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a). Introduction b). Quenched calculations c). Calculations with 2 light dynamical quarks d). (2+1) QCD LATTICE QCD SIMULATIONS, SOME RECENT RESULTS (END OF 2006) ITEP 7 February 2007

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INTRODUCTION starting from Lagrangian Main Problems: starting from Lagrangian (1)obtain hadron spectrum, (2)calculate various matrix elements, (3) describe phase transitions, and phase diagram (4) explain confinement of color

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INTRODUCTION The main difficulty is the absence of analytical methods, the interactions are strong and only computer simulations give results starting from the first principles. The force between quark and antiquark is 12 tones

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INTRODUCTION Methods Imaginary time t→it Space-time discretization Thus we get from functional integral the statistical theory in four dimensions

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INTRODUCTION The statistical theory in four dimensions can be simulated by Monte-Carlo methods The typical multiplicities of integrals are We have to invert matrices 10 8 x 10 8 The cost of simulation of one configuration is: Improved Wilson fermions

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INTRODUCTION Three limits Lattice spacing Lattice size Quark mass Typical values Extrapolation + Chiral perturbation theory

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INTRODUCTION Example of extrapolation

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INTRODUCTION Fit on the base of the chiral perturbation theory

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SU(2) glue;SU(3) glue;2qQCD;(2+1)QCD

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SU(2) glue SU(3) glue 2qQCD (2+1)QCD Theory of color confinement Theory of chiral symmetry breaking Monopoles Vortices Instantons and calorons Localization of Dirac eigenmodes

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SU(2) glue SU(3) glue 2qQCD (2+1)QCD Theory of color confinement Theory of chiral symmetry breaking Monopoles Vortices Instantons and calorons Localization of Dirac eigenmodes

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SU(2) glue SU(3) glue 2qQCD (2+1)QCD Theory of color confinement Theory of chiral symmetry breaking Monopoles Vortices Instantons and calorons Localization of Dirac eigenmodes

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SU(2) glue SU(3) glue 2qQCD (2+1)QCD Theory of color confinement Theory of chiral symmetry breaking Monopoles Vortices Instantons and calorons Localization of Dirac eigenmodes (Anderson localistion)

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SU(2) glue SU(3) glue 2qQCD (2+1)QCD

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Study of the complicated systems: a)Structure of gluon fields inside hadron b)Nucleon-Nucleon potential Three body forces!

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SU(2) glue SU(3) glue 2qQCD (2+1)QCD Usually the teams are rather big, people

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SU(2) glue SU(3) glue 2qQCD (2+1)QCD The Nuclear Force from Lattice QCD N. Ishii, S. Aoki and T. Hatsuda; nucl-th/ ; hep-lat/ From lattice calculations (six quark matrix element) Phenomenological potential

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SU(2) glue SU(3) glue 2qQCD (2+1)QCD The Nuclear Force from Lattice QCD N. Ishii, S. Aoki and T. Hatsuda; nucl-th/ hep-lat/ Lattice calculations with m p /m r =0.595

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SU(2) glue SU(3) glue 2qQCD (2+1)QCD Viscosity of quark gluon plasma A. NakamuraA. Nakamura, S. Sakai, hep-lat/ S. Sakai RHIC result at 1.4

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SU(2) glue SU(3) glue 2qQCD (2+1)QCD Potential between two B-mesons J. Savage et al. hep-lat/

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2qQCD (2+1)QCD 2qQCD (2+1)QCD u,d,s virtual quarks

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SU(2) glue SU(3) glue 2qQCD (2+1)QCD New effect: String Breaking 2qQCD QQ Qq Qq Glue Dynamical quarks

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SU(2) glue SU(3) glue 2qQCD (2+1)QCD String Breaking (DIK collaboration) MESON

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SU(2) glue SU(3) glue 2qQCD (2+1)QCD String Breaking (DIK collaboration) BARYON

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SU(2) glue SU(3) glue 2qQCD (2+1)QCD Partition function of QCD with one flavor at temperature T is: In computer

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SU(2) glue SU(3) glue 2qQCD (2+1)QCD Types of Fermions WilsonKogut-Suskind Wilson improved Wilson nonperturbatevely improved Domain wall StaggeredOverlap 1. Quark mass ->0 2. Fast algorithms

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SU(2) glue SU(3) glue 2qQCD (2+1)QCD G. Schierholz (Trento 2006) 2006 OLD 2001 NEW 2006

