Initial conditons, equations of state and final state in hydrodynamics Hydro modelsHydro models IS, EoS, FOC and FSIS, EoS, FOC and FS ObservablesObservables Csanád Máté Eötvös University Department of Atomic Physics

11 September, 2008M. Csanád, WPCF08 Krakow2 Little vocabulary of hydrodynamics Exact solutionExact solution –Solution of hydro equations analytically, without approximation Parametric solutionParametric solution –Exact solution, that has fit parameters Hydro inspired parameterizationHydro inspired parameterization –Distribution determined at freeze-out only, their time dependence is not considered Numerical solutionNumerical solution –Solution of hydro equations numerically

11 September, 2008M. Csanád, WPCF08 Krakow3 How analytic hydro works Take hydro equations and EoSTake hydro equations and EoS Find a solutionFind a solution –Will contain parameters (like Friedmann, Schwarzschild etc.) –Will use a possible set of initial conditions Use a freeze-out conditionUse a freeze-out condition –Eg fixed proper time or fixed temperature –Generally a hyper-surface Calculate the hadron source functionCalculate the hadron source function Calculate observablesCalculate observables –E.g. spectra, flow, correlations –Straightforward calculation Hydrodynamics: Initial conditions Hydrodynamics: Initial conditions  dynamical equations  freeze-out conditions

11 September, 2008M. Csanád, WPCF08 Krakow4 Famous solutions Landau’s solution (1D, developed for p+p):Landau’s solution (1D, developed for p+p): –Accelerating, implicit, complicated, 1D –L.D. Landau, Izv. Acad. Nauk SSSR 81 (1953) 51 –I.M. Khalatnikov, Zhur. Eksp.Teor.Fiz. 27 (1954) 529 –L.D.Landau and S.Z.Belenkij, Usp. Fiz. Nauk 56 (1955) 309 Hwa-Bjorken solution:Hwa-Bjorken solution: –Non-accelerating, explicit, simple, 1D, boost-invariant –R.C. Hwa, Phys. Rev. D10, 2260 (1974) –J.D. Bjorken, Phys. Rev. D27, 40(1983) OthersOthers –Chiu, Sudarshan and Wang –Baym, Friman, Blaizot, Soyeur and Czyz –Srivastava, Alam, Chakrabarty, Raha and Sinha

11 September, 2008M. Csanád, WPCF08 Krakow5 3D solutions Nonrelativistic, spherically symmetric solutionNonrelativistic, spherically symmetric solution –P. Csizmadia, T. Csörgő, B. Lukács, nucl-th/ Relativistic, spherically symmetric solutionRelativistic, spherically symmetric solution –T. Csörgő, L. Csernai, Y. Hama, T. Kodama, nucl-th/ –Accelerationless –Hubble flow profile (flow proportional to distance) Relativistic, spherically symmetric solutionRelativistic, spherically symmetric solution –T. Csörgő, M. Nagy, M. Csanád, nucl-th/ –Accelerating –Realistic rapidity distributions (data described by it) –Advanced energy and lifetime estimate All describe expanding fireballsAll describe expanding fireballs –Sometimes: rings/shells of fire

11 September, 2008M. Csanád, WPCF08 Krakow6 Where we are Other accelerationless solutions:Other accelerationless solutions: –T. S. Biró, Phys. Lett. B 474, 21 (2000) –Yu. M. Sinyukov and I. A. Karpenko, nucl-th/ Solutions by coordinate transformations:Solutions by coordinate transformations: –S. Pratt, nucl-th/ Revival of interestRevival of interest –Bialas, Janik, Peschanski: Phys.Rev.C76:054901,2007 –Borsch, Zhdanov: SIGMA 3:116,2007 There are some exotic solutions as wellThere are some exotic solutions as well Need for solutions that are:Need for solutions that are: –explicit –simple –accelerating –relativistic –realistic / compatible with the data Buda-Lund type of solutions: each fulfilledBuda-Lund type of solutions: each fulfilled –but not simultaneously

11 September, 2008M. Csanád, WPCF08 Krakow7 A Buda-Lund type of solution For sake of simplicity, take the following nonrel. solutionFor sake of simplicity, take the following nonrel. solution –Csörgő, Akkelin, Hama, Lukács, Sinyukov, Phys.Rev.C67:034904,2003 Self similarly expanding ellipsoid, Gaussian ICSelf similarly expanding ellipsoid, Gaussian IC Flow profile: directional HubbleFlow profile: directional Hubble Equation of motion for principal axes:Equation of motion for principal axes: Freeze-out at constant temperature assumedFreeze-out at constant temperature assumed

11 September, 2008M. Csanád, WPCF08 Krakow8 Dependence on IC+EoS (nonrel) Evolution of principal axes of the ellipsoidEvolution of principal axes of the ellipsoid

11 September, 2008M. Csanád, WPCF08 Krakow9 Dependence on IC+EoS (nonrel) Evolution of expansion ratesEvolution of expansion rates

11 September, 2008M. Csanád, WPCF08 Krakow10 Dependence on IC+EoS (nonrel) Time evolution of temperatureTime evolution of temperature

11 September, 2008M. Csanád, WPCF08 Krakow11 Same in relativistic hydro Nagy, Csörgő, Csanád, Phys.Rev.C77:024908,2008, Csanád, Nagy, Csörgő, Eur.Phys.J.ST 155:19-26,2008 Same final state for different evolutions, even with viscosity (see T. Csörgő, WPCF’07)

