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Vol. 39 (2008) ACTA PHYSICA POLONICA B No 3HYDRODYNAMICS OF PARTONS IN COLLISIONS?Mikolaj Chojnackia, Wojciech Florkowskia,baH.
Vol. 39 (2008) ACTA PHYSICA POLONICA B No 3HYDRODYNAMICS OF TRANSVERSALLYTHERMALIZED PARTONS IN ULTRA-RELATIVISTICHEAVY-ION COLLISIONS?Mikolaj Chojnackia, Wojciech Florkowskia,baH. Niewodniczański Institute of Nuclear Physics, Polish Academy of SciencesRadzikowskiego 152, 31-342 Kraków, PolandbInstitute of Physics, ?wi?tokrzyska Academy?wi?tokrzyska 15, 25-406 Kielce, Poland(Received October 5, 2007)The hydrodynamic description of transversally thermalized matter, pos-sibly formed at the early stages of ultra-relativistic heavy-ion collisions, isdeveloped. The formalism is based on the thermodynamically consistentapproach with all thermodynamic variables referring to two-dimensionalobjects, the so-called transverse clusters, which are identi?ed with the par-ticles having the same rapidity. The resulting hydrodynamic equations fora single cluster have the form of the two-dimensional hydrodynamic equa-tions of the perfect ?uid. Since the clusters do not perform any work inthe longitudinal direction, their energy is completely transformed and usedto generate strong radial and elliptic ?ows that turn out to be compatiblewith the experimental data.PACS numbers: 25.75.–q, 25.75.Dw, 25.75.Ld1. IntroductionThe data collected at RHIC indicates that matter produced in ultra-relativistic heavy-ion collisions behaves like an almost perfect fluid [1, 2].This observation triggers many new developments of relativistic hydrody-namics of perfect and viscous fluids [3–18]. On the other hand, the hydro-dynamic picture is still challenged by the two serious problems. The firstone refers to an unexpected short thermalization scale that is required todescribe the measured asymmetry of the transverse flow, the second one isconnected with the difficulty to explain the measured HBT radii (the so-called HBT puzzle).? This investigation was partly supported by the Polish Ministry of Science and HigherEducation grants Nos. N202 153 32/4247 and N202 034 32/0918.(721)722 M. Chojnacki, W. FlorkowskiRecently, it was proposed that these difficulties can be avoided if at theearly stages of the evolution of matter produced in ultra-relativistic heavy-ion collisions the hydrodynamic description applies only to the transversedegrees of freedom while the longitudinal expansion may be to large extentregarded as the motion of independent clusters [19]. In this paper we explainin more detail and develop the concept of Ref. [19]. The idea of purelytransverse equilibration has been analyzed previously by Heinz and Wongin Ref. [20] with the conclusion that it cannot be realistic since it doesnot lead to the large elliptic flow found in the corresponding 3-dimensional(3D) hydrodynamic calculations. Our conclusions differ substantially fromthose reached in [20] because of at least two reasons: Firstly, we comparethe results of our model calculations to the present data rather than toother hydrodynamic calculations. Secondly, we use a different technicalimplementation of the concept of transverse thermalization and longitudinalfree-streaming.The formalism introduced in [19] and developed in this paper is based onthe thermodynamically consistent approach where all thermodynamic vari-ables refer to two-dimensional objects. We call them transverse clusters andidentify with the particles having the same rapidity. The resulting hydrody-namic equations for a single cluster, i.e. for a fixed value of rapidity, havethe form of the 2-dimensional (2D) hydrodynamic equations of perfect fluid.Since the clusters do not perform any work in the longitudinal direction,their energy is completely transformed and used to generate