2008-04-21M. Csanád, T. Csörg ő, M.I. Nagy New analytic results in hydrodynamics UTILIZING THE FLUID NATURE OF QGP M. Csanád, T. Csörg ő, M. I. Nagy ELTE.

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M. Csanád, T. Csörg ő, M.I. Nagy New analytic results in hydrodynamics UTILIZING THE FLUID NATURE OF QGP M. Csanád, T. Csörg ő, M. I. Nagy ELTE MTA KFKI RMKI Budapest, Hungary Hydrodynamics at RHIC and QCD EOS Workshop, BNL, USA April 21, 2008

M. Csanád, T. Csörg ő, M.I. Nagy High temperature superfluidity at RHIC! All “realistic” hydrodynamic calculations for RHIC fluids to date have assumed zero viscosity  = 0 →  perfect fluid a conjectured quantum limit: P. Kovtun, D.T. Son, A.O. Starinets, hep-th/ How “ordinary” fluids compare to this limit? (4  ) η /s > 10 P. KovtunD.T. SonA.O. Starinetshep-th/ RHIC’s perfect fluid (4  ) η /s ~1 ! T > 2 Terakelvin The hottest & most perfect fluid ever made… (4  R. Lacey et al., Phys.Rev.Lett.98:092301,2007

M. Csanád, T. Csörg ő, M.I. Nagy Relativistic hydrodynamics Energy-momentum tensor: Relativistic Euler equation: Energy conservation: Charge conservation: Consequence is entropy conservation:

M. Csanád, T. Csörg ő, M.I. Nagy Context Renowned exact solutions Landau-Khalatnikov solution : dn/dy ~ Gaussian Hwa solution (PRD 10, 2260 (1974)) - Bjorken  0 estimate (1983) Chiu, Sudarshan and Wang: plateaux Baym, Friman, Blaizot, Soyeur and Czyz: finite size parameter  Srivastava, Alam, Chakrabarty, Raha and Sinha: dn/dy ~ Gaussian Revival of interest: Buda-Lund model + exact solutions, Biró, Karpenko+Sinyukov, Pratt (2007), Bialas+Janik+Peschanski, Borsch+Zhdanov (2007) New simple solutions Evaluation of measurables Rapidity distribution Advanced initial energy density HBT radii Advanced life-time estimation

M. Csanád, T. Csörg ő, M.I. Nagy Goal Need for solutions that are: explicit simple accelerating relativistic realistic / compatible with the data : lattice QCD EoS ellipsoidal symmetry (spectra, v 2, v 4, HBT) finite dn/dy Report on a new class that satisfies each of these criteria but not simultaneously M.I. Nagy, T. Cs., M. Csanád, arXiv: v1, PRC77: (2008) arXiv: v1 T. Cs, M. I. Nagy, M. Csanád, arXiv:nucl-th/ v4, PLB (2008)arXiv:nucl-th/ v4 M. Csanád, M. I. Nagy, T. Cs, arXiv: v3 [nucl-th] EPJ A (2008)arXiv: v3

M. Csanád, T. Csörg ő, M.I. Nagy Self-similar, ellipsoidal solutions Publication (for example): T. Csörg ő, L.P.Csernai, Y. Hama, T. Kodama, Heavy Ion Phys. A 21 (2004) 73 3D spherically symmetric HUBBLE flow: No acceleration : Define a scaling variable for self-similarly expanding ellipsoids: EoS : (massive) ideal gas Scaling function (s) can be chosen freely. Shear and bulk viscous corrections in NR limit : known analytically.

