Introduction
Rapidity correlations constrain the longitudinal structure of particle production beyond the information contained in the inclusive multiplicity density. Forward-backward analyses measure multiplicities of charged particles in two separated rapidity or pseudorapidity windows. Varying their separation probes the correlation range, whereas varying their widths probes the accumulation of correlations within a finite acceptance. An additional azimuthal separation resolves structure that a full-azimuth analysis integrates over [1 , 2 ] . Resonance decays, jets, and fragmentation contribute to short-range correlations. Correlations over larger rapidity intervals are sensitive to longitudinally extended sources, initial-state fluctuations, and fluctuations in the number of particle-producing interactions. These mechanisms have been investigated in string models, wounded-constituent models, early-stage color-field descriptions, and event generators [3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 ] .
A conventional measure is the forward-backward (FB) multiplicity correlation coefficient. For multiplicities n F n_F n F n B n_B n B
b c o r r = cov ( n F , n B ) Var ( n F ) = ⟨ n F n B ⟩ − ⟨ n F ⟩ ⟨ n B ⟩ ⟨ n F 2 ⟩ − ⟨ n F ⟩ 2 . (1) b_{\mathrm{corr}} = \frac{\operatorname{cov}(n_F,n_B)}{\operatorname{Var}(n_F)} = \frac{\langle n_F n_B\rangle-\langle n_F\rangle\langle n_B\rangle}{\langle n_F^2\rangle-\langle n_F\rangle^2}. \tag*{(1)} b corr = Var ( n F ) cov ( n F , n B ) = ⟨ n F 2 ⟩ − ⟨ n F ⟩ 2 ⟨ n F n B ⟩ − ⟨ n F ⟩ ⟨ n B ⟩ . (1) Here and below brackets denote an average over the selected events. This quantity is the slope of the linear regression of n B n_B n B n F n_F n F p ˉ p \bar{p}p p ˉ p [11 ] , by STAR in p p pp pp s N N = 200 \sqrt{s_{\mathrm{NN}}}=200 s NN = 200 [12 ] , by ALICE in p p pp pp s = 0.9 \sqrt{s}=0.9 s = 0.9 [1 ] , and by ATLAS in p p pp pp [13 ] .
An independent-source model provides a useful baseline for interpreting these measurements. Consider S S S μ F \mu_F μ F μ B \mu_B μ B S S S ω [ X ] = Var ( X ) / ⟨ X ⟩ \omega[X]=\operatorname{Var}(X)/\langle X\rangle ω [ X ] = Var ( X ) / ⟨ X ⟩ [2 ]
b c o r r = cov ( μ F , μ B ) + ⟨ μ F ⟩ ⟨ μ B ⟩ ω [ S ] Var ( μ F ) + ⟨ μ F ⟩ 2 ω [ S ] . (2) b_{\mathrm{corr}}=\frac{\operatorname{cov}(\mu_F,\mu_B)+\langle\mu_F\rangle\langle\mu_B\rangle\omega[S]}{\operatorname{Var}(\mu_F)+\langle\mu_F\rangle^2\omega[S]}. \tag*{(2)} b corr = Var ( μ F ) + ⟨ μ F ⟩ 2 ω [ S ] cov ( μ F , μ B ) + ⟨ μ F ⟩ ⟨ μ B ⟩ ω [ S ] . (2) In Ref. [2 ] this expression was related to the string pair correlation function. In particular, for two equal narrow pseudorapidity windows of width δ η \delta\eta δ η μ 0 \mu_0 μ 0
b c o r r ≃ μ 0 δ η [ ω [ S ] + Λ ( Δ η ) ] 1 + μ 0 δ η [ ω [ S ] + Λ ( 0 ) ] . (3) b_{\mathrm{corr}}\simeq\frac{\mu_0\delta\eta\left[\omega[S]+\Lambda(\Delta\eta)\right]}{1+\mu_0\delta\eta\left[\omega[S]+\Lambda(0)\right]}. \tag*{(3)} b corr ≃ 1 + μ 0 δ η [ ω [ S ] + Λ ( 0 ) ] μ 0 δ η [ ω [ S ] + Λ ( Δ η ) ] . (3) Here Δ η \Delta\eta Δ η Λ \Lambda Λ b c o r r b_{\mathrm{corr}} b corr ω [ S ] \omega[S] ω [ S ] b c o r r b_{\mathrm{corr}} b corr
The need to separate intrinsic fluctuations from variations of the system size has a long history. The Φ \Phi Φ [14 ] was designed to preserve the singlesource fluctuation signal under an independent superposition of sources. In Ref. [15 ] this idea was developed further and the general class of strongly intensive quantities was introduced. These quantities are independent of both the mean volume and volume fluctuations in the specified statistical model, or of the mean number and number fluctuations of independent identical sources. In Ref. [16 ] it was explained how to introduce a common normalization with a unit independent-particle reference and a zero no-fluctuation reference.
The application of these measures to multiplicities in separated rapidity windows was proposed in Ref. [17 ] . For disjoint windows, the normalized second-order quantities are
Σ [ n F , n B ] = ⟨ n B ⟩ ω [ n F ] + ⟨ n F ⟩ ω [ n B ] − 2 cov ( n F , n B ) ⟨ n F ⟩ + ⟨ n B ⟩ , (4) \Sigma[n_F,n_B] = \frac{\langle n_B\rangle\omega[n_F] + \langle n_F\rangle\omega[n_B] - 2\operatorname{cov}(n_F,n_B)}{\langle n_F\rangle+ \langle n_B\rangle}, \tag*{(4)} Σ [ n F , n B ] = ⟨ n F ⟩ + ⟨ n B ⟩ ⟨ n B ⟩ ω [ n F ] + ⟨ n F ⟩ ω [ n B ] − 2 cov ( n F , n B ) , (4) Δ [ n F , n B ] = ⟨ n B ⟩ ω [ n F ] − ⟨ n F ⟩ ω [ n B ] ⟨ n B ⟩ − ⟨ n F ⟩ . (5) \Delta[n_F,n_B] = \frac{\langle n_B\rangle\omega[n_F] - \langle n_F\rangle\omega[n_B]}{\langle n_B\rangle- \langle n_F\rangle}. \tag*{(5)} Δ [ n F , n B ] = ⟨ n B ⟩ − ⟨ n F ⟩ ⟨ n B ⟩ ω [ n F ] − ⟨ n F ⟩ ω [ n B ] . (5) Both cancel the source-number contribution present in Eq. (2) , under the same source assumptions:
Σ [ n F , n B ] = Σ [ μ F , μ B ] , (6) \Sigma[n_F,n_B] = \Sigma[\mu_F,\mu_B], \tag*{(6)} Σ [ n F , n B ] = Σ [ μ F , μ B ] , (6) Δ [ n F , n B ] = Δ [ μ F , μ B ] . (7) \Delta[n_F,n_B] = \Delta[\mu_F,\mu_B]. \tag*{(7)} Δ [ n F , n B ] = Δ [ μ F , μ B ] . (7) The normalization of Δ \Delta Δ Σ \Sigma Σ
This approach has been adopted in experimental FB analyses. NA61/SHINE studied Σ [ n F , n B ] \Sigma[n_F,n_B] Σ [ n F , n B ] p p pp pp [18 , 19 ] , including its dependence on the separation of the observation windows. ALICE results for p p pp pp [20 , 21 , 22 ] showed that Σ \Sigma Σ b c o r r b_{\mathrm{corr}} b corr Σ \Sigma Σ
Preparatory model and feasibility studies have also explored these observables for future measurements at lower collision energies. Calculations for MPD at NICA and for FAIR examined strongly intensive observables within the accessible rapidity and momentum ranges, including detector response and reconstruction [23 , 24 ] .
The source-level interpretation can also be applied to strongly intensive observables. In Ref. [25 ] Σ \Sigma Σ p p pp pp [26 ] . Further string-cluster and interacting string calculations examined the energy and event-selection dependencies [27 , 28 , 29 ] .
With several source types, the relative contributions can change between event classes. Cancellation that is present for a fixed identical-source law does not remove this variation. A complementary example is provided by wounded-nucleon and wounded-quark calculations [30 , 31 ] . These studies relate Σ \Sigma Σ Σ \Sigma Σ
Higher-order observables are of particular interest because pair correlations do not determine the full joint multiplicity distribution. Production laws with the same Σ \Sigma Σ Δ \Delta Δ [32 ] . It was suggested that solving them is a way to construct higher-order strongly intensive measures. The third-order quantity Γ \Gamma Γ [32 ] was investigated for FB multiplicities and tested with the PYTHIA event generator [33 ] in Ref. [34 ] . A general construction of strongly intensive cumulants at arbitrary order was also presented in Ref. [35 ] . These results establish that the method need not be restricted to variances and covariances.
The question addressed here is how to retain all independent higher-order information while removing the dependence on the source-count distribution and keeping a common normalization. We construct a complete basis J i j J_{ij} J ij n n n n n n
Section Section II defines the observables and compares them with cumulant ratios. Section § 1.2 develops analytical particle production models, and Section Section IV presents the PYTHIA calculations and conditional ALICE scenarios. Section § 2.3 discusses the scope and limitations of the results.
Appendix Appendix A presents the all-order construction and completeness proof. Appendix Appendix B collects the moment and cumulant conversion formulas. Appendix Appendix C provides the evaluation prescription and symmetric-window limit for the pair-only string model. Appendix Appendix D documents the PYTHIA tune settings and auxiliary acceptance-correction procedure. Finally, Appendix Appendix § 2.1.4 gives the explicit window cumulants and evaluation formulas for the fitted Gaussian correlation hierarchy.
Definitions and the independent-source assumption
Let n F n_F n F n B n_B n B [32 ] throughout: P i j P_{ij} P ij R i j R_{ij} R ij Q k Q_k Q k S S S P 10 = ⟨ n F ⟩ P_{10}=\langle n_F\rangle P 10 = ⟨ n F ⟩ P 20 = Var ( n F ) P_{20}=\operatorname{Var}(n_F) P 20 = Var ( n F ) P 11 = cov ( n F , n B ) P_{11}=\operatorname{cov}(n_F,n_B) P 11 = cov ( n F , n B ) n F n_F n F n B n_B n B
In the independent-source model all sources share the same emission law into both windows, and this law does not depend on their number in the event. The cumulant generating functions then obey
P ( x , y ) = Q ( R ( x , y ) ) , P ( x , y ) = ∑ i + j ≥ 1 P i j x i y j i ! j ! . (8) P(x,y)=Q(R(x,y)),\qquad P(x,y)=\sum_{i+j\geq1}P_{ij}\frac{x^iy^j}{i!j!}. \tag*{(8)} P ( x , y ) = Q ( R ( x , y )) , P ( x , y ) = i + j ≥ 1 ∑ P ij i ! j ! x i y j . (8) The functions P P P R R R Q Q Q S S S Table I summarizes the notation used below. Differentiation gives the first- and second-order relations
Table I. List of notations.
P 10 = Q 1 R 10 , P 01 = Q 1 R 01 , P 20 = Q 1 R 20 + Q 2 R 10 2 , P 11 = Q 1 R 11 + Q 2 R 10 R 01 , P 02 = Q 1 R 02 + Q 2 R 01 2 . (9) \begin{aligned}
P_{10}&=Q_1R_{10},\\
P_{01}&=Q_1R_{01},\\
P_{20}&=Q_1R_{20}+Q_2R_{10}^2,\\
P_{11}&=Q_1R_{11}+Q_2R_{10}R_{01},\\
P_{02}&=Q_1R_{02}+Q_2R_{01}^2. \tag*{(9)}
\end{aligned} P 10 P 01 P 20 P 11 P 02 = Q 1 R 10 , = Q 1 R 01 , = Q 1 R 20 + Q 2 R 10 2 , = Q 1 R 11 + Q 2 R 10 R 01 , = Q 1 R 02 + Q 2 R 01 2 . (9) At third order,
P 30 = Q 1 R 30 + 3 Q 2 R 20 R 10 + Q 3 R 10 3 , P 21 = Q 1 R 21 + Q 2 ( R 20 R 01 + 2 R 11 R 10 ) + Q 3 R 10 2 R 01 , P 12 = Q 1 R 12 + Q 2 ( R 02 R 10 + 2 R 11 R 01 ) + Q 3 R 10 R 01 2 , P 03 = Q 1 R 03 + 3 Q 2 R 02 R 01 + Q 3 R 01 3 . (10) \begin{aligned}
P_{30}&=Q_1R_{30}+3Q_2R_{20}R_{10}+Q_3R_{10}^3,\\
P_{21}&=Q_1R_{21}+Q_2(R_{20}R_{01}+2R_{11}R_{10})+Q_3R_{10}^2R_{01},\\
P_{12}&=Q_1R_{12}+Q_2(R_{02}R_{10}+2R_{11}R_{01})+Q_3R_{10}R_{01}^2,\\
P_{03}&=Q_1R_{03}+3Q_2R_{02}R_{01}+Q_3R_{01}^3. \tag*{(10)}
\end{aligned} P 30 P 21 P 12 P 03 = Q 1 R 30 + 3 Q 2 R 20 R 10 + Q 3 R 10 3 , = Q 1 R 21 + Q 2 ( R 20 R 01 + 2 R 11 R 10 ) + Q 3 R 10 2 R 01 , = Q 1 R 12 + Q 2 ( R 02 R 10 + 2 R 11 R 01 ) + Q 3 R 10 R 01 2 , = Q 1 R 03 + 3 Q 2 R 02 R 01 + Q 3 R 01 3 . (10) At fourth order,
P 40 = Q 1 R 40 + Q 2 ( 4 R 30 R 10 + 3 R 20 2 ) + 6 Q 3 R 20 R 10 2 + Q 4 R 10 4 , P 31 = Q 1 R 31 + Q 2 ( R 30 R 01 + 3 R 21 R 10 + 3 R 20 R 11 ) + 3 Q 3 ( R 20 R 10 R 01 + R 11 R 10 2 ) + Q 4 R 10 3 R 01 , P 22 = Q 1 R 22 + Q 2 ( 2 R 21 R 01 + 2 R 12 R 10 ) + Q 2 ( R 20 R 02 + 2 R 11 2 ) + Q 3 ( R 20 R 01 2 + R 02 R 10 2 ) + 4 Q 3 R 11 R 10 R 01 + Q 4 R 10 2 R 01 2 , P 13 = Q 1 R 13 + Q 2 ( R 03 R 10 + 3 R 12 R 01 + 3 R 02 R 11 ) + 3 Q 3 ( R 02 R 10 R 01 + R 11 R 01 2 ) + Q 4 R 10 R 01 3 , P 04 = Q 1 R 04 + Q 2 ( 4 R 03 R 01 + 3 R 02 2 ) + 6 Q 3 R 02 R 01 2 + Q 4 R 01 4 . (11) \begin{aligned}
P_{40}&=Q_1R_{40}+Q_2(4R_{30}R_{10}+3R_{20}^2)+6Q_3R_{20}R_{10}^2+Q_4R_{10}^4,\\
P_{31}&=Q_1R_{31}+Q_2(R_{30}R_{01}+3R_{21}R_{10}+3R_{20}R_{11})\\
&\quad+3Q_3(R_{20}R_{10}R_{01}+R_{11}R_{10}^2)+Q_4R_{10}^3R_{01},\\
P_{22}&=Q_1R_{22}+Q_2(2R_{21}R_{01}+2R_{12}R_{10})\\
&\quad+Q_2(R_{20}R_{02}+2R_{11}^2)+Q_3(R_{20}R_{01}^2+R_{02}R_{10}^2)\\
&\quad+4Q_3R_{11}R_{10}R_{01}+Q_4R_{10}^2R_{01}^2,\\
P_{13}&=Q_1R_{13}+Q_2(R_{03}R_{10}+3R_{12}R_{01}+3R_{02}R_{11})\\
&\quad+3Q_3(R_{02}R_{10}R_{01}+R_{11}R_{01}^2)+Q_4R_{10}R_{01}^3,\\
P_{04}&=Q_1R_{04}+Q_2(4R_{03}R_{01}+3R_{02}^2)+6Q_3R_{02}R_{01}^2+Q_4R_{01}^4. \tag*{(11)}
\end{aligned} P 40 P 31 P 22 P 13 P 04 = Q 1 R 40 + Q 2 ( 4 R 30 R 10 + 3 R 20 2 ) + 6 Q 3 R 20 R 10 2 + Q 4 R 10 4 , = Q 1 R 31 + Q 2 ( R 30 R 01 + 3 R 21 R 10 + 3 R 20 R 11 ) + 3 Q 3 ( R 20 R 10 R 01 + R 11 R 10 2 ) + Q 4 R 10 3 R 01 , = Q 1 R 22 + Q 2 ( 2 R 21 R 01 + 2 R 12 R 10 ) + Q 2 ( R 20 R 02 + 2 R 11 2 ) + Q 3 ( R 20 R 01 2 + R 02 R 10 2 ) + 4 Q 3 R 11 R 10 R 01 + Q 4 R 10 2 R 01 2 , = Q 1 R 13 + Q 2 ( R 03 R 10 + 3 R 12 R 01 + 3 R 02 R 11 ) + 3 Q 3 ( R 02 R 10 R 01 + R 11 R 01 2 ) + Q 4 R 10 R 01 3 , = Q 1 R 04 + Q 2 ( 4 R 03 R 01 + 3 R 02 2 ) + 6 Q 3 R 02 R 01 2 + Q 4 R 01 4 . (11) We seek combinations independent of Q k Q_k Q k
m = P 10 + P 01 , α = P 10 / m , β = 1 − α . (12) m=P_{10}+P_{01},\qquad\alpha=P_{10}/m,\qquad\beta=1-\alpha. \tag*{(12)} m = P 10 + P 01 , α = P 10 / m , β = 1 − α . (12) Thus α \alpha α β \beta β α + β = 1 \alpha+\beta=1 α + β = 1 m > 0 m>0 m > 0 0 < α , β < 1 0<\alpha,\beta<1 0 < α , β < 1
The known strongly intensive observables
In this notation, the conventional second- and third-order strongly intensive observables are
Σ = β 2 P 20 + α 2 P 02 − 2 α β P 11 m α β , (13) \Sigma=\frac{\beta^2P_{20}+\alpha^2P_{02}-2\alpha\beta P_{11}}{m\alpha\beta}, \tag*{(13)} Σ = m α β β 2 P 20 + α 2 P 02 − 2 α β P 11 , (13) Δ = β 2 P 20 − α 2 P 02 m α β ( β − α ) , (14) \Delta=\frac{\beta^2P_{20}-\alpha^2P_{02}}{m\alpha\beta(\beta-\alpha)}, \tag*{(14)} Δ = m α β ( β − α ) β 2 P 20 − α 2 P 02 , (14) Γ = β 3 P 30 − 3 α β 2 P 21 + 3 α 2 β P 12 − α 3 P 03 m α β ( β − α ) . (15) \Gamma=\frac{\beta^3P_{30}-3\alpha\beta^2P_{21}+3\alpha^2\beta P_{12}-\alpha^3P_{03}}{m\alpha\beta(\beta-\alpha)}. \tag*{(15)} Γ = m α β ( β − α ) β 3 P 30 − 3 α β 2 P 21 + 3 α 2 β P 12 − α 3 P 03 . (15) These definitions agree with Refs. [16 , 17 , 34 ] . The independent-particle reference used here assigns each particle in the union of the windows independently to the forward or backward window, with probabilities α \alpha α β \beta β
The Δ \Delta Δ Γ \Gamma Γ β ≠ α \beta\ne\alpha β = α F ↔ B F\leftrightarrow B F ↔ B Σ \Sigma Σ
Σ = P 20 − P 11 P 10 . (16) \Sigma=\frac{P_{20}-P_{11}}{P_{10}}. \tag*{(16)} Σ = P 10 P 20 − P 11 . (16) The fourth-order observable
A direct fourth-order extension of the zero-mean contrast X = β μ F − α μ B X=\beta\mu_F-\alpha\mu_B X = β μ F − α μ B Γ \Gamma Γ 3 Q 2 κ 2 ( β μ F − α μ B ) 2 3Q_2\kappa_2(\beta\mu_F-\alpha\mu_B)^2 3 Q 2 κ 2 ( β μ F − α μ B ) 2
T = 1 3 m { m 2 Σ 2 ( P 20 P 10 2 + P 11 P 10 P 01 + P 02 P 01 2 ) − P 40 P 01 2 P 10 2 − P 04 P 10 2 P 01 2 + 4 P 31 P 01 P 10 + 4 P 13 P 10 P 01 − 6 P 22 } . (17) \mathcal{T}=\frac{1}{3m}\left\{m^2\Sigma^2\left(\frac{P_{20}}{P_{10}^2}+\frac{P_{11}}{P_{10}P_{01}}+\frac{P_{02}}{P_{01}^2}\right)-\frac{P_{40}P_{01}^2}{P_{10}^2}-\frac{P_{04}P_{10}^2}{P_{01}^2}+4\frac{P_{31}P_{01}}{P_{10}}+4\frac{P_{13}P_{10}}{P_{01}}-6P_{22}\right\}. \tag*{(17)} T = 3 m 1 { m 2 Σ 2 ( P 10 2 P 20 + P 10 P 01 P 11 + P 01 2 P 02 ) − P 10 2 P 40 P 01 2 − P 01 2 P 04 P 10 2 + 4 P 10 P 31 P 01 + 4 P 01 P 13 P 10 − 6 P 22 } . (17) T \mathcal{T} T F ↔ B F \leftrightarrow B F ↔ B P i j = P j i P_{ij}=P_{ji} P ij = P j i Υ \Upsilon Υ
Υ = ( P 20 − P 11 ) 2 ( 4 P 20 + 2 P 11 ) 3 P 10 3 − P 40 − 4 P 31 + 3 P 22 3 P 10 . (18) \Upsilon= \frac{(P_{20}-P_{11})^2(4P_{20}+2P_{11})}{3P_{10}^3} - \frac{P_{40}-4P_{31}+3P_{22}}{3P_{10}}. \tag*{(18)} Υ = 3 P 10 3 ( P 20 − P 11 ) 2 ( 4 P 20 + 2 P 11 ) − 3 P 10 P 40 − 4 P 31 + 3 P 22 . (18) The complete basis
The composition equations admit additional invariants beyond Σ \Sigma Σ Δ \Delta Δ Γ \Gamma Γ Υ \Upsilon Υ [32 , 35 ] . Here we require the new observables to have the reference values of Sec. II A—unity for independent particles and zero for fixed multiplicities—and no singularities at equal positive means.
