A Monte Carlo Study of Erraticity Behavior in Nucleus-Nucleus Collisions at High Energies

It is demonstrated using Monte Carlo simulation that in different nucleus$-$nucleus collision samples, the increase of the fluctuation of event factorial moments with decreasing phase space scale, called erraticity, is still dominated by the statistical fluctuations. This result does not depend on the Monte Carlo models. Nor does it depend on the concrete conditions, e.g. the collision energy, the mass of colliding nuclei, the cut of phase space, etc.. This means that the erraticity method is sensitive to the appearance of novel physics in the central collisions of heavy nuclei.

It is generally believed that through the collision of heavy nuclei at ultra-high energies big systems with very high energy density [1] might be produced. In these systems novel phenomena, such as colour deconfinement [2], chiral-symmetry restoration [3], discrete-symmetry spontaneous-breaking [4], etc., are expected to be present and different events might be governed by different dynamics. With this goal in mind, the event-by-event (E-by-E) study of high energy collisions has attracted more and more attention [5].
A well known example of E-by-E fluctuation is the dynamics of self-similar cascade, which results in a fractal system, and the dynamical probability-distribtion fluctuates E-by-E [6]. Such kind of self-similar dynamical fluctuations can be studied by means of the method of normalized factorial moments (nfm) [6]. The latter are defined as where a region ∆ in 1-, 2-or 3-dimensional phase space is divided into M cells, n m is the multiplicity in the mth cell, and · · · denotes vertically averaging over the event sample, N is the number of events in the sample. If self-similar dynamical fluctuations exist, the nfm will possess an anomalous scaling property with the diminishing of phase space scale (or increasing of partition number M ), Recently the predicted anomalous scaling of nfm, Eq.(3), has been successfully observed in experiments [7] [8]. (For a review see [9]). In Eq.(1) the vertical average · · · over the event sample precedes the horizontal average (1/M ) M m=1 (· · ·) over the M bins. The nfm defined in this way is sometimes refered to as vertically averaged factorial moment and denoted by F Alternatively, one can also reverse the order of the two average processes, i.e. doing the horizontal average first, and define horizontally averaged factorial moment as It can be shown that if the vertical nfm has the anomalous scaling property, Eq.(3), then the horizontal nfm will have the same property. Note that in the definition Eq.(5) of horizontal nfm an average over the event sample has been made for the event normalized factorial moment F where n m is the multiplicity in the mth cell of that event. Therefore, it is natural to ask the question: How about the E-by-E fluctuation of efm F (e) q ? Cao and Hwa [10] propose to quantify this fluctuation by the normalized moments of F (e) q . If C p,q has a power law behavior as the division number M goes to infinity then the phenomenon is referred to as erraticity, and is characterized by the slope µ q of ψ q (p) at p = 1 which is called entropy index. Define then the entropy index µ q can be calculated through The usefulness of erraticity, or entropy index, in the study of E-by-E fluctuation is limited by the fact that this behaviour is dominated by statistical fluctuations when the multiplicity is low [11]. Only for high multiplicity events, as for example in the central collisions of heavy nuclei, the "entropy index" coming from statistical fluctuations becomes very small and the dynamical effect can be expected to show up [12].
In the present letter this problem is studied in some detail using the Monte Carlo generators Fritiof and Venus. It will be shown that within the framework of these models the statistical fluctuations still dominate the erraticity behaviour of central nuclear collisions, even though the multiplicity is as high as several hundreds to several thousands. What is interesting is that this dominance of statistical fluctuations does not depend on the model used. Neither does it depend on any physical condition, e.g. the collision energy, the mass of the colliding nuclei, the cut of phase space, etc. This means that the erraticity method has the peculiar property that it is able to filter out all the concrete physical conditions used in data analysis and therefore may be used as a sensitive signal for the appearance of novel physics.
