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T. Csörgő, M. Csanád and Y. Hama MTA KFKI RMKI, Budapest, Hungary

from new solutions of Navier-Stokes eq uation s. Scaling properties of HBT radii and v 2. T. Csörgő, M. Csanád and Y. Hama MTA KFKI RMKI, Budapest, Hungary ELTE University, Budapest, Hungary USP, Sao Paulo, Brazil. Introduction: Observed scaling of spectra, elliptic flow and HBT radii

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T. Csörgő, M. Csanád and Y. Hama MTA KFKI RMKI, Budapest, Hungary

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  1. from new solutions of Navier-Stokes equations Scaling properties of HBT radii and v2 T. Csörgő, M. Csanád and Y. Hama MTA KFKI RMKI, Budapest, Hungary ELTE University, Budapest, Hungary USP, Sao Paulo, Brazil Introduction: • Observed scaling of spectra, elliptic flow and HBT radii • but what are the viscous corrections? Hydrodynamical scalingobserved in RHIC/SPS data Appear in beautiful, exact family of solutions of fireball hydro non-relativistic, perfect and viscous exact solutions relativistic, perfect, accelerating solutions Their application to data analysis at RHIC energies -> Buda-Lund Exact results: what can (or can not) be learned from data?

  2. An observation: Inverse slopes T of pt distribution increase ~ linearly with mass: T = T0 + m<ut>2 Increase is stronger in more head-on collisions. Suggests collective radial flow, local thermalization and hydrodynamics Nu Xu, NA44 collaboration, Pb+Pb @ CERN SPS T. Cs. and B. Lörstad, hep-ph/9509213 Successfully predicted by Buda-Lund hydro model (T. Cs et al, hep-ph/0108067) PHENIX, Phys. Rev. C69, 034909 (2004)

  3. Buda-Lund & exact hydrodynamics Relativistic solutions w/o acceleration Hwa Bjorken Hubble Ellipsoidal Buda-Lund Axial Buda-Lund Perfect non-relativistic solutions Relativistic solutions w/ acceleration Dissipative non-relativistic solutions data

  4. Input from lattice: EoS of QCD Matter • Tc=176±3 MeV (~2 terakelvin) • (hep-ph/0511166) • at  = 0, a cross-over • Aoki, Endrődi, Fodor, Katz, Szabó • hep-lat/0611014 Tc Lattice QCD EoS for hydro:p(,T) but in RHIC region:p~p(T) cs2 = p/e = cs2(T) = 1/(T) This EoS is in the Buda-Lund family of exact hydrodynamical solutions! Old idea: Quark Gluon Plasma Paradigm shift:Liquid of quarks

  5. Notation for fluid dynamics Non-relativistic dynamics t: time, r: coordinate 3-vector, r = (rx, ry, rz), m: mass, (t,r) dependent variables n: number density, : entropy density, p: pressure, : energy density, T: temperature, v: velocity 3-vector, v = (vx, vy, vz)

  6. Non-rel perfect fluid dynamics Equations of nonrelativistic hydro: local conservation of charge: continuity momentum: Euler energy EoS needed: Perfect fluid: 2 equivalent definitions, term used by PDG # 1: no bulk and shear viscosities, and no heat conduction. # 2: T = diag(e,-p,-p,-p) in the local rest frame. Ideal fluid: ambiguously defined term, discouraged #1: keeps its volume, but conforms to the outline of its container #2: an inviscid fluid

  7. Dissipative, Navier-Stokesfluids Navier-Stokes equations: dissipative, nonrelativistic: EoS needed: Shear and bulk viscosity, heat conductivity:

  8. Parametric perfecthydro solutions Ansatz: the density n (and T and ) depend on coordinates only through a scale parameter „s” • T. Cs. Acta Phys. Polonica B37 (2006), hep-ph/0111139 Principal axis of ellipsoid: (X,Y,Z) = (X(t), Y(t), Z(t)) Density=const on ellipsoids. Directional Hubble flow. g(s): arbitrary scaling function. Notation: n ~ (s), T ~ (s) etc.