The construction uses two functions: the diagonal restriction D ( v ) = P ( v , v ) D(v)=P(v,v) D ( v ) = P ( v , v ) n F + n B n_F+n_B n F + n B
h ( t , u ) = D − 1 ( P ( β t + u , − α t + u ) ) , (19) h(t,u)=D^{-1}\left(P(\beta t+u,-\alpha t+u)\right), \tag*{(19)} h ( t , u ) = D − 1 ( P ( β t + u , − α t + u ) ) , (19) where the weights are fixed to the ensemble fractions α = P 10 / m \alpha=P_{10}/m α = P 10 / m β = P 01 / m \beta=P_{01}/m β = P 01 / m m u mu m u multiplicity asymmetry
X = β n F − α n B , (20) X=\beta n_F-\alpha n_B, \tag*{(20)} X = β n F − α n B , (20) whose first cumulant vanishes by construction:
⟨ X ⟩ = β α m − α β m = 0 , (21) \langle X\rangle=\beta\alpha m-\alpha\beta m=0, \tag*{(21)} ⟨ X ⟩ = β α m − α β m = 0 , (21) Since D ′ ( 0 ) = m ≠ 0 D'(0)=m\ne0 D ′ ( 0 ) = m = 0 P = Q ∘ R P=Q\circ R P = Q ∘ R Q Q Q D D D (19) and cancels there. Normalized derivatives of e h e^h e h Σ \Sigma Σ J i j J_{ij} J ij (72) . Appendix A gives the coefficient recursion and its inverse.
Through fourth order the complete list is
J 20 = Σ , (22) J_{20}=\Sigma, \tag*{(22)} J 20 = Σ , (22) J 11 = Σ + β ( P 20 + P 11 ) − α ( P 11 + P 02 ) m α β , (23) J_{11}=\Sigma+\frac{\beta(P_{20}+P_{11})-\alpha(P_{11}+P_{02})}{m\alpha\beta}, \tag*{(23)} J 11 = Σ + m α β β ( P 20 + P 11 ) − α ( P 11 + P 02 ) , (23) J 30 = β 3 P 30 − 3 α β 2 P 21 + 3 α 2 β P 12 − α 3 P 03 m α β + 2 α Σ , (24) J_{30}=\frac{\beta^3P_{30}-3\alpha\beta^2P_{21}+3\alpha^2\beta P_{12}-\alpha^3P_{03}}{m\alpha\beta}+2\alpha\Sigma, \tag*{(24)} J 30 = m α β β 3 P 30 − 3 α β 2 P 21 + 3 α 2 β P 12 − α 3 P 03 + 2 α Σ , (24) J 21 = β 2 P 30 + ( β 2 − 2 α β ) P 21 + ( α 2 − 2 α β ) P 12 + α 2 P 03 m α β + ( 1 − P 20 + 2 P 11 + P 02 m ) Σ , (25) J_{21}=\frac{\beta^2P_{30}+(\beta^2-2\alpha\beta)P_{21}+(\alpha^2-2\alpha\beta)P_{12}+\alpha^2P_{03}}{m\alpha\beta}+\left(1-\frac{P_{20}+2P_{11}+P_{02}}{m}\right)\Sigma, \tag*{(25)} J 21 = m α β β 2 P 30 + ( β 2 − 2 α β ) P 21 + ( α 2 − 2 α β ) P 12 + α 2 P 03 + ( 1 − m P 20 + 2 P 11 + P 02 ) Σ , (25) J 12 = Σ + β P 30 + ( 2 β − α ) P 21 + ( β − 2 α ) P 12 − α P 03 m α β + 2 ( 1 − P 20 + 2 P 11 + P 02 m ) ( J 11 − Σ ) . (26) J_{12}=\Sigma+\frac{\beta P_{30}+(2\beta-\alpha)P_{21}+(\beta-2\alpha)P_{12}-\alpha P_{03}}{m\alpha\beta}+2\left(1-\frac{P_{20}+2P_{11}+P_{02}}{m}\right)(J_{11}-\Sigma). \tag*{(26)} J 12 = Σ + m α β β P 30 + ( 2 β − α ) P 21 + ( β − 2 α ) P 12 − α P 03 + 2 ( 1 − m P 20 + 2 P 11 + P 02 ) ( J 11 − Σ ) . (26) Every member is dimensionless, equals one for independent particles, and vanishes when both multiplicities are fixed. All remain defined at equal positive means. When α ≠ β \alpha\ne\beta α = β
J 11 = Σ + β − α 2 α β ( Δ − Σ ) , (27) J_{11}=\Sigma+\frac{\beta-\alpha}{2\alpha\beta}(\Delta-\Sigma), \tag*{(27)} J 11 = Σ + 2 α β β − α ( Δ − Σ ) , (27) J 30 = ( β − α ) Γ + 2 α Σ . (28) J_{30}=(\beta-\alpha)\Gamma+2\alpha\Sigma. \tag*{(28)} J 30 = ( β − α ) Γ + 2 α Σ. (28) We use Υ \Upsilon Υ J 40 J_{40} J 40
J 40 = β 4 P 40 − 4 α β 3 P 31 + 6 α 2 β 2 P 22 − 4 α 3 β P 13 + α 4 P 04 m α β + 3 α β ( 1 − P 20 + 2 P 11 + P 02 m ) Σ 2 + 3 α β Σ . (29) J_{40}=\frac{\beta^{4}P_{40}-4\alpha\beta^{3}P_{31}+6\alpha^{2}\beta^{2}P_{22}-4\alpha^{3}\beta P_{13}+\alpha^{4}P_{04}}{m\alpha\beta}+3\alpha\beta\left(1-\frac{P_{20}+2P_{11}+P_{02}}{m}\right)\Sigma^{2}+3\alpha\beta\Sigma. \tag*{(29)} J 40 = m α β β 4 P 40 − 4 α β 3 P 31 + 6 α 2 β 2 P 22 − 4 α 3 β P 13 + α 4 P 04 + 3 α β ( 1 − m P 20 + 2 P 11 + P 02 ) Σ 2 + 3 α β Σ. (29) The replacement preserves completeness because
J 40 = − 3 α β Υ + 3 α β Σ 2 + 3 α β Σ + 3 α β ( β − α ) ( J 11 − Σ ) Σ 2 + ( 1 − 3 α β ) Σ 3 . (30) J_{40}=-3\alpha\beta\Upsilon+3\alpha\beta\Sigma^{2}+3\alpha\beta\Sigma+3\alpha\beta(\beta-\alpha)(J_{11}-\Sigma)\Sigma^{2}+(1-3\alpha\beta)\Sigma^{3}. \tag*{(30)} J 40 = − 3 α β Υ + 3 α β Σ 2 + 3 α β Σ + 3 α β ( β − α ) ( J 11 − Σ ) Σ 2 + ( 1 − 3 α β ) Σ 3 . (30) The coefficient of Υ \Upsilon Υ α β ≠ 0 \alpha\beta\ne0 α β = 0
J 31 = β 3 P 40 + ( β 3 − 3 α β 2 ) P 31 + ( 3 α 2 β − 3 α β 2 ) P 22 + ( 3 α 2 β − α 3 ) P 13 − α 3 P 04 m α β + ( 1 − P 20 + 2 P 11 + P 02 m ) [ J 30 − 2 α Σ + 3 α β Σ ( J 11 − Σ ) ] + 2 α Σ . (31) J_{31}=\frac{\beta^{3}P_{40}+(\beta^{3}-3\alpha\beta^{2})P_{31}+(3\alpha^{2}\beta-3\alpha\beta^{2})P_{22}+(3\alpha^{2}\beta-\alpha^{3})P_{13}-\alpha^{3}P_{04}}{m\alpha\beta}+\left(1-\frac{P_{20}+2P_{11}+P_{02}}{m}\right)\left[J_{30}-2\alpha\Sigma+3\alpha\beta\Sigma(J_{11}-\Sigma)\right]+2\alpha\Sigma. \tag*{(31)} J 31 = m α β β 3 P 40 + ( β 3 − 3 α β 2 ) P 31 + ( 3 α 2 β − 3 α β 2 ) P 22 + ( 3 α 2 β − α 3 ) P 13 − α 3 P 04 + ( 1 − m P 20 + 2 P 11 + P 02 ) [ J 30 − 2 α Σ + 3 α β Σ ( J 11 − Σ ) ] + 2 α Σ. (31) J 22 = β 2 P 40 + ( 2 β 2 − 2 α β ) P 31 + ( β 2 + α 2 − 4 α β ) P 22 + ( 2 α 2 − 2 α β ) P 13 + α 2 P 04 m α β + 2 ( 1 − P 20 + 2 P 11 + P 02 m ) [ J 21 + α β ( J 11 − Σ ) 2 ] + [ − 1 + 2 P 20 + 2 P 11 + P 02 m − P 30 + 3 P 21 + 3 P 12 + P 03 m ] Σ . (32) J_{22}=\frac{\beta^{2}P_{40}+(2\beta^{2}-2\alpha\beta)P_{31}+(\beta^{2}+\alpha^{2}-4\alpha\beta)P_{22}+(2\alpha^{2}-2\alpha\beta)P_{13}+\alpha^{2}P_{04}}{m\alpha\beta}+2\left(1-\frac{P_{20}+2P_{11}+P_{02}}{m}\right)\left[J_{21}+\alpha\beta(J_{11}-\Sigma)^{2}\right]+\left[-1+2\frac{P_{20}+2P_{11}+P_{02}}{m}-\frac{P_{30}+3P_{21}+3P_{12}+P_{03}}{m}\right]\Sigma. \tag*{(32)} J 22 = m α β β 2 P 40 + ( 2 β 2 − 2 α β ) P 31 + ( β 2 + α 2 − 4 α β ) P 22 + ( 2 α 2 − 2 α β ) P 13 + α 2 P 04 + 2 ( 1 − m P 20 + 2 P 11 + P 02 ) [ J 21 + α β ( J 11 − Σ ) 2 ] + [ − 1 + 2 m P 20 + 2 P 11 + P 02 − m P 30 + 3 P 21 + 3 P 12 + P 03 ] Σ. (32) J 13 = Σ + β P 40 + ( 3 β − α ) P 31 + 3 ( β − α ) P 22 + ( β − 3 α ) P 13 − α P 04 m α β + 3 ( 1 − P 20 + 2 P 11 + P 02 m ) ( J 12 − Σ ) + 3 [ 2 P 20 + 2 P 11 + P 02 m − P 30 + 3 P 21 + 3 P 12 + P 03 m − 1 ] ( J 11 − Σ ) . (33) J_{13}=\Sigma+\frac{\beta P_{40}+(3\beta-\alpha)P_{31}+3(\beta-\alpha)P_{22}+(\beta-3\alpha)P_{13}-\alpha P_{04}}{m\alpha\beta}+3\left(1-\frac{P_{20}+2P_{11}+P_{02}}{m}\right)(J_{12}-\Sigma)+3\left[2\frac{P_{20}+2P_{11}+P_{02}}{m}-\frac{P_{30}+3P_{21}+3P_{12}+P_{03}}{m}-1\right](J_{11}-\Sigma). \tag*{(33)} J 13 = Σ + m α β β P 40 + ( 3 β − α ) P 31 + 3 ( β − α ) P 22 + ( β − 3 α ) P 13 − α P 04 + 3 ( 1 − m P 20 + 2 P 11 + P 02 ) ( J 12 − Σ ) + 3 [ 2 m P 20 + 2 P 11 + P 02 − m P 30 + 3 P 21 + 3 P 12 + P 03 − 1 ] ( J 11 − Σ ) . (33) Appendix A gives a constructive proof of the following theorem:
Theorem 1. Through order n n n α \alpha α J i j J_{ij} J ij i ≥ 1 i\ge1 i ≥ 1 j ≥ 0 j\ge0 j ≥ 0 2 ≤ i + j ≤ n 2\le i+j\le n 2 ≤ i + j ≤ n (8) . For n ≥ 2 n\ge2 n ≥ 2 n n n n n n n ( n + 1 ) / 2 − 1 n(n+1)/2-1 n ( n + 1 ) /2 − 1
All J i j J_{ij} J ij J 20 = Σ J_{20}=\Sigma J 20 = Σ J 11 J_{11} J 11 Σ \Sigma Σ Δ \Delta Δ (27) . From third order on, the observables involve cumulants of new orders. The theorem concerns formal cumulant series and holds on the domain m α β ≠ 0 m\alpha\beta\ne0 m α β = 0
Operationally, changing the source-count distribution at fixed source emission law leaves α \alpha α J i j J_{ij} J ij α \alpha α J i j J_{ij} J ij
Comparison with asymmetry cumulants
Higher-order fluctuation observables are particularly useful in the search for the quantum chromodynamics critical point: their critical contributions can have a stronger dependence on the correlation length than the variance [36 , 37 ] . Experiments therefore measure cumulants of net electric charge and net-proton number, together with factorial cumulants (integrated connected particle correlations) of the underlying multiplicities [38 , 39 ] . Net charge is constructed as N + − N − N_{+}-N_{-} N + − N − N + N_{+} N + N − N_{-} N − N p − N p ˉ N_{p}-N_{\bar{p}} N p − N p ˉ
The same construction provides a mathematical analogy with the forward-backward multiplicities n F n_{F} n F n B n_{B} n B X = β n F − α n B X=\beta n_{F}-\alpha n_{B} X = β n F − α n B ( n F − n B ) / ( n F + n B ) (n_{F}-n_{B})/(n_{F}+n_{B}) ( n F − n B ) / ( n F + n B ) Q Q Q X X X Q 2 Q_{2} Q 2 m m m X X X
We now take cumulants of X X X α \alpha α β \beta β
κ 1 ( X ) = β P 10 − α P 01 = 0 , κ 2 ( X ) = β 2 P 20 − 2 α β P 11 + α 2 P 02 , κ 3 ( X ) = β 3 P 30 − 3 α β 2 P 21 + 3 α 2 β P 12 − α 3 P 03 , κ 4 ( X ) = β 4 P 40 − 4 α β 3 P 31 + 6 α 2 β 2 P 22 − 4 α 3 β P 13 + α 4 P 04 . (34) \begin{aligned}
\kappa_{1}(X)&=\beta P_{10}-\alpha P_{01}=0,\\
\kappa_{2}(X)&=\beta^{2}P_{20}-2\alpha\beta P_{11}+\alpha^{2}P_{02},\\
\kappa_{3}(X)&=\beta^{3}P_{30}-3\alpha\beta^{2}P_{21}+3\alpha^{2}\beta P_{12}-\alpha^{3}P_{03},\\
\kappa_{4}(X)&=\beta^{4}P_{40}-4\alpha\beta^{3}P_{31}+6\alpha^{2}\beta^{2}P_{22}\\
&\quad-4\alpha^{3}\beta P_{13}+\alpha^{4}P_{04}. \tag*{(34)}
\end{aligned} κ 1 ( X ) κ 2 ( X ) κ 3 ( X ) κ 4 ( X ) = β P 10 − α P 01 = 0 , = β 2 P 20 − 2 α β P 11 + α 2 P 02 , = β 3 P 30 − 3 α β 2 P 21 + 3 α 2 β P 12 − α 3 P 03 , = β 4 P 40 − 4 α β 3 P 31 + 6 α 2 β 2 P 22 − 4 α 3 β P 13 + α 4 P 04 . (34) In particular κ 2 ( X ) / m = α β Σ \kappa_{2}(X)/m=\alpha\beta\Sigma κ 2 ( X ) / m = α β Σ κ 3 ( X ) / m = α β ( J 30 − 2 α Σ ) \kappa_{3}(X)/m=\alpha\beta(J_{30}-2\alpha\Sigma) κ 3 ( X ) / m = α β ( J 30 − 2 α Σ )
For the source contrast X s = β μ F − α μ B X_{s}=\beta\mu_{F}-\alpha\mu_{B} X s = β μ F − α μ B ⟨ X s ⟩ = 0 \langle X_{s}\rangle=0 ⟨ X s ⟩ = 0 m = Q 1 m s m=Q_{1}m_{s} m = Q 1 m s m s m_{s} m s
κ 1 ( X ) = 0 , κ 2 ( X ) = Q 1 κ 2 ( X s ) , κ 3 ( X ) = Q 1 κ 3 ( X s ) , κ 4 ( X ) = Q 1 κ 4 ( X s ) + 3 Q 2 κ 2 ( X s ) 2 . (35) \begin{aligned}
\kappa_{1}(X)&=0,\\
\kappa_{2}(X)&=Q_{1}\kappa_{2}(X_{s}),\\
\kappa_{3}(X)&=Q_{1}\kappa_{3}(X_{s}),\\
\kappa_{4}(X)&=Q_{1}\kappa_{4}(X_{s})+3Q_{2}\kappa_{2}(X_{s})^{2}. \tag*{(35)}
\end{aligned} κ 1 ( X ) κ 2 ( X ) κ 3 ( X ) κ 4 ( X ) = 0 , = Q 1 κ 2 ( X s ) , = Q 1 κ 3 ( X s ) , = Q 1 κ 4 ( X s ) + 3 Q 2 κ 2 ( X s ) 2 . (35) Division by m m m Q 1 Q_{1} Q 1
For a signed, noninteger weighted quantity such as X X X log { ( 1 + z ) X } \log\{(1+z)^{X}\} log {( 1 + z ) X } z z z f n ( X ) f_{n}(X) f n ( X )
f 1 ( X ) = κ 1 ( X ) = 0 , f 2 ( X ) = κ 2 ( X ) − κ 1 ( X ) = κ 2 ( X ) , f 3 ( X ) = κ 3 ( X ) − 3 κ 2 ( X ) + 2 κ 1 ( X ) = κ 3 ( X ) − 3 κ 2 ( X ) , f 4 ( X ) = κ 4 ( X ) − 6 κ 3 ( X ) + 11 κ 2 ( X ) − 6 κ 1 ( X ) = κ 4 ( X ) − 6 κ 3 ( X ) + 11 κ 2 ( X ) . (36) \begin{aligned}
f_{1}(X)&=\kappa_{1}(X)=0,\\
f_{2}(X)&=\kappa_{2}(X)-\kappa_{1}(X)=\kappa_{2}(X),\\
f_{3}(X)&=\kappa_{3}(X)-3\kappa_{2}(X)+2\kappa_{1}(X)\\
&=\kappa_{3}(X)-3\kappa_{2}(X),\\
f_{4}(X)&=\kappa_{4}(X)-6\kappa_{3}(X)+11\kappa_{2}(X)-6\kappa_{1}(X)\\
&=\kappa_{4}(X)-6\kappa_{3}(X)+11\kappa_{2}(X). \tag*{(36)}
