Factorial Cumulant Moments Research Articles

Statistical moments in superposition models and strongly intensive measures

First, we present a concise glossary of formulas for composition of standard, cumulant, factorial, and factorial cumulant moments in superposition (compound) models, where final particles are created via independent emission from a collection of sources. Explicit mathematical formulas for the composed moments are given to all orders. We discuss the composition laws for various types of moments via the generating-function methods and list the formulas for the unfolding of the unwanted fluctuations. Second, the technique is applied to the difference of the scaled multiplicities of two particle types. This allows us for a systematic derivation and a simple algebraic interpretation of the so-called strongly intensive fluctuation measures. With the help of the formalism we obtain several new strongly intensive measures involving higher-rank moments. The reviewed as well as the new results may be useful in investigations of mechanisms of particle production and event-by-event fluctuations in high-energy nuclear and hadronic collisions, and in particular in the search for signatures of the QCD phase transition at a finite baryon density.

Measurements of charged hadron fluctuations in pseudo-rapidity bins in 16O-AgBr at 60 A GeV and 32S-AgBr at 200 A GeV

We have measured the hadron-hadron correlation in the multiparticle production process of 16O-AgBr interactions at 60A GeV and 32S-AgBr interactions at 200A GeV in the forward and backward zone of the pseudo-rapidity space using the normalized factorial cumulant moment method. The experimental results have been compared with the Monte Carlo data generated according to the independent emission model. The results reveal that the observed correlation between the produced hadrons is purely dynamical. Correlation is found to be stronger in the backward zone than that in the forward zone of the pseudo-rapidity space for both interactions. Factorial cumulant moments for the experimental data show a power law rise with decreasing phase space bin size. The results obtained from the experimental data have also been compared with those obtained from analyzing FRITIOF data. Though FRITIOF data itself show the presence of a measured correlation among the hadrons, in most of the cases the values of the factorial cumulant moments are much less than the experimental values and they remain constant with phase space bin size. The study might hint towards the Bose-Einstein correlation as the origin of the observed behavior of the measured correlation. The failure of FRITIOF data to reproduce the exact experimental behavior in terms of factorial cumulant moment also supports this notion as the model does not take the BE correlation into account.

Genuine pion–pion correlations in heavy-ion collisions

The search for genuine two-particle dynamical correlations among the pions produced in 16O–AgBr interactions at 60 AGeV and 32S–AgBr interactions at 200 AGeV (super proton synchrotron data) has been carried out by the method of normalized factorial cumulant moments in pseudo-rapidity (η), azimuthal angle (ϕ) and two-dimensional anisotropic (η–ϕ) phase space. The analysis reveals the presence of genuine two-particle dynamical correlations among the pions in 16O–AgBr and 32S–AgBr interactions in both one and two dimensions. Stronger correlation has been observed in higher dimensional space. Correlations between the pions are found to decrease with the increase of projectile mass and energy. Experimental results have been compared with the FRITIOF data sample in the pseudo-rapidity space. In azimuthal angle space, events simulated by generating random numbers (Monte Carlo Simulation) have been used for the purpose of comparison. The FRITIOF data and the data of the simulated interactions have failed to reproduce the experimental results. The experimental results have also been compared with the event sample generated by the UrQMD model in both pseudo-rapidity and azimuthal angle space. Like the FRITIOF model, the UrQMD model is also unable to replicate the experimental results. The variation of normalized cumulant moment (K2) against the number of bin M has been fitted with a function K2 = aMb in both one and two dimensions. No fitting of this kind has been possible in the case of the FRITIOF data, UrQMD data and the MC simulated data.

Genuine two particle correlations of target fragments in nuclear interactions

The search for genuine two-particle dynamical correlations among the target evaporated slow particles in 12C-AgBr and 24Mg-AgBr interactions at 4.5 AGeV and 28Si-AgBr interactions at 14.5 AGeV in the emission angle space has been carried out using the method of normalized factorial cumulant moments. The analysis reveals the presence of genuine two-particle dynamical correlations among the target fragments in 12C-AgBr, 24Mg-AgBr, and 28Si-AgBr interactions. The correlations among the target evaporated slow particles are found to decrease with an increase in mass and energy of the projectile beam.