strong radialand elliptic flows that turn out to be compatible with the experimental data.In contrast to Ref. [20] our approach conserves entropy and, therefore, de-scribes perfect fluid.An attractive feature of our approach are short space and time scalescharacterizing the system at the moment when the substantial elliptic flowis formed. We expect that this feature may help to solve the HBT puzzle andthe problem of early equilibration. The verification of this point requires,however, further developments, e.g. an implementation of the hadronizationscheme which transforms partons into the observed hadrons.The paper is organized as follows: In Sections 2–4 we introduce thephase-space distribution function, its moments, and the equations of trans-verse hydrodynamics, respectively. The moments of the distribution functiondefine the particle density current, the energy-momentum tensor, and theentropy current. The hydrodynamic equations are obtained from the con-servation laws for the energy and momentum. In Sections 5 and 6 we discussour initial conditions and the Cooper–Frye formula. In Sections 7 and 8 wepresent our results and conclusions. Finally, in the Appendix the technicaldetails of our calculations are given.Hydrodynamics of Transversally Thermalized Partons in . . . 7232. Ansatz for the phase-space distribution functionIn our approach we assume the following factorization of the phase-space distribution function f(x, p) into the longitudinal and transverse part(see Fig. 1)f(x, p) = f‖ geq(τ, η,?→x ,?→⊥ p ⊥) , (1)whereδ(η ? y)f‖ = n0δ(p‖t? Ez) = n0 . (2)τm⊥Here the standard definitions of the energy and longitudinal momentum interms of the rapidity have been usedE = m⊥ coshy , p‖ = m⊥ sinh y . (3)Similarly, the spacetime coordinates t and z have been expressed in termsof the proper time τ and spacetime rapidity ηt = τ coshη , z = τ sinh η . (4)We note that the form of the distribution (1)–(2) is very much similar tothat used by Heinz and Wong in Ref. [20], the only formal difference is thatthe factor n0/m⊥ in Eq. (1) is replaced in Ref. [20] by the factor τ0. Withour ansatz all thermodynamic variables which appear in the formalism areconsistently defined as two-dimensional quantities and the resulting equa-tions describe perfect fluid. This is different from Ref. [20] where the authorstreat their approach as non-equilibrium viscous dynamics which generatesentropy.Fig. 1. Visualization of the ansatz (1)–(2). Particles having the same longitudinalvelocity form transverse clusters. Their dynamics is governed by the equations of2D hydrodynamics of perfect ?uid. The rapidity distribution of clusters is de?nedby the function f‖.724 M. Chojnacki, W. FlorkowskiIn Eq. (2) the dimensionless normalization parameter n0 describes thedensity of clusters in rapidity. The role of this parameter may be comparedwith the role played by the time scale parameter τ0 introduced in Ref. [20].Heinz and Wong interpret τ0 as the initial time for the hydrodynamic evo-lution. A closer inspection of their approach indicates, however, that thisparameter plays a role of the global normalization having no relation withthe initial time. The hydrodynamic equations of Ref. [20] do not contain theτ0 parameter. The hydrodynamic evolution may start at arbitrary value ofthe initial time, let us call it τ = τinit, and the final results are independentof the choice that has been made for τinit.The equilibrium distribution geq has the form of the two-dimensionalequilibrium distribution function convoluted with the transverse flow. Forsimplicity we use the Boltzmann statistics and neglect the chemical poten-tial. In this case we have ( )m⊥ u?→0 ? p · ?→⊥ u ⊥geq = exp ? , (5)T √where the transverse mass m⊥ is defined by the formula m = m2⊥ + p2⊥.The transverse flow u? has the structureu? = (u0, ux, uy, 0) = (u ,?