\end{aligned} f 1 ( X ) f 2 ( X ) f 3 ( X ) f 4 ( X ) = κ 1 ( X ) = 0 , = κ 2 ( X ) − κ 1 ( X ) = κ 2 ( X ) , = κ 3 ( X ) − 3 κ 2 ( X ) + 2 κ 1 ( X ) = κ 3 ( X ) − 3 κ 2 ( X ) , = κ 4 ( X ) − 6 κ 3 ( X ) + 11 κ 2 ( X ) − 6 κ 1 ( X ) = κ 4 ( X ) − 6 κ 3 ( X ) + 11 κ 2 ( X ) . (36) These quantities are mathematically well defined but do not count connected correlations of distinct particles carrying the weights β \beta β − α -\alpha − α log { ( 1 + β z ) n F ( 1 − α z ) n B } \log\{(1+\beta z)^{n_{F}}(1-\alpha z)^{n_{B}}\} log {( 1 + β z ) n F ( 1 − α z ) n B } [40 ] . We denote them by F n F_{n} F n ( n F , n B ) (n_{F},n_{B}) ( n F , n B ) f i j f_{ij} f ij
F 1 = β f 10 − α f 01 = 0 , F 2 = β 2 f 20 − 2 α β f 11 + α 2 f 02 , F 3 = β 3 f 30 − 3 α β 2 f 21 + 3 α 2 β f 12 − α 3 f 03 , F 4 = β 4 f 40 − 4 α β 3 f 31 + 6 α 2 β 2 f 22 − 4 α 3 β f 13 + α 4 f 04 . (37) \begin{aligned}
F_{1}&=\beta f_{10}-\alpha f_{01}=0,\\
F_{2}&=\beta^{2}f_{20}-2\alpha\beta f_{11}+\alpha^{2}f_{02},\\
F_{3}&=\beta^{3}f_{30}-3\alpha\beta^{2}f_{21}+3\alpha^{2}\beta f_{12}-\alpha^{3}f_{03},\\
F_{4}&=\beta^{4}f_{40}-4\alpha\beta^{3}f_{31}+6\alpha^{2}\beta^{2}f_{22}\\
&\quad-4\alpha^{3}\beta f_{13}+\alpha^{4}f_{04}. \tag*{(37)}
\end{aligned} F 1 F 2 F 3 F 4 = β f 10 − α f 01 = 0 , = β 2 f 20 − 2 α β f 11 + α 2 f 02 , = β 3 f 30 − 3 α β 2 f 21 + 3 α 2 β f 12 − α 3 f 03 , = β 4 f 40 − 4 α β 3 f 31 + 6 α 2 β 2 f 22 − 4 α 3 β f 13 + α 4 f 04 . (37) The same structure holds because F 1 ( X s ) = 0 F_{1}(X_{s})=0 F 1 ( X s ) = 0 P i j P_{ij} P ij f i j f_{ij} f ij F 2 = κ 2 ( X ) − m α β = m α β ( Σ − 1 ) F_{2}=\kappa_{2}(X)-m\alpha\beta=m\alpha\beta(\Sigma-1) F 2 = κ 2 ( X ) − m α β = m α β ( Σ − 1 ) n F , n B n_{F},n_{B} n F , n B F n = 0 F_{n}=0 F n = 0 n ≥ 2 n\geq2 n ≥ 2 f n ( X ) f_{n}(X) f n ( X )
For an unweighted multiplicity or a net multiplicity, commonly used ordinary ratios (when the denominators are nonzero) include κ 2 / κ 1 \kappa_{2}/\kappa_{1} κ 2 / κ 1 κ 3 / κ 2 \kappa_{3}/\kappa_{2} κ 3 / κ 2 κ 4 / κ 2 \kappa_{4}/\kappa_{2} κ 4 / κ 2 [39 ] . For the proton count N p N_{p} N p f 2 ( N p ) / f 1 ( N p ) f_{2}(N_{p})/f_{1}(N_{p}) f 2 ( N p ) / f 1 ( N p ) f 3 ( N p ) / f 1 ( N p ) f_{3}(N_{p})/f_{1}(N_{p}) f 3 ( N p ) / f 1 ( N p ) [41 ] . f 4 ( N p ) / f 1 ( N p ) f_{4}(N_{p})/f_{1}(N_{p}) f 4 ( N p ) / f 1 ( N p ) f 1 ( N p ) = ⟨ N p ⟩ > 0 f_{1}(N_{p})=\langle N_{p}\rangle>0 f 1 ( N p ) = ⟨ N p ⟩ > 0 X X X κ 1 \kappa_1 κ 1 f 1 f_1 f 1 F 1 F_1 F 1 κ 3 / κ 2 \kappa_3/\kappa_2 κ 3 / κ 2 F 3 / F 2 F_3/F_2 F 3 / F 2
κ 4 ( X ) κ 2 ( X ) = κ 4 ( X s ) κ 2 ( X s ) + 3 Q 2 Q 1 κ 2 ( X s ) , (38) \frac{\kappa_4(X)}{\kappa_2(X)}=\frac{\kappa_4(X_s)}{\kappa_2(X_s)}+3\frac{Q_2}{Q_1}\kappa_2(X_s), \tag*{(38)} κ 2 ( X ) κ 4 ( X ) = κ 2 ( X s ) κ 4 ( X s ) + 3 Q 1 Q 2 κ 2 ( X s ) , (38) F 4 F 2 = F 4 , s o u r c e F 2 , s o u r c e + 3 Q 2 Q 1 F 2 , s o u r c e . (39) \frac{F_4}{F_2}=\frac{F_{4,\mathrm{source}}}{F_{2,\mathrm{source}}}+3\frac{Q_2}{Q_1}F_{2,\mathrm{source}}. \tag*{(39)} F 2 F 4 = F 2 , source F 4 , source + 3 Q 1 Q 2 F 2 , source . (39) Table II summarizes the distinction between the normalized invariants and common ratios in the zero-mean direction.
Table II. Dependence on source-count fluctuations for X = β n F − α n B X=\beta n_{\mathrm{F}}-\alpha n_{\mathrm{B}} X = β n F − α n B F n F_n F n
Several procedures aim to reduce this dependence on Q Q Q [42 ] : cumulants are evaluated in narrow bins of a reference multiplicity and then averaged with event-count weights. Fixing the reference multiplicity, however, does not in general fix the number of sources. Residual size fluctuations may remain within each bin. Moreover, conditioning on a reference multiplicity correlated with the measured particles can modify the dynamical fluctuations themselves. CBWC is not a general algebraic cancellation of Q Q Q [43 ] .
The same algebra can be applied to two species counts, such as proton and antiproton multiplicities, provided the independent identical-source assumptions hold. It does not by itself identify the resulting invariants with net-baryon or net-charge susceptibilities. Global conservation and acceptance effects require separate treatment: reduced factorial cumulants provide complementary acceptance tests [44 ] , and the relation between coordinate-space and momentum-space fluctuations depends on the dynamics [45 ] . When the antiproton mean vanishes, the present normalization is undefined. A small nonzero mean can also make estimation unstable because the normalizations divide by powers of the window means. A pair of more abundant species can instead be used to study higher-order chemical fluctuations [46 , 47 , 48 , 49 ] where the same set of J i j J_{ij} J ij
Analytical models
We now evaluate the hierarchy in analytically tractable emission models. For sources with a common single-source law R R R R R R Q Q Q J i j J_{ij} J ij Σ \Sigma Σ Δ \Delta Δ Γ \Gamma Γ Υ \Upsilon Υ
Consider a random number S S S a = R 10 = ⟨ μ F ⟩ > 0 a=R_{10}=\langle\mu_F\rangle>0 a = R 10 = ⟨ μ F ⟩ > 0 b = R 01 = ⟨ μ B ⟩ > 0 b=R_{01}=\langle\mu_B\rangle>0 b = R 01 = ⟨ μ B ⟩ > 0 a a a b b b α = a / ( a + b ) \alpha=a/(a+b) α = a / ( a + b ) β = b / ( a + b ) \beta=b/(a+b) β = b / ( a + b ) R 10 = Q 1 a R_{10}=Q_1a R 10 = Q 1 a m = Q 1 ( a + b ) m=Q_1(a+b) m = Q 1 ( a + b ) S S S
Independent Poisson emission into two windows
First suppose that the two multiplicities of a single source are independent Poisson variables. Their cumulant generating function is
R ( x , y ) = a ( e x − 1 ) + b ( e y − 1 ) . (40) R(x,y)=a(e^x-1)+b(e^y-1). \tag*{(40)} R ( x , y ) = a ( e x − 1 ) + b ( e y − 1 ) . (40) Thus R i 0 = a R_{i0}=a R i 0 = a R 0 j = b R_{0j}=b R 0 j = b i , j ≥ 1 i,j\ge1 i , j ≥ 1 P 11 = a b Var ( S ) P_{11}=ab\,\operatorname{Var}(S) P 11 = ab Var ( S ) J i j J_{ij} J ij e u ( a e β t + β e − α t ) e^u(ae^{\beta t}+\beta e^{-\alpha t}) e u ( a e β t + β e − α t )
J i j = 1 ( i ≥ 1 , i + j ≥ 2 ) , J_{ij}=1 \qquad(i\ge1,\ i+j\ge2), J ij = 1 ( i ≥ 1 , i + j ≥ 2 ) , Σ = Δ = Γ = Υ = 1. (41) \Sigma=\Delta=\Gamma=\Upsilon=1. \tag*{(41)} Σ = Δ = Γ = Υ = 1. (41) The statement for J i j J_{ij} J ij Δ \Delta Δ Γ \Gamma Γ α ≠ β \alpha\ne\beta α = β
Independent negative-binomial emission
The Poisson model returns unity at every order, so any departure of the observables from one must originate in the emission law itself. The negative binomial is one of the simplest such laws and a standard parametrization of multiplicity fluctuations. Keep the two multiplicities of a source independent, but replace each Poisson variable by an independent negative-binomial variable with means a , b a,b a , b k A , k B k_A,k_B k A , k B
R ( x , y ) = − k A log [ 1 − a k A ( e x − 1 ) ] − k B log [ 1 − b k B ( e y − 1 ) ] . (42) R(x,y)=-k_A\log\left[1-\frac{a}{k_A}(e^x-1)\right]
-k_B\log\left[1-\frac{b}{k_B}(e^y-1)\right]. \tag*{(42)} R ( x , y ) = − k A log [ 1 − k A a ( e x − 1 ) ] − k B log [ 1 − k B b ( e y − 1 ) ] . (42) There is no shared fluctuating intensity between the two windows at the single-source level in this model. In particular, all mixed R i j R_{ij} R ij i , j > 0 i,j>0 i , j > 0
To give compact explicit answers, we write z A = a / k A z_A=a/k_A z A = a / k A z B = b / k B z_B=b/k_B z B = b / k B
p 2 ( z ) = 1 + z , p 3 ( z ) = 1 + 3 z + 2 z 2 , p 4 ( z ) = 1 + 7 z + 12 z 2 + 6 z 3 . (43) \begin{aligned}
p_2(z)&=1+z,\\
p_3(z)&=1+3z+2z^2,\\
p_4(z)&=1+7z+12z^2+6z^3. \tag*{(43)}
\end{aligned} p 2 ( z ) p 3 ( z ) p 4 ( z ) = 1 + z , = 1 + 3 z + 2 z 2 , = 1 + 7 z + 12 z 2 + 6 z 3 . (43) We abbreviate p n A = p n ( z A ) p_{nA}=p_n(z_A) p n A = p n ( z A ) p n B = p n ( z B ) p_{nB}=p_n(z_B) p n B = p n ( z B ) w n = α p n A + β p n B w_n=\alpha p_{nA}+\beta p_{nB} w n = α p n A + β p n B n = 2 , 3 n=2,3 n = 2 , 3 R n 0 = a p n A R_{n0}=ap_{nA} R n 0 = a p n A
R 0 n = b p n B R_{0n}=b p_n^{\mathrm{B}} R 0 n = b p n B w n w_n w n n n n a a a b b b k A k_{\mathrm{A}} k A k B k_{\mathrm{B}} k B
The conventional quantities and Υ \Upsilon Υ
Σ = β p 2 A + α p 2 B = 1 + β z A + α z B , \Sigma=\beta p_{2\mathrm{A}}+\alpha p_{2\mathrm{B}}=1+\beta z_{\mathrm{A}}+\alpha z_{\mathrm{B}}, Σ = β p 2 A + α p 2 B = 1 + β z A + α z B , Δ = β p 2 A − α p 2 B β − α , \Delta=\frac{\beta p_{2\mathrm{A}}-\alpha p_{2\mathrm{B}}}{\beta-\alpha}, Δ = β − α β p 2 A − α p 2 B , Γ = β 2 p 3 A − α 2 p 3 B β − α , \Gamma=\frac{\beta^{2}p_{3\mathrm{A}}-\alpha^{2}p_{3\mathrm{B}}}{\beta-\alpha}, Γ = β − α β 2 p 3 A − α 2 p 3 B , Υ = Σ 3 − β 3 p 4 A − α 3 p 4 B 3 α β . (44) \Upsilon=\frac{\Sigma^{3}-\beta^{3}p_{4\mathrm{A}}-\alpha^{3}p_{4\mathrm{B}}}{3\alpha\beta}. \tag*{(44)} Υ = 3 α β Σ 3 − β 3 p 4 A − α 3 p 4 B . (44) All remaining coordinates through fourth order are given explicitly by
J 11 = Σ + p 2 A − p 2 B , J 30 = β 2 p 3 A − α 2 p 3 B + 2 α Σ , J 21 = β p 3 A + α p 3 B + ( 1 − w 2 ) Σ , J 12 = Σ + p 3 A − p 3 B + 2 ( 1 − w 2 ) ( p 2 A − p 2 B ) , J 40 = β 3 p 4 A + α 3 p 4 B + 3 α β ( 1 − w 2 ) Σ 2 + 3 α β Σ , J 31 = β 2 p 4 A − α 2 p 4 B + 2 α Σ + ( 1 − w 2 ) [ β 2 p 3 A − α 2 p 3 B + 3 α β Σ ( p 2 A − p 2 B ) ] , J 22 = β p 4 A + α p 4 B + 2 ( 1 − w 2 ) [ J 21 + α β ( p 2 A − p 2 B ) 2 ] + ( − 1 + 2 w 2 − w 3 ) Σ , J 13 = Σ + p 4 A − p 4 B + 3 ( 1 − w 2 ) ( J 12 − Σ ) + 3 ( 2 w 2 − w 3 − 1 ) ( p 2 A − p 2 B ) . (45) \begin{aligned}
J_{11}&=\Sigma+p_{2\mathrm{A}}-p_{2\mathrm{B}},\\
J_{30}&=\beta^{2}p_{3\mathrm{A}}-\alpha^{2}p_{3\mathrm{B}}+2\alpha\Sigma,\\
J_{21}&=\beta p_{3\mathrm{A}}+\alpha p_{3\mathrm{B}}+(1-w_{2})\Sigma,\\
J_{12}&=\Sigma+p_{3\mathrm{A}}-p_{3\mathrm{B}}+2(1-w_{2})(p_{2\mathrm{A}}-p_{2\mathrm{B}}),\\
J_{40}&=\beta^{3}p_{4\mathrm{A}}+\alpha^{3}p_{4\mathrm{B}}+3\alpha\beta(1-w_{2})\Sigma^{2}+3\alpha\beta\Sigma,\\
J_{31}&=\beta^{2}p_{4\mathrm{A}}-\alpha^{2}p_{4\mathrm{B}}+2\alpha\Sigma\\
&\quad+(1-w_{2})[\beta^{2}p_{3\mathrm{A}}-\alpha^{2}p_{3\mathrm{B}}+3\alpha\beta\Sigma(p_{2\mathrm{A}}-p_{2\mathrm{B}})],\\
J_{22}&=\beta p_{4\mathrm{A}}+\alpha p_{4\mathrm{B}}+2(1-w_{2})[J_{21}+\alpha\beta(p_{2\mathrm{A}}-p_{2\mathrm{B}})^{2}]\\
&\quad+(-1+2w_{2}-w_{3})\Sigma,\\
J_{13}&=\Sigma+p_{4\mathrm{A}}-p_{4\mathrm{B}}+3(1-w_{2})(J_{12}-\Sigma)\\
&\quad+3(2w_{2}-w_{3}-1)(p_{2\mathrm{A}}-p_{2\mathrm{B}}). \tag*{(45)}
\end{aligned} J 11 J 30 J 21 J 12 J 40 J 31 J 22 J 13 = Σ + p 2 A − p 2 B , = β 2 p 3 A − α 2 p 3 B + 2 α Σ , = β p 3 A + α p 3 B + ( 1 − w 2 ) Σ , = Σ + p 3 A − p 3 B + 2 ( 1 − w 2 ) ( p 2 A − p 2 B ) , = β 3 p 4 A + α 3 p 4 B + 3 α β ( 1 − w 2 ) Σ 2 + 3 α β Σ , = β 2 p 4 A − α 2 p 4 B + 2 α Σ + ( 1 − w 2 ) [ β 2 p 3 A − α 2 p 3 B + 3 α β Σ ( p 2 A − p 2 B )] , = β p 4 A + α p 4 B + 2 ( 1 − w 2 ) [ J 21 + α β ( p 2 A − p 2 B ) 2 ] + ( − 1 + 2 w 2 − w 3 ) Σ , = Σ + p 4 A − p 4 B + 3 ( 1 − w 2 ) ( J 12 − Σ ) + 3 ( 2 w 2 − w 3 − 1 ) ( p 2 A − p 2 B ) . (45) J 40 J_{40} J 40 Υ \Upsilon Υ (30) and are alternative coordinates. These formulas show that zero inter-window correlation within a source does not imply unit strongly intensive observables: correlations among particles in the same window contribute.
