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Lattice Perturbation Theory from Monte Carlo Simulation Kit Yan Wong (Glasgow University) H. Trottier (Simon Fraser University) R. Woloshyn (TRIUMF) C. Davies (Glasgow University)

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Motivation: we need perturbation theory example 1. determination of [Q. Mason et al., PRL95, 052002 (2005)] 1/15 0.1150.1170.119 world average: 0.1187 0.002 convert to measure this in simulation solve for need c 1, c 2, …

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example 2. quark masses [Q. Mason et al., hep-ph/0511160] [M. Nobes & H. Trottier, hep-lat/0509128] 2/15 (MeV) lattice PDG need connection between bare mass and MS mass Motivation: we need perturbation theory

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Lattice PT is Difficult! some of the 2-loop diagrams in the expansion for a Wilson loop: taken from Q. Masons Ph.D Thesis (Cornell Electronic Library) 3/15 particularly true for highly improved actions and higher-order calculations

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so … we need another method Monte Carlo Simulations at Weak Couplings 4/15

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Weak Coupling Simulations 1. do several simulations at weak couplings weak couplings perturbative phase weak couplings? ~ 9.5-80.0 ( s ~ 0.1-0.01) 2. fit the data points with an expansion in s 5/15

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Weak Coupling Simulations example c 1 : intercept c 2 : slope c 3 : curvature 1. 9 simulations at weak couplings 9 data points ( V,-lnW R,T ) 2. fit the points to 6/15 W 2,2

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Weak Coupling Simulations many calculations have been done before: Dimm, Lepage & Mackenzie [Nucl. Phys. B (Proc. Suppl.), 42, 403 (1995)] 1 st -order mass renormalization for Wilson fermions Trottier, Shakespeare, Lepage & Mackenzie [PRD, 65, 094502 (2002)] 3 rd -order Wilson loops for the Wilson plaquette action Hart, Horgan & Storoni [PRD, 70, 034501 (2004)] tadpole factor in pure gauge theories quenched calculations here: unquenched (MILC) important cross-check of the 2-loop expansions used in the s calculation 7/15

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Difficulties need very accurate measurements: statistical errors ~ 10 -5 – 10 -6 ~ s 3 why? want to probe the 3 rd -order coefficients challenge: tight control of systematic errors, both in simulations and analysis tunneling between Z 3 center phases fitting and truncation errors zero momentum modes step size errors in simulation equations autocorrelation inaccuracy in matrix inversions 8/15

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Simulation Parameters MILC action, 9 couplings 0 9.50.08380.1278 11.00.07230.1016 13.50.05890.0741 16.00.04970.0619 19.00.04190.0506 24.00.03320.0382 32.00.02490.0271 47.00.01690.0183 80.00.00990.0101 input from analytic PT 0 9/15 (not necessary) use 2-loop expansion of the plaquette

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Results 0.88(26)-0.599(19)1.2342(4)0.85(16)-0.595(3)1.2341(0)2x3 0.58(20)-0.641(15)1.1499(4)0.59(9)-0.643(2)1.1499(0)2x2 0.37(19)-0.599(14)0.9845(3)0.38(6)-0.595(1)0.9845(0)1x3 0.20(18)-0.644(13)0.9251(3)0.23(5)-0.646(9)0.9252(0)1x2 c3c3 c2c2 c1c1 c3c3 c2c2 c1c1 Loop Perturbation TheoryMonte Carlo Simulation 0.82(15)-0.594(7)2x3 0.59(14)-0.642(7)2x2 0.39(13)-0.600(6)1x3 0.26(12)-0.649(5)1x2 c3c3 c2c2 Loop Fit with c 1 fixed to PT values 0.84(8)2x3 0.61(6)2x2 0.28(5)1x3 0.21(4)1x2 c3c3 Loop Both c 1, c 2 fixed 10/15

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Results – 2x2 Loop k1k1 k3k3 k2k2 k 1 = k 3 = k 2 = 11/15

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provides an important cross-check of the 2-loop expansions used in the s calculation 12/15

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Results – NRQCD NRQCD action: ; dispersion relation c 1 : intercept c 2 : slope c 3 : curvature aM 0 = 8.0 zero energymass renormalization zero energy: mass renormalization: c 1 : intercept c 2 : slope c 3 : curvature 13/15

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Results – NRQCD zero energymass renormalization 2.773(7)2.7824.0 2.648(5)2.6555.0 2.565(5)2.5706.0 2.447(2)2.4508.0 MCPT M0aM0a 0.798(10)0.7704.0 0.842(9)0.8205.0 0.849(7)0.8306.0 0.867(6)0.8508.0 MCPT M0aM0a c1:c1: Predictions for c 2 and c 3 : 7.4(1.4)-2.13(7)4.0 7.3(1.4)-2.02(7)5.0 6.3(1.4)-1.91(7)6.0 5.9(1.4)-1.98(8)8.0 c3c3 c2c2 M0aM0a -0.28(6.7)-0.91(56)4.0 0.7(6.1)-0.53(45)5.0 0.5(5.3)-0.33(42)6.0 -0.2(4.9)-0.34(37)8.0 c3c3 c2c2 M0aM0a 14/15

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Conclusions 1. numerical approach is simple; give accurate results 2. Wilson loops: impressive agreement important for the s calculation 3. in progress: HIST, mNRQCD 15/15

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