Hurst Exponent and Event-by-Event Fluctuations in Relativistic Nucleus–Nucleus Collisions

Abstract

A joint study of multi-particle pseudo-rapidity correlations and event-by-event fluctuations in the distributions of secondary particles and fragments of the target nucleus and the projectile nucleus was carried out in order to search for correlated clusters of secondary particles. An analysis of the collisions of the sulfur nucleus with photoemulsion nuclei at an energy of 200 A·GeV is presented based on experimental data obtained at the SPS at CERN. The analysis of multi-particle correlations was performed using the Hurst method. A detailed analysis of each individual event showed that in events of complete destruction of a projectile nucleus with a high multiplicity of secondary particles, long-distance multi-particle pseudo-rapidity correlations are observed. The distribution of average pseudo-rapidity in such events differs significantly from others, as it is much narrower, and its average value is noticeably shifted towards lower values <η>.

1. Introduction

Collisions of relativistic heavy ions provide a unique opportunity to obtain new information about the structure of matter at the smallest space–time intervals and to study nuclear matter at extreme densities and temperatures [1,2,3,4]. According to modern concepts, if the density of the nuclear matter formed during the interaction is high enough, the hadron substance transits into the state of quark–gluon plasma (QGP), in which quarks are in a quasi-free state [5,6,7,8,9]. In such collisions, nuclear matter heats up and contracts in a very short time (a few fm/s). At moderate temperatures, nucleons are excited into baryon resonances, which decay with the emission of mesons, and at higher temperatures, baryon–antibaryon pairs are formed. The first evidence of the experimental discovery of QGP was obtained at the Relativistic Heavy Ion Collider (RHIC, Brookhaven, NY, USA) [10,11,12,13]. Recently, a hypothesis about the existence of a mixed phase of nuclear matter, including not only quarks and gluons, but also hadrons, has been put forward and theoretically substantiated. This has stimulated intensive experimental research [14,15,16,17].

To search for the phase transition of nuclear matter into the quark–gluon phase, the analysis of fluctuations and correlations in the distributions of secondary particles is often used, as the scattering of secondary particles from a fireball of nuclear matter has a collective character [18,19,20]. Fluctuations in collision geometry (about which direct experimental information is usually absent) form a large background, complicating the identification of the quark–gluon phase [21,22,23]. If the collision is central, then the number of interacting nucleons is at the maximum. In a peripheral collision, the overlap of the nuclei is partial, and the resulting fireball of quark–gluon matter expands asymmetrically [24,25,26]. To isolate fluctuations associated with the geometry of a nucleus–nucleus collision, studies of event-by-event fluctuations have recently become very popular [27,28,29,30,31]. It is assumed that a detailed analysis of the data for each individual nucleus–nucleus interaction will make it possible to detect effects associated with the phase transition in those events in which the necessary conditions for the formation of QGP were met.

Additional information about the geometry of a nucleus–nucleus collision can be obtained from analysis of fragments of interacting nuclei. In the inelastic interaction of nuclei, some of the nucleons are participants; that is, they interact with at least one nucleon of the opposite nucleus. The remaining fraction are spectators; that is, they are nucleons that do not participate in the interaction. The nucleon participants produce secondary particles, which can then be observed in detection systems. Fragments of colliding nuclei consist of spectator nucleons [32]. In general, the less overlap between interacting nuclei, the greater the total charge of the fragments that should be detected. Thus, to accurately estimate the parameters of the initial state of interaction, fragmentation analysis is required. However, in experiments with colliding beams, there are “dead zones” in which information about the fragments is not available. First of all, “dead zones” are located in a tube in which colliding beams move. In experiments with a fixed target, there is no such problem. Therefore, experiments with a fixed target have significant advantages over experiments with colliding beams when estimating the collision geometry [33,34]. The mode with a fixed target also allows us to research rare processes and to study the necessary parameters for the analysis of cosmic ray data [35,36,37].

In this work, an analysis of multi-particle correlations and event-by-event pseudo-rapidity fluctuations was carried out to search for correlated clusters of secondary particles. Fragments of the target nucleus and the projectile nucleus were analyzed, as well as the distribution of secondary particles emitted from the interaction region. The results were processed using the Hurst method [38]. This method made it possible to separate events in which some dynamic multi-particle correlations were observed versus events in which there were no multi-particle correlations. This method allows one to estimate the “strength” and “length” of multi-particle correlations in the pseudo-rapidity distribution of secondary particles [39,40].

