Lattice based Equation(s) of State and its (their) effect(s) on the hydrodynamical evolution

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1 Lattice based Equation(s) of State and its (their) effect(s) on the hydrodynamical evolution Pasi Huovinen J. W. Goethe Universität, Frankfurt Quantifying the properties of Hot QCD matter June 11, 1, Institute for Nuclear Theory in collaboration with Peter Petreczky at BNL arxiv:91:51

2 QCD equation of state lattice QCD (Karsch & Laermann, hep-lat/355): 1 1 ε/t RHIC ε SB /T 1 1 SPS 3 flavour flavour +1-flavour T c = (173 +/- 15) MeV ε c ~.7 GeV/fm 3 LHC T [MeV] EoS from first principles - until recently, seldom used in hydro calculations P. INT, June 11, 1 1/19/

3 EoS by hotqcd collaboration Bazavov et al. arxiv: [hep-lat] (ε-3p)/t T [MeV] Tr asqtad: N τ = p: N τ = Tr ε SB /T ε/t : N τ = 3p/T : N τ = T [MeV] evaluate interaction measure (ǫ 3P)/T obtain pressure via P T P T T = What is P(T )? What is (ǫ(t ) 3P(T ))/T? How good is lattice below T c? dt ǫ 3P T T 5 P. INT, June 11, 1 /19/

4 Equation of state below T c Bazavov et al arxiv: (ε-3p)/t asqtad: N τ = p: N τ = Tr 1 T [MeV] Lattice EoS Hadron Resonance Gas EoS P. INT, June 11, 1 3/19/

5 Hadron Resonance Gas model EoS of interacting hadron gas well approximated by non-interacting gas of hadrons and resonances P(T) = i d 3 p p 3E f(p,t) valid when - interactions mediated by resonances Prakash & Venugopalan, NPA5, 71 (199): experimental phase shifts Gerber & Leutwyler, NPB31, 37 (199): chiral perturbation theory HRG good approximation at low temperatures lattice should reproduce HRG at T 1 1 MeV practical problem: how to convert fluid to particles? energy conservation iff EoS is the same before and after freeze-out P. INT, June 11, 1 /19/

6 Hadrons on lattice Hadron masses depend on lattice cutoff i.e. on temperature: E.g. for pseudoscalar mesons a i psx+b i psx m ps i = m ps + 1 r 1 (1+c i psx) β i x = (a/r 1 ) a = 1 N τ T + 1 pseudoscalar mesons on lattice HRG with lattice mass spectrum? P. INT, June 11, 1 5/19/

7 Hadronic fluctuations i.e. baryon number, strangeness and charge susceptibilities χ x = 1 lnz VT 3 (µ x /T) = 1 P T µ, x where µ x = µ B,µ S or µ Q.35.3 χ B 1 χ S p asqtad.. p asqtad T [MeV] T [MeV] Lattice masses fluctuations in resonance gas and lattice similar P. INT, June 11, 1 /19/

8 Interaction measure 1 (ε-3p)/t p asqtad T [MeV] very little room for modifications in hadron gas BUT, what is physical mass spectrum? conservative estimate: free particle masses P. INT, June 11, 1 7/19/

9 Phenomenological EoS T < T sw : HRG interaction measure (black) T > T sw : Lattice interaction measure (red) - lattice N τ = data, Cheng et al. Phys. Rev. D 77, 1511 () ǫ and P overshoot Stefan-Boltzmann limit! Interaction measure too large, but where? P. INT, June 11, 1 /19/

10 Interaction measure 1 1 Cheng et al ( ) (ε-3p)/t p: N τ = Tr T [MeV] Bazavov et al ( 9) (ε-3p)/t asqtad: N τ = p: N τ = T [MeV] Tr peak region sensitive to N τ P. INT, June 11, 1 9/19/

11 Procedure for EoS HRG below T 1 19 MeV Parametrize lattice using: ǫ 3P T = d T + d T + c 1 T n 1 + c T n Require that: ǫ 3P T, T d dt ǫ 3P T, T d ǫ 3P dt T T s T=MeV T 3 = (9% 95%) s SB T are continuous = T, d, c 1, c fixed χ fit to lattice above T = 5 MeV P. INT, June 11, 1 1/19/

12 Interaction measure 1 p, N = t p, N = t asqtad, N = t asqtad, N = t s95 s9 (e-3p)/t T (MeV) For the 95% s SB limit we get T = 171. MeV, d =.5, d = , c 1 = , c = , n 1 =, n = 9 P. INT, June 11, 1 11/19/

