F. Karsch for USQCD, LQCD II p. 1/27. Lattice QCD at High Temperature and Density. Frithjof Karsch for USQCD Brookhaven National Laboratory

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1 F. Karsch for USQCD, LQCD II p. 1/27 Lattice QCD at High Temperature and Density Frithjof Karsch for USQCD Brookhaven National Laboratory

2 F. Karsch for USQCD, LQCD II p. 2/27 Towards A New State of Matter Interferometry ππ,... Κ, Λ, Ω,... Last sca tering QCD, QGP, LGT and HIC Temperature 120 MeV resonance gas Smeared ρ c/2 elliptic flow chiral symmetry restoration hydrodynamics 170 MeV Strange abundances no ψ ' phase transition 190 MeV Where lattice calculations do/will contribute to the development of theoretical concepts and the analysis of experimental observables γ? no χ c 230 MeV deconfinement no J/ψ thermalization jets, jet quenching Little Bang T c, ǫ c phase diagram in the (T, µ B )-plane; µ 0 : RHIC & LHC µ > 0 : chiral critical point low energy RHIC FAIR@GSI

3 F. Karsch for USQCD, LQCD II p. 2/27 Towards A New State of Matter Interferometry ππ,... Κ, Λ, Ω,... Last sca tering QCD, QGP, LGT and HIC 120 MeV resonance gas Smeared ρ Strange abundances γ? no ψ ' 190 MeV no χ c 230 MeV deconfinement no J/ψ Temperature c/2 elliptic flow chiral symmetry restoration hydrodynamics 170 MeV phase transition thermalization jets, jet quenching Little Bang The hot questions deconfinement == χ-sym. restoration? low-t physics == hadron resonance gas? T> T c : Is the QGP a perfect fluid? do we understand the non-perturbative structure of the QGP? LQCD topics QCD equation of state; µ B = 0 and µ B > 0 the QCD transition transport coefficients spatial string tension

4 F. Karsch for USQCD, LQCD II p. 2/27 Towards A New State of Matter Interferometry ππ,... Κ, Λ, Ω,... Last sca tering QCD, QGP, LGT and HIC 120 MeV resonance gas Smeared ρ Strange abundances γ? no ψ ' 190 MeV no χ c 230 MeV deconfinement no J/ψ Temperature c/2 elliptic flow chiral symmetry restoration hydrodynamics 170 MeV phase transition thermalization jets, jet quenching Little Bang The hot questions deconfinement == χ-sym. restoration? low-t physics == hadron resonance gas? T> T c : Is the QGP a perfect fluid? do we understand the non-perturbative structure of the QGP? LQCD topics QCD equation of state; µ B = 0 and µ B > 0 the QCD transition transport coefficients spatial string tension

5 Phase diagram of strongly interacting matter RHIC I/II & LHC LGT at vanishing chemical potential RHIC at low energy LGT at non zero chemical potential early universe temperature T cooling of a plasma created in a HIC want detailed studies of baryon number (charge,..) fluctuations ~170 MeV µ q 0 RHIC@(200 GeV/A) hadron gas quark-gluon plasma deconfined, χ-symmetric µ q 100 MeV RHIC@( 10 GeV/A)?? confined, χ-sb want accurate T c, ǫ c,... determination to make contact to HI-phenomenology color superconductor few times nuclear matter density µ baryon chemical potential neutron stars F. Karsch for USQCD, LQCD II p. 3/27

6 F. Karsch for USQCD, LQCD II p. 4/27 Major LGT Thermodynamics projects BNL-Columbia-RIKEN and Bielefeld: p4-improved staggered fermions, µ = 0 and µ > 0, T c, EoS, phase diagram, N τ = 4 and QCDOC at BNL (partly funded through the LQCD project), apenext at Bielefeld and BlueGene/L at NYCCS (BNL) hotqcd: BNL-Columbia-RIKEN, MILC, LLNL, LANL: p4-improved and asqtad staggered fermions, µ = 0 EoS and T c, N τ = BlueGene/L at Livermore and NYCCS; pilot project on T c with domain wall fermions Budapest-Wuppertal: standard staggered fermions with stout smearing, EoS, T c ; N τ = 4 10 PC-cluster ALICE, graphics cards WHOT-QCD: Tokyo-Tsukuba-BNL: Wilson fermions, µ = 0 and µ > 0, T c and phase diagram, N τ = 4 and BlueGene/L at KEK finite-t terminology: N τ = 1/Ta, a lattice spacing, N τ temporal lattice extent