11 September, 2008M. Csanád, WPCF08 Krakow12 Conjectured EoS dependence of  0 Relativistic, accelerating solution → describe dn/d Relativistic, accelerating solution → describe dn/d  Energy density modified compared to BjörkenEnergy density modified compared to Björken With  f /  0 = 10, c s = 0.35 [nucl-ex/ ], correction to   is about 2.9×With  f /  0 = 10, c s = 0.35 [nucl-ex/ ], correction to   is about 2.9×   = 14.5 GeV/fm 3 in 200 GeV, 0-5 %Au+Au at RHIC   = 14.5 GeV/fm 3 in 200 GeV, 0-5 %Au+Au at RHIC

11 September, 2008M. Csanád, WPCF08 Krakow13 Predictions of the Buda-Lund models Hydro predicts scaling (even viscous)Hydro predicts scaling (even viscous) What does a scaling mean?What does a scaling mean? –See Hubble’s law – or Newtonian gravity: –Data collapse Collective, thermal behavior →Collective, thermal behavior → Loss of information Spectra slopes:Spectra slopes: Elliptic flow:Elliptic flow: HBT radii:HBT radii:

11 September, 2008M. Csanád, WPCF08 Krakow14 Elliptic flow Prediction of 2003: scaling variablePrediction of 2003: scaling variable If plotted against ‘w’, data collapse:If plotted against ‘w’, data collapse: –From 20 to 200 GeV –All centralities –Pion, kaon, proton –p t and  dependence Prediction:Prediction: Csanád, Csörgő, Lörstad, Ster et al. nucl-th/

11 September, 2008M. Csanád, WPCF08 Krakow15 Prediction for HBT radii Exact hydro result (nonrel shown)Exact hydro result (nonrel shown) Correlation radii = geometrical  thermalCorrelation radii = geometrical  thermal –Harmonic squared sum: 1/R 2 corr = 1/R 2 geom + 1/R 2 therm Geom.: R geom = XGeom.: R geom = X Thermal:Thermal: Hubble-profile → X th =Y thHubble-profile → X th =Y th R out  R side  R longR out  R side  R long Correlation radii Geometrical radii Thermal radii

11 September, 2008M. Csanád, WPCF08 Krakow16 Azimuthal HBT and elliptic flow AsHBT data describedAsHBT data described Both governed by asymmetriesBoth governed by asymmetries – a s : coordinate-space –  2 : momentum-space – v 2 depends only on  2 Csanád, Tomasik, CsörgőCsanád, Tomasik, Csörgő Eur. Phys. J. A 37,111 (2008)

11 September, 2008M. Csanád, WPCF08 Krakow17 Azimuthal HBT and elliptic flow Simultaneous descriptionSimultaneous description Slopes as before (slide 12)Slopes as before (slide 12) Elliptic flow as before (slide 13)Elliptic flow as before (slide 13) Correlation radiiCorrelation radii Asymmetry parameters used:Asymmetry parameters used:  2 =0.17, a s =0.997 Csanád, Tomasik, CsörgőCsanád, Tomasik, Csörgő Eur. Phys. J. A 37,111 (2008)

11 September, 2008M. Csanád, WPCF08 Krakow18 Prediction for kaon HBT Transverse mass scaling → same curve for pions and kaons if plotted versus m tTransverse mass scaling → same curve for pions and kaons if plotted versus m t Other models?Other models? K

11 September, 2008M. Csanád, WPCF08 Krakow19 Beyond hydro: long source tails HRC reproduces HBT (hydro as well)HRC reproduces HBT (hydro as well) But also long tails in two-pion source!But also long tails in two-pion source! Anomalous diffusion (rescattering)Anomalous diffusion (rescattering) This goes beyond hydroThis goes beyond hydro –Hydro: regular m t scaling –Lévy-tails important here! Tail depends on m.f.p.,Tail depends on m.f.p., thus the cross-section –Kaons: lowest cross- section → heaviest tail T. Humanic, Int. J. Mod. Phys. E (2006) Csörgő, Braz.J.Phys.37: ,2007

11 September, 2008M. Csanád, WPCF08 Krakow20 The HBT test Models with acceptable results:Models with acceptable results: –nucl-th/ Multiphase Trasport model (AMPT) ‏ Z. Lin, C. M. Ko, S. Pal –nucl-th/ Hadron cascade model T. Humanic –hep-ph/ Family of Buda-Lund hydro models T. Csörgő, B. Lörstad, A. Ster –hep-ph/ Cracow (single freeze-out, thermal) W. Broniowski, W. Florkowski –nucl-ex/ Blast wave model F. Retiére for STAR – boost invariant rel. hydro, W. Broniowski, M. Chojnacki, W. Broniowski, M. Chojnacki, W. Florkowski, A. Kisiel

11 September, 2008M. Csanád, WPCF08 Krakow21 Conclusions Several types of hydro modelsSeveral types of hydro models –Success in spectra and flow –Few describe v 2 (  ) or HBT Hadronic final state: combination of IC, EoS and FCHadronic final state: combination of IC, EoS and FC –Penetrating probes required Similarities of successful models?Similarities of successful models? –Gaussian IC, Hubble flow etc. –Compare Hubble-coefficients in models! –Search for decisive tests!

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