Higher-order strongly intensive observables for forward-backward multiplicity correlations

Technical Summary: Higher-order strongly intensive observables for forward-backward multiplicity correlations

Problem Statement
Forward-backward (FB) multiplicity correlations are a primary tool for probing the longitudinal structure of particle production in high-energy collisions. However, standard measures, such as the correlation coefficient $b_{corr}$, are complicated by fluctuations in the number of emitting sources (e.g., strings or wounded constituents). These source-number fluctuations ($\omega[S]$) obscure the intrinsic emission dynamics of individual sources. While "strongly intensive" observables (specifically $\Sigma$ and $\Delta$ at second order) were previously developed to cancel the contribution of source-number fluctuations under the assumption of independent identical sources, existing frameworks were limited to second-order variances and covariances. Higher-order information (three- and four-particle correlations) is crucial because different production laws can share the same second-order moments but possess distinct higher-order structures. The challenge addressed is how to construct a complete set of higher-order observables that retain all independent information while removing dependence on the source-count distribution and maintaining a common normalization.

Methodology
The paper constructs a normalized hierarchy of strongly intensive observables, denoted $J_{ij}$, for two disjoint windows (forward $F$ and backward $B$) containing particle counts $n_F$ and $n_B$.

1. Theoretical Framework: The analysis relies on the independent-source model where an event consists of $S$ independent, identical sources. The event cumulant generating function $P(x,y)$ is related to the single-source generating function $R(x,y)$ and the source-count generating function $Q(v)$ via the composition $P(x,y) = Q(R(x,y))$.
2. Construction of Invariants: The author defines a mean fraction $\alpha = \langle n_F \rangle / (\langle n_F \rangle + \langle n_B \rangle)$ and a total mean $m$. They introduce a transformation involving the diagonal restriction $D(v) = P(v,v)$ and a series $h(t,u)$ designed to isolate the source-count independent components.
3. Recursive Definition: A complete basis of $n$ algebraically independent observables is constructed at each order $n$. These observables are derived from normalized derivatives of the series $e^h$ and combinations of lower-order terms.
   • Order 2: Recovers the known observables $\Sigma$ and $\Delta$.
   • Order 3: Introduces $J_{30}, J_{21}, J_{12}$.
   • Order 4: Introduces $\Upsilon$ (a specific combination of fourth-order cumulants) and $J_{31}, J_{22}, J_{13}$.
4. Normalization Properties: Every observable in the hierarchy is normalized to:
   • Unity for statistically independent particles (Poisson emission).
   • Zero when both window multiplicities are fixed (no fluctuations).
   • Finite values even when the mean multiplicities in the two windows are equal ($\alpha = \beta$), a condition where traditional $\Delta$ and $\Gamma$ become singular.
5. Analytical and Numerical Validation:
   • Analytical Models: The hierarchy is evaluated for independent Poisson emission (yielding unity), independent negative-binomial emission (yielding deviations from unity due to intra-window correlations), and a string model with an exponential pair correlation function.
   • PYTHIA Simulations: Calculations were performed using PYTHIA 8.318 for $pp$ collisions at $\sqrt{s} = 0.9$ and $13$ TeV. Three samples were used: Monash tune, Trieste 2 tune (incorporating string close-packing), and Monash at 13 TeV.
   • Fitting Procedure: A conditional fitting approach was employed. Lower-order parameters (density profile and pair correlation amplitude) were fitted to mean multiplicities, $\Sigma$, and $\Delta$. These were then held fixed to predict higher-order observables under a "pair-only" closure (setting connected three- and four-particle densities to zero). Discrepancies were then tested by sequentially adding fitted three- and four-particle amplitudes.

Key Contributions

• Complete Basis Construction: The paper provides explicit expressions and an all-order recursion for a complete set of $n$ new algebraically independent strongly intensive observables at order $n$. This extends the strongly intensive concept beyond second-order moments.
• Regularization at Equal Means: The construction ensures that all observables remain well-defined and finite even when the forward and backward window means are equal, overcoming a limitation of previous asymmetry-based observables.
• Theorem of Completeness: The author proves that the constructed set $\{J_{ij}\}$, along with the mean fraction $\alpha$, forms a complete set of rational invariants for the invertible outer composition of the independent-source model.
• Explicit Higher-Order Formulas: The paper derives explicit formulas for the fourth-order observables, including the complex combination $\Upsilon$ and the canonical coordinates $J_{31}, J_{22}, J_{13}$.

Results

• Analytical Models:
  • For independent Poisson sources, all $J_{ij} = 1$.
  • For independent negative-binomial sources, deviations from unity arise solely from within-window correlations, demonstrating that non-Poisson emission alone generates non-trivial higher-order signals.
  • In the string model with exponential pair correlations, the hierarchy separates from conventional observables even without genuine three- or four-particle dynamics, as lower-order correlations contribute to higher-order cumulants.
• PYTHIA Calculations:
  • The observables show sensitivity to window geometry (widths and separation) and event selection (centrality classes).
  • The Trieste 2 tune (with close-packing) yields larger $\Sigma$ values than the Monash tune, consistent with string-fusion mechanisms, though the higher-order coordinates also respond to the tune change.
  • Energy dependence ($\sqrt{s} = 0.9$ vs. $13$ TeV) affects the magnitude of the observables but not their qualitative dependence on window separation.
• Conditional Fits:
  • A "pair-only" model (fitting density and pair correlations, setting higher-order connected densities to zero) fails to describe the PYTHIA higher-order data, particularly at 13 TeV.
  • Introducing fitted three-particle amplitudes ($a_3$) significantly reduces the discrepancy in third-order observables.
  • Adding four-particle amplitudes ($a_4$) further improves the description of fourth-order observables.
  • The study demonstrates that substantial parts of the higher-order discrepancy can be accounted for by specific connected three- and four-particle terms, though the underlying correlation functions are not uniquely determined by the data alone.
• ALICE Application: Preliminary ALICE measurements at 13 TeV were used to construct conditional scenarios. By transferring amplitude ratios ($r_3, r_4$) from the PYTHIA Monash 13 TeV fit, the author generated predictions for higher-order ALICE observables under different assumptions about the presence of three- and four-particle correlations.

Significance and Claims
The paper claims to provide a systematic framework for comparing information across successive cumulant orders in multiparticle production while rigorously removing the dependence on source-number fluctuations.

• Interpretation: The author emphasizes that a deviation of $J_{ij}$ from unity is not a direct measurement of a connected $(i+j)$-particle density. Instead, it reflects the combined effect of lower-order correlations and genuine higher-order dynamics. Identifying specific contributions requires a validated lower-order baseline.
• Limitations: The paper is modest about its quantitative claims. It notes that the interpretation is limited by:
  • Residuals in the lower-order fits (especially at 13 TeV).
  • The lack of propagated uncertainties in the sequential fitting procedure.
  • The assumption that the selected ensemble admits independent identical sources (violated by source mixtures or global conservation).
  • The reliance on preliminary, graphically extracted ALICE data and generator-based acceptance corrections.
• Future Utility: The hierarchy offers a benchmark for testing correlation models. If future measurements of third- and fourth-order coordinates in the same windows match the predictions of a hierarchy constrained only by second-order data, it would validate the assumed correlation structure. Conversely, deviations would test the validity of the assumed hierarchy or the presence of new dynamical mechanisms.

The work establishes that strongly intensive measures need not be restricted to variances and covariances, providing a robust algebraic tool for isolating intrinsic emission dynamics from volume fluctuations in high-energy physics.