We start from the study of Pb-Pb collisions. Two samples are generated using Fritiof for the incident energies 158 and 500 A GeV, each consisting of 10 000 events. The phase space regions used for the study of erraticity behaviour are listed in the first 3 rows of Table I. The collisions are central in the sense that the impact parameters lie between 0 and 0.5 fm.  In order to elliminate the effect of non-flat average distribution, the phase space variables y, p t , ϕ are transformed into the corresponding cumulant forms [13] X y , X pt , X ϕ as usual. After the transformation, the phase space regions of all three X a (a = y, p t , ϕ) become [0,1].
In  Table I. It can be seen from the figures that the log-log plots of C p,2 versus M 3D have similar shape for all the cases but only with different scales. This means that erraticity exists in all the cases with different strength, characterized by the different values of entropy index µ. A regularity that can easily be observed from Table I is that the entropy index µ decreases with increasing average mutiplicity N .
The dependence of µ 2 on N is plotted in Fig.3. The full line in this figure is the result of pure statistical fluctuations taken from Ref. [12]. Our results lie well above this line, which seems to indicate that some dynamical effect shows up. However, this conclusion cannot be drawn because the full line was obtained from the pure-statistical-fluctuation model in one-dimensional phase space [12], while our results are for 3-dimensional case. In order to make a faithful comparison between the results from the Fritiof generator and the pure-statistical-fluctuation case, we construct models of pure statistical fluctuations in 1-, 2-and 3-dimensions, respectively. For illustration, consider the 2-D model. Let X a and X b denote the two (cumulant) variables. For each particle in an event take two random numbers distributed uniformly in the region [0,1] as the values of X a and X b of this particle. Repeating N times, the X a and X b values of all the N particles in the event are determined and a Monte Carlo event, containing only statistical fluctuations, is obtained. Constructing in this way N events, the C p,q and Σ q can be calculated. Note that, by construction, for the characterization of each particle in the 1-, 2-, 3-D models we need 1, 2, 3 random numbers, respectively. Therefore, the "degree of randomness" is higher and the entropy index µ q should be larger for   In order to check whether this conclusion depends on the projectile and target nuclei and/or on the event generator used, similar analysis is carried out for various colliding systems at different incident energies using both Fritiof and Venus event generators.   Tables II and III. Full stars are from Gaussian-alpha model. The resulting average multiplicity N and entropy index µ 2 are listed in Tables II, III  and Fig.4. Also listed in the tables are the colliding nuclei, the incident energy, the particle type and the rapidity region used in the analysis. The p t and ϕ regions in all cases are [0,10] and [0, 2π], respectively. The impact parameter takes a value between 0 and 0.5 fm.
It can be seen from Fig.4 that µ 2 versus N from both Fritiof and Venus Monte Carlo simulations fits very well to that expected from the 3-D pure-statistical-fluctuation model, independent of the event generator, colliding nuclei, incident energy, particle type and phase space region used in the calculation. This means that, in the framework of Fritiof and/or Venus event generators, even in the central collision of heavy nuclei at energies up to 200 A GeV, the statistical fluctuations still dominate the erraticity behaviour. No dynamical fluctuation can be observed through erraticity analysis.
This disappointing fact, however, provides us a possibility to signal the appearance of novel physics. The point is that, within the framework of traditional high energy nuclear physics the dominance of statistical fluctuations in a given physical process does not depend on the concrete conditions, e.g. the collision energy, the mass of colliding nuclei, the cut of phase space, etc.. This dominance will disappear and the observed erraticity will deviate from that of pure statistical fluctuations only if the events of the studied sample are coming from some new kind of physical processes. For illustration, we plot in Fig's. 3 and 4 the results from the Gaussian-alpha model proposed in Ref. [12] as stars. It can clearly be seen that they do not lie on any of the three curves in these figures. Therefore, we conclude that erraticity method has the peculiar property that it is able to filter out all the concrete physical conditions used in data analysis and is sensitive to the appearance of novel physics in the central collisions of heavy nuclei.