→0 u ⊥, 0) ,u20 ??→u 2⊥ = 1 , (6)which explicitly underlines its transverse character (vanishing longitudinalcomponent, uz = 0).3. Moments of the phase-space distribution functionBy calculating the appropriate momentum integrals of the distributionfunction one obtains the particle current N?, the energy-momentum tensorT ?ν , and the entropy current S?. Treating particles as massless with theansatz (1) one finds ∫? dy d2p⊥ ? δ(y ? η)N = n0νg p geq(2pi)2 τm⊥n 20νgT= ?∫ U , (7)2piτ?ν dy d2p⊥ ? ν δ(y ? η)T = n0νg p p geq(2pi)2 τm⊥n0ν3gT= (3U?Uν ? g?ν ? V ?V ν) , (8)2piτHydrodynamics of Transversally Thermalized Partons in . . . 725∫? dy d2p⊥ ? δ(y ? η)S = ?n0νg p g2 eq (ln geq ? 1)(2pi) τm⊥3n0νgT2= U? . (9)2piτHere we have introduced the degeneracy factor νg connected with the densityof states in the transverse space. Treating our system as dominated by gluonswe assume that νg = 16. In Eqs. (7)–(9) the 3D hydrodynamic flow U? hasthe structureU? = (cosh η u0, ux, uy, sinh η u0) . (10)We note that the correct normalization of the transverse flow u? leads di-rectly to the correct normalization of the four-vector U?, namely U?U? = 1.We also note that the energy-momentum tensor includes an extra contri-bution which does not appear in the standard hydrodynamics. The extraterm is proportional to the product V ?V ν , where the four vector V ? has thestructureV ? = (sinh η, 0, 0, cosh η) . (11)The four vector V ? is spacelike, V ?V? = ?1, and its origin is connected withspecial role of the longitudinal direction in our case, at η = 0 it takes theform V ? = (0, 0, 0, 1). With the presence of the term proportional to V ?V ν ,the energy-momentum tensor is traceless, T ?? = 0, as required for masslessparticles. The matrix form of the energy-momentum tensor is given in theAppendix.The appearance of the extra terms in the energy-momentum tensor wasalready discussed in Ref. [20]. We stress that our structure is simpler thanthat discussed by Heinz and Wong and this fact is connected with the iden-tification of the two-dimensional thermodynamic properties of the system.With our ansatz for the distribution function, the time components of thecurrents (7)–(9) reduce in the rest frame of the system to the two-dimensionalthermodynamic densities divided by the proper time. In particular, in theframe where U? = (1, 0, 0, 0) we find0 n0 n0 n0N = n2 , T00 = ε , S02 = s2 , (12)τ τ τwhere the appropriate tw∫o-dimensional densities are defined by the equationsd2p ν T 2gn2 = νg g∫ 2 eq= ,(2pi) 2pid2p ν T 3gε2 = νg p∫ 2 ⊥geq = ,(2pi) pid2p 3ν T 2gs2 = ?νg geq ( ln geq ? 1) = . (13)(2pi)2 2pi726 M. Chojnacki, W. FlorkowskiThose definitions may be supplemented by the definition of pressure, whichis obtained from the thermodynamic relationε2 + P2 = Ts2 . (14)A simple calculation givesνgT3 ε2P2 = = n2T = . (15)2pi 2The last equality, in agreement with the expectations for two-dimensionalsystems, yields the sound velocity2 1cs = . (16)2We note that the factor 1/τ in Eqs. (12) is of pure kinematic origin anddescribes the decrease of three dimensional densities due to the increasingdistance between the transverse clusters. At midrapidity we have dz = τ dyand the rapidity densities per unit transverse area, dA = dxdy, aredN dE dS= n0n2 , = n0ε2 , = n0s2 . (17)dAdy dAdy dAdyEqs. (17) are natural for 2D systems, and one can see that rapidity densitiesper unit transverse area change only if the temperature decreases, i.e., onlyif the transverse flow is present. We note that this type of behavior iscompletely different from the scenario assumed in the Bjorken model.The temperature dependence of our 2D thermodynamic variables is dif-ferent from the scaling of the energy density ε3 ? T4 and pressure P 43 ? Tthat was used in Ref. [20]. The relations ε3 ? T4, P3 ? T4, and