Lattice QCD based equation of state at finite baryon density

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1 Lattice QCD based equation of state at finite baryon density Pasi Huovinen J. W. Goethe Universität & Frankfurt Institute for Advanced Studies Hydrodynamics for Strongly Coupled Fluids May 12, 214, ECT*, Trento, Italia in collaboration with Peter Petreczky and Christian Schmidt funded by BMBF under contract no. 6FY992

2 From physics to phenomenology: Equation of state for modeling heavy ion collisions P. ECT*, May 12, 214 1/28

3 To close the equations of motion for relativistic fluid Dn = n µ u µ Dǫ = (ǫ+p) µ u ν (ǫ+p)du µ = µ P, we need an equation of state P = P(T,{µ i }) or P = P(ǫ,{n i }) For partons and hadrons obtain using QCD = lattice QCD P. ECT*, May 12, 214 2/28

4 lattice QCD: Budapest-Wuppertal collaboration, arxiv: : Trace anomaly ǫ 3P T 4 P. ECT*, May 12, 214 3/28

5 lattice QCD: Budapest-Wuppertal collaboration, arxiv: : obtain pressure via P T 4 P T 4 = T dt ǫ 3P T T 5 P. ECT*, May 12, 214 3/28

6 Practical problem: we observe particles not fluid have to convert fluid to particles Usual Cooper-Frye approach E dn dp 3 = Σ fo dσ µ p µ f(x,p u) conserves energy and momentum iff the EoS is the same before and after particlization Hadron Resonance Gas P. ECT*, May 12, 214 4/28

7 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 p2 3E f(p,t) valid when - interactions mediated by resonances - resonances have zero width Prakash & Venugopalan, NPA546, 718 (1992): experimental phase shifts Gerber & Leutwyler, NPB321, 387 (1989): chiral perturbation theory HRG good approximation at low temperatures lattice should reproduce HRG at T MeV P. ECT*, May 12, 214 5/28

8 Lattice vs. hadron resonance gas lattice QCD: Budapest-Wuppertal collaboration, arxiv: : (e-3p)/t T [MeV] BW HRG P. ECT*, May 12, 214 6/28

9 Lattice vs. hadron resonance gas lattice QCD: Budapest-Wuppertal collaboration, arxiv: : (e-3p)/t BW HRG T [MeV] HRG up to MeV for consistent switching to cascade P. ECT*, May 12, 214 6/28

10 Lattice vs. hadron resonance gas lattice QCD: Budapest-Wuppertal collaboration, arxiv: : (e-3p)/t BW HRG T [MeV] deviation from HRG at 167 MeV vs. T c 155 MeV P. ECT*, May 12, 214 6/28

11 Lattice vs. hadron resonance gas lattice QCD: Budapest-Wuppertal collaboration, arxiv: : (e-3p)/t BW HRG T [MeV] P. ECT*, May 12, 214 6/28

12 Temperature [MeV] Nuclear phase diagram K ~17 ǫ.7 GeV/fm 3 Quark gluon plasma? Tricritical point? Hadrons : baryons p,n,... mesons π,k,... SC phases? Everyday world n = /m 3 ǫ.16 GeV/fm 3 n ~5 1 Baryon density 1 3 (n q n q ) P. ECT*, May 12, 214 7/28

13 Taylor expansion for pressure where P T 4 = Σ i,jc ij (T) ( µb T ) i ( µs ) j, c ij (T) = 1 i j P i!j! (µ B /T) i (µ S /T) j T 4, T i.e. moments of baryon number and strangeness fluctuations and correlations an EoS based on lattice calculations of these? P. ECT*, May 12, 214 8/28

14 Continuum extrapolated second order coefficients (also c 11 ):.2 c 2.5 c HISQ stout T (MeV).2.1 HISQ stout T (MeV) HISQ: hotqcd collaboration, Phys. Rev. D 86, 3459 (212) stout: Budapest-Wuppertal collaboration, JHEP 121, 138 (212) Are first coefficients enough? Fourth and sixth order coefficients using p4 action with N τ = 4 large discretization effects? P. ECT*, May 12, 214 9/28

15 Hadrons on lattice 16 pseudoscalar mesons on lattice Hadron masses depend on lattice cutoff i.e. on temperature: E.g. for pseudoscalar mesons on asqtad calculations a i psx+b i psx 2 m 2 ps i = m 2 ps + 1 r 2 1 (1+c i psx) β i x = (a/r 1 ) 2 a = 1 N τ T P. ECT*, May 12, 214 1/28