  9. Family of perfecthydro solutions T. Cs. Acta Phys. Polonica B37 (2006) hep-ph/0111139 Volume is V = XYZ  = (T) exact solutions: T. Cs, S.V. Akkelin, Y. Hama, B. Lukács, Yu. Sinyukov, hep-ph/0108067, Phys.Rev.C67:034904,2003 or see the sols of Navier-Stokes later. The dynamics is reduced to ordinary differential equations for the scales X,Y,Z: PARAMETRIC solutions. Ti: constant of integration Many hydro problemscan be easilyillustrated and understoodon the equivalent problem:a classical potential motion of a mass-point in (a shot)! Note: temperature scaling function (s) arbitrary! (s) depends on (s). -> FAMILY of solutions.

  10. Initial boundary conditions From the new family of exact solutions, the initial conditions: Initial coordinates: (nuclear geometry + time of thermalization) Initial velocities: (pre-equilibrium+ time of thermalization) Initial temperature: Initial density: Initial profile function: (energy deposition and pre-equilibrium process) Initial profile = const of motion = final, observable profile!

  11. Final (freeze-out) boundary conditions From the new exact hydro solutions, the quantities that determine soft hadronic observables: Freeze-out temperature: (from small corrections to HBT radii) Final coordinates: (from small corrections to HBT radii) Final velocities: (from slopes of particle spectra) Final density: (enters as normalization factor) Final profile function: (= initial profile function! from solution)

  12. Role of the Equation of States: The time evolution(trajectory) depends on a „potential term” throughp= 1/cs2, related to the speed of sound: g Time evolution of the scales (X,Y,Z) follows a classic potential motion. Scales at freeze out determine the hadronic observables. Info on history LOST! No go theorem - constraints on initial conditions (information on spectra, elliptic flow of penetrating probels) indispensable. The arrow hits the target, but can one determine gravity from this information??

  13. Illustration, (in)dependence on EOS The initial conditions and the EoS can covary so that the freeze-out distributions are unchanged (T/m = 180/940)

  14. Initial and Freeze-out conditions: Different initial conditions lead to same freeze-out conditions. Ambiguity! Penetrating probes radiate through the time evolution!

  15. Family of viscoushydro solutions T. Cs.,Y. Hama in preparation Volume is V = XYZ Similar tohep-ph/0108067 The dynamics is reduced to non-conservative equations of motion for the parameters X,Y,Z: n <-> s, m cancels from new terms: depends on h/s and z/s

  16. Dissipative, heat conductive hydro solutions T. Cs. and Y. Hama, in preparation Introduction of ‘kinematic’ heat conductivity: Navier-Stokes, for small heat conduction, solved by the directional Hubble ansatz! Only new eq. from the energy equation: Asymptotic (large t) role of heat conduction - same order of magnitude (1/t2) as bulk viscosity (1/t2) - shear viscosity term is one order of magnitude smaller (1/t3) - valid only for nearly constant densities, - destroys self-similarity of the solution (if hot spots)

  17. Illustration, bulk and shear viscosity The initial conditions and the EoS can covary even in viscous case so that exactly the same freeze-out distributions (T/m = 180/940,h/n = 0.1 and z/n = 0.1)

  18. Role of shear and bulk viscosity: The time evolution(trajectory) depends on the fricition of the air (velocity dependent terms)! nS, nB Time evolution of scales (X,Y,Z) is modified in case of viscosity/friction. Scales at freeze out determine the hadronic observables. Info on history is LOST in the viscous case too! No go theorem - constraints on initial conditions (information from penetrating probes) indispensable. The arrow hits the target, but can one determine air friction from this information??