M. Csanád, T. Csörg ő, M.I. Nagy Some general remarks Hydrodynamics= Initial conditions  dynamical equations  freeze-out conditions Exact solution = formulas solve hydro without approximation Parametric solution = shape parameters introduced, time dependence given by ordinary coupled diff. eqs. Hydro inspired parameterization = shape parameters determined only at the freeze-out their time dependence is not considered Report on new class of exact, parametric solution of relativistic hydro M.I. Nagy, T. Cs., M. Csanád, arXiv: v1, PRC77: (2008) arXiv: v1 T. Cs, M. I. Nagy, M. Csanád, arXiv:nucl-th/ v4, PLB (2008)arXiv:nucl-th/ v4 M. Csanád, M. I. Nagy, T. Cs, arXiv: v3 [nucl-th] EPJ A (2008)arXiv: v3 Initial conditions: pressure and velocity on  =  0 = const EoS:  - B =  (p+B) c s 2 = 1/  Freeze-out condition: T= T f (  = 0), local simultaneity, n = u

M. Csanád, T. Csörg ő, M.I. Nagy New, simple, exact solutions If  = d = 1, general solution is obtained, for initial conditions. It is STABLE ! ARBITRARY initial conditions. It is STABLE ! Possible cases (one row of the table is one solution): Hwa-Bjorken, Buda-Lund type New, accelerating, d dimension d dimensional with p=p( ,  ) (thanks T. S. Biró) Special EoS, but general velocity Nagy,CsT, Csanád: Nagy,CsT, Csanád: arXiv: v1

M. Csanád, T. Csörg ő, M.I. Nagy New simple solutions Different final states from similar initial states are reached by varying

M. Csanád, T. Csörg ő, M.I. Nagy New simple solutions Similar final states from different initial states are reached by varying

M. Csanád, T. Csörg ő, M.I. Nagy Rapidity distribution Rapidity distribution from the 1+1 dimensional solution, for  > 1. T f : slope parameter.

M. Csanád, T. Csörg ő, M.I. Nagy Pseudorapidity distributions BRAHMS data fitted with the analytic formula of Additionally: y  η transformation

M. Csanád, T. Csörg ő, M.I. Nagy BRAHMS rapidity distribution BRAHMS dn/dy data fitted with the analytic formula

M. Csanád, T. Csörg ő, M.I. Nagy Advanced energy density estimate Fit result:  > 1 Flows accelerate: do work initial energy density is higher than Bjorken’s Work and acceleration. FYI: For  > 1 (accelerating) flows, both factors > 1

M. Csanád, T. Csörg ő, M.I. Nagy Advanced energy density estimate Correction depends on timescales, dependence is: With a typical  f /  0 of ~8-10, one gets a correction With a typical  f /  0 of ~8-10, one gets a correction factor of 2!

M. Csanád, T. Csörg ő, M.I. Nagy Conjecture: EoS dependence of  0 Four constraints 1)  Bj is independent of EoS (  = 1 case) 2) c s 2 = 1 case is solved for any  > 0.5 2) c s 2 = 1 case is solved for any  > 0.5 Corrections due to respect these limits. 3) c s 2 dependence of  is known in NR limit 3) c s 2 dependence of  is known in NR limit arXiv:hep- ph/ v2 arXiv:hep- ph/ v2 4) Numerical hydro results, 4) Numerical hydro results, e.g. K. Morita, arXiv:nucl-th/ v2arXiv:nucl-th/ v2 Conjectured formula – given by the principle of Occam’s razor: Using = 1.18, c s = 0.35,  f /  0 = 10, we get c s /  Bj = 2.9 Using = 1.18, c s = 0.35,  f /  0 = 10, we get  c s /  Bj = 2.9 in 200 GeV, 0-5 % Au+Au at RHIC  0 = 14.5 GeV/fm 3 in 200 GeV, 0-5 % Au+Au at RHIC

M. Csanád, T. Csörg ő, M.I. Nagy Conjectured EoS dependence of  0 Using = 1.18, and  f /  0 = 10 as before and c s = 0.35, [PHENIX, we get c s /  Bj = 2.9 and c s = 0.35, [PHENIX, arXiv:nucl-ex/ v1 ] we get  c s /  Bj = 2.9 arXiv:nucl-ex/ v1 in 200 GeV, 0-5 % Au+Au at RHIC  0 = 14.5 GeV/fm 3 in 200 GeV, 0-5 % Au+Au at RHIC