2. Experimental Details

The following experimental data were used for the analysis: 837 inelastic interactions of sulfur nuclei, S, with a nuclear emulsion (Em, NIKFI BR-2) at 200 A·GeV, obtained at the Super Proton Synchrotron (SPS) at the European Center for Nuclear Research (CERN) [41], and 924 inelastic interactions of S+Em 3.7 A·GeV obtained at the Synchrophasotron at the Joint Institute for Nuclear Research (JINR, Dubna, Russia) [42].

According to the generally accepted methodology used in emulsion experiments, all charged secondary particles were classified as fragments of the target and the projectile nuclei, as well as secondary particles from the interaction region (s-particles) [43,44,45].

Each track that has an imprint in the emulsion corresponds to one particle or fragment. For each track, the azimuthal, Ψ, and polar, Θ, angles were measured. For s-particles, the pseudo-rapidity was calculated using the formula η = −ln [tan Θ/2]. The charge was measured for each fragment of the projectile nucleus.

The method of nuclear emulsions is unique in that it allows one to study in each interaction not only fluctuations in the distribution of secondary particles but also fragments of the target nucleus and the projectile nucleus. Such informative studies are possible due to the high spatial resolution of the nuclear emulsion and the ability to observe particle interactions in the 4π geometry of the experiment.

3. Estimation of Target Nucleus and the Collision Geometry

For a more accurate assessment of the geometry of the collision of nuclei, it is necessary to analyze fragments of the colliding nuclei. A schematic representation of a nuclear interaction is shown in Figure 1a.

Spectator nucleons, which form fragments of the projectile and the target nuclei, are indicated by the white circles. Participating nucleons are marked by the darker shaded circles. Based on geometric concepts, during peripheral collisions, one multi-charged fragment (A) of the projectile nucleus should be detected. Moreover, the less overlap between interacting nuclei, the greater the total charge of the fragments should be. In strong central interactions, the absence of multi-charged fragments can be observed; that is, complete destruction of the projectile nucleus can be observed [46].

The NIKFI BR-2 emulsion contains heavy silver and bromine (AgBr) nuclei (25.5%), light hydrogen nuclei (39.2%), and carbon, nitrogen, and oxygen (CNO) nuclei (35.3%). Therefore, it is necessary to separate events that occurred with different targets. For this purpose, the dependence of the number of fragments from the target nucleus, N_h, on the multiplicity of s-particles, n_s, is often used (Figure 1b). The separation of events with heavy targets was carried out according to the criterion N_h > 8, as the heaviest nucleus among the light nuclei of the emulsion, oxygen, has a charge of 8.

4. Projectile Nucleus Fragmentation and Multiplicity Fluctuations

To understand the possible connection between the features of multi-particle production and the parameters of fragmentation, multiplicity distributions in events with different numbers of projectile nucleus fragments were analyzed. Figure 2 shows the multiplicity distributions of secondary particles n_s in central events with complete destruction of the projectile nucleus (N_f = 0) and in peripheral events with one multi-charged fragment (N_f = 1) in interactions of S+Em 200 A·GeV and 3.7 A·GeV.

From Figure 2a, it can be seen that, as expected, the probability of the occurrence of high-multiplicity events with N_f = 1 is significantly less than the probability of low-multiplicity events. At the same time, for events with N_f = 0, an unexpected result was obtained, presented in Figure 2b.

The multiplicity distribution in events with N_f = 0 has a clear two-peak structure, separated by the n_s = 200 level. In this case, the probability of events of high multiplicity (for example, with n_s = 400) practically coincides with the probability of the occurrence of events of low multiplicity (for example, with n_s = 40).

For comparison, Figure 2c,d show the multiplicities of s-particles for interactions of S+Em 3.7 A·GeV with different numbers of multi-charged fragments N_f. The histogram in Figure 2c is very similar to the histogram shown in Figure 2a. At the same time, the multiplicity distribution of events with N_f = 0, presented in Figure 2d, differs significantly from those in Figure 2b, as the number of events with a high multiplicity of s-particles is considerably lower than the number of events with low multiplicity. Therefore, it depends on the energy of the projectile nucleus.