A useful width-dependent specialization assumes a homogeneous source with a = ρ δ η A a=\rho\,\delta\eta_{\mathrm{A}} a = ρ δ η A b = ρ δ η B b=\rho\,\delta\eta_{\mathrm{B}} b = ρ δ η B k A = κ δ η A k_{\mathrm{A}}=\kappa\,\delta\eta_{\mathrm{A}} k A = κ δ η A k B = κ δ η B k_{\mathrm{B}}=\kappa\,\delta\eta_{\mathrm{B}} k B = κ δ η B z A = z B = z = ρ / κ z_{\mathrm{A}}=z_{\mathrm{B}}=z=\rho/\kappa z A = z B = z = ρ / κ α = δ η A / ( δ η A + δ η B ) \alpha=\delta\eta_{\mathrm{A}}/(\delta\eta_{\mathrm{A}}+\delta\eta_{\mathrm{B}}) α = δ η A / ( δ η A + δ η B )
Σ = Δ = J 11 = J 12 = J 13 = 1 + z . (46) \Sigma=\Delta=J_{11}=J_{12}=J_{13}=1+z. \tag*{(46)} Σ = Δ = J 11 = J 12 = J 13 = 1 + z . (46) Independent strings with the two-particle exponential pair function
The two preceding models emit into the windows independently and therefore cannot depend on the window separation at fixed widths. The minimal extension that produces such a dependence keeps the source translationally invariant but gives its correlations a finite range.
Γ = 1 + 3 z + 2 z 2 , \Gamma=1+3z+2z^{2}, Γ = 1 + 3 z + 2 z 2 , J 21 = ( 1 + z ) 2 , J_{21}=(1+z)^{2}, J 21 = ( 1 + z ) 2 , J 30 = ( β − α ) ( 1 + 3 z + 2 z 2 ) + 2 α ( 1 + z ) , J_{30}=(\beta-\alpha)(1+3z+2z^{2})+2\alpha(1+z), J 30 = ( β − α ) ( 1 + 3 z + 2 z 2 ) + 2 α ( 1 + z ) , J 31 = ( β − α ) ( 1 + 6 z + 9 z 2 + 4 z 3 ) + 2 α ( 1 + z ) , J_{31}=(\beta-\alpha)(1+6z+9z^{2}+4z^{3})+2\alpha(1+z), J 31 = ( β − α ) ( 1 + 6 z + 9 z 2 + 4 z 3 ) + 2 α ( 1 + z ) , J 22 = ( 1 + z ) 2 ( 1 + 2 z ) , J_{22}=(1+z)^{2}(1+2z), J 22 = ( 1 + z ) 2 ( 1 + 2 z ) , Υ = ( 1 + z ) 3 3 α β − 1 − 3 α β 3 α β ( 1 + 7 z + 12 z 2 + 6 z 3 ) . (47) \Upsilon=\frac{(1+z)^{3}}{3\alpha\beta}-\frac{1-3\alpha\beta}{3\alpha\beta}(1+7z+12z^{2}+6z^{3}). \tag*{(47)} Υ = 3 α β ( 1 + z ) 3 − 3 α β 1 − 3 α β ( 1 + 7 z + 12 z 2 + 6 z 3 ) . (47) Within the same specialization, the equal-mean limit (α = β = 1 2 \alpha=\beta=\frac{1}{2} α = β = 2 1 Δ \Delta Δ Γ \Gamma Γ
Σ = J 11 = J 12 = J 13 = J 30 = J 31 = 1 + z . (48) \Sigma=J_{11}=J_{12}=J_{13}=J_{30}=J_{31}=1+z. \tag*{(48)} Σ = J 11 = J 12 = J 13 = J 30 = J 31 = 1 + z . (48) J 21 = Σ 2 , J_{21}=\Sigma^{2}, J 21 = Σ 2 , J 22 = Σ 2 ( 2 Σ − 1 ) , J_{22}=\Sigma^{2}(2\Sigma-1), J 22 = Σ 2 ( 2Σ − 1 ) , Υ = − 2 Σ 3 + 6 Σ 2 − Σ 3 . (49) \Upsilon=\frac{-2\Sigma^{3}+6\Sigma^{2}-\Sigma}{3}. \tag*{(49)} Υ = 3 − 2 Σ 3 + 6 Σ 2 − Σ . (49) Neither the Poisson nor the negative-binomial model depends on the window separation at fixed widths and fixed source parameters. Such a dependence requires inter-window correlations or a position-dependent source law, as in the string model below. The contrast between the two models is the point of this subsection: Poisson emission gives unity at all orders, whereas merely non-Poisson independent emission already moves every observable away from unity through within-window correlations.
Following the single-string formulation of Ref. [25 ] , let the one-particle density be constant, λ 1 ( η ) = μ 0 \lambda_1(\eta)=\mu_0 λ 1 ( η ) = μ 0 Λ ( η 1 − η 2 ) = Λ 0 exp ( − ∣ η 1 − η 2 ∣ / ℓ ) \Lambda(\eta_1-\eta_2)=\Lambda_0\exp(-|\eta_1-\eta_2|/\ell) Λ ( η 1 − η 2 ) = Λ 0 exp ( − ∣ η 1 − η 2 ∣/ ℓ )
Λ ( η 1 − η 2 ) = λ 2 ( η 1 , η 2 ) λ 1 ( η 1 ) λ 1 ( η 2 ) − 1. (50) \Lambda(\eta_1-\eta_2)=\frac{\lambda_2(\eta_1,\eta_2)}{\lambda_1(\eta_1)\lambda_1(\eta_2)}-1. \tag*{(50)} Λ ( η 1 − η 2 ) = λ 1 ( η 1 ) λ 1 ( η 2 ) λ 2 ( η 1 , η 2 ) − 1. (50) For a constant one-particle density,
Λ ( η 1 − η 2 ) = λ 2 ( η 1 , η 2 ) μ 0 2 − 1. (51) \Lambda(\eta_1-\eta_2)=\frac{\lambda_2(\eta_1,\eta_2)}{\mu_0^2}-1. \tag*{(51)} Λ ( η 1 − η 2 ) = μ 0 2 λ 2 ( η 1 , η 2 ) − 1. (51) Higher-order cumulants additionally require the corresponding multiparticle densities. The standard moment–cumulant expansion sums over partitions of the particles into disjoint groups [50 ] . Explicitly,
λ 3 μ 0 3 = 1 + Λ 12 + Λ 13 + Λ 23 + C 3 ( 1 , 2 , 3 ) μ 0 3 , \frac{\lambda_3}{\mu_0^3}=1+\Lambda_{12}+\Lambda_{13}+\Lambda_{23}+\frac{C_3(1,2,3)}{\mu_0^3}, μ 0 3 λ 3 = 1 + Λ 12 + Λ 13 + Λ 23 + μ 0 3 C 3 ( 1 , 2 , 3 ) , λ 4 μ 0 4 = 1 + ∑ i < j Λ i j + Λ 12 Λ 34 + Λ 13 Λ 24 + Λ 14 Λ 23 + ∑ i < j < k C 3 ( i , j , k ) μ 0 3 + C 4 ( 1 , 2 , 3 , 4 ) μ 0 4 . (52) \frac{\lambda_4}{\mu_0^4}=1+\sum_{i<j}\Lambda_{ij}+\Lambda_{12}\Lambda_{34}+\Lambda_{13}\Lambda_{24}+\Lambda_{14}\Lambda_{23}+\frac{\sum_{i<j<k}C_3(i,j,k)}{\mu_0^3}+\frac{C_4(1,2,3,4)}{\mu_0^4}. \tag*{(52)} μ 0 4 λ 4 = 1 + i < j ∑ Λ ij + Λ 12 Λ 34 + Λ 13 Λ 24 + Λ 14 Λ 23 + μ 0 3 ∑ i < j < k C 3 ( i , j , k ) + μ 0 4 C 4 ( 1 , 2 , 3 , 4 ) . (52) Here λ n \lambda_n λ n n n n Λ i j = Λ ( η i − η j ) \Lambda_{ij}=\Lambda(\eta_i-\eta_j) Λ ij = Λ ( η i − η j ) C 3 C_3 C 3 C 4 C_4 C 4
The source cumulants follow from integrals of the inclusive densities. For disjoint windows F F F B B B
L i j = ∫ F i × B j λ i + j ( η 1 , … , η i + j ) d η 1 ⋯ d η i + j . (53) L_{ij}=\int_{F^i\times B^j}\lambda_{i+j}(\eta_1,\ldots,\eta_{i+j})\,\mathrm{d}\eta_1\cdots\mathrm{d}\eta_{i+j}. \tag*{(53)} L ij = ∫ F i × B j λ i + j ( η 1 , … , η i + j ) d η 1 ⋯ d η i + j . (53) The conversion from the integrated densities L i j L_{ij} L ij
As a pair-only closure through fourth order, set C 3 = C 4 = 0 C_3=C_4=0 C 3 = C 4 = 0 C 2 = μ 0 2 Λ C_2=\mu_0^2\Lambda C 2 = μ 0 2 Λ
λ 3 μ 0 3 = 1 + Λ 12 + Λ 13 + Λ 23 , \frac{\lambda_3}{\mu_0^3}=1+\Lambda_{12}+\Lambda_{13}+\Lambda_{23}, μ 0 3 λ 3 = 1 + Λ 12 + Λ 13 + Λ 23 , λ 4 μ 0 4 = 1 + ∑ i < j Λ i j + Λ 12 Λ 34 + Λ 13 Λ 24 + Λ 14 Λ 23 . (54) \frac{\lambda_4}{\mu_0^4}=1+\sum_{i<j}\Lambda_{ij}+\Lambda_{12}\Lambda_{34}+\Lambda_{13}\Lambda_{24}+\Lambda_{14}\Lambda_{23}. \tag*{(54)} μ 0 4 λ 4 = 1 + i < j ∑ Λ ij + Λ 12 Λ 34 + Λ 13 Λ 24 + Λ 14 Λ 23 . (54) Vanishing connected factorial densities above second order do not imply vanishing higher-order J i j J_{ij} J ij
U F = 2 μ 0 2 Λ 0 ℓ [ w F − ℓ ( 1 − e − w F / ℓ ) ] , U_F=2\mu_0^2\Lambda_0\ell\left[w_F-\ell\left(1-e^{-w_F/\ell}\right)\right], U F = 2 μ 0 2 Λ 0 ℓ [ w F − ℓ ( 1 − e − w F / ℓ ) ] , U B = 2 μ 0 2 Λ 0 ℓ [ w B − ℓ ( 1 − e − w B / ℓ ) ] , U_B=2\mu_0^2\Lambda_0\ell\left[w_B-\ell\left(1-e^{-w_B/\ell}\right)\right], U B = 2 μ 0 2 Λ 0 ℓ [ w B − ℓ ( 1 − e − w B / ℓ ) ] , U F B = μ 0 2 Λ 0 ℓ 2 e − η g a p / ℓ ( 1 − e − w F / ℓ ) ( 1 − e − w B / ℓ ) . (55) U_{FB}=\mu_0^2\Lambda_0\ell^2e^{-\eta_{\mathrm{gap}}/\ell}\left(1-e^{-w_F/\ell}\right)\left(1-e^{-w_B/\ell}\right). \tag*{(55)} U F B = μ 0 2 Λ 0 ℓ 2 e − η gap / ℓ ( 1 − e − w F / ℓ ) ( 1 − e − w B / ℓ ) . (55) where w F , w B w_F,w_B w F , w B η g a p ≥ 0 \eta_{\mathrm{gap}}\ge0 η gap ≥ 0 Δ η = η g a p + ( w F + w B ) / 2 \Delta\eta=\eta_{\mathrm{gap}}+(w_F+w_B)/2 Δ η = η gap + ( w F + w B ) /2
R 10 = a = μ 0 w F , R 01 = b = μ 0 w B , R 20 = a + U F , R 02 = b + U B , R 11 = U F B , R 30 = a + 3 U F , R 03 = b + 3 U B , R 21 = U F B , R 12 = U F B , R 40 = a + 7 U F , R 04 = b + 7 U B , R 31 = U F B , R 22 = U F B , R 13 = U F B . (56) \begin{aligned}
R_{10}&=a=\mu_0w_F, & R_{01}&=b=\mu_0w_B,\\
R_{20}&=a+U_F, & R_{02}&=b+U_B,\\
R_{11}&=U_{FB},\\
R_{30}&=a+3U_F, & R_{03}&=b+3U_B,\\
R_{21}&=U_{FB}, & R_{12}&=U_{FB},\\
R_{40}&=a+7U_F, & R_{04}&=b+7U_B,\\
R_{31}&=U_{FB}, & R_{22}&=U_{FB}, & R_{13}&=U_{FB}. \tag*{(56)}
\end{aligned} R 10 R 20 R 11 R 30 R 21 R 40 R 31 = a = μ 0 w F , = a + U F , = U F B , = a + 3 U F , = U F B , = a + 7 U F , = U F B , R 01 R 02 R 03 R 12 R 04 R 22 = b = μ 0 w B , = b + U B , = b + 3 U B , = U F B , = b + 7 U B , = U F B , R 13 = U F B . (56) and for low-order strongly intensive observables we get:
Σ = 1 + b 2 U F + a 2 U B − 2 a b U F B a b ( a + b ) . (57) \Sigma=1+\frac{b^2U_F+a^2U_B-2abU_{FB}}{ab(a+b)}. \tag*{(57)} Σ = 1 + ab ( a + b ) b 2 U F + a 2 U B − 2 ab U F B . (57) Δ = 1 + b 2 U F − a 2 U B a b ( b − a ) . (58) \Delta=1+\frac{b^2U_F-a^2U_B}{ab(b-a)}. \tag*{(58)} Δ = 1 + ab ( b − a ) b 2 U F − a 2 U B . (58) Γ = 1 + 3 ( b 3 U F − a 3 U B − a b ( b − a ) U F B ) a b ( a + b ) ( b − a ) . (59) \Gamma=1+\frac{3\left(b^3U_F-a^3U_B-ab(b-a)U_{FB}\right)}{ab(a+b)(b-a)}. \tag*{(59)} Γ = 1 + ab ( a + b ) ( b − a ) 3 ( b 3 U F − a 3 U B − ab ( b − a ) U F B ) . (59) Appendix C gives a compact evaluation prescription for the other J i j J_{ij} J ij [25 ] : μ 0 Λ 0 = 0.73 \mu_0\Lambda_0=0.73 μ 0 Λ 0 = 0.73 ℓ = 1.52 \ell=1.52 ℓ = 1.52 Figure 1 shows results for three combinations of window widths: ( w F , w B ) = ( 0.2 , 0.2 ) (w_F,w_B)=(0.2,0.2) ( w F , w B ) = ( 0.2 , 0.2 ) ( 0.4 , 0.2 ) (0.4,0.2) ( 0.4 , 0.2 ) ( 0.4 , 0.05 ) (0.4,0.05) ( 0.4 , 0.05 )
Figure 1. Strongly intensive observables through fourth order as a function of the distance between window centers. Curves are obtained directly from the ordinary source cumulants in Eq. (56) , with the exponential pair parameters μ 0 Λ 0 = 0.73 \mu_0\Lambda_0 = 0.73 μ 0 Λ 0 = 0.73 ℓ = 1.52 \ell= 1.52 ℓ = 1.52 Σ = J 20 \Sigma= J_{20} Σ = J 20 Δ \Delta Δ Γ \Gamma Γ Υ \Upsilon Υ J 11 J_{11} J 11 J 30 J_{30} J 30 J 21 J_{21} J 21 J 12 J_{12} J 12 J 40 J_{40} J 40 J 31 J_{31} J 31 J 22 J_{22} J 22 J 13 J_{13} J 13
At fixed widths, Δ \Delta Δ Δ η \Delta\eta Δ η Δ \Delta Δ Γ \Gamma Γ Σ \Sigma Σ J 21 J_{21} J 21 J 22 J_{22} J 22
PYTHIA calculations
Three samples with a common generator version
The analytical examples isolate particular contributions to the observables. PYTHIA implements a more detailed particle-production model in which these contributions occur together. We use three samples generated with PYTHIA 8.318 [33 ] : Monash at s = 900 \sqrt{s}=900 s = 900
We compare the Monash results with the Trieste 2 tune, which incorporates string close-packing effects into PYTHIA [51 , 52 ] . In this description, overlapping strings modify the effective color field experienced during fragmentation, making the string tension dependent on the surrounding string environment. This provides a phenomenological realization of the broader idea that overlapping strings need not fragment independently. A related physical motivation underlies the string-fusion model, in which overlapping color strings
form sources with modified particle-production properties [53 ] . The comparison is therefore of interest for testing how strongly intensive multiplicity observables respond to changes in string fragmentation associated with overlap. The two descriptions implement this idea differently, and the Monash–Trieste 2 comparison also includes other differences between the tunes. It should therefore not be interpreted as an isolated measurement of the close-packing contribution.