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SU(2) glue SU(3) glue 2qQCD (2+1)QCD G. Schierholz (2006) (Trento)

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SU(2) glue SU(3) glue 2qQCD (2+1)QCD G. Schierholz (2006) (Trento)

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SU(2) glue SU(3) glue 2qQCD (2+1)QCD G. Schierholz (Trento 2006)

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SU(2) glue SU(3) glue 2qQCD (2+1)QCD Phase diagram (F.Karsch) Four plenary talks at Lattice 2006! Color superconductivity in ultra-dense quark matter. Mark G. Alford; hep-lat/ Lattice QCD at finite density. C. Schmidt; hep-lat/ C. Schmidt Recent progress in finite temperature lattice QCD. Urs M. Heller; hep-lat/ Urs M. Heller QCD phase diagram: an overview. M.A. Stephanov; hep-lat/ M.A. Stephanov

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SU(2) glue SU(3) glue 2qQCD (2+1)QCD Phase diagram THEORY 3 2-nd or 1-st order for m=0? Di Giacomo –first order (2006) First order (Pisarski, Wilczek)

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SU(2) glue SU(3) glue 2qQCD (2+1)QCD Phase diagram: numerical calculations are very difficult, since we have a complex Monte-Carlo weight COMPLEX

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SU(2) glue SU(3) glue 2qQCD (2+1)QCD Phase diagram: numerical calculations are very difficult, since we have a complex Monte-Carlo weight Various numerical tricks: analytical continuations, m->im QCD critical point in T- m plane RED – RHIC experiment BLACK – phenomenological models GREEN – Lattice calculations M.A. StephanovM.A. Stephanov; hep-lat/

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SU(2) glue SU(3) glue 2qQCD (2+1)QCD Pure glue SU(3) F. Karsch Two flavor QCD, clover improved Wilson fermions C.Bernard (2005) C.Bernard (2005) DIK collaboration (2005) DIK collaboration (2005) Two flavor QCD, improved staggered fermions F.Karsch (2000) F.Karsch (2000) Three flavor QCD, improved staggered fermions! F.Karsch (2000) F.Karsch (2000) Critical temperature, m =0

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SU(2) glue SU(3) glue 2qQCD (2+1)QCD Tc by DIK (DESY-ITEP-Kanazawa) collaboration V.G. Bornyakov, M.N. Chernodub, Y. Mori, S.M. Morozov, Y. Nakamura, M.I. Polikarpov, G. Schierholz, A.A. Slavnov, H. Stüben, T. Suzuki (2006) Russian (JSCC) supercomputer M15000

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SU(2) glue SU(3) glue 2qQCD (2+1)QCD Tc by DIK (DESY-ITEP-Kanazawa) collaboration V.G. Bornyakov, M.N. Chernodub, Y. Mori, S.M. Morozov, Y. Nakamura, M.I. Polikarpov, G. Schierholz, A.A. Slavnov, H. Stüben, T. Suzuki (2006)

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SU(2) glue SU(3) glue 2qQCD (2+1)QCD DIK RESULTS

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SU(2) glue SU(3) glue 2qQCD (2+1)QCD Plasma thermodynamics Free energy density energy, entropy, velocity of sound,. pressure energy, entropy, velocity of sound,. pressure

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SU(2) glue SU(3) glue 2qQCD (2+1)QCD Plasma thermodynamics, example: pressure F. Karsch ( )

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SU(2) glue SU(3) glue 2qQCD (2+1)QCD Quark condensate F.Karsch et al.

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(2+1)QCD JLQCD, CP-PACS The description of the meson mass spectrum is good, but not excellent for lattice QCD with two dynamical quarks

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(2+1)QCD JLQCD, CP-PACS The description of the meson mass spectrum is good, but not excellent for lattice QCD with two dynamical quarks f meson mass vs lattice spacing (the mass of the s-quark is fitted from the mass of the K meson)

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(2+1)QCD JLQCD, CP-PACS Almost three years of gauge field trajectories generation at Earth Simulator; Lattice spacial volume is (2 fm)^3, a=0.07, 0.1, 0.12 fm

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(2+1)QCD JLQCD, CP-PACS RESULTS

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(2+1)QCD JLQCD, CP-PACS RESULTS

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(2+1)QCD MILC configurations, staggered dynamical fermions, NPLQCD Collaboration Hyperon-Nucleon phase shifts (hep-lat/ )

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Instead of Conclusions I did not discuss a number of important topics Formfactors Heavy-Light mesons Heavy – Heavy mesons and many others

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