Event-by-event fluctuations of transverse momentum and multiparticle clusters in relativistic heavy-ion collisions

FLUCTUATIONS AND CORRELATIONS

Event-by-event fluctuations of transverse momentum and multiparticle clusters in relativistic heavy-ion collisions^*

W. Florkowski^{I, II}; W. Broniowski^{I, II}; B. Hiller^III; P. Bozek^{II, IV}

^IInstitute of Physics, Swietokrzyska Academy, ul. Swietokrzyska 15, PL-25406 Kielce, Poland

^IIThe H. Niewodniczanski Institute of Nuclear Physics, Polish Academy of Sciences, PL-31342 Kraków, Poland

^IIICentro de Física Teórica, Departamento de Física, University of Coimbra, P-3004-516 Portugal

^IVRzeszów University, PL-35959 Poland

ABSTRACT

We analyze the event-by-event fluctuations of mean transverse momentum measured recently by the PHENIX and STAR Collaborations at RHIC. We argue that the observed scaling of strength of dynamical fluctuations with the inverse number of particles can be naturally explained by formation of multiparticle clusters.

Keywords: Relativistic heavy-ion collisions; Event-by-event fluctuations; Particle correlations

I. INTRODUCTION

This paper is based on our recent analysis [1] of the event-by-event fluctuations of the mean transverse-momentum measured by PHENIX [2] and STAR [3, 4] Collaborations at RHIC. The discussed new data has extended our knowledge of the phenomena studied intensely in recent years [5-20]. One of the central issues in this field is whether the p_T fluctuations at intermediate momenta result from jets [21, 22]. In this paper we argue that the observed scaling of strength of dynamical fluctuations may be naturally explained by formation of multiparticle clusters. From our point of view, the explanation of the data in terms of quenched jets represents only one possible option, other physical mechanisms leading to clustering also explain the data.

Our reasoning is based on the following three observations: i) the mean and the variance of the inclusive momentum distribution are practically constant at low centrality parameters (c = 0-30%),

ii) the variance of the average momenta for the mixed events is practically equal to the variance of the inclusive distribution divided by the average multiplicity,

finally, iii) the difference of the experimental and mixed-event variances of mean p_T, denoted as , scales as inverse multiplicity

As mentioned above, an explanation of this scaling can be provided by clustering in the expansion velocity. Moreover, the extracted value of s_dyn is large at the expected scale provided by the variance of p_T, which indicates that the clusters should contain at least several particles in order to combinatorically enhance the magnitude to the observed level. This point is supported by the numerical calculations of the values of coming from the resonance decays and from thermal clusters in statistical models of heavy-ion collisions, which show that these values are very small [1, 23].

II. SIMPLE WAY OF DOING STATISTICAL ANALYSIS

In this Section we present elementary statistical considerations which form the basis of our treatment of the fluctuations. We use the notation where the letter p denotes , p_i is the value of p for the ith particle, and M = p_i/n is the mean transverse momentum in an event of multiplicity n. Consider events of multiplicity n and transverse momenta p₁, p₂, ...,p_n. The multiplicity n and the momenta are varying randomly from event to event. The probability density of occurrence of a given momentum configuration is P(n) r _n(p₁,...,p_n), where P(n) is the multiplicity distribution and r _n(p₁,...,p_n) is the conditional probability distribution of occurrence of p₁,...,p_n in accepted events, provided we have the multiplicity n. Note that in general r depends functionally on n, as indicated by the subscript. The normalization is

The marginal probability densities are defined as

with k = 1,...,n-1. These are also normalized to unity, as follows from Eq. (4). Since the number of arguments distinguishes the marginal distributions , in the following we drop the superscript (n-k). Further, we introduce the following definitions

The subscript n indicates that the averaging is taken in samples of multiplicity n. We note in passing that the commonly used inclusive distributions are related to the marginal probability distributions in the following way:

which are normalized to á n ñ and á n(n-1) ñ , respectively. For the variable M = p_i / n we find immediately

III. ANALYSIS OF THE PHENIX DATA

At first we discuss the PHENIX measurement [2] of the event-by-event fluctuations of the mean transverse momentum at = 130 GeV. We use the experimental fact that the variance of the momentum distribution and its mean are independent of centrality (in the considered range), which allows us to replace the quantities á p ñ_n by á M ñ and á p²ñ_n - by at the average multiplicity, denoted as . In this way we get

In the mixed events, by construction, particles are not correlated, hence the covariance term in Eq. (11) vanishes and

where in the last equality we have used the fact that the distribution P(n) is narrow and expanded 1/n = 1/[ á n ñ +(n- á n ñ )] to second order in (n- á n ñ ). Comparison made in Table I, see the 8th and 9th rows, shows that formula (12) works at the 1-2% level. In addition, since s _{p, á n ñ} is not altered by the event mixing procedure, subtracting (12) from (11) yields

IV. CLUSTER EXPLANATION

The scaling of the dynamical fluctuations with á n ñ , see the last row of Table I and Eq. (3), may be regarded as a constraint restricting the behavior of the covariance term. For instance, one can immediately conclude that if all particles were correlated to each other, cov _{á n ñ} (p_i,p_j) would be proportional to the number of pairs, and s_dyn would not depend on á n ñ at large multiplicities. A natural explanation of the scaling (3) comes from the cluster model, depicted in Fig. 1. The system is assumed to have N_cl clusters, each containing (on the average) r particles. Below we keep r = const. for simplicity (a more general case is discussed in Ref. [1]). The particles are correlated if and only if they belong to the same cluster, where the covariance per pair is 2 cov* . The number of correlated pairs within a cluster is r(r-1)/2. Some particles may be unclustered, hence the ratio of clustered to all particles is á N_clñ r / á n ñ = a . If all particles are clustered then a = 1. With these assumptions Eq. (13) becomes

which complies to the scaling (3). An immediate conclusion here is that the ratio a cannot depend on á n ñ (in the considered centrality range, c = 0-30%) in order for the scaling to hold.