Each sample contains 150 million generated inelastic p p pp pp
The generation uses SoftQCD:inelastic=on with proton beams and Tune:pp=14. Trieste 2 is applied on top of this baseline, including its colour-reconnection and close-packing settings [51 ] . Appendix D lists the complete override card and the corresponding Monash values.
INEL> 0 > 0 > 0 ∣ η ∣ < 1 |\eta| < 1 ∣ η ∣ < 1 > 0 > 0 > 0 V = N A + N C V = N_{\mathrm{A}} + N_{\mathrm{C}} V = N A + N C N A N_{\mathrm{A}} N A 2.8 < η < 5.1 2.8 < \eta< 5.1 2.8 < η < 5.1 N C N_{\mathrm{C}} N C − 3.7 < η < − 1.7 -3.7 < \eta< -1.7 − 3.7 < η < − 1.7 p T p_{\mathrm{T}} p T [54 ] , not a simulated detector amplitude.
The measurement windows, in contrast, count particles with 0.2 < p T < 2 0.2 < p_{\mathrm{T}} < 2 0.2 < p T < 2 Δ η = η F − η B \Delta\eta= \eta_{\mathrm{F}} - \eta_{\mathrm{B}} Δ η = η F − η B w F w_{\mathrm{F}} w F w B w_{\mathrm{B}} w B η g a p = Δ η − ( w F + w B ) / 2 \eta_{\mathrm{gap}} = \Delta\eta- (w_{\mathrm{F}} + w_{\mathrm{B}})/2 η gap = Δ η − ( w F + w B ) /2
Statistical uncertainties are estimated using the jackknife method [55 ] , dividing each sample into 150 equal-sized blocks and omitting one block at a time. These errors describe event sampling, not generator or detector systematics.
Scans over window geometry and collision energy
The scans probe how the observables depend on the position and widths of the two windows. In the geometry of Figure 2 , the backward window is fixed at η B = − 0.775 \eta_{\mathrm{B}} = -0.775 η B = − 0.775 w B = 0.05 w_{\mathrm{B}} = 0.05 w B = 0.05 w F = 0.1 w_{\mathrm{F}} = 0.1 w F = 0.1 0.4 0.4 0.4 Δ η ≤ 1.4 \Delta\eta\le1.4 Δ η ≤ 1.4 Figure 3 , the windows are centered at ± Δ η / 2 \pm\Delta\eta/2 ± Δ η /2 0.2 / 0.1 0.2/0.1 0.2/0.1 0.4 / 0.2 0.4/0.2 0.4/0.2 Σ \Sigma Σ
Figure 2. Pseudorapidity scan: Monash and Trieste 2 PYTHIA 8.318 at p p pp pp s = 900 \sqrt{s}=900 s = 900 > > > 0.2 < p T < 2 0.2<p_{\mathrm{T}}<2 0.2 < p T < 2 η B = − 0.775 \eta_{\mathrm{B}}=-0.775 η B = − 0.775 Σ \Sigma Σ Δ \Delta Δ Γ \Gamma Γ J 11 J_{11} J 11 J 30 J_{30} J 30 J 21 J_{21} J 21 J 12 J_{12} J 12 Υ \Upsilon Υ J 31 J_{31} J 31 J 22 J_{22} J 22 J 13 J_{13} J 13
Figure 3. Monash/Trieste 2 comparison in PYTHIA 8.318 at p p pp pp s = 900 \sqrt{s}=900 s = 900 > 0 >0 > 0 0.2 < p T < 2 0.2 < p_{\mathrm{T}} < 2 0.2 < p T < 2 ± Δ η / 2 \pm\Delta\eta/2 ± Δ η /2 Δ η ≤ 1.4 \Delta\eta\le1.4 Δ η ≤ 1.4 Σ \Sigma Σ Δ \Delta Δ Γ \Gamma Γ J 11 J_{11} J 11 J 30 J_{30} J 30 J 21 J_{21} J 21 J 12 J_{12} J 12 Υ \Upsilon Υ J 31 J_{31} J 31 J 22 J_{22} J 22 J 13 J_{13} J 13
The Monash and Trieste 2 curves in these figures use the same acceptance and event selection. The differences between the curves therefore reflect the specified settings within PYTHIA 8.318 rather than the generator release. The Trieste 2 tune gives larger Σ \Sigma Σ [27 ] . The higher-order coordinates also respond to the tune change, which does not isolate the contribution of close packing.
Figure 4 compares the two collision energies with the same asymmetric pseudorapidity windows as in Figure 3 . Changing s \sqrt{s} s I j j I_{jj} I j j Δ η \Delta\eta Δ η
Figure 4. Energy comparison: Monash 8.318 at s = 900 \sqrt{s}=900 s = 900 p p pp pp > 0 >0 > 0 0.2 < p T < 2 0.2 < p_{\mathrm{T}} < 2 0.2 < p T < 2 Σ \Sigma Σ Δ \Delta Δ Γ \Gamma Γ J 11 J_{11} J 11 J 30 J_{30} J 30 J 21 J_{21} J 21 J 12 J_{12} J 12 Υ \Upsilon Υ J 31 J_{31} J 31 J 22 J_{22} J 22 J 13 J_{13} J 13
Classes selected with the V0M-like activity estimator
We evaluate the observables in percentile classes of the activity estimator V V V > 0 > 0 > 0 Figure 5 shows Σ \Sigma Σ Figure 6 compares the regular higher-order basis in asymmetric windows at 900 GeV.
Figure 5. Class dependence of Σ \Sigma Σ > > > < p T < < p_{\mathrm{T}} < < p T <
Figure 6. Class-dependent comparison of Monash and Trieste 2, PYTHIA 8.318, in p p pp pp s = 900 \sqrt{s}=900 s = 900 > 0 >0 > 0 ± Δ η / 2 \pm\Delta\eta/2 ± Δ η /2 0.2 < p T < 2 0.2 < p_{\mathrm{T}} < 2 0.2 < p T < 2 Σ \Sigma Σ Δ \Delta Δ Γ \Gamma Γ J 11 J_{11} J 11 J 30 J_{30} J 30 J 21 J_{21} J 21 J 12 J_{12} J 12 Υ \Upsilon Υ J 31 J_{31} J 31 J 22 J_{22} J 22 J 13 J_{13} J 13
Σ \Sigma Σ Σ ≈ 1 \Sigma\approx1 Σ ≈ 1 Σ − 1 ≈ 0.06 \Sigma- 1 \approx0.06 Σ − 1 ≈ 0.06 Δ η = 1.4 \Delta\eta= 1.4 Δ η = 1.4 Σ − 1 ≈ 0.12 \Sigma- 1 \approx0.12 Σ − 1 ≈ 0.12
Across much of the higher-order basis at 900 GeV (Figure 6 ), the 0–1% class lies above the 70–100% class, while the tune changes generally produce smaller shifts. This ordering is not universal for every coordinate and separation. Several higher-order points overlap within their statistical errors or reverse the tune ordering. The observable Δ \Delta Δ
Selection on V0 activity can alter both the source-count distribution and the emission law itself. Strongly intensive observables cancel only the former, and only while all sources share a common emission law. A remaining class dependence therefore reflects a change in the source composition, in the source emission, or in correlations induced by the selection. The class scans alone cannot separate these mechanisms.
Joint pseudorapidity and azimuthal separations
Azimuthal separation resolves structure that the full-azimuth scans average over. To expose it, we split the azimuth into two sectors of width π / 4 \pi/4 π /4 w F = 0.4 w_{\mathrm{F}}=0.4 w F = 0.4 w B = 0.2 w_{\mathrm{B}}=0.2 w B = 0.2 Δ η \Delta\eta Δ η Δ ϕ \Delta\phi Δ ϕ
Figure 7 shows the Monash maps at 900 GeV and Figure 8 at 13 TeV for the displayed strongly intensive coordinates as functions of Δ η \Delta\eta Δ η Δ ϕ \Delta\phi Δ ϕ Δ \Delta Δ Γ \Gamma Γ Figure 2 and Figure 3 . All surfaces rise with Δ η \Delta\eta Δ η Δ ϕ = 0 \Delta\phi=0 Δ ϕ = 0 Δ ϕ = π \Delta\phi=\pi Δ ϕ = π Δ η \Delta\eta Δ η Δ ϕ \Delta\phi Δ ϕ
Figure 7. PYTHIA 8.318 Monash results for p p pp pp s = 900 \sqrt{s}=900 s = 900 > 0 >0 > 0 0.2 < p T < 2 0.2<p_{\mathrm{T}}<2 0.2 < p T < 2 0.4 / 0.2 0.4/0.2 0.4/0.2 π / 4 \pi/4 π /4 Δ η \Delta\eta Δ η Δ ϕ \Delta\phi Δ ϕ 0 ≤ Δ ϕ ≤ π 0\leq\Delta\phi\leq\pi 0 ≤ Δ ϕ ≤ π Δ ϕ = 0 \Delta\phi=0 Δ ϕ = 0 Σ \Sigma Σ J 11 J_{11} J 11 J 30 J_{30} J 30 J 21 J_{21} J 21 J 12 J_{12} J 12 Υ \Upsilon Υ J 31 J_{31} J 31 J 22 J_{22} J 22 J 13 J_{13} J 13
Figure 8. PYTHIA 8.318 Monash results for p p pp pp s = 13 \sqrt{s}=13 s = 13 > 0 >0 > 0 0.2 < p T < 2 0.2 < p_{\mathrm{T}} < 2 0.2 < p T < 2 0.4 / 0.2 0.4/0.2 0.4/0.2 π / 4 \pi/4 π /4 Δ η \Delta\eta Δ η Δ ϕ \Delta\phi Δ ϕ 0 ≤ Δ ϕ ≤ π 0 \leq\Delta\phi\leq\pi 0 ≤ Δ ϕ ≤ π Δ ϕ = 0 \Delta\phi=0 Δ ϕ = 0 Σ \Sigma Σ J 11 J_{11} J 11 J 30 J_{30} J 30 J 21 J_{21} J 21 J 12 J_{12} J 12 Υ \Upsilon Υ J 31 J_{31} J 31 J 22 J_{22} J 22 J 13 J_{13} J 13
already seen in the one-dimensional scans. At 13 TeV the surfaces flatten in Δ ϕ \Delta\phi Δ ϕ Δ η \Delta\eta Δ η
The surfaces connect the evaluated geometries without smoothing, and the calculated half-plane 0 ≤ Δ ϕ ≤ π 0 \leq\Delta\phi\leq\pi 0 ≤ Δ ϕ ≤ π Δ ϕ = 0 \Delta\phi= 0 Δ ϕ = 0 Figure 2 –Figure 4 do not quantify the uncertainties for these narrower azimuthal sectors, so the surfaces are used here to illustrate qualitative trends.
Fitting the density and testing a pair-only closure
We test whether a specified single-source density and pair correlation function can jointly describe the PYTHIA mean multiplicities, Σ \Sigma Σ Δ \Delta Δ
We use the even profile
f ( η ) = 1 + c 2 η 2 + c 4 η 4 , λ 1 ( η ) = μ 0 f ( η ) , (60) f(\eta) = 1 + c_{2}\eta^{2} + c_{4}\eta^{4}, \qquad\lambda_{1}(\eta) = \mu_{0}f(\eta), \tag*{(60)} f ( η ) = 1 + c 2 η 2 + c 4 η 4 , λ 1 ( η ) = μ 0 f ( η ) , (60) as a local parametrization in the central region ∣ η ∣ ≤ 0.8 |\eta| \le0.8 ∣ η ∣ ≤ 0.8 ρ e v e n t ( η ) = ρ 0 f ( η ) \rho_{\mathrm{event}}(\eta) = \rho_{0}f(\eta) ρ event ( η ) = ρ 0 f ( η ) ρ 0 = ⟨ S ⟩ μ 0 \rho_{0} = \langle S\rangle\mu_{0} ρ 0 = ⟨ S ⟩ μ 0
For the connected pair correlation we adopt the Gaussian form
Λ ( η 1 , η 2 ) = Λ 0 exp [ − ( η 1 − η 2 ) 2 2 ℓ 2 ] . (61) \Lambda(\eta_{1},\eta_{2}) = \Lambda_{0}\exp\left[-\frac{(\eta_{1}-\eta_{2})^{2}}{2\ell^{2}}\right]. \tag*{(61)} Λ ( η 1 , η 2 ) = Λ 0 exp [ − 2 ℓ 2 ( η 1 − η 2 ) 2 ] . (61) This is a separate model assumption from the exponential kernel considered in § 1.2 . The comparison below tests this specified Gaussian model.
The joint fit uses windows of widths 0.2 and 0.1, centered at η F = Δ η / 2 \eta_{\mathrm{F}} = \Delta\eta/2 η F = Δ η /2 η B = − Δ η / 2 \eta_{\mathrm{B}} = -\Delta\eta/2 η B = − Δ η /2 ⟨ n F ⟩ \langle n_{\mathrm{F}}\rangle ⟨ n F ⟩ ⟨ n B ⟩ \langle n_{\mathrm{B}}\rangle ⟨ n B ⟩ Σ \Sigma Σ Δ \Delta Δ Δ η \Delta\eta Δ η ρ 0 \rho_{0} ρ 0 c 2 c_{2} c 2 c 4 c_{4} c 4 a 2 = μ 0 Λ 0 a_{2} = \mu_{0}\Lambda_{0} a 2 = μ 0 Λ 0 ℓ \ell ℓ
Figure 9 displays the fits, and Table III lists their parameters. The reported PYTHIA χ 2 \chi^{2} χ 2 Δ \Delta Δ Δ η ≤ 1.0 \Delta\eta\le1.0 Δ η ≤ 1.0 ⟨ n B ⟩ / 0.1 \langle n_{\mathrm{B}}\rangle/0.1 ⟨ n B ⟩ /0.1
Table III. Parameters of the joint density and pair fit. The identifiable normalization combinations are ρ 0 = ⟨ S ⟩ μ 0 \rho_0 = \langle S\rangle\mu_0 ρ 0 = ⟨ S ⟩ μ 0 a 2 = μ 0 Λ 0 a_2 = \mu_0\Lambda_0 a 2 = μ 0 Λ 0
With the fitted density and pair kernel held fixed, setting the connected three- and four-particle densities to zero determines all observables through fourth order without additional parameters. Figure 10 –Figure 12 compare these pair-only predictions with PYTHIA. The three comparisons use the same 0.2/0.1 windows and center separations up to 1.4, with INEL> > > < p T < < p_{\mathrm{T}} < < p T <
Figure 10. Pair-only predictions (curves) compared with PYTHIA 8.318 results (points) for Monash at s = 900 \sqrt{s}=900 s = 900 p p pp pp > 0 >0 > 0 0.2 < p T < 2 0.2<p_{\mathrm{T}}<2 0.2 < p T < 2 Σ \Sigma Σ Δ \Delta Δ J 11 J_{11} J 11 Γ \Gamma Γ J 30 J_{30} J 30 J 21 J_{21} J 21 J 12 J_{12} J 12 Υ \Upsilon Υ J 40 J_{40} J 40 J 31 J_{31} J 31 J 22 J_{22} J 22 J 13 J_{13} J 13
Figure 12. Pair-only predictions (curves) compared with PYTHIA 8.318 results (points) for Monash at s = 13 \sqrt{s}=13 s = 13 p p pp pp > 0 >0 > 0 0.2 < p T < 2 0.2<p_{\mathrm{T}}<2 0.2 < p T < 2 Σ \Sigma Σ Δ \Delta Δ J 11 J_{11} J 11 Γ \Gamma Γ J 30 J_{30} J 30 J 21 J_{21} J 21 J 12 J_{12} J 12 Υ \Upsilon Υ J 40 J_{40} J 40 J 31 J_{31} J 31 J 22 J_{22} J 22 J 13 J_{13} J 13
These comparisons test the higher-order consequences of the specified density and pair kernel. A discrepancy may reflect nonzero connected three- or four-particle correlations, an inadequate lower-order parametrization, or both. In particular, the substantial lower-order residuals for Monash at 13 TeV prevent attributing its higher-order discrepancies exclusively to missing connected correlations. The next subsection tests the effect of adding specified three- and four-particle terms while retaining this lower-order baseline.
Conditional fits of connected three- and four-particle terms
We extend the pair-only model by introducing connected three- and four-particle densities with two additional amplitudes. The lower-order parameters are held fixed at the values in Table III . We adopt the symmetric
Gaussian hierarchy
C n ( η 1 , … , η n ) = μ 0 a n ∏ r = 1 n f ( η r ) × exp [ − ∑ r < s ( η r − η s ) 2 n ℓ 2 ] (62) C_n(\eta_1,\ldots,\eta_n)=\mu_0a_n\prod_{r=1}^{n}f(\eta_r)\times\exp\left[-\frac{\sum_{r<s}(\eta_r-\eta_s)^2}{n\ell^2}\right] \tag*{(62)} C n ( η 1 , … , η n ) = μ 0 a n r = 1 ∏ n f ( η r ) × exp [ − n ℓ 2 ∑ r < s ( η r − η s ) 2 ] (62) where f ( η ) = 1 + c 2 η 2 + c 4 η 4 f(\eta)=1+c_2\eta^2+c_4\eta^4 f ( η ) = 1 + c 2 η 2 + c 4 η 4
For n = 2 n=2 n = 2 (62) is consistent with the normalized pair kernel through
C 2 ( η 1 , η 2 ) = λ 1 ( η 1 ) λ 1 ( η 2 ) Λ ( η 1 , η 2 ) , λ 1 ( η ) = μ 0 f ( η ) , (63) C_2(\eta_1,\eta_2)=\lambda_1(\eta_1)\lambda_1(\eta_2)\Lambda(\eta_1,\eta_2), \qquad\lambda_1(\eta)=\mu_0 f(\eta), \tag*{(63)} C 2 ( η 1 , η 2 ) = λ 1 ( η 1 ) λ 1 ( η 2 ) Λ ( η 1 , η 2 ) , λ 1 ( η ) = μ 0 f ( η ) , (63) with a 2 = μ 0 Λ 0 a_2 = \mu_0\Lambda_0 a 2 = μ 0 Λ 0 ℓ \ell ℓ μ 0 = 1 \mu_0 = 1 μ 0 = 1 μ 0 \mu_0 μ 0 § 2.1.4 .
