Describing transverse dynamics and space-time evolution at RHIC in a hydrodynamic model with statistical hadronization
aa r X i v : . [ nu c l - t h ] S e p Describing transverse dynamics and space-time evolution at RHICin a hydrodynamic model with statistical hadronization
W. Florkowski a , b , W. Broniowski a , b , M. Chojnacki b , A. Kisiel c , d a Institute of Physics, Jan Kochanowski University, 25-406 Kielce, Poland b The H. Niewodnicza´nski Institute of Nuclear Physics, Polish Academy of Sciences, 31-342 Krak´ow, Poland c Faculty of Physics, Warsaw University of Technology, 00-661 Warsaw, Poland d Physics Department, CERN, CH-1211 Geneve 23, Switzerland
Abstract
A hydrodynamic model coupled to the statistical hadronization code Therminator is used to studya set of observables in the soft sector at RHIC. A satisfactory description of the p ⊥ -spectra andelliptic flow is obtained, similarly to other hydrodynamic models. With the Gaussian initialconditions the transverse femtoscopic radii are also reproduced, providing a possible solution ofthe RHIC HBT puzzle.
1. Intoduction
The consistent description of various features of the soft hadron production in the nucleus-nucleus collisions at the Relativistic Heavy-Ion Collider (RHIC) is a well known problem [1].The so called RHIC HBT puzzle [1, 2, 3, 4] refers to the di ffi culty of simultaneous descriptionof the hadronic transverse-momentum spectra, the elliptic flow coe ffi cient v , and the Hanbury-Brown–Twiss (HBT) interferometry data in various approaches including hydrodynamics [5, 6,7, 8, 9].Recently, we have constructed a hydrodynamic code [10, 11] that is coupled to the statisticalhadronization model Therminator [12]. Within this approach a successful uniform description ofthe soft hadronic RHIC data has been accomplished [13]. The main ingredients of our approachare the following: i) We use a realistic equation of state that interpolates between the latticeQCD results at high temperatures and the hadron-gas results at lower temperatures [14]. ii)
Thesingle freeze-out scenario including all well established resonance states is assumed [15]. Thisassumption leads to shorter emission times and helps to reproduce the ratio R out / R side . iii) Theuse of Therminator allows for the use of two-particle methods in the evaluation of the correlationfunctions (with and without the Coulomb corrections). iv)
The early starting / thermalization time τ i = .
25 fm is assumed. This helps to develop fast the transverse flow and shortens the evolutiontime. v) The initial conditions for the energy density in the transverse plane are taken in theGaussian form [13] ε ( x ⊥ ) = ε i exp − x a − y b ! . (1)The values of the width parameters a and b depend on the centrality class The Gaussian profilesare obtained with the Monte-Carlo Glauber simulations done with Glissando [16]. vi) The initialconditions include fluctuations of the initial eccentricity [9, 17].
Preprint submitted to Nuclear Physics A November 10, 2018 . Results
In Fig. 1 we show our model results for the transverse-momentum spectra and compare themwith the PHENIX data [18] for Au + Au collisions at the top RHIC energy of √ s NN =
200 GeV.One observes a very good agreement between the model predictions and the data. The modelresults have been obtained with the initial central temperature T i =
460 MeV and the final / freeze-out temperature T f =
145 MeV. These are the two main parameters of our approach. The valuesof the width parameters follow from the Monte-Carlo Glauber simulations and for the centralityclass 20-30% we have found a = .
00 fm and b = .