  19. Scaling predictions for (viscous) fluid dynamics - Slope parameters increase linearly with mass - Elliptic flow is a universal function its variable w is proportional to transverse kinetic energy and depends on slope differences. Gaussian approx: Inverse of the HBT radii increase linearly with mass analysis shows that they are asymptotically the same Relativistic correction: m -> mt hep-ph/0108067, nucl-th/0206051

  20. Hydro scaling of slope parameters Buda-Lund hydro prediction: Exact non-rel. hydro PHENIX data:

  21. Universal hydro scaling of v2 Black line: Theoretically predicted, universal scaling function from analytic works on perfect fluid hydrodynamics: hep-ph/0108067, nucl-th/0310040 nucl-th/0512078

  22. Hydro scaling of Bose-Einstein/HBT radii 1/R2eff=1/R2geom+1/R2thrm and 1/R2thrm ~mt intercept is nearly 0, indicating 1/RG2 ~0, thus (x)/T(x) = const! reason for success of thermal models @ RHIC! same slopes ~ fully developed, 3d Hubble flow Rside/Rout ~ 1 Rside/Rlong ~ 1 Rout/Rlong ~ 1 1/R2side ~ mt 1/R2out ~ mt 1/R2long ~ mt

  23. Summary • Buda-Lund model • Was a parameterization • Is an interpolator btwn analytic, exact hydro solutions with • Lattice QCD EOS • Shear and Bulk viscosity (NR), heat conductivity (NR) • Relativistic acceleration • Scaling predictions of viscous hydrodynamics • Scaling properties of slope parameters do not change • Scaling properties of elliptic flow do not change • Scaling properties of HBT radii are the same • If shear and bulk viscosities are present in Navier-Stokes eqs. • Asymptotic analysis: • Initially: shear > bulk > perfect fluid effects • At late times: perfect fluid > bulk > shear effects

  24. Back-up Slides

  25. Understanding hydro results Hydro problem equivalent to potential motion (a shot)! Hydro:Shot of an arrow: Desription of data Hitting the target Initial conditions (IC) Initial position and velocity Equations of state Strength of the potential Freeze-out (FC) Position of the target Data constrain EOS Hitting the target tells the potential (?) Different IC yields same FC Different archers can hit the target EoS and IC can co-vary IC and potential co-varied Universal scaling of v2 F/ma = 1 Viscosity effectsDrag force of air numerical hydro fails (HBT) Arrow misses the target (!)

  26. Some analytic Buda-Lund results HBT radii widths: Slopes, effective temperatures: Flow coefficients are universal:

  27. Role of initial temperature profiles Determines density profile! Examples of density profiles - Fireball - Ring of fire - Embedded shells of fire Exact integrals of hydro Scales expand in time Time evolution of the scales (X,Y,Z) - potential motion. Scales at freeze out -> observables. - info on history LOST! No go theorem: - constraintson initial conditions (penetrating probels) indispensable.

  28. #2: Our last work: inflation at RHIC nucl-th/0206051

  29. Suppression of high pt particle production in Au+Au collisions at RHIC 1st milestone: new phenomena

  30. 2nd milestone: new form of matter d+Au: no suppression Its not the nuclear effect on the structure functions Au+Au: new form of matter !

  31. 3rd milestone: Top Physics Story 2005 http://arxiv.org/abs/nucl-ex/0410003 PHENIX White Paper: second most cited in nucl-ex during 2006

  32. 4th Milestone: A fluid of quarks v2 for the D follows that of other mesons v2 for the φ follows that of other mesons Strange and even charm quarks participate in the flow

  33. Precision Probes central Ncoll = 975  94 • This one figure encodes rigorous control of systematics • in four different measurements over many orders of magnitude = =

  34. Motion Is Hydrodynamic z y x • When does thermalization occur? • Strong evidence that final state bulk behavior reflects the initial state geometry • Because the initial azimuthal asymmetrypersists in the final statedn/d ~ 1 + 2v2(pT) cos (2) + ... 2v2

  35. The “Flow” Knows Quarks baryons mesons • The “fine structure” v2(pT) for different mass particles shows good agreement with ideal (“perfect fluid”) hydrodynamics • Scaling flow parameters by quark content nq resolves meson-baryon separation of final state hadrons

  36. From fluid expansion to potential motion Dynamics of pricipal axis: Canonical coordinates, canonical momenta: Hamiltonian of the motion (for EoS cs2 = 1/ = 2/3): The role of initial boundary conditions, EoS and freeze-out in hydro can be understood from potential motion! i

