Initial condition for hydrodynamics, partonic free streaming, and the uniform description of soft observables at RHIC
Wojciech Broniowski, Mikolaj Chojnacki, Wojciech Florkowski, Adam Kisiel
aa r X i v : . [ nu c l - t h ] J a n Initial condition for hydrodynamics, partonic free streaming, and the uniformdescription of soft observables at RHIC ∗ Wojciech Broniowski,
Mikolaj Chojnacki, Wojciech Florkowski,
1, 2 and Adam Kisiel
3, 4 The H. Niewodnicza´nski Institute of Nuclear Physics,Polish Academy of Sciences, PL-31342 Krak´ow, Poland Institute of Physics, ´Swi¸etokrzyska Academy, ul. ´Swi¸etokrzyska 15, PL-25406 Kielce, Poland Faculty of Physics, Warsaw University of Technology, PL-00661 Warsaw, Poland Department of Physics, Ohio State University, 1040 Physics Research Building,191 West Woodruff Ave., Columbus, OH 43210, USA (Dated: January 28, 2008)We investigate the role of the initial condition used for the hydrodynamic evolution of the systemformed in ultra-relativistic heavy-ion collisions and find that an appropriate choice motivated by themodels of early-stage dynamics, specifically a simple two-dimensional Gaussian profile, leads to auniform description of soft observables measured in the Relativistic Heavy-Ion Collider (RHIC). Inparticular, the transverse-momentum spectra, the elliptic-flow, and the Hanbury-Brown–Twiss cor-relation radii, including the ratio R out /R side as well as the dependence of the radii on the azimuthalangle (azHBT), are properly described. We use the perfect-fluid hydrodynamics with a realisticequation of state based on lattice calculations and the hadronic gas at high and low temperatures,respectively. We also show that the inclusion of the partonic free-streaming in the early stage allowsto delay the start of the hydrodynamical description to comfortable times of the order of 1 fm/c.Free streaming broadens the initial energy-density profile, but generates the initial transverse andelliptic flow. The data may be described equally well when the hydrodynamics is started early, orwith a delay due to partonic free-streaming. PACS numbers: 25.75.-q, 25.75.Dw, 25.75.LdKeywords: relativistic heavy-ion collisions, hydrodynamics, partonic free-streaming, statistical models,transverse-momentum spectra, elliptic flow, femtoscopy, Hanbury-Brown–Twiss correlations, RHIC, LHC
The notorious difficulties in simultaneous descriptionof various features of the soft hadron production in thenucleus-nucleus collisions at RHIC, which occur withinthe standard approach consisting of partonic, hydrody-namic, and hadronic stages, are well known [1]. In partic-ular, one of the so called RHIC puzzles [1, 2, 3, 4] refersto problems in reconciling the large value of the ellip-tic flow coefficient, v , with the Hanbury-Brown–Twiss(HBT) interferometry in numerous approaches includinghydrodynamics [5, 6, 7, 8, 9]. In most existing analy-ses, the large value of v prefers long evolution times ofthe system, while the HBT radii indicate that this timeshould be short. Typically, the hydrodynamic evolutionis initiated from an initial profile generated by Glauber-like models, with the initial temperature serving as a freeparameter.In this Letter we show that the choice of the initialcondition for hydrodynamics is a very important elementfor the proper description of the data. Ideally, the ini-tial density and flow profiles should be provided by theearly partonic dynamics, for instance the Color GlassCondensate (CGC) [10, 11]. In practice, however, thetheory of the partonic stage carries some uncertainty inits parameters, moreover, there may exist other effects,see e.g. [12, 13], which influence the early dynamics. It ∗ Supported in part by the Polish Ministry of Science and HigherEducation, grants N202 153 32/4247 and N202 034 32/0918. is then practical to use some simple parameterization ofthe initial profile. Here we investigate boost-invariantsystems, which approximate well the RHIC collisions atmid-rapidity. We take the following Gaussian parame-terization for the initial density profile of the system inthe transverse plane ( x , y ) at the initial proper time τ = 0 .
