Lattice QCD at high temperature Péter Petreczky Physics Department and RIKEN-BNL EFT in Particle and Nuclear Physics, KITPC, Beijing August 19, 2009 Introduction : Lattice QCD at T>0 Equation of state calculations Effective field theory approach to high temperature thermodynamics vs. lattice results Chiral and deconfinement transition Taylor expansion at finite chemical potential and fluctuations of conserved charges Summary

Deconfinement at high temperature and density Hadron Gas Transition Quark Gluon Plasma (QGP) temperature and/or density Why this is interesting ? : basic properties of strong interaction astrophysical (compact stars) cosmological consequences ( Early Universe few microseconds after Big Bang ) LQCD LQCD based effective models : PNJL, …

Relativistic Heavy Ion Collisions RBC-Bielefeld and HotQCD collaborations 40% QCD T>0 50% QCD T>0 24racks, 1 rack = 1024 processors 18 racks 1 rack= 2048 processors

Bulk particle spectra Thermal photons and dileptons Lattice QCD at T>0 and RHIC Heavy quark bound states Spatial correlation functions, heavy quark potential Temporal correlation functions, spectral function, transport coefficients Transition temperature, equation of state, susceptibilities LQCDLQCD RHIC (STAR) effective models : hydrodynamics, HRG, PNJL, PQM

evolution operator in imaginary time Finite Temperature QCD and its Lattice Formulation Integral over functions Lattice integral with very large (but finite) dimension ( > ) Costs : Monte-Carlo Methodssign problem ? improved discretization schemes are needed : p4, asqtad RHMC algorithm => x30 improvement

Lattice results on the trace of energy momentum tensor huge peak in the interaction measure for For non-interacting gas of quarks and gluons (Stefan-Boltzmann limit): For weakly interacting quarks and gluons HotQCD, arXiv:

Lattice results on the trace of energy momentum tensor deviations from the hadron resonance gas (HRG) at low T are due to unphysical quark masses and discretization error These deviations can be understood if the quark mass dependence and a-dependence of hadron masses in the HRG model is taken into account

Deconfinement : entropy, pressure and energy density rapid change in the number of degrees of freedom at T= MeV: => deconfinement deviation from ideal gas limit is about 10% at high T consistent with pert. th. no large discretization errors in the pressure and energy density at high T free gas of quarks and gluons = 18 quark+18 anti-quarks +16 gluons =52 light d.o.f meson gas = 3 light d.o.f.

3D effective theory for high temperature QCD High temperature  weak coupling => separation of scales : Integrate out the highest energy scale => 3D effective theory (EQCD) The parameters can be calculated in perturbation theory in terms of Appelquist, Pisarski, PRD 23 (81) 2305; Nadkarni, PRD 23 (83) 917; T. Reisz, ZPC 53 (92)169 ; Braaten, Nieto, PRD 53 (96) 3421 Kajantie et al., NPB 503 (97) 357 the effective theory is confining and non-perturbative at scale mass gap  inverse chromo-magnetic screening length pressure cannot be calculated in the loop expansion beyond Linde, PLB 96 (1980) 289

Thermodynamics at high temperature weak coupling calculations tend to agree with lattice at high T, at lower temperature non-perturative effects could be significant Braaten, Nieto, PRD 51 (95) 6990 Kajantie et al., NPB 503 (97) 357; PRL 86 (01) 10 Laine, Schröder, PRD 73 (06) good agreement between lattice and resummed perturbative (NLA) calculations of the entropy Rebhan, arXiv:hep-ph/ Blaizot et al, PRL 83 (99) 2906 a constant non-perturbative term is not present in the entropy density

Spatial string tension at T>0 non-perturbative calculated perturbatively Laine, Schröder, JHEP 0503:067,2005

Spatial correlators at T>0

Deconfinement and color screening free energy of a static quark large in confined phase ~ 500MeV zero in the deconfined phase free energy of static quark anti-quark pair shows Debye screening at high temperatures order parameter melting of bound states of heavy quarks important input for effective models (e.g. PQM, PNJL)

Deconfinement and chiral symmetry restoration masses of opposite parity mesons become equal Chiral symmetry restoration manifest itself in the spectrum of meson screening masses rapid decrease in the chiral condensate happens in the T-region where entropy density increases Renormalized chiral condensate

QCD thermodynamics at non-zero chemical potential Taylor expansion : hadronic quark Fluctuation of conserved quantum numbers at zero baryon density : probe of deconfinement and chiral aspects of the QCD transitions at zero density Physics at non-zero baryon density: Isentropic EoS radius of convergence, critical end-point

Deconfinement : fluctuations of conserved charges baryon number electric charge strange quark number Ideal gas of quarks : conserved charges are carried by massive hadrons conserved charges carried by light quarks

Deconfinement : fluctuations of conserved charges baryon number electric charge strange quark number Ideal gas of quarks : conserved charges are carried by massive hadrons conserved charges carried by light quarks enhanced fluctuations due to nearby critical point

Fluctuations in the hadron resonance gas model Cheng et al., arXiv: Kurtosis : ratio of the quartic fluctuations to quadratic fluctuations, can be studied also experimentally, see e.g. Schuster, arXiv: reasonable agreement with HRG at low T rapid change from hadronic to quark degrees of freedom ( deconfinement) Hadron resonance gas (HRG) can be used as a reference at low temperatures

Fluctuations of conserved charges at high T 1)Strangeness fluctuations are suppressed at low T 2) For T>300MeV no strangeness suppression 3) In the intermediate T-region strangeness fluctuations are also suppressed but can be understood in effective PQM model: Schaefer et al, PRD76 (07) Schaefer, Wagner, PRD79 (09) The quark number susceptibilities for T>300MeV agree with resummed petrurbative predictions A. Rebhan, arXiv:hep-ph/ Blaizot et al, PLB 523 (01) 143 and are in contrrast with AdS/CFT expectations Teaney, PRD 74 (06)

Critical end-point and isentropic equation of state If all expansion coefficients are positive there is a singularity for real The largest temperature for which all expansion coefficients are positive provides an estimate for Radius of convergence at provides an estimate for Using Taylor expansion one can calculate the entropy density at finite and the set of which corresponds to constant ratio of entropy to baryon number

Summary Simulations of lattice QCD on massively parallel computers show that at temperatures MeV strongly interacting matter undergoes a transition to a new state QGP characterized by deconfinement and chiral symmetry restoration Calculations of thermodynamic quantities, pressure, energy density, entropy density, fluctuations of conserved charges can be done controlled systematic errors above the transition and provide evidence that the relevant degrees of freedom are quarks and gluons T>300MeV (LHC): weakly coupled region T<300MeV (RHIC): strongly coupled region 3D effective theory (EQCD) can described thermodynamic quantities and spatial correlation functions in high temperature QCD It is possible to extend the lattice calculations to finite baryon density using Taylor expansion, which in addition provides information on fluctuations of conserved charges relevant for event-by-event fluctuations in RHIC and insight into microscopic picture of QGP needed to formulate effective models of QGP

Back-up: Deconfinement and chiral transition stout : Budapest-Wuppertal Group, Aoki et al., PLB 643 (06) 46; arXiv: no qualitative change, but significant shift of the transition region toward smaller T talk by Zoltán Fodor, parallel session 6B, Friday Renormalized Polyakov loopRenormalized chiral condensate stout action is optimized to reduce the effect of flavor symmetry breaking, but not the quark the quark dispersion relation 5MeV, quark mass 6MeV, continuum extrapolation