  37. Role of initial temperature profile • Initial temperature profile = arbitrary positive function • Infinitly rich class of solutions • Matching initial conditions for the density profile • T. Cs. Acta Phys. Polonica B37 (2006) 1001, hep-ph/0111139 • Homogeneous temperature  Gaussian density • Buda-Lund profile: Zimányi-Bondorf-Garpman profile:

  38. Illustrations of exact hydro results • Propagate the hydro solution in time numerically:

  39. Principles for Buda-Lund hydro model • Analytic expressions for all the observables • 3d expansion, local thermal equilibrium, symmetry • Goes back to known exact hydro solutions: • nonrel, Bjorken, and Hubble limits, 1+3 d ellipsoids • but phenomenology, extrapolation for unsolved cases • Separation of the Core and the Halo • Core: perfect fluid dynamical evolution • Halo: decay products of long-lived resonances • Missing links: phenomenology needed • search for accelerating ellipsoidal rel. solutions • first accelerating rel. solution: nucl-th/0605070

  40. A useful analogy • Core  Sun • Halo  Solar wind • T0,RHIC ~ 210 MeV  T0,SUN ~ 16 million K • Tsurface,RHIC ~ 100 MeV  Tsurface,SUN  ~6000 K Fireball at RHIC  our Sun

  41. Buda-Lund hydro model The general form of the emission function: Calculation of observables with core-halo correction: Assuming profiles for flux, temperature, chemical potential and flow

  42. The generalized Buda-Lund model The original model was for axial symmetry only, central coll. In its general hydrodynamical form: Based on 3d relativistic and non-rel solutions of perfect fluid dynamics: Have to assume special shapes: Generalized Cooper-Frye prefactor: Four-velocity distribution: Temperature: Fugacity:

  43. Buda-Lund model is based on fluid dynamics • First formulation: parameterization • based on the flow profiles of • Zimanyi-Bondorf-Garpman non-rel. exact sol. • Bjorken rel. exact sol. • Hubble rel. exact sol. • Remarkably successfull in describing • h+p and A+A collisions at CERN SPS and at RHIC • led to the discovery of an incredibly rich family of • parametric, exact solutions of • non-relativistic, perfect hydrodynamics • imperfect hydro with bulk + shear viscosity + heat conductivity • relativistic hydrodynamics, finite dn/d and initial acceleration • all cases: with temperature profile ! • Further research: relativistic ellipsoidal exact solutions • with acceleration and dissipative terms

  44. Scaling predictions: Buda-Lund hydro - Slope parameters increase linearly with transverse mass - Elliptic flow is same universal function. - Scaling variable w is prop. to generalized transv. kinetic energy and depends on effective slope diffs. Inverse of the HBT radii increase linearly with mass analysis shows that they are asymptotically the same Relativistic correction: m -> mt hep-ph/0108067, nucl-th/0206051

  45. Scaling and scaling violations Universal hydro scaling breaks where scaling with number of VALENCE QUARKS sets in, pt ~ 1-2 GeV Fluid of QUARKS!! R. Lacey and M. Oldenburg, proc. QM’05 A. Taranenko et al, PHENIX: nucl-ex/0608033

  46. Exact scaling laws of NR hydro - Slope parameters increase linearly with mass - Elliptic flow is a universal function and variable w is proportional to transverse kinetic energy and depends on slope differences. Inverse of the HBT radii increase linearly with mass analysis shows that they are asymptotically the same Relativistic correction: m -> mt hep-ph/0108067, nucl-th/0206051

  47. Hydro scaling of elliptic flow G. Veres, PHOBOS data, proc QM2005 Nucl. Phys. A774 (2006) in press

  48. Hydro scaling of v2 and dependence PHOBOS, nucl-ex/0406021

  49. Universal scaling and v2(centrality,) PHOBOS, nucl-ex/0407012

  50. Universal v2 scaling and PID dependence PHENIX, nucl-ex/0305013

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