ε3 = 2P3,when used in equilibrium, lead to the contradictory results for the value ofthe sound velocity. The first two yield s3 ? T3 and c2s = 1/3, whereas thethird one gives c2s = 1/2. Consequently, the approach presented in Ref. [20]should be interpreted as an effective description of viscous dynamics, whilein our approach the transverse dynamics is reduced to 2D hydrodynamicsof perfect fluid with c2s = 1/2.The large value of the sound velocity leads, as expected, to much stifferthan usual equation of state, and favors formation of the strong elliptic flow.The most important effect responsible for the formation of the strong flow is,however, the lack of the interaction between the transverse clusters. Sincethe energy of the clusters is not reduced by the work done in the longitudinaldirection, it is exclusively transformed and used to generate strong radial andelliptic flows.Hydrodynamics of Transversally Thermalized Partons in . . . 7274. Transverse hydrodynamicsThe hydrodynamic equations are obtained from the energy and momen-tum conservation laws? T ?ν? = 0 , (18)where the energy-momentum tensor is given by Eq. (8). We have checkedthat Eqs. (18) imply the entropy conservation law??S? = 0 , (19)with the entropy current defined by Eq. (9). The connection between Eqs. (18)and Eq. (19) is the same as in the standard 3D hydrodynamics of perfectfluid where the entropy conservation law ? ??S = 0 is obtained by the pro-jection of the energy-momentum conservation laws on the four-velocity ofthe fluid, i.e., from the formula U ?νν??T = 0.The use of our form of the energy-momentum tensor in Eq. (18) leads tothe following three equations? ? ?(rs2u0) + (rs2u0v cosα) + (s2u0v sinα) = 0 ,?τ ?r ?φ? ? ?(rTu(0v) + r cosα )(Tu0) + sinα (Tu0) = 0 ,?τ ?r ?φdα v sinα ?T cosα ?TTu20v + ? sinα + = 0 . (20)dτ r ?r r ?φThe first equation in (20) is just the entropy conservation, compare Eq. (19).The second equation in (20) follows from the symmetric linear combinationU1? T?1? + U2??T?2 = 0, while the third equation results from the asym-metric linear combination U ? T ?12 ? ? U?21??T = 0. In Eqs. (20) we usedthe cylindrical coordinat√es (y)r = x2 + y2 , φ = tan?1 . (21)( x )?1/2The quantity v is the transverse flow, u0 = 1? v2 , and α is thedynamical angle describing deviation of the direction of the flow from theradial directionvx = v cos(α + φ) , vy = v sin(α + φ) , (22)see Fig. 2. The differential operator d/dτ is the total time derivative definedby the formulad ? ? v sinα ?= + v cosα + . (23)dτ ?τ ?r r ?φ728 M. Chojnacki, W. FlorkowskiFig. 2. Geometrical de?nition of the dynamical angle α. In our approach, insteadof the two components of the transverse ?ow we use the magnitude of the ?ow vand the angle α. Both v and α are functions of τ, r and φ.The structure of Eqs. (20) is similar to the structure of the equations de-scribing the boost-invariant 3D hydrodynamic expansion, see [21] and [22]? ? ? rs3u0(rs3u0) + (rs3u0v cosα) + (s3u0v sinα) = ? ,?τ ?r ?φ τ? ? ?(rTu(0v) + r cosα )(Tu0) + sinα (Tu0) = 0 ,?τ ?r ?φTu2dα v sinα ?T cosα ?T0v + ? sinα + = 0 . (24)dτ r ?r r ?φThey differ from Eqs. (20) by the presence of the term rs3u0/τ on theright-hand side of the first equation in (24) and by the different form ofthe entropy density. We recall that the three dimensional entropy densitys3 of massless particles with ν internal degrees of freedom is given by theexpression2s3 = νpi2T 3 . (25)45The main physical difference between Eqs. (20) and (24) resides in the term(rs3u0)/τ leading to the decrease of the entropy and energy densities evenin the case where the transverse flow is absent. The physical origin of thisterm is the presence of the work which is done in the longitudinal direction.In the case of the 2D expansion such a term is not present and the depositedcollision energy is used only to produce the transverse flow.We note that (20) is valid even if the temperature T , the transverseflow v, its direction α, and the normalization factor n0 depend on the space-time rapidity η. The structure of the energy-momentum tensor (8) impliesHydrodynamics of Transversally Thermalized Partons in . . . 