59 fm. p T @ GeV D d N (cid:144) H d y2 Π p T dp T L PHENIX ž (cid:143)!!!!!!!!!! s NN =
200 GeVc = - % T i = @ MeV D T f = @ MeV D Τ i = @ fm D Π + K + p Figure 1: The transverse-momentum spectra of pions, kaons and protons for the centrality class 20-30%. The modelresults with the Gaussian initial conditions are plotted as functions of the transverse momentum and compared to the datafrom [18]. p T @ GeV D v PHENIX ž (cid:143)!!!!!!!!!! s NN =
200 GeVc = - % c = - % T i = @ MeV D T f = @ MeV D Τ i = @ fm D Π + + K + p Figure 2: The elliptic flow coe ffi cient v for the centrality class 20-40%. The model results with the Gaussian initialconditions are plotted as functions of the transverse momentum and compared to the data from [19]. Model parametersare the same as in Fig. 1. In Fig. 2 we show our model results for the elliptic flow coe ffi cient v . The same values of theparameters have been used as in the calculation of the spectra. The pion + kaon data are very wellreproduced, while the model results for the protons slightly overshoot the data. This behaviormay be attributed to the lack of hadron rescattering in the final state in our approach.Our HBT results are shown in Fig. 3. Again, the same values of the input parameters havebeen used. We observe good agreement between the data [20] and the model calculations. Inparticular, the ratio R out / R side is very well reproduced.2 R ou t @ f m D STAR ž (cid:143)!!!!!!!!! s NN =
200 GeVc = - % R s i d e @ f m D R l ong @ f m D Π T i = @ MeV D T f = @ MeV D Τ i = @ fm D k T @ MeV D R ou t (cid:144) R s i d e Figure 3:
The pion HBT radii R side , R out , R long , and the ratio R out / R side for the centrality class 20-30%. Theresults of the model calculation with the Gaussian initial conditions (lines) are compared to the data from[20] (dots). Model parameters are the same as in Fig. 1. Encouraged by the success of reproducing the azimuthally averaged HBT radii, we havealso calculated the azimuthally sensitive femtoscopic observables for di ff erent centralities andaverage transverse momenta [21]. The summary of our results is shown in Fig. 4. The modelresults are compared to the experimental STAR data [22]. For each centrality, associated herewith the number of participants N part on the horizontal axis, we plot the experimental points(filled dots) and the model results (empty symbols). The points from top to bottom correspondto k T contained in the bins of 0.15-0.25 GeV, 0.25-0.35 GeV, and 0.35-0.6 GeV. The top panelsshow the radii squared averaged over the φ angle, from left to right, R , , R , , and R , .The bottom panels show the magnitude of the allowed oscillations divided by R , , which isthe adopted convention used in presenting the experimental data.We conclude with the statement that the consistent and uniform description of the soft hadronicdata at RHIC may be achieved within the hydrodynamic approach if a proper choice of the initialprofile and a realistic equation of state are used. We note that a similar conclusion has been alsoreached recently in Ref. [23] (see also [24]) where the fast building of the transverse flow that isrequired for the correct description of the HBT radii is achieved with the very early start of thehydrodynamics ( τ i = ff ects. However, the v coe ffi cientis not evaluated in [23]. Acknowledgments
This work was supported in part by the Polish Ministry of Science and Higher Education,grant N202 034 32 / art N50 100 150 200 250 300 350 ou t , R part N50 100 150 200 250 300 350 i d e , R part N50 100 150 200 250 300 350 l ong , R part N50 100 150 200 250 300 350 i d e , / R ou t , R −0.14−0.12−0.1−0.08−0.06−0.04−0.020 part N50 100 150 200 250 300 350 i d e , / R i d e , R −0.0200.020.040.060.080.1 part N50 100 150 200 250 300 350 i d e , / R ou t − s i d e , R Figure 4:
Results for the RHIC HBT radii and their azimuthal oscillations. For each value of N part on thehorizontal axis we plot the experimental points (filled symbols) and the model results (empty symbols). Thepoints from top to bottom at each plot correspond to k T contained in the bins 0.15-0.25 GeV (circles), 0.25-0.35 GeV (squares), and 0.35-0.6 GeV(triangles). The top panels show R , , R , , and R , , the bottompanels the magnitude of the allowed oscillations divided conventionally by R , . Data from Ref. [22].Model parameters for di ff erent centralities are given in [21]. References [1] U. W. Heinz, P. F. Kolb, hep-ph / B36 (2005) 187.[3] M. A. Lisa, S. Pratt, R. Soltz, U. Wiedemann, Ann. Rev. Nucl. Part. Sci., (2005) 357.[4] P. Huovinen, P. V. Ruuskanen, Ann. Rev. Nucl. Part. Sci., (2006) 163.[5] U. W. Heinz, P. F. Kolb, Nucl. Phys., A702 (2002) 269.[6] T. Hirano, K. Morita, S. Muroya, C. Nonaka, Phys. Rev.,
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