5. Fluctuations of the Average Pseudo-Rapidity of Secondary Particles

When the nuclei are not completely overlapped, the resulting fireball expands unevenly. To study fluctuations depending on the collision geometry, the average pseudo-rapidity <η> was calculated. Then, the distribution of the average pseudo-rapidity over the events was constructed. The results are presented in Figure 3.

For comparison, in Figure 3a–d, a fit with a Gaussian function is superimposed for the complete distribution over the average pseudo-rapidity of s-particles, normalized to the number of events. As it can be seen from Figure 3d, the distribution of the average pseudo-rapidity of s-particles in events with N_f = 0 and a multiplicity n_s ≥ 200 differs significantly from other distributions presented in this figure.

First, this distribution is noticeably narrower than other distributions. The Root Mean Square (RMS) of the distribution presented in Figure 3d is 0.1782, and in Figure 3c, it is 0.4712. In other distributions, the RMS is considerably higher. Second, the mean of the distribution presented in Figure 3d is shifted towards low values, <η> = 3.129, in contrast to Figure 3c, with <η> = 3.680.

Thus, in collisions of S+Em at 200 A·GeV, events of complete destruction of the projectile nucleus with high multiplicity were identified, leading to the emergence of a flux of secondary particles in a narrow range of average pseudo-rapidity and vastly shifted towards low values of <η>.

6. Pseudo-Rapidity Correlations

For the identification of correlated groups of secondary particles, an analysis was carried out using the Hurst method [38,39,40,47].

6.1. Method

The classical Hurst method was developed to analyze correlations in time sequences [30]. If some analyzed sequence ξ(z_i) is uncorrelated, the sum of fluctuations on a sufficiently large investigated interval of variation of the sequence ξ_i (i >> 1) will tend to zero. A correlated signal can be detected if ∑(ξ(z_i) − <ξ>) is significantly different from zero.

To quantitatively characterize the “strength” of correlations, the ratio of two quantities is used, which are the standard deviation (S) and the range (R), which is defined as the difference between the largest and smallest accumulated deviation from the average. To calculate the range, the accumulated fluctuations X(m,k) of the sequence are calculated relative to the average value <ξ>:

$$X\left(m,k\right)=\sum _{i=1}^m\left[\xi \left(z_i\right)-<\xi >\right],1\le i\le m\le k,$$

(1)

where

$$<\xi >={\displaystyle \frac{1}{k}}\sum _{i=1}^k\xi \left(z_i\right),$$

(2)

We will call the difference between the maximum and the minimum accumulated deviation the range:

$$R\left(k\right)=\underset{1\le m\le k}{\underbrace{\max X(m,k)}}-\underset{1\le m\le k}{\underbrace{\min X\left(m,k\right)}},$$

(3)

Next, we normalize this value to the standard deviation:

$$S\left(k\right)={\left[{\displaystyle \frac{1}{k}}\sum _{i=1}^k{\left[\xi \left(z_i\right)-<\xi >\right]}^2\right]}^{1/2},$$

(4)

It was shown in [38] that the normalized range, defined as

H (k) = R (k)/S (k),

(5)

obeys the relation

$$H\left(k\right)={\left(ak\right)}^h,$$

(6)

where h and a are two free parameters (h is the so-called correlation index or Hurst index). Moreover, if the sequence ξ(z_i) represents white noise, i.e., a completely uncorrelated signal, then h = 0.5. The case 0.5 < h < 1 indicates the presence of correlations in the system (for a fully correlated signal, h = 1) [38].