729that all partial derivatives with respect to η are multiplied by the expres-sions which vanish in the case η = y. Consequently, Eqs. (20) are valid forany value of the rapidity, and they should be solved, with the correspondinginitial condition, independently for each value of η. This property showsexplicitly that our system is not boost-invariant and may indeed be treatedas a superposition of the independent transverse clusters.The change of the initial entropy density is nicely described by the globalconservation laws which follow from Eqs. (20) and (24). The integration ofthe first equation in (20) over the transverse spacetime coordinates yields∫∞ ∫2pidr r dφT 2(τ, r, φ)u0(τ, r, φ) = const . (26)0 0On the other hand, for the 3D boost-invariant case one finds∫∞ ∫2pi3 constdr r dφT (τ, r, φ)u0(τ, r, φ) = . (27)τ0 0For 2D hydrodynamic expansion, see Eq. (26), the initial thermal energy maydecrease only at the expense of increasing transverse flow. For 3D boost-invariant expansion, see Eq. (27), even without the transverse expansion thetemperature drops, as is well known from the Bjorken model.We solve Eqs. (20), and also for the comparison Eqs. (24), using thetechnique presented in Ref. [22] which is a direct extension of the methodproposed in Ref. [23]. This method satisfies very accurately the conservationlaws (26) and (27). We find this agreement as an important check of ournumerical scheme.5. Initial conditionsThe hydrodynamic equations (20) are three equations for three unknownfunctions: T , v, and α. At the initial time τ = τinit the transverse flow vis zero. We also set the angle α to be zero at τ = τinit. On the other handthe temperature profile at τ = τinit is not trivial and its asymmetry in thetransverse plane generates the elliptic flow.Similarly to other hydrodynamic calculations we assume that the initialenergy density at the transverse position point ?→x ⊥ is proportional to thewounded-nucleon density ρWN at this point, namely3 ?→?→ νT ( x ⊥)ε2 ( x ⊥) = ∝ ρ (?→WN x ⊥) . (28)pi730 M. Chojnacki, W. FlorkowskiWe note that the assumption (28) used by us for a 2D system is equivalent tothe assumption s3 ∝ ρWN used in 3D hydrodynamic codes. In our practicalcalculations Eq. (28) takes the form[ρ ?→]1/3?→ WN ( x ⊥)T (τinit, x ⊥) = Ti , (29)ρWN (0)where the parameter Ti is the initial central temperature and the wounded-nucleon density is obtained from t?he formula [24]( ??→ ) [ ( ?→ )]A?→ b? ?ρ ( x ) = T +?→σ bWN ⊥ A x ?1? 1? T ? +?→⊥ A x ⊥2 ? A 2 ??( ?→ )? [ (?→ )]Ab ?+T ? +?→σ bA x ⊥ ?1? 1? TA +?→x ⊥ ? . (30)2 A 2In Eq. (30) σ = 40mb is the total nucleon–nucleon cross section and TA (x, y)is the nucleus thickness function ∫TA(x, y) = dz ρ (x, y, z) . (31)Here ρ(r) is the nuclear density profile given by the Woods–Saxon functionwith the conventional choice of parameters used for the gold nucleus:ρ = 0.17 fm?30 ,r0 = (1.12A1/3 ? 0.86A?1/3) fm ,a = 0.54 fm , A = 197 . (32)The value of the impact parameter in Eq. (30) depends on the centralityclass considered in the calculations.6. Cooper–Frye formulaTo calculate the transverse-momentum spectra at a certain value of thefinal temperature T 1f we use the st∫andard Cooper–Frye prescriptiondN n0νg ? δ(η ? y)= dΣ p? geq , (33)dyd2p⊥ (2pi)2 τm⊥1 Since we concentrate on the evolution of partons this temperature cannot be directlyinterpreted as the freeze-out temperature.Hydrodynamics of Transversally Thermalized Partons in . . . 