac{1}{C_n} = \frac{\langle p_T \rangle^n}{\langle p_T^n \rangle}$$

The third-order scaled moment is

$$C_3 = \frac{\langle p_T^3 \rangle}{\langle p_T \rangle^3}, \quad D_3 = \frac{\langle p_T \rangle^3}{\langle p_T^3 \rangle}$$

For joint event-by-event multiplicity observables, define for two windows (forward, $F$, and backward, $B$):

• Mean: $P_{10} = \langle N_F \rangle$, $P_{01} = \langle N_B \rangle$
• Covariances: $P_{20}$, $P_{11}$, $P_{02}$
• Third-order cumulants: $P_{30}$, $P_{21}$, $P_{12}$, $P_{03}$

A canonical third-order strongly intensive observable (denoted $\Delta_3$ or, for rapidity correlations, $\Omega_3$) is:

$$\Delta_3 = \frac{P_{30}}{P_{10}^3} - 3\,\frac{P_{21}}{P_{10}^2\,P_{01}} + 3\,\frac{P_{12}}{P_{10}\,P_{01}^2} - \frac{P_{03}}{P_{01}^3}$$

and

$$\Omega_3 = \frac{P_{10} P_{01}}{P_{01}^2 - P_{10}^2} \left[ \frac{P_{01} P_{30}}{P_{10}^2} - 3\,\frac{P_{21}}{P_{10}} + 3\,\frac{P_{12}}{P_{01}} - \frac{P_{10} P_{03}}{P_{01}^2} \right]$$

Each observable is dimensionless and reduces to a fixed “baseline” value in independent, Poissonian superposition models (0810.1989, Andronov, 12 Nov 2025, Broniowski et al., 2017).

2. Volume Independence and Strong Intensivity

The hallmark of a strongly intensive observable is exact cancellation of volume and volume-fluctuation contributions, thereby isolating dynamics intrinsic to particle production sources:

• For moments constructed from single-particle spectra (e.g., $D_3$), all relevant moments ($\langle p_T\rangle$, $\langle p_T^3\rangle$) are intensive under independent emission, and their combinations remain insensitive to the overall multiplicity or system size.
• For event-by-event cumulant constructs (e.g., $\Delta_3$ or $\Omega_3$), the generating-function formalism in superposition models shows that specific combinations of joint cumulants retain identical functional form when recomputed at the single-source level, with all source-number cumulant dependence canceled. Explicitly, for the scaled difference of multiplicities $$\hat N_- = \frac{N_a}{\langle N_a\rangle} - \frac{N_b}{\langle N_b\rangle}$$, one has

$Q_1^2 \kappa_3(\hat N_-) = \kappa_3(\hat n_-)$

for any event sample, so the third-order cumulant, normalized appropriately, is strongly intensive (Broniowski et al., 2017).

3. Dynamical Sensitivity and Theoretical Motivation

Third-order strongly intensive observables are especially sensitive to non-Gaussian structures—triplet correlations and skewness—arising from critical phenomena or the emergence of collectivity:

• Second-order quantities (such as $\Sigma$) probe only two-point correlations and cannot isolate effects from genuine higher-order (e.g., three-particle) clustering.
• Near a critical point in QCD, theory predicts that higher-order cumulants (especially third and fourth order) of conserved charges or event-by-event observables can exhibit faster scaling, sign changes, and nonmonotonic structures that are not visible in second-order statistics.
• In string or cluster models, the emergence of multi-string or multi-cluster correlations leads to nonzero third-order single-source cumulants ($R_{30}, R_{21}, ...$) that only third-order strongly intensive observables can access.
• For transverse-momentum observables, higher-order scaled moments such as $D_3$ or $C_3$ are acutely sensitive to the tail and curvature changes in the spectrum hypothesized to arise when the system crosses a phase boundary or percolation threshold (0810.1989, Andronov, 12 Nov 2025).

4. Practical Measurement and Experimental Strategies

The construction and measurement of third-order strongly intensive observables proceeds as follows.

