Early dynamics of transversally thermalized matter
aa r X i v : . [ nu c l - t h ] A p r Quark Matter 2008, Jaipur, India, February 2008
Early dynamics of transversally thermalized matter ‡ A. Bialas , , M. Chojnacki , W. Florkowski , M. Smoluchowski Institute of Physics, Jagellonian University, 30-059 Krak´ow,Poland The H. Niewodnicza´nski Institute of Nuclear Physics, Polish Academy ofSciences, 31-342 Krak´ow, Poland Institute of Physics, Jan Kochanowski University, 25-406 Kielce, PolandE-mail:
Abstract.
We argue that the idea that the parton system created in relativisticheavy-ion collisions is formed in a state with transverse momenta close tothermodynamic equilibrium and its subsequent dynamics at early times isdominated by pure transverse hydrodynamics of the perfect fluid is compatiblewith the data collected at RHIC. This scenario of early parton dynamics may helpto solve the problem of early equilibration.PACS numbers: 25.75.-q, 25.75.Dw, 25.75.Ld
It is now commonly accepted that the evolution of matter created in heavy-ion collisions at RHIC energies is most successfully described by hydrodynamics ofthe perfect fluid [1, 2, 3]. In particular, the hydrodynamical approach reproducesreasonably well the particle transverse-momentum spectra and the elliptic flowcoefficient v . Very recently, it has been also shown that the hydrodynamic modelcan describe consistently the HBT radii [4].On the other hand, the hydrodynamic picture is challenged by a serious problem.To reproduce correctly the data, the hydrodynamic evolution should start at thetime well below 1 fm after the collision takes place. Such fast equilibration timeis not easy to achieve with elastic perturbative cross-sections, hence the success ofthe hydrodynamic approach inevitably leads to the puzzle of early thermalization.This puzzle was widely discussed and several exotic mechanisms were proposed for itssolution [5, 6], however no one was yet accepted as fully satisfactory.In the present paper we analyze the possibility that, at the early stages of high-energy collisions, the hydrodynamic evolution applies only to transverse degrees offreedom of the partonic system. The longitudinal dynamics is essentially described bypartonic free-streaming (we consider here the midrapidity region, y ≈ ‡ Research supported by the Polish Ministry of Science and Higher Education grants: 1 P03B 04529 (2005-2008), N202 153 32/4247 (2007-2009) and N202 034 32/0918 (2007-20010). arly dynamics of transversally thermalized matter Figure 1.
Transverse-momentum spectra of π + measured by PHENIX inthe centrality class 30-40% (solid line) [17] and the model spectra of gluonsfor various choices of T i and T f . The lowest dash dotted curve correspondsto T i = 200 MeV and T f = 180 MeV. The highest dash dotted curve wasobtained for T i = 300 MeV and T f = 180 MeV. The four almost parallellines represent our results for T i = 250 MeV and for four final values ofthe temperature: T f = 200 (long dashed line), 180 (dash dotted line),160 (dashed line) and 140 (dotted line) MeV. Note that the original π + experimental spectra collected at √ s NN = 200 GeV have been multipliedby a factor of 3 to account for the total hadron multiplicity. In our approach the partonic system may be treated as a superposition ofindependent transverse clusters. The clusters are formed by the particles having thesame value of the rapidity. The space-time evolution of the system is described by thehydrodynamic equations introduced in Ref. [9]. They are based on the energy andmomentum conservation laws ∂ µ T µν = 0 , (1)where the energy-momentum tensor is defined by the formula T µν = n ν g T πτ (3 U µ U ν − g µν − V µ V ν ) . (2)In Eq. (2) the parameter n is the density of clusters in rapidity space, ν g is thedegeneracy factor (we assume that our system is dominated by gluons, hence we use ν g = 16), T is the temperature, and τ = √ t − z . The four vectors U µ and V µ aredefined by the equations: U µ = ( u cosh η, u x , u y , u sinh η ) , (3) V µ = (sinh η, , , cosh η ) , (4)where η is the spacetime rapidity, and u , u x , u y are the components of the four-velocityof the fluid in the rest-frame of a cluster, where y = η = 0, u µ = (cid:0) u , ~u ⊥ , (cid:1) = (cid:0) u , u x , u y , (cid:1) . (5) arly dynamics of transversally thermalized matter Figure 2.