731where the hypersurface Σ is defined by the condition Tf = const. Forcylindrically asymmetric collisions and midrapidity, y = 0, the transverse-momentum spectrum has the following expansion in the azimuthal angle ofthe emitted particlesdN dN= (1 + 2v2(p⊥) cos(2φp) + . . .) . (34)dyd2p⊥ dy 2pip⊥ dp⊥Eq. (34) defines the elliptic flow coefficient v2, which may be calculated from(34) as the asymmetry of the momentum spectrum1 fN (p⊥, φp = 0)? fN (ppi⊥, φp = )v2(p2⊥) = pi , (35)2 fN (p⊥, φp = 0) + fN (p⊥, φp = )2with fN being a shorthand notation for dN/(dyd2p⊥).We are of the opinion that for the essentially 2D expansion, the freeze-out criterion cannot involve the 3D energy density as it was proposed in [20].Application of a 3D freeze-out criterion to a superposition of 2D systems im-plies that the freeze-out is triggered, in the artificial way, by the increase ofthe relative distance between the 2D parts. With those remarks in mind weadopt a different strategy; we present our results for different final tempera-tures and check if the experimental data can be successfully reproduced forone of them.In our calculations we use the following parameterization of the hyper-surface Σ,t = d (φ, ζ, η) sin ζ cosh η , z = d (φ, ζ, η) sin ζ sinh η ,x = d (φ, ζ, η) cos ζ cosφ , y = d (φ, ζ, η) cos ζ sinφ , (36)which a√lso yields √τ = t2?z2 = d (φ, ζ, η) sin ζ , ρ= x2+y2 = d (φ, ζ, η) cos ζ . (37)At any given value of the spacetime rapidity η the position of the point onthe hypersurface is defined by the two angles, φ and ζ, and the distanceto the origin of the coordinate system, d (φ, ζ, η), see Fig. 3. The angle φis the standard azimuthal angle in the y–x plane, while the angle ζ is theazimuthal angle in the τ–ρ plane. With the standard definition of the four-momentum in terms of the rapidity and transverse momentum, and withthe standard definition of the element of the hypersurface dΣ? in terms ofthe totally antisymmetric tensor ε?αβγ , namelydxα dxβ dxγdΣ? = ε?αβγ dη dφ dζ , (38)dη dφ dζ732 M. Chojnacki, W. FlorkowskiFig. 3. Geometric interpretation of the parameters used to de?ne the shape of thehypersurface corresponding to a constant value of the ?nal temperature, T = Tf .we find the explicit f[orm of the Cooper–Frye integration measuredΣ? p? = d 2 sin ζ d cos ζ (m⊥ sin ζ cosh (η ? y) + p⊥ cos ζ cos (φ? φp))?d+ cos ζ (?m⊥ cos ζ cosh (η ? y) + p⊥ sin ζ cos (φ? φp))?ζ?d+ p⊥ sin (φ? φp)?φ ]?d+ cot ζ m⊥ sinh (η ? y) dηdφdζ . (39)?ηEq. (37) and the presence of the delta function in Eq. (33) imply that theintegration measure for massless particles appears always in a simple com-binationdΣ p?[?δ (η ? y) = d d cos ζ (sin ζ + cos ζ cos (φ? φp))τp⊥?d+ cos ζ (? cos ζ]+ sin ζ cos (φ? φp))?ζ?d+ sin (φ? φp) δ (η ? y) dηdφdζ. (40)?φThis form is used in Eq. (33) to calculate the transverse-momentum spectra.The interesting feature of this formula is that the derivative of the distanced with respect to the spacetime rapidity η has disappeared. This meansthat for each value of the rapidity (and spacetime rapidity) one calculatesthe spectra using the function d defined exactly for this value of rapidity.Such a function is provided by the hydrodynamic code which should beexecuted also for exactly the same value of rapidity (treated as a parameter).Hydrodynamics of Transversally Thermalized Partons in . . . 