25 fm, n ( x , y ) = exp (cid:18) − x a − y b (cid:19) . (1)The values of the a and b width parameters depend onthe centrality and are obtained by matching to the resultsfor h x i and h y i from GLISSANDO [14], which implementsthe shape fluctuations of the system [15]. The values forcentrality classes used in this work are collected in Ta-ble I. The profile (1) determines the energy-density pro-file, from which the temperature profile is obtained [16].The initial central temperature, which may depend onthe centrality, is denoted by T i and is a free parameterof our approach.The hydrodynamics used is inviscid, baryon-free, andboost-invariant. The equation of state is taken to be asrealistic, as possible. We use the lattice QCD simula-tions of Ref. [20] at high temperatures, T >
170 MeV,
TABLE I: Shape parameters for various centrality classes. c [%] 0-5 0-20 20-30 20-40 a [fm] 2.65 2.41 1.94 1.78 b [fm] 2.90 2.78 2.52 2.45 FIG. 1: (Color online) The transverse-momentum spectraof pions, kaons and protons for the centrality class c =0-5%(the upper panel) and c =20-30% (the middle panel), and thetransverse-momentum dependence of the elliptic flow coeffi-cient v for the centrality class c =20-40% (the lower panel).The darker (lighter) lines/bands describe the model results forthe case without (with) free-streaming. The PHENIX dataare taken from Refs. [17, 18]. the hadronic gas at T <
170 MeV, and a smooth interpo-lation in the vicinity of 170 MeV, as described in Ref. [16].According to recent knowledge, no first-order transitionis implemented, but a smooth cross-over. The hydrody-namic equations are solved by the method of Ref. [21].The accuracy of the method is tested with the entropyconservation, satisfied at the relative level of 10 − or bet-ter. At the temperature T f = 145 MeV (model param-eter) the system freezes and hadrons (stable and reso-nances) are generated according to the Cooper-Frye for-malism. Possible elastic rescattering processes amongthese hadrons are neglected, thus they stream freely, withthe resonances decaying on the way. This stage is simu- FIG. 2: (Color online) The transverse-momentum dependenceof the HBT radii R side (a), R out (b), R long (c), and the ratio R out /R side (d) for central collisions. The darker (lighter) linesdescribe the results without (with) free-streaming. The STARdata are from Ref. [19]. lated with THERMINATOR [23]. With the freeze-out hyper-surfaces generated in this work, the collision rate afterfreeze-out is not very large. For the obtained hypersur-faces the number of pionic trajectory crossings at thedistance corresponding to the cross section for the pioncollisions is about 1.5-1.7. Hence the single-freeze-outscenario [24] seems to be a fairly good approximation forthe present case. The use of hadronic afterburners forelastic collisions has been described, e.g., in [25, 26, 27].Our results for central and mid-peripheral collisionswith the centrality classes adjusted to the availablePHENIX [17, 18] and STAR [19] data are shown in Figs. 1and 2 (darker lines/bands). We note a uniform agree-ment for all soft phenomena studied. In particular, thetransverse-momentum spectra, the pionic elliptic-flow,and the HBT radii, including the ratio R out /R side , aredescribed within 10% or better. The HBT results forperipheral collisions are of similar quality as in Fig. 2.The discussed results have been obtained with an earlystart of hydrodynamics, at the proper time τ = 0 .
25 fm.It is unlikely that the system should equilibrate so early.Next, we argue that one may delay the starting point ofhydrodynamics to realistic times by the inclusion of thepartonic free-streaming between the initial proper time τ = 0 .
25 fm when the partons are formed and some latertime, τ , when hydrodynamics starts. Similar ideas havebeen described by Sinyukov et al. in Refs. [28, 29]. Thusthe global picture is as follows: early phase (CGC) gener-ating partons at time τ – partonic free streaming until τ – hydrodynamic evolution until freeze-out at temperature T f – free streaming of hadrons and decay of resonances.Massless partons are formed at the initial proper time τ = p t − z and move along straight lines at thespeed of light until the proper time when free streamingends, τ = √ t − z . We introduce the space-time ra-pidities η = log t − z t + z and η = log t − zt + z . Elementarykinematics [28] links the positions of a parton on theinitial and final hypersurfaces and its four-momentum p µ = ( p T cosh Y, p T cos φ, p T sin φ, p T sinh Y ), where Y and p T are the parton’s rapidity and transverse momentum: τ sinh( η − Y ) = τ sinh( η − Y ) , (2) x = x + ∆ cos φ, y = y + ∆ sin φ, ∆ = t − t cosh Y = τ cosh( Y − η ) − q τ + τ sinh ( Y − η ) . Thus the phase-space density of partons at the propertimes τ and τ are related, d N ( τ ) dY d p T dηdxdy = Z dη dx dy d N ( τ ) dY d p T dη dx dy × δ ( η − Y − arcsinh[ ττ sinh( η − Y )]) × (3) δ ( x − x − ∆ cos φ ) δ ( y − y − ∆ sin φ ) . It is reasonable to assume a factorized boost-invariantform of the initial distribution of partons, d N ( τ ) dY d p T dη dx dy = n ( x , y ) F ( Y − η , p T ) , (4)where n is the transverse density of Eq. (1). Whenthe emission profile F is focused near Y = η , for in-stance F ∼ exp[ − ( Y − η ) / (2 σ )], with σ ∼
1, and if τ ≫ τ , then the kinematic condition (2) transforms itinto F ∼ exp (cid:16) − arcsinh h ττ sin( Y − η ) i / (2 σ ) (cid:17) which isso sharply peaked that effectively F ∼ δ ( Y − η ). ThenEq. (3) yields d N ( τ ) dY d p T dηdxdy = n ( x − ∆ τ cos φ, y − ∆ τ sin φ ) × δ ( Y − η ) f ( p T ) . (5)where ∆ τ = τ − τ and f ( p T ) is the transverse momentumdistribution. The energy-momentum tensor at the proper FIG. 3: (Color online) Sections of the energy-density profile ǫ (gaussian-like curves) normalized to unity at the origin, andof the velocity profile v = p v x + v y (curves starting at theorigin), cut along the x axis (solid lines) and y -axis (dashedlines). The initial profile is from Eq. (1) for centrality 20-40%at τ = 025 fm. The ǫ profiles are for τ = τ = 0 .