The amplitudes are fitted sequentially. First, a 3 a_3 a 3 J 30 J_{30} J 30 J 21 J_{21} J 21 J 12 J_{12} J 12 a 3 a_3 a 3 a 4 a_4 a 4 Υ \Upsilon Υ J 31 J_{31} J 31 J 22 J_{22} J 22 J 13 J_{13} J 13
Table IV reports the fitted amplitudes and nominal fit diagnostics. The quoted amplitude errors condition on all preceding fit parameters. They do not include uncertainty propagated from earlier stages and are not errors from a simultaneous fit of the entire hierarchy.
Table IV. Conditional higher-order amplitudes and nominal fit diagnostics. Each arrow compares the objective with the new amplitude fixed to zero and with that amplitude fitted. The third-order baseline is the pair-only model. The fourth-order baseline already includes the fitted a 3 a_3 a 3 a 4 a_4 a 4
Adding a single three-particle amplitude reduces the nominal third-order objective by more than an order of magnitude in every sample. The fitted amplitudes are similar for the two tunes at 900 GeV. For Monash, a 3 a_3 a 3 a 3 a_3 a 3 a 4 a_4 a 4
Figure 13 –Figure 15 display the resulting higher-order observables. All three comparisons use pseudorapidity windows of widths 0.2/0.1, centered at ± Δ η / 2 \pm\Delta\eta/2 ± Δ η /2 > > > 0.2 < p T < 2 0.2 < p_T < 2 0.2 < p T < 2
Figure 13. Sequential fits with connected three- and four-particle terms (curves) compared with PYTHIA 8.318 results (points) for Monash at s = 900 \sqrt{s}=900 s = 900 p p pp pp > 0 >0 > 0 0.2 < p T < 2 0.2 < p_{\mathrm{T}} < 2 0.2 < p T < 2 Σ \Sigma Σ Δ \Delta Δ J 11 J_{11} J 11 Γ \Gamma Γ J 30 J_{30} J 30 J 21 J_{21} J 21 J 12 J_{12} J 12 Υ \Upsilon Υ J 40 J_{40} J 40 J 31 J_{31} J 31 J 22 J_{22} J 22 J 13 J_{13} J 13
Figure 15. Sequential fits with connected three- and four-particle terms (curves) compared with PYTHIA 8.318 results (points) for Monash at s = 13 \sqrt{s}=13 s = 13 p p pp pp > 0 >0 > 0 0.2 < p T < 2 0.2 < p_{\mathrm{T}} < 2 0.2 < p T < 2 Σ \Sigma Σ Δ \Delta Δ J 11 J_{11} J 11 Γ \Gamma Γ J 30 J_{30} J 30 J 21 J_{21} J 21 J 12 J_{12} J 12 Υ \Upsilon Υ J 40 J_{40} J 40 J 31 J_{31} J 31 J 22 J_{22} J 22 J 13 J_{13} J 13
The improvement demonstrates that the specified three- and four-particle terms account for a substantial part of the higher-order discrepancy. It does not uniquely determine the underlying connected densities: their shapes and common length are imposed, and residual discrepancies remain.
Application to ALICE at 13 TeV
We combine the published ALICE pseudorapidity density at 13 TeV [56 ] with preliminary measurements of Σ \Sigma Σ [20 ] to construct conditional predictions for the higher-order observables. The density and Σ \Sigma Σ
The published ALICE pseudorapidity density, measured over the full p T p_T p T [56 ] , is converted to 0.2 < p T < 2 0.2 < p_T < 2 0.2 < p T < 2
The preliminary Σ \Sigma Σ
An unweighted linear least-squares fit to eight corrected density-bin averages in ∣ η ∣ < 0.8 |\eta| < 0.8 ∣ η ∣ < 0.8
ρ 0 = 5.271 , c 2 = 0.0834 , c 4 = − 0.0109. (64) \rho_0 = 5.271,\qquad c_2 = 0.0834,\qquad c_4 = -0.0109. \tag*{(64)} ρ 0 = 5.271 , c 2 = 0.0834 , c 4 = − 0.0109. (64) With this profile fixed, an unweighted fit of the Gaussian pair model to 11 preliminary Σ \Sigma Σ Δ η = 0.4 , 0.5 , … , 1.4 \Delta\eta= 0.4, 0.5,\ldots,1.4 Δ η = 0.4 , 0.5 , … , 1.4
a 2 = 0.769 , ℓ = 0.522. (65) a_2 = 0.769,\qquad\ell= 0.522. \tag*{(65)} a 2 = 0.769 , ℓ = 0.522. (65) The root-mean-square residual in Σ \Sigma Σ Figure 16 shows the acceptance correction and the two fits. Published density errors are scaled for display but are not used as fit weights. No experimental errors are assigned to the graphically extracted Σ \Sigma Σ
Figure 16. Acceptance correction and lower-order fits for the ALICE comparison at s = 13 \sqrt{s}=13 s = 13 p T p_{\mathrm{T}} p T 0.2 < p T < 2 0.2 < p_{\mathrm{T}} < 2 0.2 < p T < 2 Σ − 1 \Sigma- 1 Σ − 1
With this parametrization, the ALICE inputs give a larger a 2 a_2 a 2 ℓ \ell ℓ a 2 = 0.499 a_2 = 0.499 a 2 = 0.499 ℓ = 0.651 \ell= 0.651 ℓ = 0.651 a 2 = μ 0 Λ 0 a_2 = \mu_0\Lambda_0 a 2 = μ 0 Λ 0 Λ 0 \Lambda_0 Λ 0 μ 0 \mu_0 μ 0
Figure 9. Joint Gaussian fit to PYTHIA 8.318 mean multiplicities, Σ \Sigma Σ Δ \Delta Δ p p pp pp > 0 >0 > 0 0.2 < p T < 2 0.2 < p_{\mathrm{T}} < 2 0.2 < p T < 2 0.2 / 0.1 0.2/0.1 0.2/0.1 ⟨ n B ⟩ / 0.1 \langle n_{\mathrm{B}}\rangle/0.1 ⟨ n B ⟩ /0.1 η B = − Δ η / 2 \eta_{\mathrm{B}}=-\Delta\eta/2 η B = − Δ η /2 Σ \Sigma Σ Δ \Delta Δ Δ \Delta Δ Δ η ≤ 1.0 \Delta\eta\le1.0 Δ η ≤ 1.0
To specify the higher connected terms, we transfer the ratios
r 3 = a 3 a 2 2 ≃ 9.44 , r 4 = a 4 a 2 3 ≃ 85.5 (66) r_3 = \frac{a_3}{a_2^2} \simeq9.44,\qquad r_4 = \frac{a_4}{a_2^3} \simeq85.5 \tag*{(66)} r 3 = a 2 2 a 3 ≃ 9.44 , r 4 = a 2 3 a 4 ≃ 85.5 (66) from the conditional Monash fits at 13 TeV. These ratios determine the ALICE scenario amplitudes
a 3 t r ≡ a 3 , A L I C E = r 3 a 2 , A L I C E 2 , a_{3}^{\mathrm{tr}}\equiv a_{3,\mathrm{ALICE}}=r_{3}a_{2,\mathrm{ALICE}}^{2}, a 3 tr ≡ a 3 , ALICE = r 3 a 2 , ALICE 2 , a 4 t r ≡ a 4 , A L I C E = r 4 a 2 , A L I C E 3 . (67) a_{4}^{\mathrm{tr}}\equiv a_{4,\mathrm{ALICE}}=r_{4}a_{2,\mathrm{ALICE}}^{3}. \tag*{(67)} a 4 tr ≡ a 4 , ALICE = r 4 a 2 , ALICE 3 . (67) The superscript tr denotes amplitudes determined by this transfer prescription, not fitted to higher-order ALICE measurements. The prescription is an additional model assumption, rather than a consequence of strong intensivity. Only the amplitude ratios are transferred from Monash. The connected densities C 3 C_3 C 3 C 4 C_4 C 4 (62) with the profile and pair length fitted to the ALICE inputs.
Figure 11. Pair-only predictions (curves) compared with PYTHIA 8.318 results (points) for Trieste 2 at s = 900 \sqrt{s}=900 s = 900 p p pp pp > 0 >0 > 0 0.2 < p T < 2 0.2 < p_{\mathrm{T}} < 2 0.2 < p T < 2 Σ \Sigma Σ Δ \Delta Δ J 11 J_{11} J 11 Γ \Gamma Γ J 30 J_{30} J 30 J 21 J_{21} J 21 J 12 J_{12} J 12 Υ \Upsilon Υ J 40 J_{40} J 40 J 31 J_{31} J 31 J 22 J_{22} J 22 J 13 J_{13} J 13
Figure 17 compares three scenarios:
Figure 17. Conditional ALICE-constrained scenarios for p p pp pp s = 13 \sqrt{s}=13 s = 13 > 0 >0 > 0 0.2 < p T < 2 0.2<p_T<2 0.2 < p T < 2 ± Δ η / 2 \pm\Delta\eta/2 ± Δ η /2 Σ \Sigma Σ J 11 J_{11} J 11 J 30 J_{30} J 30 J 21 J_{21} J 21 J 12 J_{12} J 12 Υ \Upsilon Υ J 40 J_{40} J 40 J 31 J_{31} J 31 J 22 J_{22} J 22 J 13 J_{13} J 13
the pair-only closure, a 3 = a 4 = 0 a_{3}=a_{4}=0 a 3 = a 4 = 0
a nonzero three-particle amplitude is present, a 3 = a 3 t r a_{3}=a_{3}^{\mathrm{tr}} a 3 = a 3 tr a 4 = 0 a_{4}=0 a 4 = 0
both higher-order amplitudes are present, a 3 = a 3 t r a_{3}=a_{3}^{\mathrm{tr}} a 3 = a 3 tr a 4 = a 4 t r a_{4}=a_{4}^{\mathrm{tr}} a 4 = a 4 tr
All three share the same fitted one- and two-particle inputs. Introducing C 3 C_3 C 3 C 4 C_4 C 4 C 3 C_3 C 3
These curves are conditional scenarios, not measurements of the ALICE connected densities C 3 C_3 C 3 C 4 C_4 C 4
Figure 14. Sequential fits with connected three- and four-particle terms (curves) compared with PYTHIA 8.318 results (points) for Trieste 2 at s = 900 \sqrt{s}=900 s = 900 p p pp pp > 0 >0 > 0 0.2 < p T < 2 0.2<p_{\mathrm{T}}<2 0.2 < p T < 2 Σ \Sigma Σ Δ \Delta Δ J 11 J_{11} J 11 Γ \Gamma Γ J 30 J_{30} J 30 J 21 J_{21} J 21 J 12 J_{12} J 12 Υ \Upsilon Υ J 40 J_{40} J 40 J 31 J_{31} J 31 J 22 J_{22} J 22 J 13 J_{13} J 13
Discussion and conclusions
We have constructed a normalized hierarchy of strongly intensive observables for two particle counts in the independent identical-source model. The hierarchy contains n n n n n n α \alpha α m α β ≠ 0 m\alpha\beta\ne0 m α β = 0
The analytical models clarify the interpretation of the higher orders. Independent non-Poisson emission and a formal pair-only closure already generate nontrivial higher-order observables. Consequently, a departure of J i j J_{ij} J ij ( i + j ) (i+j) ( i + j )
The PYTHIA study uses three samples of 150 million generated events: Monash and Trieste 2 at 900 GeV and Monash at 13 TeV, all with version 8.318. Window and event-class scans illustrate the dependence of the invariant coordinates on acceptance and particle-production dynamics. The Gaussian model of the density and pair observables gives a conditional baseline for the higher orders, but its substantial lower-order residuals at 13 TeV prevent attributing the higher-order discrepancies exclusively to C 3 C_3 C 3 C 4 C_4 C 4
The immediate limitations on quantitative interpretation are the lower-order model residuals and the absence of uncertainty propagation through the sequential fits. The ALICE application additionally depends on preliminary graphically extracted second-order input, assumed acceptance matching, and a generator-based acceptance correction. Experimental, fit, and transfer uncertainties must be propagated before its scenarios can be used for quantitative discrimination. A separate question for future data is whether direct higher-order measurements support the specified correlation hierarchy and the transfer of r 3 r_3 r 3 r 4 r_4 r 4
The strongly intensive property holds only while the selected ensemble admits independent identical sources whose emission law does not depend on the source count. Source mixtures, global conservation, and selection-induced correlations may violate this condition. Detector efficiency and finite-sample bias also require a separate analysis. Within its stated domain, the hierarchy provides a systematic means of comparing the information in successive cumulant orders while removing the dependence on the source-count distribution. Measurements of the third- and fourth-order coordinates in the same windows would test whether a hierarchy constrained at second order describes the additional invariant information.
Acknowledgments
The author acknowledges helpful conversations with Semyon Yurchenko.
The author acknowledges Saint-Petersburg State University for a research project ID: 153212394.
The author acknowledges the use of AI-based tools, including ChatGPT-6, to improve grammar, enhance clarity, and check mathematical statements. All revisions assisted by AI were carefully reviewed to ensure that the intended meaning and scientific content remain unchanged. The author takes full responsibility for the published content.
