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Lattice QCD (INTRODUCTION) DUBNA WINTER SCHOOL 1-2 FEBRUARY 2005

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2 Main Problems Starting from Lagrangian (1) obtain hadron spectrum, (2) describe phase transitions, (3) explain confinement of color http://www.claymath.org/Millennium_Prize_Problems/

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3 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 The force between quark and antiquark is 12 tones

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

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5 The statistical theory in four dimensions can be simulated by Monte-Carlo methods The typical multiplicities of integrals are 10 6 -10 8 We have to invert matrices 10 6 x 10 6 The cost of simulation of one configuration is:

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

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7 Chiral limit Quark masses Pion mass Exp

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8 Nucleon mass extrapolation Fit on the base of the chiral perturbation theory

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9 Spectrum

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10 Earth Simulator Earth Simulator Based on the NEC SX architecture, 640 nodes, each node with 8 vector processors (8 Gflop/s peak per processor), 2 ns cycle time, 16GB shared memory. – Total of 5104 total processors, 40 TFlop/s peak, and 10TB memory. Based on the NEC SX architecture, 640 nodes, each node with 8 vector processors (8 Gflop/s peak per processor), 2 ns cycle time, 16GB shared memory. – Total of 5104 total processors, 40 TFlop/s peak, and 10TB memory. It has a single stage crossbar (1800 miles of cable) 83,000 copper cables, 16 GB/s cross section bandwidth. It has a single stage crossbar (1800 miles of cable) 83,000 copper cables, 16 GB/s cross section bandwidth. 700 TB disk space, 1.6 PB mass store Area of computer = 4 tennis courts, 3 floors

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Lattice QCD at finite temperature and density (INTRODUCTION) DUBNA WINTER SCHOOL 2-3 FEBRUARY 2006

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12 Finite Temperature in Field Theory In imaginary time the partition function which defines the field theory is: The action is:

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13 QCD at Finite Temperature Partition function of QCD with one flavor at temperature T is: The action is: In computer

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14 Types of Fermions

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15 Types of Fermions WilsonKogut-Suskind Wilson improved Wilson nonperturbatevely improved Domain wall StaggeredOverlap 1. Quark mass ->0 2. Fast algorithms

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16 Approximations to real QCD Quenched approximation (no fermion loops), gauge group SU(2), SU(3) Dynamical fermions, the realistic situation, heavy s quark and 5Mev u and d quarks will be available on computers in 2015(?).

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17 Types of Algorithms 1. Hybrid Monte Carlo + Molecular dynamics, leap-frog 2. Local Boson Algorithm 3. Pseudofermionic Hybrid Monte Carlo 3. Two step multyboson 4. Polynomial Hybrid Monte Carlo

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19 Earth simulator

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20 Earth simulator

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21 Earth simulator

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22 Earth simulator

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23 Earth simulator

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24 How to find quark gluon plasma? ORDER PARAMETERS

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25 Example

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26 Lattice calculation by DIK group Polyakov loop susceptibility clower improved Wilson fermions Polyakov loop susceptibility clower improved Wilson fermions 16**3*8 lattice 16**3*8 lattice

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27 Quark condensate Fermion condensate vs. T, F.Karsch et al.

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28 Phase diagram m q -T, one flavor

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29 Quark mass dependence of Tc

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30 Three quarks

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31 Critical temperature for pure glue and for various dynamical quarks

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32 PureSU(3)glue(simplestcase)

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33 Temperature of the phase transition Pure glue SU(3) F. Karsch

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35 Temperature of the phase transition 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)

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37 Temperature of the phase transition 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)

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39 Temperature of the phase transition 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)

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40 Example of extrapolation (DIK 2005) Russian (JSCC) supercomputer M1000

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41 Plasma thermodynamics Free energy density energy, entropy, velosity of sound,. pressure energy, entropy, velosity of sound,. pressure

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42 Example: pressure F. Karsch (2001-2005)

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43 FULL PROBLEM

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44 Non zero chemical potential QCD partition function at finite temperature and chemical potential At finite chemical potential the fermionic determinant is not positively defined! Thus we have no interpretation of the partition function as probability wait (big difficulties with MC).

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45 Анекдот

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46 Mu-T diagramme, example of calculation (weighted expantion on mu) C.R. Aallton et al. (2005)

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48 Full diagram (theory) C.R. Aallton et al. (2005)

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50 Literature H. Satz hep-ph/0007209 F. Karsch hep-lat/0106019 C.R. Allton et al. hep-lat/0504011 F. Karsch hep-lat/0601013 DIK (DESY-ITEP-Kanazawa) collaboration hep-lat/0509122, hep-lat/0401014

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52 Special thanks to VALERY SCHEKOLDIN (photo )

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