733We thus see that the Cooper–Frye prescription is consistent with the physicalpicture of non-interacting transversally expanding 2D clusters. Moreover,the system under study is not necessarily boost-invariant.7. ResultsIn Fig. 4 we show the PHENIX pion spectra for the centrality classes20–30% and 60–70% [25], while in Fig. 5 we show the PHENIX data onthe elliptic flow for the centrality classes 20–40% and 40–60% [26]. In bothcases the data (solid lines) are compared with our hydrodynamic calcula-tions (dashed lines) done for the appropriate values of the impact parame-Fig. 4. Transverse-momentum spectra for the centrality c = 20–30% (upper part)and c = 60–70% (lower part). The PHENIX experimental results [25] (solid lines)are compared to the model calculations (dashed lines). The initial central tem-perature, Ti = 250MeV, is the same in all cases. The four di?erent values of the?nal temperature are considered: Tf = 200, 180, 160 and 140MeV. As discussedin the text in more detail, the spectra are to large extent independent of the ?naltemperature (the dashed lines overlap). The normalization factor n0 = 1 was used.734 M. Chojnacki, W. Florkowskiter. Following the PHENIX study performed within the Glauber model [25]we used the values: b = 7.25 fm for c = 20–30%, b = 7.94 fm for c =20–40%, b = 10.26 fm for c = 40–60% and b = 11.69 fm for c = 60–70%.The experimental spectrum of pions shown in Fig. 4 is the spectrum of pos-itive pions multiplied by a factor of 3. This is done to account for the totalhadron multiplicity.Fig. 5. The elliptic ?ow coe?cient v2 for the centrality c = 20–40% (upper part)and c = 40–60% (lower part). The solid lines represent the PHENIX data for pionsand kaons [26] (the width indicates the experimental error). The dashed lines showour model results. The values of the initial and ?nal temperatures are the same asin Fig. 4.Our results were obtained for the initial central temperature Ti =250 MeV and for four different values of the final temperature Tf = 200,180, 160 and 140MeV. Similarly to the results obtained by Heinz and Wongwe observe that a lower than usual (i.e., lower than for 3D expansion) initialtemperature is required to describe correctly the slope of the experimentalspectrum. This effect is related to the generation of stronger transverse flowHydrodynamics of Transversally Thermalized Partons in . . . 735in the case of 2D expansion. In agreement with Ref. [20] we also find thatthe shape of the spectrum is quite insensitive to the final temperature sincethe lowering of the temperature in the distribution function is compensatedby the increase of the magnitude of the transverse flow. This property canbe understood in our case as an effect of the exact conservation of the trans-verse energy and entropy. Since the entropy current is proportional to theparticle current, the particle number is also conserved. Combing the conser-vation laws with the fact that particles have fixed rapidity we find that theaverage transverse momentum must be exactly conserveddE /⊥ dN dE⊥= = 〈p⊥〉 = const . (41)dy dy dNThe conservations of the average transverse momentum and the particlenumber do not allow for substantial changes of the slope of the spectrum.This behavior is shown in Fig. 4, where the spectra for different final tem-peratures are shown to be very much similar.The results presented in Fig. 4 were obtained with the normalizationfactor n0 = 1. The correct normalization for the peripheral collisions maybe obtained by the small reduction of n0, while the correct normalization forthe central collisions requires larger values of n0. Dependence of the normal-ization factor on the centrality reflects the experimental fact that the totalmultiplicity grows faster with centrality than the number of the woundednucleons. Frequently, this effect is understood as the extra contribution fromthe binary collisions. A possible explanation of such a centrality dependenceis also offered by the model of wounded quarks and diquarks [27, 28]. Sincesuch effects are not included in the form of our initial condition (28), differentvalues of n0 should be applied for different values of the centrality.In Fig. 5 we show our main results concerning the elliptic flow coefficient.The very striking observation is that the elliptic flow is large and for