16 3 MeV shift.2 c 2.2 c cut=1.7gev cut=1.9gev p4, N τ =4.1.5 cut=1.7gev cut=1.9gev p4, N τ =4 N τ =8, HISQ T [MeV] T [MeV] P. ECT*, May 12, /28

17 Parametrization c ij (T) = a ij1 ˆT n ij1 + a ij2 ˆT n ij2 + a ij3 ˆT n ij3 + a ij4 ˆT n ij4 + a ij5 ˆT n ij5 + a ij6 ˆT n ij6 +c SB ij, where n kij are integers with 1 < n kij < 23, and ˆT = T T s R, with T s =.1 or GeV, and R =.5 or.15 GeV. P. ECT*, May 12, /28

18 Constraints: c ij (T sw ) = c HRG ij (T sw ) d dt c ij(t sw ) = d dt chrg ij (T sw ) d 2 dt 2c ij(t sw ) = d2 dt 2cHRG ij (T sw ) d 3 dt 3c ij(t sw ) = d3 dt 3cHRG ij (T sw ) at T sw = 16 MeV for second order coefficients T sw = 155 MeV for fourth and sixth order coefficients 3rd derivative to quarantee smooth behaviour of speed of sound: c 2 s d2 dt 2c ij P. ECT*, May 12, /28

19 .2 c 2.5 c HISQ stout HG T (MeV).2.1 HISQ stout HG T (MeV) P. ECT*, May 12, /28

20 .6 c 4 p4, N τ =4.2 c p4, N τ =4.2.1 T [MeV] T [MeV] P. ECT*, May 12, /28

21 P/T P [GeV/fm -3 ] T [MeV] µ B [MeV] 3 P. ECT*, May 12, /28

22 P/T P [GeV/fm -3 ] T [MeV] µ B [MeV] 3 P. ECT*, May 12, /28

23 Speed of sound c s T [MeV] µ B = P. ECT*, May 12, /28

24 Speed of sound c s T [MeV] µ B = s/n B = 4 P. ECT*, May 12, /28

25 Speed of sound c s T [MeV] µ B = s/n B = 4 s/n B = 1 P. ECT*, May 12, 214 2/28

26 Speed of sound c s T [MeV] µ B = s/n B = 4 s/n B = 1 s/n B = 65 P. ECT*, May 12, /28

27 Speed of sound c s T [MeV] µ B = s/n B = 4 s/n B = 1 s/n B = 65 s/n B = 4 P. ECT*, May 12, /28

28 Speed of sound c s T [MeV] µ B = 2th order 4th order 6th order each correction smaller than previous expansion under control 4th order essential at low temperatures! P. ECT*, May 12, /28

29 Temperature [MeV] Nuclear phase diagram K ~17 ǫ.7 GeV/fm 3 Quark gluon plasma? Tricritical point? Hadrons : baryons p,n,... mesons π,k,... SC phases? Everyday world n = /m 3 ǫ.16 GeV/fm 3 n ~5 1 Baryon density 1 3 (n q n q ) P. ECT*, May 12, /28

30 p T -spectra at SPS dn/dyp T dp T [GeV -2 ] h - p-p bag model lattice.1 NA p T [GeV] harder EoS, more transverse flow, flatter spectra P. ECT*, May 12, /28

31 p T -spectra at SPS dn/dyp T dp T [GeV -2 ] h - p-p bag model lattice.1 NA p T [GeV] T fo 12 MeV (bag) 13 MeV (lattice) P. ECT*, May 12, /28

32 v 2 at SPS (b = 7 fm) π - v bag model lattice p T [GeV] p T fo 12 MeV (bag) 13 MeV (lattice) P. ECT*, May 12, /28

33 Conclusions Even continuum extrapolated lattice results require some fiddling to be useful for hydrodynamical modeling deviations from HRG around T c? EoS at finite baryon densities based on lattice QCD calculations of baryon number and strangeness fluctuations and correlations effect on flow when compared to bag model EoS tiny at SPS and (some?) RHIC low energy scan energies P. ECT*, May 12, /28

34 Backups P. ECT*, May 12, /28

35 c c HISQ stout HG T (MeV) P. ECT*, May 12, 214 3/28

36 c 13 c 13 p4, N τ = c 4 p4, N τ = T [MeV] T [MeV] P. ECT*, May 12, /28

37 v 2 at SPS (b = 7 fm) π - v bag model lattice p p T [GeV] T fo 12 MeV (both) P. ECT*, May 12, /28