The particle emitter size dependence on energy in e+e− annihilations into hadrons

The nearly energy independent hadron emitter dimension r, measured in e+e− annihilation in the energy range 10 to 91 GeV via the Bose–Einstein correlation of two identical charged pions, is shown to be well accounted for by choosing the hadron jets as independent pion sources. To this end the known normalised factorial cumulant moments dependence on particle sources is adapted to the Bose–Einstein correlation formalism to yield a relation between r and these sources. This approach is also able to account for the measured r values obtained for the Z0 decays into two and three hadron jets. Finally the estimated r value of the hadronic ϒ(9.46) decay via three gluons is expected to be higher by about 6 to 11% over that predicted for its one photon hadronic decay mode.

Factorial cumulant moments and correlation integrals in pp collisions at 400 GeV/c

The pseudorapidity distribution of charged particles produced in pp collisions at 400 GeV/c was measured by using LEBC films offered by CERN NA27 collaboration. The scaled factorial cumulant moments have been calculated. The results show that the second order cumulants have positive values, while the cumulants of higher order are consistent with zero except for the situation ofnch ≥ 4 events, where the third-order cumulants have positive values beyond the statistical uncertainties. It means that the observed increase of the higherorder factorial momentsFq is almost due to the dynamical two-particle correlation in pp collisions at 400 GeV/c. The cumulant moments also have been calculated by star correlation integrals. It significantly reduced the statistical errors, especially for higher order cumulants. From Monte Carlo events with the same single particle spectrum and no correlations, we observed that for broad mixed-multiplicity distributions, a significant part ofK2 is coming from the single-particle fluctuation due to the fluctuating multiplicity.

Properties of factorial cumulant to factorial moment ratio

It is shown that the ratio of factorial cumulant moments to factorial moments for a multiplicity distribution truncated in the tail reveals oscillations in sign similar to those observed in experimental data. It is suggested that this effect be taken into account in the analysis of data in order to obtain correct physical information on the multiplicity distributions.

Factorial Cumulant Moments and Short Range Correlation in P-P Collisions at 400 GeV/c

The pseudorapidity distribution of charged particles produced in p-p collisions at 400 GeV/c was measured by using the Lexan bubble chamber films offered by the CERN NA27 collaboration. The scaled factorial cumulant moments have been calculated. The results show that the second order cumulants have positive values, while the cumulants of higher order are consistent with zero except for the case with nch ⩾ 4 events, where the third-order cumulants have small positive values beyond the statistical uncertainties. From Monte Carlo events with the same single particle spectrum and no correlations, we observed that for broad mixed-multiplicity distributions, a significant part of K2 is coming from the single-particle fluctuation due to the fluctuation multiplicity.

Intermittency and correlations in 16O+Ag/Br interactions at 2.1 GeV/nucleon.

An increase of the scaled factorial moments with decreasing bin size of phase space---the phenomenon popularly known as intermittency---is a common feature in multiparticle production in relativistic high energy interactions. The data of $^{16}\mathrm{O}$ + Ag/Br interactions at 2.1 GeV/nucleon also revealed the same feature which, usually, is identified as the revelation of singularity structure of correlation functions. In this article, we extend our analysis of $^{16}\mathrm{O}$ + Ag/Br interactions at 2.1 GeV/nucleon data in terms of other tools (giving correlations) such as the scaled factorial correlators, the factorial cumulant moments, and the split-bin correlation functions. The study reveals that the information on correlations of our particle production data are contained in two-particle dynamical correlations only and that the two-particle dynamical correlations are due to a resonancelike production mechanism.