lowervalues of the final temperature it even exceeds the data. The origin ofthis effect is the genuine two-dimensional hydrodynamic expansion whichtransforms the initial thermal energy exclusively into the transverse flow. Weobserve that for peripheral collisions the elliptic flow coefficient v2 becomescompatible with the data already at the temperature as high as 200MeV.For more central collisions the experimental values are recovered at smallerbut still high values of Tf .736 M. Chojnacki, W. Florkowski8. ConclusionsIn this paper we have shown that the idea that the system created inultra-relativistic heavy-ion collisions undergoes the hydrodynamic expan-sion only in the transverse direction is compatible with the experimentaldata. With a suitable choice of the initial and final temperatures one isable to describe the measured hadron spectra and the elliptic flow coeffi-cient v2. Clearly, further investigations and developments of such a simpleidea should be performed to address more subtle aspects of hadron produc-tion. In particular, the present description should be supplemented with thehadronization model.We thank Professor Andrzej Bialas for his inspirations to continue thisresearch and for the critical reading of the manuscript.Appendix AThe standard definition of the energy-momentum tensor as the secondmoment of the distribution function in the momentum space yieldsn ν T 30 gT ?ν = t?ν ,2piτwhere?(sinhx =( sh x, cosh) x = ch x) ( )?? ch2η 2u20 + u2⊥ chη 3u0ux chη 3u0uy chηshη 2u2 2? 0+ u⊥?ν ? chη 3u0ux (1 + 3uxux) 3uxuy shη 3u0u ?t = ( x ? .chη 3u0uy ) 3uyux (1 + 3uyuy) sh(η 3u0uy ) ?chηshη 2u2 + u2 shη 3u u shη 3u u sh2η 2u2 + u20 ⊥ 0 x 0 y 0 ⊥This expression may be rewritten in the compact form, presented in thelower line in Eq. (8), if the definitions of the four-vectors U? and V ? areused.REFERENCES[1] U.W. Heinz, nucl-th/0512051.[2] E. Shuryak, Nucl. Phys. A750, 64 (2005).[3] D. Teaney, Phys. Rev. C68, 034913 (2003).[4] T. Hirano, M. Gyulassy, Nucl. Phys. A769, 71 (2006).[5] T. Hirano, Nucl. Phys. A774, 531 (2006).Hydrodynamics of Transversally Thermalized Partons in . . . 737[6] T. Hirano, U.W. Heinz, D. Kharzeev, R. Lacey, Y. Nara, Phys. Lett. B636,299 (2006).[7] Y. Hama et al., Nucl. Phys. A774, 169 (2006).[8] K.J. Eskola, H. Honkanen, H. Niemi, P.V. Ruuskanen, S.S. Rasanen, Phys.Rev. C72, 044904 (2005).[9] U.W. Heinz, H. Song, A.K. Chaudhuri, Phys. Rev. C73, 034904 (2006).[10] C. Nonaka, S.A. Bass, Nucl. Phys. A774, 873 (2006).[11] R. Andrade, F. Grassi, Y. Hama, T. Kodama, J. Socolowski, Phys. Rev. Lett.97, 202302 (2006).[12] T. Koide, G.S. Denicol, P. Mota, T. Kodama, hep-ph/0609117.[13] C. Nonaka, S.A. Bass, Phys. Rev. C75, 014902 (2007).[14] T. Hirano, U.W. Heinz, D. Kharzeev, R. Lacey, Y. Nara, nucl-th/0701075.[15] R. Baier, P. Romatschke, U.A. Wiedemann, Nucl. Phys. A782, 313 (2007).[16] P. Huovinen, P.V. Ruuskanen, nucl-th/0605008.[17] L.M. Satarov, I.N. Mishustin, A.V. Merdeev, H. Stoecker, hep-ph/0611099.[18] R. Baier, P. Romatschke, U.A. Wiedemann, Phys. Rev. C73, 064903 (2006).[19] A. Bialas, M. Chojnacki, W. Florkowski, Phys. Lett. B, in printarXiv:0708.1076 [nucl-th].[20] U. Heinz, S.M.H. Wong, Phys. Rev. C66, 014907 (2002).[21] A. Dyrek, W. Florkowski, Acta Phys. Pol. B 15, 653 (1984).[22] M. Chojnacki, W. Florkowski, Phys. Rev. C74, 034905. (2006)[23] G. Baym, B.L. Friman, J.P. Blaizot, M. Soyeur, W. Czyz, Nucl. Phys. A407,541 (1983).[24] A. Bialas, M. Bleszynski, W. Czyz, Nucl. Phys. B111, 461 (1976).[25] S. Adler et al. [PHENIX Collaboration], Phys. Rev. C69, 034909 (2004).[26] S. Adler et al. [PHENIX Collaboration], Phys. Rev. Lett. 91, 182301 (2003).[27] A. Bialas, A. Bzdak, Phys. Lett. B649, 263 (2007).[28] A. Bialas, A. Bzdak, Acta Phys. Pol. B 38, 159 (2007).
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