All-order construction and completeness
We work with formal cumulant series truncated at total degree n ≥ 2 n \ge2 n ≥ 2 m α β ≠ 0 m_{\alpha\beta} \ne0 m α β = 0
Invariant series and recursion
Under the independent-source assumptions of Sec. II, P = Q ∘ R P = Q \circ R P = Q ∘ R Q ′ ( 0 ) = ⟨ S ⟩ > 0 Q'(0) = \langle S\rangle> 0 Q ′ ( 0 ) = ⟨ S ⟩ > 0 D ( v ) = P ( v , v ) D(v) = P(v,v) D ( v ) = P ( v , v ) h h h
D ( h ( t , u ) ) = P ( β t + u , − α t + u ) . (68) D(h(t,u)) = P(\beta t + u, -\alpha t + u). \tag*{(68)} D ( h ( t , u )) = P ( β t + u , − α t + u ) . (68) Since D ′ ( 0 ) = m ≠ 0 D'(0) = m \ne0 D ′ ( 0 ) = m = 0 m u mu m u h = u + O ( ( t , u ) 2 ) h = u + O((t,u)^2) h = u + O (( t , u ) 2 ) h ( 0 , u ) = u h(0,u) = u h ( 0 , u ) = u P = Q ∘ R P = Q \circ R P = Q ∘ R (68) and cancellation of the invertible series Q Q Q
R ( h , h ) = R ( β t + u , − α t + u ) . (69) R(h,h) = R(\beta t + u, -\alpha t + u). \tag*{(69)} R ( h , h ) = R ( β t + u , − α t + u ) . (69) The weights also agree at event and source levels because P 10 = Q 1 R 10 P_{10} = Q_1R_{10} P 10 = Q 1 R 10 P 01 = Q 1 R 01 P_{01} = Q_1R_{01} P 01 = Q 1 R 01 h h h
To compute h h h n n n h = u h = u h = u r = 2 , … , n r = 2,\ldots,n r = 2 , … , n
h ← h + [ P ( β t + u , − α t + u ) − D ( h ) ] r m . (70) h \leftarrow h + \frac{\left[P(\beta t + u,-\alpha t + u)-D(h)\right]_r}{m}. \tag*{(70)} h ← h + m [ P ( β t + u , − α t + u ) − D ( h ) ] r . (70) Here [ ⋅ ] r [\cdot]_r [ ⋅ ] r r r r h h h r r r D ( h ) D(h) D ( h ) m m m D D D r + 1 r+1 r + 1
Normalization
For the independent-particle reference, let the union multiplicity N N N α , β \alpha,\beta α , β K K K N N N P ( x , y ) = K ( log ( α e x + β e y ) ) P(x,y) = K(\log(\alpha e^x + \beta e^y)) P ( x , y ) = K ( log ( α e x + β e y )) D ( v ) = K ( v ) D(v) = K(v) D ( v ) = K ( v )
e h ( t , u ) = e u ( α e β t + β e − α t ) . (71) e^{h(t,u)} = e^u(\alpha e^{\beta t}+\beta e^{-\alpha t}). \tag*{(71)} e h ( t , u ) = e u ( α e β t + β e − α t ) . (71) Its derivatives are α β [ β i − 1 + ( − 1 ) i α i − 1 ] \alpha\beta[\beta^{i-1}+(-1)^i\alpha^{i-1}] α β [ β i − 1 + ( − 1 ) i α i − 1 ] j j j i = 1 i=1 i = 1 i i i i ≥ 1 i \ge1 i ≥ 1 j ≥ 0 j \ge0 j ≥ 0 2 ≤ i + j ≤ n 2 \le i+j \le n 2 ≤ i + j ≤ n
J i j = i ! j ! α β [ t i u j ] e h + ζ i Σ , J_{ij} = \frac{i!j!}{\alpha\beta}[t^i u^j]e^h + \zeta_i\Sigma, J ij = α β i ! j ! [ t i u j ] e h + ζ i Σ , ζ i = 1 − β i − 1 − ( − 1 ) i α i − 1 . (72) \zeta_i = 1-\beta^{i-1}-(-1)^i\alpha^{i-1}. \tag*{(72)} ζ i = 1 − β i − 1 − ( − 1 ) i α i − 1 . (72) The bracket denotes an ordinary monomial coefficient. First obtain Σ = 2 [ t 2 u 0 ] e h / ( α β ) \Sigma= 2[t^2u^0]e^h/(\alpha\beta) Σ = 2 [ t 2 u 0 ] e h / ( α β ) ζ 2 = 0 \zeta_2 = 0 ζ 2 = 0 J 20 = Σ J_{20} = \Sigma J 20 = Σ J i j = 1 J_{ij}=1 J ij = 1 h = u h=u h = u Σ = 0 \Sigma=0 Σ = 0 J i j = 0 J_{ij}=0 J ij = 0 α β \alpha\beta α β h h h e h e^h e h
Inverse construction and completeness
Prescribe α \alpha α J i j J_{ij} J ij n n n β = 1 − α \beta=1-\alpha β = 1 − α α β ≠ 0 \alpha\beta\ne0 α β = 0 Σ = J 20 \Sigma=J_{20} Σ = J 20
E ( t , u ) = e u + α β ∑ i ≥ 1 , j ≥ 0 2 ≤ i + j ≤ n ( J i j − ζ i Σ ) t i u j i ! j ! . (73) E(t,u)=e^u+\alpha\beta\sum_{\substack{i\ge1,\ j\ge0\\2\le i+j\le n}}(J_{ij}-\zeta_i\Sigma)\frac{t^i u^j}{i!j!}. \tag*{(73)} E ( t , u ) = e u + α β i ≥ 1 , j ≥ 0 2 ≤ i + j ≤ n ∑ ( J ij − ζ i Σ ) i ! j ! t i u j . (73) All series are truncated above degree n n n h = log E h=\log E h = log E h ( 0 , u ) = u h(0,u)=u h ( 0 , u ) = u u u u 2 [ t 2 u 0 ] E / ( α β ) = Σ 2[t^2u^0]E/(\alpha\beta)=\Sigma 2 [ t 2 u 0 ] E / ( α β ) = Σ D ( v ) = ∑ k = 1 n d k v k / k ! D(v)=\sum_{k=1}^n d_kv^k/k! D ( v ) = ∑ k = 1 n d k v k / k ! d 1 ≠ 0 d_1\ne0 d 1 = 0 t = x − y t=x-y t = x − y u = α x + β y u=\alpha x+\beta y u = α x + β y
P ( x , y ) = D ( h ( x − y , α x + β y ) ) ( m o d ( x , y ) n + 1 ) . (74) P(x,y)=D(h(x-y,\alpha x+\beta y))\pmod{(x,y)^{n+1}}. \tag*{(74)} P ( x , y ) = D ( h ( x − y , α x + β y )) ( mod ( x , y ) n + 1 ) . (74) Then P ( v , v ) = D ( v ) P(v,v)=D(v) P ( v , v ) = D ( v ) P 10 = α d 1 P_{10}=\alpha d_1 P 10 = α d 1 P 01 = β d 1 P_{01}=\beta d_1 P 01 = β d 1 α \alpha α J i j J_{ij} J ij
For fixed α , h \alpha,h α , h P 2 = ( D 2 ∘ D 1 − 1 ) ∘ P 1 P_2=(D_2\circ D_1^{-1})\circ P_1 P 2 = ( D 2 ∘ D 1 − 1 ) ∘ P 1 P ~ = F ∘ P \widetilde P=F\circ P P = F ∘ P F ′ ( 0 ) F'(0) F ′ ( 0 ) D ~ = F ∘ D \widetilde D=F\circ D D = F ∘ D α \alpha α
h ~ = ( F ∘ D ) − 1 ∘ F ∘ D ∘ h = h . (75) \widetilde h=(F\circ D)^{-1}\circ F\circ D\circ h=h. \tag*{(75)} h = ( F ∘ D ) − 1 ∘ F ∘ D ∘ h = h . (75) Thus α \alpha α J i j J_{ij} J ij d k d_k d k α \alpha α J i j J_{ij} J ij
At degree r ≥ 2 r\ge2 r ≥ 2 r r r i ≥ 1 i\ge1 i ≥ 1 i + j = r i+j=r i + j = r n n n n ( n + 1 ) / 2 − 1 n(n+1)/2-1 n ( n + 1 ) /2 − 1 α \alpha α
Second-order example
The first recursion step is
h ( t , u ) = u + A 2 m t 2 + B m t u + O ( ( t , u ) 3 ) , h(t,u)=u+\frac{A}{2m}t^2+\frac{B}{m}tu+O((t,u)^3), h ( t , u ) = u + 2 m A t 2 + m B t u + O (( t , u ) 3 ) , A = β 2 P 20 − 2 α β P 11 + α 2 P 02 , A=\beta^2P_{20}-2\alpha\beta P_{11}+\alpha^2P_{02}, A = β 2 P 20 − 2 α β P 11 + α 2 P 02 , B = β ( P 20 + P 11 ) − α ( P 11 + P 02 ) . (76) B=\beta(P_{20}+P_{11})-\alpha(P_{11}+P_{02}). \tag*{(76)} B = β ( P 20 + P 11 ) − α ( P 11 + P 02 ) . (76) Exponentiation preserves the t 2 t^2 t 2 t u tu t u J 20 = A / ( m α β ) = Σ J_{20}=A/(m\alpha\beta)=\Sigma J 20 = A / ( m α β ) = Σ J 11 = Σ + B / ( m α β ) J_{11}=\Sigma+B/(m\alpha\beta) J 11 = Σ + B / ( m α β )
Coefficient form of the composition law
For a zero-constant series G ( x , y ) = ∑ a + b ≥ 1 g a b x a y b / ( a ! b ! ) G(x,y)=\sum_{a+b\geq1}g_{ab}x^ay^b/(a!b!) G ( x , y ) = ∑ a + b ≥ 1 g ab x a y b / ( a ! b !)
B i j , k ( g ) = i ! j ! k ! [ x i y j ] G ( x , y ) k . (77) B_{ij,k}(g)=\frac{i!j!}{k!}[x^iy^j]G(x,y)^k. \tag*{(77)} B ij , k ( g ) = k ! i ! j ! [ x i y j ] G ( x , y ) k . (77) The Faà di Bruno composition law [32 , 57 ] then gives
P i j = ∑ k = 1 i + j Q k B i j , k ( R ) , (78) P_{ij}=\sum_{k=1}^{i+j}Q_kB_{ij,k}(R), \tag*{(78)} P ij = k = 1 ∑ i + j Q k B ij , k ( R ) , (78) i ! j ! [ t i u j ] e h = ∑ k = 1 i + j B i j , k ( h ) , (79) i!j![t^iu^j]e^h=\sum_{k=1}^{i+j}B_{ij,k}(h), \tag*{(79)} i ! j ! [ t i u j ] e h = k = 1 ∑ i + j B ij , k ( h ) , (79) where the second line uses h a b = ∂ t a ∂ u b h ( 0 , 0 ) h_{ab}=\partial_t^a\partial_u^b h(0,0) h ab = ∂ t a ∂ u b h ( 0 , 0 )
Moment and cumulant conversions
These identities specify the conversions used in the main text without listing every component. Write s ( i , p ) s(i,p) s ( i , p ) S ( i , p ) S(i,p) S ( i , p ) s ( 0 , 0 ) = S ( 0 , 0 ) = 1 s(0,0)=S(0,0)=1 s ( 0 , 0 ) = S ( 0 , 0 ) = 1
Joint factorial cumulants
Substituting x = log ( 1 + z F ) x=\log(1+z_F) x = log ( 1 + z F ) y = log ( 1 + z B ) y=\log(1+z_B) y = log ( 1 + z B )
f i j = ∑ p = 0 i ∑ q = 0 j s ( i , p ) s ( j , q ) P p q , i + j ≥ 1 , (80) f_{ij}=\sum_{p=0}^{i}\sum_{q=0}^{j}s(i,p)s(j,q)P_{pq},\qquad i+j\geq1, \tag*{(80)} f ij = p = 0 ∑ i q = 0 ∑ j s ( i , p ) s ( j , q ) P pq , i + j ≥ 1 , (80) where P 00 = 0 P_{00}=0 P 00 = 0
f 20 = P 20 − P 10 , f_{20}=P_{20}-P_{10}, f 20 = P 20 − P 10 , f 31 = P 31 − 3 P 21 + 2 P 11 , f_{31}=P_{31}-3P_{21}+2P_{11}, f 31 = P 31 − 3 P 21 + 2 P 11 , f 22 = P 22 − P 21 − P 12 + P 11 . (81) f_{22}=P_{22}-P_{21}-P_{12}+P_{11}. \tag*{(81)} f 22 = P 22 − P 21 − P 12 + P 11 . (81) The inverse conversion uses S ( i , p ) S ( j , q ) S(i,p)S(j,q) S ( i , p ) S ( j , q ) s ( i , p ) s ( j , q ) s(i,p)s(j,q) s ( i , p ) s ( j , q )
Inclusive densities, moments, and source cumulants
For disjoint windows, the integrals L i j L_{ij} L ij (53) are joint falling-factorial moments. Setting L 00 = 1 L_{00}=1 L 00 = 1
M i j = ∑ p = 0 i ∑ q = 0 j S ( i , p ) S ( j , q ) L p q . (82) M_{ij}=\sum_{p=0}^{i}\sum_{q=0}^{j}S(i,p)S(j,q)L_{pq}. \tag*{(82)} M ij = p = 0 ∑ i q = 0 ∑ j S ( i , p ) S ( j , q ) L pq . (82) Thus M 10 = a M_{10}=a M 10 = a M 01 = b M_{01}=b M 01 = b M 22 = L 22 + L 21 + L 12 + L 11 M_{22}=L_{22}+L_{21}+L_{12}+L_{11} M 22 = L 22 + L 21 + L 12 + L 11
M n ( x , y ) = ∑ 0 ≤ i + j ≤ n M i j x i y j i ! j ! , M 00 = 1. (83) \mathcal{M}_n(x,y)=\sum_{0\leq i+j\leq n}M_{ij}\frac{x^iy^j}{i!j!},\qquad M_{00}=1. \tag*{(83)} M n ( x , y ) = 0 ≤ i + j ≤ n ∑ M ij i ! j ! x i y j , M 00 = 1. (83) The source cumulants through order n n n
R i j = i ! j ! [ x i y j ] ∑ k = 1 n ( − 1 ) k − 1 k ( M n − 1 ) k , (84) R_{ij}=i!j![x^iy^j]\sum_{k=1}^{n}\frac{(-1)^{k-1}}{k}(\mathcal{M}_n-1)^k, \tag*{(84)} R ij = i ! j ! [ x i y j ] k = 1 ∑ n k ( − 1 ) k − 1 ( M n − 1 ) k , (84) for 1 ≤ i + j ≤ n 1\leq i+j\leq n 1 ≤ i + j ≤ n
R 22 = M 22 − 2 a M 12 − 2 b M 21 − M 20 M 02 − 2 M 11 2 + 2 a 2 M 02 + 2 b 2 M 20 + 8 a b M 11 − 6 a 2 b 2 . (85) \begin{aligned}
R_{22}={}&M_{22}-2aM_{12}-2bM_{21}-M_{20}M_{02}\\
&-2M_{11}^2+2a^2M_{02}+2b^2M_{20}\\
&+8abM_{11}-6a^2b^2. \tag*{(85)}
\end{aligned} R 22 = M 22 − 2 a M 12 − 2 b M 21 − M 20 M 02 − 2 M 11 2 + 2 a 2 M 02 + 2 b 2 M 20 + 8 ab M 11 − 6 a 2 b 2 . (85) Setting n = 4 n=4 n = 4 H i j H_{ij} H ij (101) gives the cumulants directly and avoids the intermediate ordinary moments.
Pair-only string model and its symmetric limit
The pair-only model of § 1.2 C is specified through fourth order by a = μ 0 w F a=\mu_0w_F a = μ 0 w F b = μ 0 w B b=\mu_0w_B b = μ 0 w B U F , U B , U F B U_F,U_B,U_{FB} U F , U B , U F B (55) . A compact way to generate all required source cumulants is
R ( x , y ) = a z F + b z B + 1 2 U F z F 2 + U F B z F z B + 1 2 U B z B 2 + O ( ( x , y ) 5 ) , R(x,y)=az_F+bz_B+\frac{1}{2}U_Fz_F^2+U_{FB}z_Fz_B+\frac{1}{2}U_Bz_B^2+O((x,y)^5), R ( x , y ) = a z F + b z B + 2 1 U F z F 2 + U F B z F z B + 2 1 U B z B 2 + O (( x , y ) 5 ) , z F = e x − 1 , z B = e y − 1. (86) z_F=e^x-1,\qquad z_B=e^y-1. \tag*{(86)} z F = e x − 1 , z B = e y − 1. (86) The closure C 3 = C 4 = 0 C_{3}=C_{4}=0 C 3 = C 4 = 0 (56) .
For arbitrary widths, use the general algebraic prescription in Appendix § 2.1.4 below. Insert these derivatives into Eqs. (103) and (104) , then use Eqs. (107) and (108) . Those algebraic formulas apply to any source cumulants and do not require the Gaussian kernel used to evaluate the integrals in Appendix § 2.1.4 . This prescription specifies all nine canonical coordinates without separate polynomial expansions. The alternative coordinate Υ \Upsilon Υ
For the homogeneous source considered here, equal window widths imply a = b a=b a = b U F = U B U_{F}=U_{B} U F = U B Δ \Delta Δ Γ \Gamma Γ (14) and (15) . All nine canonical J i j J_{ij} J ij Υ \Upsilon Υ
J 20 = J 11 = J 30 = J 12 = J 31 = J 13 = Σ , (87) J_{20}=J_{11}=J_{30}=J_{12}=J_{31}=J_{13}=\Sigma, \tag*{(87)} J 20 = J 11 = J 30 = J 12 = J 31 = J 13 = Σ , (87) where
Σ = 1 + U F − U F B a . (88) \Sigma=1+\frac{U_{F}-U_{FB}}{a}. \tag*{(88)} Σ = 1 + a U F − U F B . (88) The remaining three canonical coordinates and the alternative observable Υ \Upsilon Υ
J 21 = 1 + 2 ( U F − U F B ) a − U F 2 − U F B 2 a 2 , J_{21}=1+\frac{2(U_{F}-U_{FB})}{a}-\frac{U_{F}^{2}-U_{FB}^{2}}{a^{2}}, J 21 = 1 + a 2 ( U F − U F B ) − a 2 U F 2 − U F B 2 , J 22 = 1 + 4 ( U F − U F B ) a − 5 ( U F 2 − U F B 2 ) a 2 + 2 ( U F + U F B ) 2 ( U F − U F B ) a 3 . (89) \begin{aligned}
J_{22}&=1+\frac{4(U_{F}-U_{FB})}{a}-\frac{5(U_{F}^{2}-U_{FB}^{2})}{a^{2}}\\
&\quad+\frac{2(U_{F}+U_{FB})^{2}(U_{F}-U_{FB})}{a^{3}}. \tag*{(89)}
\end{aligned} J 22 = 1 + a 4 ( U F − U F B ) − a 2 5 ( U F 2 − U F B 2 ) + a 3 2 ( U F + U F B ) 2 ( U F − U F B ) . (89) and
J 40 = 1 + 7 ( U F − U F B ) 4 a − 3 ( U F 2 − U F B 2 ) 2 a 2 − 3 ( U F + U F B ) ( U F − U F B ) 2 4 a 3 , Υ = 1 + 5 ( U F − U F B ) 3 a + 4 U F ( U F − U F B ) a 2 + 2 ( U F − U F B ) 2 ( 2 U F + U F B ) 3 a 3 . (90) \begin{aligned}
J_{40}&=1+\frac{7(U_{F}-U_{FB})}{4a}-\frac{3(U_{F}^{2}-U_{FB}^{2})}{2a^{2}}\\
&\quad-\frac{3(U_{F}+U_{FB})(U_{F}-U_{FB})^{2}}{4a^{3}},\\
\Upsilon&=1+\frac{5(U_{F}-U_{FB})}{3a}+\frac{4U_{F}(U_{F}-U_{FB})}{a^{2}}\\
&\quad+\frac{2(U_{F}-U_{FB})^{2}(2U_{F}+U_{FB})}{3a^{3}}. \tag*{(90)}
\end{aligned} J 40 Υ = 1 + 4 a 7 ( U F − U F B ) − 2 a 2 3 ( U F 2 − U F B 2 ) − 4 a 3 3 ( U F + U F B ) ( U F − U F B ) 2 , = 1 + 3 a 5 ( U F − U F B ) + a 2 4 U F ( U F − U F B ) + 3 a 3 2 ( U F − U F B ) 2 ( 2 U F + U F B ) . (90) Production settings and auxiliary acceptance correction
Trieste 2 settings
Table Table V specifies every entry applied after Tune:pp=14 to obtain the Trieste 2 sample. Entries that keep the baseline value are included so that the complete card is explicit.
Table V. Complete Trieste 2 override card and the corresponding Monash 8.318 values. Both samples start from Tune:pp=14.
Auxiliary acceptance correction
The published ALICE pseudorapidity density, measured over the full p T p_{\mathrm{T}} p T [56 ] , is converted to the target acceptance using an auxiliary sample of one million generated Monash PYTHIA 8.318 events. Full- and restricted-p T p_{\mathrm{T}} p T > > >
The preliminary Σ \Sigma Σ [20 ] . They are not rescaled by this one-particle correction, and matching the particle and event selections remains a separate assumption.
Explicit window cumulants and observables for the fitted Gaussian hierarchy
This appendix evaluates the Gaussian hierarchy of Eq. (62) through fourth order. We consider disjoint windows F = [ η F − , η F + ] F=[\eta_{F}^{-},\eta_{F}^{+}] F = [ η F − , η F + ] B = [ η B − , η B + ] B=[\eta_{B}^{-},\eta_{B}^{+}] B = [ η B − , η B + ]
Window integrals of the connected densities
For the one-particle density λ 1 ( η ) = μ 0 f ( η ) \lambda_{1}(\eta)=\mu_{0}f(\eta) λ 1 ( η ) = μ 0 f ( η ) f ( η ) = 1 + c 2 η 2 + c 4 η 4 f(\eta)=1+c_{2}\eta^{2}+c_{4}\eta^{4} f ( η ) = 1 + c 2 η 2 + c 4 η 4
a = R 10 = μ 0 [ η + c 2 η 3 3 + c 4 η 5 5 ] η F − η F + , (91) a=R_{10}=\mu_{0}\left[\eta+\frac{c_{2}\eta^{3}}{3}+\frac{c_{4}\eta^{5}}{5}\right]_{\eta_{F}^{-}}^{\eta_{F}^{+}}, \tag*{(91)} a = R 10 = μ 0 [ η + 3 c 2 η 3 + 5 c 4 η 5 ] η F − η F + , (91) b = R 01 = μ 0 [ η + c 2 η 3 3 + c 4 η 5 5 ] η B − η B + . (92) b=R_{01}=\mu_{0}\left[\eta+\frac{c_{2}\eta^{3}}{3}+\frac{c_{4}\eta^{5}}{5}\right]_{\eta_{B}^{-}}^{\eta_{B}^{+}}. \tag*{(92)} b = R 01 = μ 0 [ η + 3 c 2 η 3 + 5 c 4 η 5 ] η B − η B + . (92) For nonnegative integers i , j i,j i , j 2 ≤ n = i + j ≤ 4 2\leq n=i+j\leq4 2 ≤ n = i + j ≤ 4
H i j = ∫ F i × B j C n ( η 1 , … , η n ) d η 1 ⋯ d η n . (93) H_{ij}=\int_{F^{i}\times B^{j}}C_{n}(\eta_{1},\ldots,\eta_{n})\,\mathrm{d}\eta_{1}\cdots\mathrm{d}\eta_{n}. \tag*{(93)} H ij = ∫ F i × B j C n ( η 1 , … , η n ) d η 1 ⋯ d η n . (93) These are integrated connected factorial densities, rather than inclusive densities or ordinary cumulants. They are determined by the model parameters and introduce no additional fit parameters. Their conversion to ordinary source cumulants is given in the next subsection.