M. Csanád, T. Csörg ő, M.I. Nagy Advanced life-time estimate Life-time estimation: for Hwa-Bjorken type of flows Makhlin & Sinyukov, Z. Phys. C 39, 69 (1988) Underestimates lifetime (Renk, CsT, Wiedemann, Pratt, … ) Advanced life-time estimate: width of dn/dy related to acceleration and work At RHIC energies: correction is about +20%

M. Csanád, T. Csörg ő, M.I. Nagy Conjectured EoS dependence of  c Using = 1.18, and c s = 0.35, we get Using = 1.18, and c s = 0.35, we get  c s /  Bj = 1.36 in 200 GeV, 0-5 % Au+Au at RHIC in 200 GeV, 0-5 % Au+Au at RHIC

M. Csanád, T. Csörg ő, M.I. Nagy Conclusions Explicit simple accelerating relativistic hydrodynamics Analytic (approximate) calculation of observables Realistic rapidity distributions; BRAHMS data well described No go theorem: same final states, different initial states New estimate of initial energy density:  c /  Bj at least RHIC dependence on c s estimated,  c /  Bj ~ 3 for c s = 0.35 Estimated work effects on lifetime: at least 20% RHIC dependence on c s estimated,  c /  Bj ~ 1.4 for c s = 0.35 A lot to do … more general EoS less symmetry, ellipsoidal solutions asymptotically Hubble-like flows

M. Csanád, T. Csörg ő, M.I. Nagy New simple solutions in 1+D dim Fluid trajectories of the 1+D dimenisonal new solution THANK YOU!

M. Csanád, T. Csörg ő, M.I. Nagy Some comments

M. Csanád, T. Csörg ő, M.I. Nagy RHIC and the Phase “Transition” The lattice tells us that collisions at RHIC map out the interesting region from  Bj ~ 5 GeV/fm 3 for flat dn/dy to  c ~ 15 GeV/fm 3 for finite dn/dy ~ from RHIC to LHC What about SPS?

M. Csanád, T. Csörg ő, M.I. Nagy Models that pass the HBT test 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/ Buda-Lund hydro model nucl-th/ T. Csörg ő, B. Lörstad, A. Ster hep-ph/ Cracow (single freeze-out, thermal) W. Broniowski, W. Florkowski nucl-ex/ Blast wave model F. Retiére for STAR arXiv: v1arXiv: v boost invariant rel. hydrodynamical solution, Gaussian IC, lattice QCD EoS, resonance decays W. Broniowski, M. Chojnacki, W. Florkowski, A. Kisiel

M. Csanád, T. Csörg ő, M.I. Nagy Comments on RHIC HBT puzzle Spectra, v2 and HBT radii described by ideal hydro + resonance decays using Gaussian initial pressure profile and directional Hubble flow W. Broniowski et al, arXiv: v1arXiv: v1

M. Csanád, T. Csörg ő, M.I. Nagy Back-up Slides

M. Csanád, T. Csörg ő, M.I. Nagy How Perfect is Perfect? Measure η /s ! Damping (flow, fluctuations, heavy quark motion) ~ η /s FLOW: Has the QCD Critical Point Been Signaled by Observations at RHIC?, R. Lacey et al., Phys.Rev.Lett.98:092301,2007 (nucl-ex/ )nucl-ex/ The Centrality dependence of Elliptic flow, the Hydrodynamic Limit, and the Viscosity of Hot QCD, H.-J. Drescher et al., (arXiv: )arXiv: FLUCTUATIONS: Measuring Shear Viscosity Using Transverse Momentum Correlations in Relativistic Nuclear Collisions, S. Gavin and M. Abdel-Aziz, Phys.Rev.Lett.97:162302,2006 (nucl-th/ )nucl-th/ DRAG, FLOW: Energy Loss and Flow of Heavy Quarks in Au+Au Collisions at √s NN = 200 GeV (PHENIX Collaboration), A. Adare et al., Phys.Rev.Lett.98:172301,2007 (nucl-ex/ )nucl-ex/ CHARM!CHARM!