The elliptic flow coefficient v as the function of the transversemomentum. The PHENIX experimental results for pions and kaons in thecentrality class 20-40% and for the collision energy √ s NN = 200 GeV [18]are compared to the model calculations with T i = 250 MeV and for fourfinal values of the temperature T f = 200 , T f ≈
180 MeV.
One of the interesting features of our approach defined by Eqs. (1) - (5) is that itnaturally leads to the entropy conservation, which takes the form ∂ µ S µ = 0 , (6)where S µ = 3 n ν g T πτ U µ . (7)Eq. (6) follows directly from the projection of Eq. (1) with the energy-momentumtensor (2) on the four-velocity U ν . More information concerning the formal aspects oftransverse hydrodynamics may be found in Refs. [10, 11]. Here we only note that ourapproach differs from the similar model by Heinz and Wong [12], where the entropyis not conserved and the system is treated as dissipative.To solve numerically the hydrodynamic equations (1) we have to specify the initialconditions, appropriate for the physical situation encountered in gold on gold collisionsat RHIC energy. In our approach we assume that the transverse profile of the initialenergy density is given by the density of wounded nucleons [13] (see also [14, 15]).This density is determined, for a given centrality, from the Glauber model. In thetheoretical calculations we consider the centrality class 20 − b ≈ T i at the origin, ~x ⊥ = 0, istaken as a free parameter.The parton spectra and the elliptic flow coefficient v are evaluated using theCooper-Frye prescription [16] dNd p ⊥ dy = n ν g (2 π ) Z d Σ µ ( x ) p µ F ( x, p ) , (8) arly dynamics of transversally thermalized matter F ( x, p ) is the underlying phase-space distribution function, see Ref. [9]. Inour calculations Σ is determined by the condition of constant temperature T = T f .The details of our calculation of the parton transverse-momenta and the elliptic flowcoefficient v are given in Ref. [10].Our results obtained for several choices of the parameters T i and T f are shown inFigs. 1 and 2. We find that with T i ∼
250 MeV the slope of the transverse momentumspectrum may be easily adjusted to the one measured for the pion spectra [17]. Onthe other hand, the experimental values for v [18] select the final-temperature value T f = 180 MeV. This value of the final temperature is reached by a system in arelatively short evolution time of about 4 fm. The short evolution time implies thatthe transverse size of the system is relatively small, smaller than that obtained fromthe HBT measurements, thus consistently leaving space for further, three-dimensionalexpansion of the system.We thus conclude that the concept of the initial transverse equilibration andhydrodynamic evolution of the partonic system is compatible with the data and theproposed model represents a possible solution the the problem of early equilibration.Very recently, the successful description of the RHIC data has been achievedin a model which assumes initial free-streaming in three space directions followedby a sudden transition to standard hydrodynamics [4]. Similarly to our arguments,that model indicates that the successful description of the data may be achievedin the approach without complete thermalization in the early stage. Thus, furtherinvestigations of early non-equilibrium dynamics (for example, see Ref. [19]) arecertainly necessary and interesting – they bring us closer to the solution of the earlythermalization puzzle. [1] E. Shuryak, Nucl. Phys. A750 (2005) 64.[2] P. F. Kolb and U. Heinz, in Quark-Gluon Plasma 3, edited by R. C. Hwa and X.-N. Wang(World Scientific, Singapore, 2004), p. 634, nucl-th/0305084.[3] P. Huovinen, in Quark-Gluon Plasma 3, edited by R. C. Hwa and X.-N. Wang (World Scientific,Singapore, 2004), p. 600, nucl-th/0305064.[4] W. Broniowski, M. Chojnacki, W. Florkowski, and A. Kisiel, arXiv:0801.4361 [nucl-th].[5] St. Mr´owczy´nski, Acta Phys. Pol.
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