6.2. Analysis Procedure

From the complete kinematic region, only the central pseudo-rapidity interval Δη = 4, the so-called pionization region, was considered. According to modern concepts, secondary particles emitting from the interaction area are concentrated in this region. For S+Em events with an energy of 200 A GeV, this interval corresponds to pseudo-rapidity values from η_min = 1.0 to η_max = 5.0. This interval Δη = 4 was divided into k parts. By counting the number of particles that filled each subinterval δη = Δη/k, where i varies from 1 to k, we obtained a sequence of numbers n_i. To analyze the relative fluctuations in individual events relative to the average pseudo-rapidity distribution, we considered the sequence

$${\xi }_i=\left({\displaystyle \frac{n_i^e}{n^e}}-{\displaystyle \frac{n_i}{n}}\right)/{\displaystyle \frac{n_i}{n}},$$

(7)

where

• $n_i^e$ is the number of particles in the i-th bin of the event under consideration;
• $n^e$ is the number of particles in this event;
• $n_i$= ∑$n_i^e$ is total number of particles in the i-th bin for all events;
• $n$ = ∑$n^e$ is the total number of particles in all events.

To study multi-particle correlations in the distribution of secondary particles, we considered the dependence of the normalized range H(dη) = R(dη)/S(dη), гдe dη = k′·δη (1 ≤ k′ ≤ k) on the value of the pseudo-rapidity interval. Taking into account the constancy of Δη, it is convenient to replace the value of the interval dη with the number of intervals k′. The first value of k′ corresponded to k = 1024, and the value of dη(k′) = Δη.

According to Equations (1)–(6), the normalized range for this pseudo-rapidity interval was calculated. Next, the sequence ξ_i was divided into two parts. Having thus obtained two independent series of “length” k′ = k/2, we calculated the value of H(k/2) for each series separately. After this, we divided each of the newly obtained series into two parts, thus obtaining four independent sequences of “length” k′ = k/4, and we calculated H(k/4). This procedure for dividing and analyzing the newly obtained series of values ξ_i continued until the number of terms in the series—remainders—became less than 16 (»1), after which the division procedure stopped. Results for H corresponding to the same value of k′ were averaged and plotted on a log–log scale as a function of k′. The Hurst exponent was determined for each event using the following formula:

$$h_{a\nu }={\displaystyle \frac{1}{i_{max}-1}}\sum _{i=1}^{i_{max}}{\displaystyle \frac{\ln \left(H\left(k_i\right)\right)-\mathrm{l}\mathrm{n}\left(H\left(k_{i-1}\right)\right)}{\mathrm{l}\mathrm{n}\left(k_i\right)-\mathrm{l}\mathrm{n}\left(k_{i-1}\right)}},$$

(8)

6.3. Results

A detailed analysis of each individual event showed that the behavior of the Hurst exponent for events with N_f = 0 and a multiplicity of n_s ≥ 200 in S+Em at 200 A·GeV interactions differs significantly from the behavior of the Hurst exponent for events with N_f = 0 and a multiplicity of n_s < 200. The results are presented in Figure 4.

The values of the Hurst exponent h~0.66 at dη~0.1 mean that in events with n_s < 200, short-distance multi-particle pseudo-rapidity correlations are detected. At the same time, in events with n_s ≥ 200, long-distance multi-particle correlations are observed. All h values are over 0.5, so all of the data are correlated. The largest correlations are shown at dη ≥ 1. These correlations indicate that the scattering of secondary particles from the interaction region was collective, and, therefore, they are highly likely a signal of the formation of the quark–gluon plasma.

7. Summary

In this work, a joint study of multi-particle pseudo-rapidity correlations and event-by-event fluctuations in the distributions of secondary particles and fragments of the target nucleus and the projectile nucleus was carried out in order to search for correlated clusters of secondary particles. An analysis of fluctuations in the distributions of secondary particles detected in events of complete destruction of the projectile nucleus in S+Em 200 A·GeV is presented based on experimental data obtained at SPS in CERN.

This study showed that the multiplicity distribution in events with complete destruction of the projectile nucleus represents a clear two-peak structure separated by the n_s = 200 level. The distribution of the average pseudo-rapidity of secondary particles in the events with a multiplicity of n_s ≥ 200 differs significantly from other distributions. First, this distribution is noticeably narrower than other distributions. Secondly, the mean of the distribution is vastly shifted towards lower values of <η>. Moreover, in these events, long-distance multi-particle correlations are observed through the Hurst method in the pseudo-rapidity distribution of secondary particles at dη ≥ 1.

These correlations indicate that the scattering of secondary particles from the interaction region was collective, and, therefore, they are highly likely a signal of the formation of the quark–gluon plasma.