Factorial cumulant moments and correlations in pp collisions at 400 GeV/c

The pseudorapidity distribution of charged particles produced in pp collisions at 400 GeV/ c was measured by using LEBC films offered by the CERN NA27 Collaboration. The scaled factorial cumulant moments have been calculated. The results shows that the second-order cumulants have positive values, while the cumulants of higher order are consistent with zero except for the situation of n ch ⩾ 4 events, where the third-order cumulants have positive values beyond the statistical uncertainties. It means that the observed increase of the higher-order factorial moments F q is almost due to the short range dynamical two-particle correlations in pp collisions at 400 GeV/ c.

Factorial moments, cumulants and correlation integrals in ?+ p andK + p interactions at 250 GeV/c

A selected sample of 59200 π+p andK+p nonsingle-diffrative interactions at\(\sqrt s \) =22 GeV is used to investigate one, two- and three-dimensional factorial moments, factorial cumulant moments, as well as correlation integrals. The rise of factorial moments and cumulants with decreasing phase-space volume is stronger when evaluated in three than in lower dimensions. Ratios of slopes are easier to obtain than the slopes themselves. Contrary to earlier findings, they turn out to depend on the dimension. The order dependence of the averaged ratios is better described by a Levy stable law solution with μ=1.6 than by Gaussian approximation of the α-model (μ=2) or a second order phase transition (μ=0), but values μ>2 inconsistent with Levy-type fluctuations are reached in a three-dimensional analysis. The multiparticle contributions to the factorial moments are calculated by means of factorial cumulant moments. A particular improvement of the method is that of correlation (or density) integrals. It leads to the conclusion that Bose-Einstein interference plays an important role in the intermittency effect, but indication is found for an interpretation alternative to the conventional view of Bose-Einstein correlations.

Systematic investigation of scaled factorial cumulant moments for nucleus-nucleus interactions.

We use scaled factorial cumulant moments to analyze pseudorapidity fluctuations in nucleus-nucleus collisions. This approach is similar to the method of scaled factorial moments except that it removes the effects of lower-order correlations upon a given moment. Significant second-order cumulants and cumulant indices (slopes with respect to bin size) are found for all of the interactions studied. These indices are found to have an inverse dependence upon average pseudorapidity particle density.Received 21 July 1992DOI:https://doi.org/10.1103/PhysRevD.47.3726©1993 American Physical Society

FRACTAL STRUCTURES AND CORRELATIONS IN HADRONIC MULTIPARTICLE DISTRIBUTIONS

Multihadron rapidity distributions exhibit highly irregular (perhaps fractal) event structure. Application of methods used in nonlinear dynamics to hadronic data is made uncertain by the relatively small number of particles in a given event. We analyze three standard methods to assess their applicability: the Hausdorff (box-counting) method, the correlation dimension, and the information dimension. The Hausdorff method can work for large multiplicities if the fractal set (strange attractor) has simple self-similar behavior. It is noted that addition of points from different events will doubtless erase sponge-like structure characteristic of the attractor. This defect is even greater for the information dimension, whose proper definition requires probabilities whose evaluation involves event averaging. For a fixed attractor, this is of no consequence; however, individual collisions differ by a "noise" effect. The correlation dimension, which has the merit of rapid computational convergence, depends only on relative rapidities |yi − yj| and therefore should not depend on overall rapidity shifts between events. Possible statistical independence of different subsets of a given partition of n particles is analyzed using factorial cumulant moments and information entropy. Additivity of constituent cumulants and entropies is characteristic of statistical independence. Conditional entropies are introduced and used to generalize conventional definitions of information entropy and its (Renyi) generalization.

Counting statistics of partial coherent light with Lorentzian spectrum.

Analytic aspects of the photon-counting distributions in quantum optics are investigated for the case where the underlying field includes a mixture of harmonic signal and thermal noise. A Lorentzian spectral profile for the thermal light is used to find simple iterative relations for the factorial cumulant moments. Explicit analytic expressions for the moments are then given to eighth order. Approximate interpolation formulas, which display proper limiting behavior at small and large correlation length, are also investigated.