7 F. Karsch for USQCD, LQCD II p. 5/27 Thermodynamics projects realized with LQCD project resources EoS, N τ = 4, QCD Thermodynamics with improved staggered fermions (BNL-Columbia-RIKEN) M. Cheng, N. H. Christ, S. Datta, J. van der Heide, C. Jung, F. Karsch, O. Kaczmarek, E. Laermann, R. D. Mawhinney, C. Miao, P. Petreczky, K. Petrov, C. Schmidt, T. Umeda, M. Cheng et al, Eur. Phys. J. C51 (2007) 75 M. Cheng et al, Phys. Rev. D77 (200) continues as hotqcd project: N τ = Dimensionally reduced QCD and the spatial screening lengths probing the non-perturbative, magnetic sector of QCD beyond perturbation theory M. Cheng et al, completed, publication in preparation Charge fluctuations and QCD phase diagram for µ > 0 non-gaussian fluctuations of baryon number, electric charge and strangeness as indicator of the QCD phase transition (BNL-Columbia-RIKEN) C. Schmidt, arxiv: (heplat), presented at Quark Matter 200, publication in preparation

8 F. Karsch for USQCD, LQCD II p. /27 Roadmap: QCD Equation of State 2000: Bielefeld N τ = 4, m π 770 MeV : LQCD-I N τ = 4,, m π 220 MeV RBC-Bielefeld: Phys. Rev. D77, (200) MILC: Phys. Rev. D75, (2007) /0: hotqcd N τ =, m π 220 MeV /09: LQCD-I N τ =, m π 150 MeV ?: LQCD-II? N τ = 12, m π 150 MeV

9 F. Karsch for USQCD, LQCD II p. 7/27 (ǫ 3p)/T 4 on LCP requires good control over T > 0 observables (action differences, chiral condensates); difficult: CPU requirement a (10 12) requires accurate determination of T = 0 scales interplay with studies of hadron spectroscopy (ε-3p)/t 4 m π 220 MeV p4: N τ =4 asqtad: N τ =4 Tr p4 vs. asqtad: overall good agreement However: We still need to... T < T c T < 2T c T > 2T c make contact to hadron gas phenomenology analyze large deviation from conformal limit (ǫ = 3p) make contact to (resummed) perturbation theory p4fat3-data: RBC-Bielefeld M. Cheng et al, PRD77, (200) asqtad data: MILC C. Bernard et al., PRD 75, (2007)

10 F. Karsch for USQCD, LQCD II p. /27 EoS: low and high T regime (ε-3p)/t 4 p4: N τ =4 asqtad: N τ =4 hadron r. gas Tr LGT vs resonance gas (ε-3p)/t 4 p4: N τ =4 asqtad: N τ = Tr LGT vs. pert. theory control over low T -regime influence approach to ideal gas limit at hight T. non-perturbative corrections: (ǫ 3p)/T 4 A + B/T 2 + C/T 4 want to make contact with HRG want to establish contact to (resummed) phenomenology perturbative QCD improved staggered fermions but still on rather coarse lattices: need N τ = (hotqcd collaboration) and 12...to control continuum limit (LQCD-II)

11 F. Karsch for USQCD, LQCD II p. 9/27..towards the cont. limit: N τ = hotqcd-collaboration PRELIMINARY (ε-3p)/t 4 Tr 0 asqtad: N τ = p4: N τ = hotqcd preliminary trace anomaly (ε-3p)/t 4 p4: N τ =4 asqtad: N τ = Tr 0 hotqcd preliminary high-t behavior [(ǫ 3p)/T 4 ] max > 200 MeV ( softest point of EoS) cut-off effects persist in the peak region high-t: good agreement between N τ = and results for T> 300 MeV low-t: small shift of transition region; better agreement with HRG model

12 F. Karsch for USQCD, LQCD II p. 10/27 EoS: low T regime (ε-3p)/t 4 asqtad: N τ = p4: N τ = hotqcd preliminary Tr LGT vs resonance gas approach to physical quark masses O(5MeV ) shift of T -scale

13 F. Karsch for USQCD, LQCD II p. 10/27 EoS: low T regime (ε-3p)/t 4 Tr (ε-3p)/t 4 Tr asqtad: N τ = p4: N τ = hotqcd preliminary LGT vs resonance gas approach to physical quark masses O(5MeV ) shift of T -scale asqtad: N τ = p4: N τ = hotqcd preliminary data, fit shifted by -5 MeV LGT shifted to phys. masses good agreement with HRG model in the transition region LQCD-project 0/09: physical quark masses, N τ =