Writing η ˉ = n − 1 ∑ r η r \bar{\eta}=n^{-1}\sum_{r}\eta_{r} η ˉ = n − 1 ∑ r η r ∑ r < s ( η r − η s ) 2 = n ∑ r ( η r − η ˉ ) 2 \sum_{r<s}(\eta_{r}-\eta_{s})^{2}=n\sum_{r}(\eta_{r}-\bar{\eta})^{2} ∑ r < s ( η r − η s ) 2 = n ∑ r ( η r − η ˉ ) 2 z z z ℓ > 0 \ell>0 ℓ > 0 z z z
∫ − ∞ ∞ e − ∑ r ( η r − z ) 2 / ℓ 2 d z = π ℓ n e − ∑ r < s ( η r − η s ) 2 / ( n ℓ 2 ) . (94) \int_{-\infty}^{\infty}e^{-\sum_{r}(\eta_{r}-z)^{2}/\ell^{2}}\,\mathrm{d}z=\frac{\sqrt{\pi}\ell}{\sqrt{n}}e^{-\sum_{r<s}(\eta_{r}-\eta_{s})^{2}/(n\ell^{2})}. \tag*{(94)} ∫ − ∞ ∞ e − ∑ r ( η r − z ) 2 / ℓ 2 d z = n π ℓ e − ∑ r < s ( η r − η s ) 2 / ( n ℓ 2 ) . (94) Substituting this identity into Eq. (93) yields
H i j = μ 0 a n n π ℓ ∫ − ∞ ∞ F F ( z ) i F B ( z ) j d z , (95) H_{ij}=\frac{\mu_0a_n\sqrt{n}}{\sqrt{\pi}\ell}\int_{-\infty}^{\infty}F_F(z)^iF_B(z)^j\,\mathrm{d}z, \tag*{(95)} H ij = π ℓ μ 0 a n n ∫ − ∞ ∞ F F ( z ) i F B ( z ) j d z , (95) F W ( z ) = ∫ η W − η W + f ( η ) e − ( η − z ) 2 / ℓ 2 d η , W ∈ { F , B } . (96) F_W(z)=\int_{\eta_W^-}^{\eta_W^+}f(\eta)e^{-(\eta-z)^2/\ell^2}\,\mathrm{d}\eta,\qquad W\in\{F,B\}. \tag*{(96)} F W ( z ) = ∫ η W − η W + f ( η ) e − ( η − z ) 2 / ℓ 2 d η , W ∈ { F , B } . (96) For the polynomial profile, the inner integral can be evaluated explicitly. Using c 0 = 1 c_0=1 c 0 = 1 s = ( η − z ) / ℓ s=(\eta-z)/\ell s = ( η − z ) / ℓ
F W ( z ) = ℓ ∑ k ∈ { 0 , 2 , 4 } c k ∑ r = 0 k ( k r ) z k − r ℓ r [ I r ( η W + − z ℓ ) − I r ( η W − − z ℓ ) ] . (97) F_W(z)=\ell\sum_{k\in\{0,2,4\}}c_k\sum_{r=0}^{k}\binom{k}{r}z^{k-r}\ell^r\left[I_r\left(\frac{\eta_W^+-z}{\ell}\right)-I_r\left(\frac{\eta_W^--z}{\ell}\right)\right]. \tag*{(97)} F W ( z ) = ℓ k ∈ { 0 , 2 , 4 } ∑ c k r = 0 ∑ k ( r k ) z k − r ℓ r [ I r ( ℓ η W + − z ) − I r ( ℓ η W − − z ) ] . (97) Here I r I_r I r I r ′ ( t ) = t r e − t 2 I_r'(t)=t^re^{-t^2} I r ′ ( t ) = t r e − t 2
I 0 ( t ) = π 2 erf ( t ) , I_0(t)=\frac{\sqrt{\pi}}{2}\operatorname{erf}(t), I 0 ( t ) = 2 π erf ( t ) , I 1 ( t ) = − 1 2 e − t 2 , I_1(t)=-\frac{1}{2}e^{-t^2}, I 1 ( t ) = − 2 1 e − t 2 , I r ( t ) = − 1 2 t r − 1 e − t 2 + r − 1 2 I r − 2 ( t ) , r = 2 , 3 , 4. (98) I_r(t)=-\frac{1}{2}t^{r-1}e^{-t^2}+\frac{r-1}{2}I_{r-2}(t),\qquad r=2,3,4. \tag*{(98)} I r ( t ) = − 2 1 t r − 1 e − t 2 + 2 r − 1 I r − 2 ( t ) , r = 2 , 3 , 4. (98) Only differences of these antiderivatives enter Eq. (97) , so their additive constants are immaterial. Equations (E5), (E7), and (E8) reduce every required connected integral through fourth order to a one-dimensional quadrature for the finite windows and the fitted nonuniform profile. The reduction is exact. Only the remaining integration over z z z
All fourteen ordinary source cumulants
For disjoint windows, the conversion from integrated connected factorial densities to ordinary cumulants fol lows from
R ( x , y ) = K ( e x − 1 , e y − 1 ) , (99) R(x,y)=K(e^x-1,e^y-1), \tag*{(99)} R ( x , y ) = K ( e x − 1 , e y − 1 ) , (99) where K K K
K ( z F , z B ) = a z F + b z B + ∑ p , q ≥ 0 2 ≤ p + q ≤ 4 H p q z F p z B q p ! q ! + O ( ( z F , z B ) 5 ) . (100) K(z_F,z_B)=az_F+bz_B+\sum_{\substack{p,q\geq0\\2\leq p+q\leq4}}H_{pq}\frac{z_F^p z_B^q}{p!q!}+O\left((z_F,z_B)^5\right). \tag*{(100)} K ( z F , z B ) = a z F + b z B + p , q ≥ 0 2 ≤ p + q ≤ 4 ∑ H pq p ! q ! z F p z B q + O ( ( z F , z B ) 5 ) . (100) Terms of higher total degree cannot contribute to ordinary cumulants through fourth order.
For a compact expression, extend the notation by H 10 = a H_{10}=a H 10 = a H 01 = b H_{01}=b H 01 = b H 00 = 0 H_{00}=0 H 00 = 0 (99) gives
R i j = ∑ p = 0 i ∑ q = 0 j { i p } { j q } H p q , 1 ≤ i + j ≤ 4 , (101) R_{ij}=\sum_{p=0}^{i}\sum_{q=0}^{j}\left\{\begin{matrix}i\\p\end{matrix}\right\}\left\{\begin{matrix}j\\q\end{matrix}\right\}H_{pq},\qquad1\leq i+j\leq4, \tag*{(101)} R ij = p = 0 ∑ i q = 0 ∑ j { i p } { j q } H pq , 1 ≤ i + j ≤ 4 , (101) where the braces denote Stirling numbers of the second kind, with { 0 0 } = 1 \left\{\begin{matrix}0\\0\end{matrix}\right\}=1 { 0 0 } = 1 § 3.3.2 .
The complete list of fourteen ordinary source cumulants is
R 10 = a , R 01 = b , R 20 = a + H 20 , R 02 = b + H 02 , R 11 = H 11 , R 30 = a + 3 H 20 + H 30 , R 03 = b + 3 H 02 + H 03 , R 21 = H 11 + H 21 , R 12 = H 11 + H 12 , R 40 = a + 7 H 20 + 6 H 30 + H 40 , R 04 = b + 7 H 02 + 6 H 03 + H 04 , R 31 = H 11 + 3 H 21 + H 31 , R 22 = H 11 + H 21 + H 12 + H 22 , R 13 = H 11 + 3 H 12 + H 13 . (102) \begin{aligned}
R_{10}&=a, & R_{01}&=b,\\
R_{20}&=a+H_{20}, & R_{02}&=b+H_{02},\\
R_{11}&=H_{11},\\
R_{30}&=a+3H_{20}+H_{30},\\
R_{03}&=b+3H_{02}+H_{03},\\
R_{21}&=H_{11}+H_{21}, & R_{12}&=H_{11}+H_{12},\\
R_{40}&=a+7H_{20}+6H_{30}+H_{40},\\
R_{04}&=b+7H_{02}+6H_{03}+H_{04},\\
R_{31}&=H_{11}+3H_{21}+H_{31},\\
R_{22}&=H_{11}+H_{21}+H_{12}+H_{22},\\
R_{13}&=H_{11}+3H_{12}+H_{13}. \tag*{(102)}
\end{aligned} R 10 R 20 R 11 R 30 R 03 R 21 R 40 R 04 R 31 R 22 R 13 = a , = a + H 20 , = H 11 , = a + 3 H 20 + H 30 , = b + 3 H 02 + H 03 , = H 11 + H 21 , = a + 7 H 20 + 6 H 30 + H 40 , = b + 7 H 02 + 6 H 03 + H 04 , = H 11 + 3 H 21 + H 31 , = H 11 + H 21 + H 12 + H 22 , = H 11 + 3 H 12 + H 13 . R 01 R 02 R 12 = b , = b + H 02 , = H 11 + H 12 , (102) The terms proportional to H 30 H_{30} H 30 H 21 H_{21} H 21 H 12 H_{12} H 12 H 03 H_{03} H 03 C 3 C_3 C 3 C 4 = 0 C_4=0 C 4 = 0 C 4 C_4 C 4
Here the general construction is applied to the single-source cumulant series R R R m = a + b m=a+b m = a + b α = a / m \alpha=a/m α = a / m β = b / m \beta=b/m β = b / m
K i j = 1 m ∂ t i ∂ u j R ( β t + u , − α t + u ) ∣ t = u = 0 . (103) K_{ij}=\left.\frac{1}{m}\partial_t^i\partial_u^j R(\beta t+u,-\alpha t+u)\right|_{t=u=0}. \tag*{(103)} K ij = m 1 ∂ t i ∂ u j R ( β t + u , − α t + u ) t = u = 0 . (103) They are the joint cumulants of the source imbalance β μ F − α μ B \beta\mu_F-\alpha\mu_B β μ F − α μ B μ F + μ B \mu_F+\mu_B μ F + μ B m m m c 2 c_2 c 2 c 4 c_4 c 4 C n C_n C n (102) .
All required directional cumulants follow from the finite binomial sum
m K i j = ∑ p = 0 i ∑ q = 0 j ( i p ) ( j q ) β p ( − α ) i − p R p + q , i + j − p − q . (104) mK_{ij}=\sum_{p=0}^{i}\sum_{q=0}^{j}\binom{i}{p}\binom{j}{q}\beta^p(-\alpha)^{i-p}R_{p+q,i+j-p-q}. \tag*{(104)} m K ij = p = 0 ∑ i q = 0 ∑ j ( p i ) ( q j ) β p ( − α ) i − p R p + q , i + j − p − q . (104) For example, m K 20 = β 2 R 20 − 2 α β R 11 + α 2 R 02 mK_{20}=\beta^2R_{20}-2\alpha\beta R_{11}+\alpha^2R_{02} m K 20 = β 2 R 20 − 2 α β R 11 + α 2 R 02 ( β ∂ x − α ∂ y ) i ( ∂ x + ∂ y ) j R (\beta\partial_x-\alpha\partial_y)^i(\partial_x+\partial_y)^jR ( β ∂ x − α ∂ y ) i ( ∂ x + ∂ y ) j R
The conventional second- and third-order observables Δ \Delta Δ Γ \Gamma Γ
Δ = β 2 R 20 − α 2 R 02 m α β ( β − α ) , (105) \Delta=\frac{\beta^2R_{20}-\alpha^2R_{02}}{m\alpha\beta(\beta-\alpha)}, \tag*{(105)} Δ = m α β ( β − α ) β 2 R 20 − α 2 R 02 , (105) Γ = K 30 α β ( β − α ) . (106) \Gamma=\frac{K_{30}}{\alpha\beta(\beta-\alpha)}. \tag*{(106)} Γ = α β ( β − α ) K 30 . (106) These normalizations require α ≠ β \alpha\ne\beta α = β
Υ = ( K 02 + ( β − α ) K 11 α β ) ( K 20 α β ) 2 + ( 1 3 α β − 1 ) ( K 20 α β ) 3 − K 40 3 α 2 β 2 . (107) \Upsilon=\left(K_{02}+\frac{(\beta-\alpha)K_{11}}{\alpha\beta}\right)\left(\frac{K_{20}}{\alpha\beta}\right)^2+\left(\frac{1}{3\alpha\beta}-1\right)\left(\frac{K_{20}}{\alpha\beta}\right)^3-\frac{K_{40}}{3\alpha^2\beta^2}. \tag*{(107)} Υ = ( K 02 + α β ( β − α ) K 11 ) ( α β K 20 ) 2 + ( 3 α β 1 − 1 ) ( α β K 20 ) 3 − 3 α 2 β 2 K 40 . (107) Evaluating Eq. (72) through fourth order gives the nine canonical coordinates:
J 20 = Σ = κ 20 α β , J 11 = κ 20 + κ 11 α β , J 30 = κ 30 + 2 α κ 20 α β , J 21 = κ 21 + ( 1 − κ 02 ) κ 20 α β , J 12 = κ 12 + 2 ( 1 − κ 02 ) κ 11 + κ 20 α β , J 40 = κ 40 + 3 ( 1 − κ 02 ) κ 20 2 α β + 3 κ 20 , J 31 = 1 α β [ κ 31 + 2 α κ 20 + ( 1 − κ 02 ) ( κ 30 + 3 κ 20 κ 11 ) ] , J 22 = 1 α β [ κ 22 + 2 ( 1 − κ 02 ) ( κ 21 + κ 11 2 ) + ( 1 − 2 κ 02 + 2 κ 02 2 − κ 03 ) κ 20 ] , J 13 = 1 α β [ κ 13 + 3 ( 1 − κ 02 ) κ 12 + κ 20 + 3 ( 1 − 2 κ 02 + 2 κ 02 2 − κ 03 ) κ 11 ] . (108) \begin{aligned}
J_{20} &= \Sigma= \frac{\kappa_{20}}{\alpha\beta},\\
J_{11} &= \frac{\kappa_{20}+\kappa_{11}}{\alpha\beta},\\
J_{30} &= \frac{\kappa_{30}+2\alpha\kappa_{20}}{\alpha\beta},\\
J_{21} &= \frac{\kappa_{21}+(1-\kappa_{02})\kappa_{20}}{\alpha\beta},\\
J_{12} &= \frac{\kappa_{12}+2(1-\kappa_{02})\kappa_{11}+\kappa_{20}}{\alpha\beta},\\
J_{40} &= \frac{\kappa_{40}+3(1-\kappa_{02})\kappa_{20}^{2}}{\alpha\beta}+3\kappa_{20},\\
J_{31} &= \frac{1}{\alpha\beta}\left[\kappa_{31}+2\alpha\kappa_{20}+(1-\kappa_{02})(\kappa_{30}+3\kappa_{20}\kappa_{11})\right],\\
J_{22} &= \frac{1}{\alpha\beta}\left[\kappa_{22}+2(1-\kappa_{02})(\kappa_{21}+\kappa_{11}^{2})+(1-2\kappa_{02}+2\kappa_{02}^{2}-\kappa_{03})\kappa_{20}\right],\\
J_{13} &= \frac{1}{\alpha\beta}\left[\kappa_{13}+3(1-\kappa_{02})\kappa_{12}+\kappa_{20}+3(1-2\kappa_{02}+2\kappa_{02}^{2}-\kappa_{03})\kappa_{11}\right]. \tag*{(108)}
\end{aligned} J 20 J 11 J 30 J 21 J 12 J 40 J 31 J 22 J 13 = Σ = α β κ 20 , = α β κ 20 + κ 11 , = α β κ 30 + 2 α κ 20 , = α β κ 21 + ( 1 − κ 02 ) κ 20 , = α β κ 12 + 2 ( 1 − κ 02 ) κ 11 + κ 20 , = α β κ 40 + 3 ( 1 − κ 02 ) κ 20 2 + 3 κ 20 , = α β 1 [ κ 31 + 2 α κ 20 + ( 1 − κ 02 ) ( κ 30 + 3 κ 20 κ 11 ) ] , = α β 1 [ κ 22 + 2 ( 1 − κ 02 ) ( κ 21 + κ 11 2 ) + ( 1 − 2 κ 02 + 2 κ 02 2 − κ 03 ) κ 20 ] , = α β 1 [ κ 13 + 3 ( 1 − κ 02 ) κ 12 + κ 20 + 3 ( 1 − 2 κ 02 + 2 κ 02 2 − κ 03 ) κ 11 ] . (108) Equations (E5), (E12), and (E14), together with Eqs. (E17) and (E18) provide an explicit finite evaluation of the observables for the fitted Gaussian hierarchy. No numerical differentiation or unspecified higher-order cumulants are required.
Setting a 3 = a 4 = 0 a_3=a_4=0 a 3 = a 4 = 0 a a a b b b U F U_F U F U B U_B U B U F B U_{FB} U F B
U F = H 20 , U B = H 02 , U F B = H 11 . (109) U_F=H_{20},\qquad U_B=H_{02},\qquad U_{FB}=H_{11}. \tag*{(109)} U F = H 20 , U B = H 02 , U F B = H 11 . (109) The integrals are evaluated here with the Gaussian kernel and fitted nonuniform profile, rather than the homogeneous exponential model of Appendix C.
Setting a 2 = a 3 = a 4 = 0 a_2=a_3=a_4=0 a 2 = a 3 = a 4 = 0 (72) then yields Σ = J i j = 1 \Sigma=J_{ij}=1 Σ = J ij = 1 Υ = 1 \Upsilon=1 Υ = 1 Δ \Delta Δ Γ \Gamma Γ α ≠ β \alpha\ne\beta α = β J i j J_{ij} J ij Υ \Upsilon Υ Δ \Delta Δ Γ \Gamma Γ
At fixed a n a_n a n ℓ \ell ℓ μ 0 \mu_0 μ 0 m m m (103) . Under the independent-source assumptions of Sec. II, fluctuations of the number of sources also leave the observables unchanged, by the outer-composition cancellation established in Appendix A.
References 