13 Interaction measure II add estimated peak to the fit s95 s9 s95p s9p (e-3p)/t T (MeV) P. INT, June 11, 1 1/19/

14 Phenomenological EoS s95 s9 s95p s9p 1 1 (e-3p)/t e/t, 3P/T p, N = t asqtad, N = t s95 s9 s95p T (MeV) T (MeV) obtain pressure via P T P T = T dt ǫ 3P T T 5 P. INT, June 11, 1 13/19/

15 Speed of sound no softening below the HRG! P. INT, June 11, 1 1/19/

16 ideal fluid, b = 7 fm keep everything fixed: τ =. fm/c, T dec = 15 MeV Effect on flow I = harder EoS, flatter spectra P. INT, June 11, 1 15/19/

17 Effect on flow II ideal hydro, Au+Au at s NN = GeV chemical equilibrium s95p: T dec = 1 MeV EoS Q: first order phase transition at T c = 17 MeV, T dec = 15 MeV P. INT, June 11, 1 1/19/

18 Chemical non-equilibrium ideal fluid, b = 7 fm keep everything fixed: τ =. fm/c, T chem = 15 MeV, T dec = 1 MeV = harder EoS, flatter spectra P. INT, June 11, 1 17/19/

19 Effect on flow III ideal hydro, Au+Au at s NN = GeV T chem = 15 MeV EoS Q: T dec = 1 MeV, s ini N bin τ =. fm/c s95p, τ =.: T dec = 1 MeV, s ini N bin, τ =. fm/c s95p, τ =.: T dec = 1 MeV, s ini N bin +N part, τ =. fm/c P. INT, June 11, 1 1/19/

20 Conclusions below T c lattice and HRG differ because of hadron mass spectrum HRG good description below T c some uncertainty in the parametrization of the EoS but it doesn t matter proton v (p T ) may or may not be sensitive to EoS details matter! EoS tables available at of_state and P. INT, June 11, 1 19/19/

21 Budapest-Wuppertal EoS Aoki et al. hep-lat/51 P(T) from expectation values of gauge action S g chiral condensates ΨΨ temperature scale? P. INT, June 11, 1 /19/

22 and its interpretation M. Chojnacki et al., arxiv:71.97 HRG speed of sound below T c lattice speed of sound above T c (parametrization by Wuppertal group) fix the scale by T c = 17 MeV [ ] T dt s(t) = s(t )exp T T c s P. INT, June 11, 1 1/19/

23 Equations of State ε/t P/T FB-Bielefeld K-Wuppertal T (MeV) FB-Bielefeld: Frankfurt-BNL interpretation of hotqcd EoS, s95p K-Wuppertal: Krakow interpretation of Wuppertal EoS P. INT, June 11, 1 /19/

24 Speed of sound c s FB-Bielefeld K-Wuppertal T (MeV) no softening below the HRG! P. INT, June 11, 1 3/19/

25 Flow anisotropy Au+Au collision at RHIC, s = GeV, b=7 fm.1.1. ε p... FB-Bielefeld K-Wuppertal ǫ p = Txx T yy T xx +T yy 1 1 τ (fm) P. INT, June 11, 1 /19/

26 v (p T ) at RHIC Au+Au collision at RHIC, s = GeV, b=7 fm v (%) p T (GeV) π p FB-Bielefeld K-Wuppertal FB-Bielefeld: T c = 1 MeV, K-Wuppertal: T c = 15 MeV P. INT, June 11, 1 5/19/

27 Other observables? HBT? τ (fm) FB-Bielefeld K-Wuppertal x (fm) not promising γ and l + l? what are the rates? P. INT, June 11, 1 /19/

28 Preliminary Bazavov and Petreczky, arxiv: (ε-3p)/t HISQ,N t = HISQ, N t = asqtad p Laine s95p-v1 T [MeV] Ε 3p T 5 3 Borsányi et al, arxiv:15.35 HRGphysical HRGdistortedstoutN t HRGdistortedasqtadN t stoutn t 1 stoutn t asqtadn t pn t T MeV P. INT, June 11, 1 7/19/

29 Conclusions below T c lattice and HRG differ because of hadron mass spectrum HRG good description below T c some uncertainty in the parametrization of the EoS but it doesn t matter proton v (p T ) may or may not be sensitive to EoS details matter! EoS tables available at of_state and P. INT, June 11, 1 /19/