The question now is whether we can use the above results to draw conclusions on effects of jets (minijets), which have been proposed as a possible explanation of the experimental data even at the considered soft momenta [21]. Jets, when fragmenting, lead to clusters in the momentum space. The resulting full covariance from jets is then N_{cl, jet}j(j-1) , where N_{cl, jet} is the number of clusters originating from jets, j is the average number of particles in the cluster, and 2 cov^j is the average covariance per pair. The total number of particles produced from jets is N_{cl, jet} j. On the other hand, the commonly accepted estimate of the dependence of N_{cl, jet}j on centrality is accounted for by the nuclear modification factor R_AA multiplied by the number of binary nucleon-nucleon collisons N_bin. Since R_AA depends on the ratio á n ñ / á n ñ _pp, where á n ñ _pp is the multiplicity in the proton-proton collisions, in a given p_T bin one finds

which complies to the scaling of Eq. (14).

V. CONCLUSIONS

The observed scaling of the dynamical fluctuations of mean transverse-momentum may be naturally explained by the cluster model. We emphasize that the scaling follows from the presence of clusters only, and is insensitive to the nature of their physical origin as long as one imposes the condition N_cl ~ á n ñ .

The explanation of the observed data in terms of quenched jets agrees with the cluster picture. However, the explanation of the centrality dependence of the p_T fluctuations in terms of jets based solely on Eq. (15) is not decisive: any mechanism leading to clusters would do, for instance the thermal clusters discussed in Ref. [1] or mechanisms of Ref. [24]. Microscopic realistic estimates of the magnitude of cov^j and j are necessary in that regard, including the interplay of the jets and medium. In short, the nature of clusters remains an open issue. For the current status of this program the reader is referred to [22, 25].

Received on 10 October, 2006

• [1] W. Broniowski, B. Hiller, W. Florkowski, and P. Bozek, Phys. Lett. B 635, 290 (2006).
• [2] K. Adcox, et al., PHENIX, Phys. Rev. C 66, 024901 (2002).
• [3] J. Adams, et al., STAR, Phys. Rev. C 71, 064906 (2005).
• [4] J. Adams, et al., STAR, Phys. Rev. C 72, 044902 (2005).
• [5] M. Gazdzicki, S. Mrowczynski, Z. Phys. C 54, 127 (1992).
• [6] L. Stodolsky, Phys. Rev. Lett. 75, 1044 (1995).
• [7] E. V. Shuryak, Phys. Lett. B 423, 9 (1998).
• [8] S. Mrowczynski, Phys. Lett. B 430, 9 (1998).
• [9] M. A. Stephanov, K. Rajagopal, and E. V. Shuryak, Phys. Rev. D 60, 114028 (1999).
• [10] S. A. Voloshin, V. Koch, and H. G. Ritter, Phys. Rev. C 60, 024901 (1999).
• [11] H. Appelshauser, et al., NA49, Phys. Lett. B 459, 679 (1999).
• [12] R. Korus, S. Mrowczynski, M. Rybczynski, and Z. Wlodarczyk, Phys. Rev. C 64, 054908 (2001).
• [13] G. Baym, H. Heiselberg, Phys. Lett. B 469, 7 (1999).
• [14] A. Bialas, V. Koch, Phys. Lett. B 456, 1 (1999).
• [15] M. Asakawa, U. W. Heinz, and B. Muller, Phys. Rev. Lett. 85, 2072 (2000).
• [16] H. Heiselberg, Phys. Rept. 351, 161 (2001).
• [17] C. Pruneau, S. Gavin, and S. Voloshin, Phys. Rev. C 66, 044904 (2002).
• [18] S. Jeon, V. Koch, in Quark-Gluon Plasma 3 eds. R.C. Hwa and X.-N. Wang, World Scientific Singapore.
• [19] S. Gavin, Phys. Rev. Lett. 92, 162301 (2004).
• [20] M. Abdel-Aziz, S. Gavin, Nucl. Phys, A 774, 623 (2006).
• [21] S. S. Adler, et al., PHENIX, Phys. Rev. Lett. 93, 092301 (2004).
• [22] Q.-J. Liu, T. A. Trainor, Phys. Lett. B 567, 184 (2003).
• [23] A. Kisiel et al., Comput. Phys. Commun. 174, 669 (2006).
• [24] J. Randrup, talk at the International Workshops on Critical Point and Onset of Deconfinement, July 3-9, 2006, Florence, Italy .
• [25] International Workshops on Critical Point and Onset of Deconfinement, July 3-9, 2006, Florence, Italy, http://hep.fi.infn.it/becattini/files/talks.html

*

Talk presented by WF at the XXXVI International Symposium on Multiparticle Dynamics, Sept. 2 - 8, 2006, Paraty, Brazil. Research supported in part by the grants: POCI/FP/63412/2005 and POCI/FP/63930/2005.

Publication Dates

• Publication in this collection
  10 Aug 2007
• Date of issue
  June 2007

History

• Received
  10 Oct 2006