14 F. Karsch for USQCD, LQCD II p. 11/27 The QCD Equation of State RBC-Bielefeld: Phys. Rev. D74, (200); D75, (2007); D77, (200) Tr ε SB/T 0 ε/t 4 3p/T 4 p4fat3 asqtad pressure N τ = RHIC LHC p/ε fit: p/ε HRG: p/ε c s 2 ε [GeV/fm 3 ] calculation of energy density and pressure as function of temperature equation of state: p(ǫ) velocity of sound: c s hotqcd: finer lattices, N τ = c 2 s at RHIC: opening angle of mach cone input for hydrodynamic modeling of Au-Au collisions at RHIC

15 need further systematic studies of bulk and shear viscosity LQCD-I 0/09 F. Karsch for USQCD, LQCD II p. 12/27 Transport coefficients and critical behavior sqgp, AdS/CFT QGP: a perfect fluid? (??) η shear /s 1/4π ρ θ (ω)/(t 4 sinh(ω/2t)) ζ/s T=1.24T c T=1.5T c ω/t spectral analysis of SU(3) gauge theory bulk viscosity H. Meyer, arxiv: [hep-lat] (2007) T/T c combine OP-expansion with (2+1)-f QCD lattice results on (ǫ 3p)/T 4 bulk viscosity FK, D. Kharzeev, K. Tuchin, arxiv: [hep-ph] (2007) direct studies of spectral functions so far only for quenched QCD without controled continuum and volume extrapolations;

16 F. Karsch for USQCD, LQCD II p. 13/27 Deconfinement and χ-symmetry The chiral phase transition (i.e. at m q = 0) is deconfining true in QCD, i.e. SU(3) + fermions in the fundamental representation SU(3) + fermions in the adjoint representation: T deconf < T χ The transition in QCD with physical quark masses is a crossover In which sense is the transition deconfining and chiral symmetry restoring? deconfinement: heavy hadrons light quarks and gluons; liberation of many new light degrees of freedom rapid change in ǫ/t 4, s/t 3,... chiral symmetry restoration: vanishing mass splittings, no new degrees of freedom minor effect on bulk thermodynamics, but rapid change of chiral condensate

17 F. Karsch for USQCD, LQCD II p. 13/27 Deconfinement and χ-symmetry The chiral phase transition (i.e. at m q = 0) is deconfining true in QCD, i.e. SU(3) + fermions in the fundamental representation SU(3) + fermions in the adjoint representation: T deconf < T χ The transition in QCD with physical quark masses is a crossover In which sense is the transition deconfining and chiral symmetry restoring? deconfinement: heavy hadrons light quarks and gluons; liberation of many new light degrees of freedom rapid change in ǫ/t 4, s/t 3,... chiral symmetry restoration: vanishing mass splittings, no new degrees of freedom minor effect on bulk thermodynamics, but rapid change of chiral condensate

18 F. Karsch for USQCD, LQCD II p. 14/27 Deconfinement renormalized Polyakov loop and strange quark number susceptibility L ren e F Q(T)/T, χ s /T 2 N 2 s L ren band: 15MeV T 195MeV Tr Tr 0 χ s /T 2 SB p4: N τ = p4, N τ = asqtad: N τ = N τ = 4, (p4): RBC-Bielefeld, PRD77, (200) N τ =, and N τ = (asqtad): hotqcd, preliminary

19 CHIRAL SYMMETRY RESTORATION: χ-condensate and susceptibility sudden change in chiral condensate is, of course, related to peaks in the (singlet) chiral susceptibility 1.0 χ tot /T 2 = 2χ dis /T 2 + χ con /T 2 l,s (T) = ψψ l,t m l m s ψψ s,t ψψ l,0 m l m s ψψ s, Tr χ disc /T 2 p4fat3 asqtad s,l asqtad p4fat3 N τ = χ tot /T p4 and asqtad: hotqcd, preliminary *p4fat3 asqtad F. Karsch for USQCD, LQCD II p. 15/27

20 χ-symmetry restoration χ-symmetry restoration: drop in condensate; peak in susceptibilities χ tot /T 2 Tr 0 p4 asqtad N τ = F. Karsch for USQCD, LQCD II p. 1/27