M. Csanád, T. Csörg ő, M.I. Nagy Landau-Khalatnikov solution Publications: 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 Implicit 1D solution with approx. Gaussian rapidity distribution Basic relations: Unknown variables: Auxiliary function: Expression of is a true „tour de force”

M. Csanád, T. Csörg ő, M.I. Nagy Landau-Khalatnikov solution Temperature distribution (animation courtesy of T. Kodama) „Tour de force” implicit solution: t=t(T,v), r=r(T,v)

M. Csanád, T. Csörg ő, M.I. Nagy Hwa-Bjorken solution The Hwa-Bjorken solution / Rindler coordinates

M. Csanád, T. Csörg ő, M.I. Nagy Hwa-Bjorken solution The Hwa-Bjorken solution / Temperature evolution

M. Csanád, T. Csörg ő, M.I. Nagy Bialas-Janik-Peschanski solution Publications: A. Bialas, R. Janik, R. Peschanski, arXiv: v1 Accelerating, expanding 1D solution interpolates between Landau and Bjorken Generalized Rindler coordinates:

M. Csanád, T. Csörg ő, M.I. Nagy Hwa-Bjorken solution Publications: R.C. Hwa, Phys. Rev. D10, 2260 (1974) J.D. Bjorken, Phys. Rev. D27, 40(1983) Accelerationless, expanding 1D simple boost-invariant solution Rindler coordinates: Boost-invariance (valid for asymptotically high energies): depends on EoS, e.g.

M. Csanád, T. Csörg ő, M.I. Nagy New simple solutions in 1+d dim The fluid lines (red) and the pseudo-orthogonal freeze-out surface (black)

M. Csanád, T. Csörg ő, M.I. Nagy Rapidity distribution Rapidity distribution from the 1+1 dimensional solution, for.

M. Csanád, T. Csörg ő, M.I. Nagy 1 st milestone: new phenomena Suppression of high p t particle production in Au+Au collisions at RHIC

M. Csanád, T. Csörg ő, M.I. Nagy 2 nd milestone: new form of matter d+Au: no suppression Its not the nuclear effect on the structure functions Au+Au: new form of matter !

M. Csanád, T. Csörg ő, M.I. Nagy 3 rd milestone: Top Physics Story PHENIX White Paper: second most cited in nucl-ex during 2006

M. Csanád, T. Csörg ő, M.I. Nagy Strange and even charm quarks participate in the flow v 2 for the φ follows that of other mesons v 2 for the D follows that of other mesons 4 th Milestone: A fluid of quarks

M. Csanád, T. Csörg ő, M.I. Nagy Predictions of the Buda-Lund model Hydro predicts scaling (even viscous) What does a scaling mean? See Hubble’s law – or Newtonian gravity: Cannot predict acceleration or height Collective, thermal behavior → Loss of information Spectra slopes: Elliptic flow: HBT radii:

M. Csanád, T. Csörg ő, M.I. Nagy data Axial Buda-Lund Ellipsoidal Buda-Lund Perfect non-relativistic solutions Relativistic solutions w/o acceleration Relativistic solutions w/ acceleration Dissipative non- relativistic solutions Hwa Bjorken Hubble What does the data tell us

M. Csanád, T. Csörg ő, M.I. Nagy BudaLund fits to 130 GeV RHIC data M. Csanád, T. Csörg ő, B. Lörstad, A. Ster, nucl-th/ , ISMD03

M. Csanád, T. Csörg ő, M.I. Nagy BudaLund fits to 200 GeV RHIC data M. Csanád, T. Csörg ő, B. Lörstad, A. Ster, nucl-th/ , QM04