Flow characteristics and strangeness production in the framework of highly-anisotropic and strongly-dissipative hydrodynamics
aa r X i v : . [ h e p - ph ] N ov Flow characteristics and strangeness production in theframework of highly-anisotropic and strongly-dissipativehydrodynamics ∗ Rados law Ryblewski
The H. Niewodnicza´nski Institute of Nuclear Physics,Polish Academy of Sciences,ul. Radzikowskiego 152, PL-31342 Krak´ow, PolandThe recently formulated model of highly-anisotropic and strongly dis-sipative hydrodynamics is used in 3+1 dimensions to describe flow char-acteristics and strangeness production in Au+Au collisions at the highestRHIC beam energy. Our results show very weak dependence on the initialmomentum anisotropy, provided the anisotropic phase lasts no longer than1 fm/c.PACS numbers: 25.75.-q, 25.75.Ld, 24.10.Nz, 25.75.Nq
1. Introduction
Soft-hadronic data collected in the ultra-relativistic heavy-ion experi-ments may be well described in the framework of the standard perfect-fluidhydrodynamics [1] or by dissipative hydrodynamics with small viscosity[2, 3, 4, 5, 6, 7]. However, the success of those approaches relies stronglyon the assumption that the produced system reaches state of local thermalequilibrium within a fraction of a fermi .Most of the microscopic models of the early stages fail to explain suchshort thermalization times [1]. This difficulty is known as the early ther-malization puzzle . One of its common solutions is the concept of a stronglycoupled quark-gluon plasma [8]. In addition, many microscopic approachesassume that the produced system exhibits initially large anisotropies in themomentum space, for example, see [9].Recently, several models have been developed [10, 11], which include theearly highly-anisotropic phase of the collisions. This has been achieved intro- ∗ Presented at
Strangeness in Quark Matter 2011 , Sept. 18–24, Cracow, Poland. We use the natural system of units where ¯ h = c = 1. (1) sqm2011 printed on July 9, 2018 ducing a pre-equilibrium stage connected with a subsequent perfect-fluid de-scription. Combination of different approaches in a single framework seems,however, not completely satisfactory. The need for a concise model whichcan describe different stages of heavy-ion collisions in the uniform way trig-gered our development of the highly-Anisotropic and strongly-DissipativeHYDROdynamics (ADHYDRO) [12, 13, 14, 15], see also [16, 17]In this paper we present our results obtained within the ADHYDROframework coupled to THERMINATOR [18, 19]. For the first time we useADHYDRO in 3+1 dimensions. We discuss flow characteristics of the emit-ted hadrons and strangeness production depending on the initial momentumanisotropy.
2. Formulation of the model
In the ADHYDRO model the evolution of the system is described bythe following equations [12] ∂ µ T µν = 0 , (1) ∂ µ σ µ = Σ , (2)which express the energy-momentum conservation and the entropy produc-tion laws, respectively. The energy-momentum tensor T µν in Eq. (1) hasthe form T µν = ( ε + P ⊥ ) U µ U ν − P ⊥ g µν − ( P ⊥ − P k ) V µ V ν , (3)which allows for the asymmetry between the longitudinal, P k , and trans-verse, P ⊥ , pressures. In the limit where the system becomes isotropic, P k = P ⊥ = P , the formula (3) reproduces the energy-momentum tensorof the perfect fluid. Similarly, the entropy production law (2) is reducedto the entropy conservation law, if we assume that the entropy source, Σ,vanishes. The four-vector U µ defines the four-velocity of the fluid and V µ is the four-vector defining the beam axis. In the general case, U µ and V µ may be parametrized in the following way U µ = ( u cosh ϑ, u x , u y , u sinh ϑ ) , (4) V µ = (sinh ϑ, , , cosh ϑ ) , (5)where u x and u y are the transverse components of the four-velocity veloc-ity field ( u ⊥ = q u x + u y and u = q u ⊥ ), ϑ is the longitudinal fluidrapidity. The parametrizations (4) and (5) satisfy simple normalization con-ditions, i.e., U = 1, V = − U · V = 0. The entropy flux σ µ in (2) isdefined by the formula σ µ = σ U µ , (6) qm2011 printed on July 9, 2018 where σ is the non-equilibrium entropy density.One can show [12] that instead of P k and P ⊥ it is more convenient to usethe entropy density σ and the anisotropy parameter x as two independentvariables (to a good approximation we have P k /P ⊥ = x − / ). Similarlyto standard hydrodynamics with vanishing baryon chemical potential, theenergy density ε introduced in Eq. (3), the entropy density σ , and theanisotropy parameter x are related through the generalized equation of state ε = ε ( σ, x ). In our model we use the following ansatz [13] ε ( x, σ ) = ε qgp ( σ ) r ( x ) , (7) P ⊥ ( x, σ ) = P qgp ( σ ) (cid:2) r ( x ) + 3 xr ′ ( x ) (cid:3) ,P k ( x, σ ) = P qgp ( σ ) (cid:2) r ( x ) − xr ′ ( x ) (cid:3) . where ε qgp and P qgp define the realistic equation of state constructed in Ref.[20]. The function r ( x ) is the pressure relaxation function characterizing theproperties of the fluid which exhibits the anisotropy x . Here we use the formintroduced in [12] r ( x ) = x − " x arctan √ x − √ x − . (8)In the isotropic case x = 1, r (1) = 1, r ′ (1) = 0, and Eq. (7) is reduced tothe equation of state used in [20].The function Σ which appears on the right-hand-side of Eq. (2) definesthe entropy production due to microscopic processes taking place in thesystem. Exactly these processes lead to thermalization of the system. Weuse the form proposed in [12]Σ( σ, x ) = (1 − √ x ) √ x στ eq , (9)where the time-scale parameter τ eq = 0 .