21 Deconfinement and χ-symmetry and bulk thermodynamics most prominent features of bulk thermodynamics are related to deconfinement χ-symmetry restoration: drop in condensate; peak in susceptibilities χ tot /T 2 Tr εsb/t Tr 0 ε/t 4 p4 asqtad N τ = 4 N τ = p4 2 asqtad do they stay closely related in the continuum limit? F. Karsch for USQCD, LQCD II p. 1/27

22 F. Karsch for USQCD, LQCD II p. 17/27 The spatial string tension Does dimensional reduction work with light quarks? Non-perturbative, vanishes in high-t perturbation theory: σs = lim ln W(R x, R y ) R x,r y R x R y σs g M g 2 f M : dim. red. pert. th. g 2 (T)T = c Mf M (g(t)), c M = 0.553(1) cm : 3-d SU(3), LGT _ 1.2 T c / Λ MS = loop 4-d SU(3), LGT T/σ s 1/ loop 0. 4d lattice, N τ 1 = T / T c M. Laine, Y. Schröder, JHEP 0503 (2005) 07

23 The spatial string tension Does dimensional reduction work with light quarks? Non-perturbative, vanishes in high-t perturbation theory: σs = lim ln W(R x, R y ) R x,r y R x R y r 0 σ s 1/2 (T) σs g M g 2 f M : dim. red. pert. th. g 2 (T)T = c Mf M (g(t)), c M = 0.553(1) cm : 3-d SU(3), LGT (2+1)-flavor QCD r 0 T N τ =4 N τ = N τ = g 2 (T) 2 α(t) 0.1 dimensional reduction works for T> 2T c - c M (almost) flavor independent - g 2 (T) shows 2-loop running c = 0.5(13) [SU(3)] c = 0.57(41) [QCD] 1 0. T/T RBC-Bielefeld, in preparation F. Karsch for USQCD, LQCD II p. 17/27

24 Isentropic Equation of State: p/ǫ S. Ejiri, F. Karsch, E. Laermann and C. Schmidt, Phys. Rev. D73 (200) C. Bernard et al. (MILC collaboration), Phys. Rev. D77 (200) µ B /T=0.24 µ B /T=1.3 µ B /T= p/ε RHIC SPS AGS (low-e RHIC) S/N B = freeze-out µ B [MeV] S/N B = ε [GeV/fm 3 ] awaits confirmation on finer lattices p/ǫ vs. ǫ shows almost no dependence on S/N B softest point: p/ǫ phenomenological EoS for T 0 < T < 2T 0 p ǫ = 1 3 ( ǫ fm 3 /GeV hotqcd (N τ = ), LQCD-I with lighter quarks, LQCD-II for a 0 ) F. Karsch for USQCD, LQCD II p. 1/27

25 F. Karsch for USQCD, LQCD II p. 19/27 Extending the phase diagram to non-vanishing chemical potential non-zero baryon number density: µ > 0 Z(V, T, µ) = DADψD ψ e S E(V,T,µ) = DAD det M(µ) e S E(V,T) three (partial) solutions for large T, small µ complex fermion determinant; long standing problem

26 Extending the phase diagram to non-vanishing chemical potential non-zero baryon number density: µ > 0 Z(V, T, µ) = DADψD ψ e S E(V,T,µ) = DAD det M(µ) e S E(V,T) T/T 0 RHIC, LHC F,K 2004 F,K 2002 T 0.2 c, Bielefeld-Swansea (N f =2) T c, Forcrand, Philipsen (N f =2) T c, Forcrand, Philipsen (N f =3) T c, Fodor, Katz (N f =2+1) µ B /T T 0 f, J.Cleymans et. al low energy RHIC, FAIR T c (µ) T c (0) : (4)(µ B/T) 2 deforcrand, Philipsen (imag. µ) (3)(µ B /T) 2 Bielefeld-Swansea (O(µ 2 ) reweighting) search for critical point establish relation between freeze-out and critical behavior F. Karsch for USQCD, LQCD II p. 19/27

27 F. Karsch for USQCD, LQCD II p. 20/27 Hadronic fluctuations at µ = 0 from Taylor expansion coefficients for µ > 0 n f = 2 + 1, m π 210 MeV: RBC-Bielefeld, preliminary Taylor expansion of bulk thermodynamics in terms of µ q,s p T 4 1 V T ln Z(V, T, µ q, µ 3 s ) = ( ) i ( ) j µq µs c i,j T T i,j expansion coefficients evaluated at µ q,s = 0 are related to fluctuations of B, S, Q at µ B,S,Q = 0: baryon number, strangeness, charge fluctuations event-by-event fluctuations at RHIC and LHC