• For $D_3$ in $p_T$:
  • Accumulate inclusive spectra for a given hadron species over broad event ensembles.
  • Compute $m_1 = \langle p_T \rangle$ and $m_3 = \langle p_T^3 \rangle$.
  • Form $D_3 = m_1^3 / m_3$ and plot as a function of collision energy or system parameter.
  • A non-smooth change, discontinuity, or kink in $D_3$ as a function of control parameter (energy, system size, centrality) is the expected signature of a critical point (0810.1989).
• For $\Omega_3$ or $\Delta_3$ in rapidity or charge correlations:
  • Specify two phase-space bins (e.g., rapidity windows), count all particles $N_F, N_B$ in each, accumulate the required moments and joint cumulants for all events.
  • Construct the observable by the explicit formula, without need for corrections for event multiplicity or centrality bin width; $\Omega_3$ remains flat under changes in system size in standard models (Andronov, 12 Nov 2025).
• For robust signal extraction, repeat measurements for various system sizes (different nuclei), rapidity intervals, and particle species. Crossing points or non-monotonic behavior across system size or energy are indicative of underlying nontrivial collective dynamics or criticality.

The table summarizes explicit forms:

Observable	Construction Domain	Formula (brief)
$D_3$	Inclusive $p_T$ spectra	$$D_3 = \frac{\langle p_T\rangle^3}{\langle p_T^3\rangle}$$
$\Omega_3$	Forward-backward bins	See above; third-order cumulants ($P_{ij}$) combined as in $\Omega_3$ formula
$\Delta_3$	Difference of multiplicity	$$\Delta_3 = P_{30}/P_{10}^3 - 3P_{21}/(P_{10}^2 P_{01}) + 3P_{12}/(P_{10}P_{01}^2) - P_{03}/P_{01}^3$$

5. Model Contexts and Baseline Behavior

In classical superposition (compound) models, each final-state particle is emitted independently from sources whose number may fluctuate event to event. In this context:

• All strongly intensive observables reduce to fixed baseline values (e.g., $\Omega_3 = 1$) for independent, Poisson-distributed sources.
• Any statistically significant deviation from these baselines signals nontrivial dynamics—correlated emission, clustering, or critical fluctuations.
• Color-string models with string fusion predict the emergence of nonzero higher-order single-source cumulants, which can be interrogated by measuring $\Omega_3$ at varying rapidity gap or window sizes (Andronov, 12 Nov 2025).

In simulated studies (e.g., with PYTHIA8/Angantyr), $\Omega_3$ remains unity over a broad domain and only deviates at large rapidity separation or system size, suggesting the onset of multi-string dynamics that modestly challenge strict intensivity.

6. Experimental and Theoretical Limitations

Key assumptions and caveats relevant to the application of third-order strongly intensive observables include:

• Superposition model validity: Sources must emit independently, and source count must be uncorrelated with single-source properties. Violations by global conservation laws or nonadditive effects compromise strict intensivity.
• Neglect of higher cumulants: The standard constructions cancel source-number cumulants up to third order; extension to fourth or higher requires analogous, more complex formulas.
• Detector and acceptance effects: Experimental cumulants must be corrected for efficiency and acceptance in a manner compatible with superposition composition to preserve strong intensivity.
• Statistical uncertainties: Higher-order fluctuations demand large event samples; statistical errors and detector non-idealities become more substantial with increasing cumulant order.
• Nonzero denominators and phase-space bin occupation: Mean particle numbers in phase-space bins must be sufficiently large to stably compute required moments.

Notwithstanding these, third-order strongly intensive observables furnish a systematic avenue for distinguishing dynamical three-body effects or critical behavior from trivial geometric and statistical backgrounds.

7. Significance and Ongoing Developments

The deployment of third-order strongly intensive observables enhances the toolkit for probing non-Poissonian statistics and collective phenomena in high-energy nuclear collisions. Their precise construction ensures that any observed nontrivial dynamics arise from underlying physics—such as the QCD critical point or multi-string interactions—rather than from mundane multiplicity or centrality variations.

Contemporary research continues to generalize the formalism to higher orders, different quantum numbers (net-charge, net-baryon), and incorporate global conservation laws and acceptance effects. These efforts aim to establish a fully volume-independent, multi-variate set of fluctuation measures, enabling robust, model-agnostic signatures of phase structure and collectivity in strongly interacting matter (0810.1989, Andronov, 12 Nov 2025, Broniowski et al., 2017).