25, 1, and2 fm, while the velocity profiles are for τ = 1 and 2 fm, allfrom bottom to top. We note that the flow is azimuthallyasymmetric and stronger along the x axis. time τ , rapidity η , and transverse position ( x, y ) is givenby the formula T µν = Z dY d p T d N ( τ ) dY d p T dηdxdy p µ p ν (6)= A Z π dφ n ( x − ∆ τ cos φ, y − ∆ τ sin φ ) × cosh η cosh η cos φ cosh η sin φ cosh η sinh η cosh η cos φ cos φ cos φ sin φ cos φ sinh η cosh η sin φ cos φ sin φ sin φ sin φ sinh η cosh η sinh η cos φ sinh η sin φ sinh η sinh η , where A is a constant from the p T integration. Due toboost invariance the further calculation may be carriedat η = 0. Next, we assume that at the proper time τ the system equilibrates rapidly . We thus use the Landaumatching condition , T µν ( x, y ) u ν ( x, y ) = ǫ ( x, y ) g µν u ν ( x, y ) , (7)which states that at each point the four-velocity of thefluid, u µ = (1 , v x , v y , / √ − v , is such that T µν is di-agonal in the local rest frame of the matter. The position-dependent eigenvalue ǫ is identified with the energy-density profile. The result of solving Eq. (7) with T µν from (6) for ∆ τ = 0 .
75 fm and the initial profile (1) for c =20-40% is shown in Fig. 3. The curves show the sec-tions along the x and y axes of ǫ and the velocity v at τ (no free streaming) and τ = 1 and 2 fm. Obviously, at τ we find ǫ ( x, y ) = 2 πAn ( x, y ). We note that, naturally,the profile spreads out as the time progresses. Impor-tantly, this effect is faster along the shorter axes, x . Thisis clearly indicated by the velocity profiles along the x and y axes. Thus the flow generated by free streamingis azimuthally asymmetric . It can also be obtained uponthe low ∆ τ and low x, y expansion, where straightforward FIG. 4: (Color online) Study of azHBT: The quantities R , /R , and R , /R , from the model (bands) andexperiment [30] (points), plotted as functions of the transversemomentum of the pion pair. algebra gives (for x ∆ τ ≪ a and y ∆ τ ≪ b ) v ( x, y ) = − ∆ τ ∇ n ( x, y ) n ( x, y ) = ∆ τ (cid:16) xa , yb , (cid:17) . (8)The results of the hydrodynamic calculation startingfrom free-streaming down to τ = 1 fm and readjustedinitial temperature at the proper time τ , followed by THERMINATOR simulations, are shown in Figs. 1 and 2with lighter curves/bands. We notice very similar resultsto the case with no free streaming, and again a properdescription of the data. Larger free-streaming times spoilthis agreement, as the flow becomes too strong. We re-mark, however, that the departure from the δ ( Y − η ) condition weakens the flow, which allows for larger val-ues of τ . The basic conclusion here is that the partonicfree streaming may be used to delay the start of hydrody-namics . The physical results are basically unaltered, asthe dispersion of the density profile, resulting in milderhydrodynamic development of flow, is accompanied bythe buildup of the initial flow .Having properly described the HBT radii, we may nowdeal “as a bonus” with the azimuthally sensitive HBTinterferometry [30] and consider the averages over theazimuthal angle, R i, ( k T ) = h R i ( k T , ϕ ) cos(2 ϕ ) i , where i = side or out ( R , long = 0). The results shown in Fig. 4display a remarkable agreement between our model andthe data (the curves with and without free streamingoverlap).In conclusion: 1) With a proper choice of the initialprofile (e.g. Gaussian) one may obtain a uniform de-scription of soft observables ( p T -spectra, v , and pionicHBT radii) at RHIC with the boost-invariant inviscid hy-drodynamics with a realistic equation of state and thussolve one of the “RHIC puzzles”. 2) Partonic free stream-ing generates initial transverse and elliptic flow. It maybe used to delay the start of the hydrodynamic phase.3) Azimuthally-sensitive HBT is described in agreementwith the data. 4) The complete treatment of resonancesis important, as is well known from statistical models.5) When other effects (viscosity, departure from boostinvariance) are incorporated, the important role of theinitial condition should not be overlooked. [1] U. W. Heinz, P. F. Kolb, hep-ph/0204061.[2] T. Hirano, Acta Phys. Polon., 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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