28 F. Karsch for USQCD, LQCD II p. 21/27 Hadronic fluctuations at µ = 0 from Taylor expansion coefficients for µ > 0 n f = 2 + 1, m π 210 MeV: RBC-Bielefeld, preliminary higher derivatives higher moments mixed derivatives correlations 2c x 2 = 2 p/t 4 (µ x /T) 2 = 1 V T 3 (δn x) 2 µ=0 = 1 V T 3 N2 x µ=0 24c x 4 = 4 p/t 4 (µ x /T) 4 = 1 V T 3 ` (δnx ) 4 3 (δn x ) 2 2 µ=0 = 1 V T 3 ` N 4 x 3 N 2 x 2 µ= 4c xy 22 = 4 p/t 4 (µ x /T) 2 (µ y /T) 2 = 1 V T 3 ˆ N 2 x N 2 y 2 N x N y 2 N 2 x N 2 y µ=0 with x = q, s N 2 q N 2 s N 2 s N 2 q N 2 s = 4cqs (cqs 11 )2 2c s 2

29 F. Karsch for USQCD, LQCD II p. 22/27 Quadratic fluctuations of baryon number charge & strangeness in (2+1)-flavor QCD vanishing chemical potentials: RBC-Bielefeld, preliminary B χ 2 Q χ 2 S χ T[MeV] SB full: N t =4 open: N t = χ Q 2 = 1 V T 3 Q2 χ B 2 = 1 V T 3 N2 B χ S 2 = 1 V T 3 N2 S smooth change of quadratic fluctuations across transition region chiral limit: χ B 2, χq 2 T T c 1 α + regular

30 F. Karsch for USQCD, LQCD II p. 23/27 Quartic fluctuations of baryon number charge & strangeness in (2+1)-flavor QCD vanishing chemical potentials: T[MeV] χ 4 B χ 4 Q χ 4 S full: N t =4 open: N t = SB RBC-Bielefeld, preliminary χ Q 4 = 1 V T 3 ( Q 4 3 Q 2 2) χ B 4 = 1 V T 3 ( N 4 B 3 N2 B 2) χ S 4 = 1 V T 3 ( N 4 S 3 N2 S 2) large light quark number & charge fluctuations across transition region chiral limit: χ B 4, χq 4 T T c α + regular

31 F. Karsch for USQCD, LQCD II p. 24/27 Ratios of quartic and quadratic fluctuations of charges in (2+1)-flavor QCD RBC-Bielefeld, preliminary baryon number fluctuation charge fluctuation <B 4 >-3<B 2 > 2 <B 2 > n f =2+1, m π =220 MeV n f =2, m π =770 MeV Resonance gas <Q 4 >-3<Q 2 > 2 <Q 2 > n f =2+1, m π =220 MeV n f =2, m π =770 MeV Resonance gas SB SB chiral limit: ratios T T c α + regular enhancement over resonance gas values may be observable in event-by-event fluctuations

32 Fluctuations of baryon number and charge densities (µ 0) baryon number density fluctuations: Bielefeld-Swansea, PRD (2003) ; MILC, PRD77 (200) th order Taylor expansion χ B / T 2 µ B /T=0.0 µ B /T=1.0 µ B /T= χ B T 3 = ( = T V susceptibilities d 2 d(µ B /T) 2 p T 4 ) T fixed ( N 2 B N B 2) - to be studied in event-by-event fluctuations - to be compared to hadron resonance gas phenomenology seeing true singular behavior as signal for a critical point requires large volumes and high order Taylor expansions not yet studied with physical quark masses and a 0!!! F. Karsch for USQCD, LQCD II p. 25/27

33 F. Karsch for USQCD, LQCD II p. 2/27 Summary: projects on future machines......next generation of thermodynamics calculations: (extension of (exploratory) studies on current Teraflops computers) Project Lattice Temp. Quark MC traj. TFlop values Masses Years EoS: µ = 0, T < 2T c , EoS: µ = 0, T < 2T c ,000 EoS DWF: µ = 0, T T c ,000 0 EoS: µ > 0, T < 0.95T c ,000 0 phase boundary µ , spectral function, quenched 12 3 N τ , transport, dynamical 4 3 N τ , Projects and CPU requirements TFlop Years sustained TFlop Years 1/5 peak TFlop Years

34 F. Karsch for USQCD, LQCD II p. 27/27