25 fm controls the rate of equili-bration . In the limit of small anisotropy Eq. (9) is consistent with thequadratic form of the entropy production in the Israel-Stewart theory. Farfrom equilibrium, hints for the form of Σ are lacking, although we mayexpect some suggestions from the AdS/CFT correspondence [21]. Thus,for large anisotropies the formula (9) should be treated as an assumptiondefining the dynamics of the system. Using this value we find that the system equilibrates within about 1 fm. sqm2011 printed on July 9, 2018
3. Initial conditions and freeze-out
In the general 3+1 case we have to solve Eqs. (1) and (2) for fiveunknown functions σ , x , u x , u y , and ϑ , which depend on the space-timecoordinates: τ, x ⊥ , and η ( τ is the proper time and η is the space-timerapidity). Since the system’s evolution is treated hydrodynamically from thevery early stages where the anisotropies are expected to be very large, we fixthe initial starting time for ADHYDRO to τ = 0 .
25 fm. Similarly to otherhydrodynamic calculations, we assume that there is no initial transverseflow, u x ( τ , x ⊥ , η ) = u y ( τ , x ⊥ , η ) = 0. For the initial longitudinal rapidityof the fluid we assume the Bjorken scaling ϑ ( τ , x ⊥ , η ) = η . We checkthree scenarios: i) the initial source is strongly oblate in the momentumspace, x ( τ , x ⊥ , η ) = 100, which corresponds to a transversally thermalizedsource, ii) the source is prolate in momentum space, x ( τ , x ⊥ , η ) = 0 . r (100) = r (0 . x ( τ , x ⊥ , η ) = 1, which gives the closest description tothe standard hydrodynamic case. The initial entropy density profile is givenby the formula σ ( τ , x ⊥ , η ) = ε − (cid:20) ε i ˜ ρ ( b, x ⊥ , η ) r ( x ( τ , x ⊥ , η )) (cid:21) , ˜ ρ ( b, x ⊥ , η ) = ρ ( b, x ⊥ , η ) ρ (0 , , , (10)where ˜ ρ is the normalized initial density of sources. The density profile ρ isgiven as the tilted source worked out in Ref. [22] and ε − is the inverse ofthe function ε gqp ( σ ). The initial energy density ε i = 107 . / fm is thesame for all three analyzed cases.The evolution is determined by the hydrodynamic equations until theentropy density drops to σ f = 1 .
79 fm − , which for x = 1 corresponds tothe temperature T f = 150 MeV. According to the single-freeze-out scenario,at this moment the abundances and momenta of particles are expectedto freeze-out and particles freely stream to detectors. The processes ofparticle production and decays of unstable resonances are described by using THERMINATOR 2 [19], which applies the Cooper-Frye formalism to generatehadrons on the freeze-out hypersurface extracted from ADHYDRO.
4. Results and conclusions
The
THERMINATOR 2 package [19] allows us the calculation of several one-and two-particle observables (such as the particle p T spectra, the directedand elliptic flow coefficients, the HBT radii, etc.). Due to limited space, inthis paper we show only the hyperon spectra and the directed flow coefficient v . More results will be presented and discussed in a separate publication.The scaling (10) helps us to keep the final particle multiplicities approx-imately the same in the three considered cases. In particular, this can be qm2011 printed on July 9, 2018 p T @ GeV D d N (cid:144) H d y2 Π p T dp T L @ G e V - D L STAR Au + Au ž S NN =
200 GeV X - W - +W + - - - - - Η v H % L STAR Au + Au ž S NN =
200 GeV c = - % Fig. 1. Transverse-momentum spectra of hiperons (left part) compared to the ex-perimental data from STAR [23] and directed flow coefficient (right part) comparedto the STAR data [24]. Theoretical lines are obtained from ADHYDRO for Au+Aucollisions at √ s NN = 200 GeV and the centrality class c = 0 − concluded from the left-hand-side of Fig. 1 where we show the transversemomentum spectra of hyperons. We observe that the stronger transverseflow in the case i) results in a bit harder spectra as compared to the caseii). Despite this fact, the spectra do not differ significantly. We observethat the model spectra of Ξ’s and Ω’s agree well with the data, while thenormalization of Λ’s is too small.The authors of Ref. [25] have shown that the v coefficient may be treatedas a probe for measuring the thermalization time, since it is very sensitive tothe early difference of pressures. On the right-hand-side of Fig. 1 we presentour results for the directed flow coefficient v . We observe small sensitivityto large initial anisotropy in the midrapidity region provided the anisotropicstage lasts not longer than 1 fm. Our results are not so much restrictive asthe results presented in [25], since we do not fix the final multiplicities butallow them to vary within the experimental errors.More results obtained in the ADHYDRO model in 3+1 dimensions willbe presented in a separate publication. We note that our previous resultsobtained in the 2+1 version [15] (a boost-invariant version) show similar,weak dependence of other physical observables on the initial pressure asym-metry. In conclusion, we state that we have found further evidence thatthere is a place for a highly-anisotropic phase at the early stages of heavy-ion collisions, provided the system reaches local thermal equilibrium before1 fm. sqm2011 printed on July 9, 2018
5. Acknowledgments
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