Energy-Momentum Tensor and Thermodynamics of Lattice Gauge Theory from Gradient Flow Masakiyo Kitazawa (Osaka U.) 北澤正清（大阪大学） for FlowQCD Collaboration.

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Presentation transcript:

Energy-Momentum Tensor and Thermodynamics of Lattice Gauge Theory from Gradient Flow Masakiyo Kitazawa (Osaka U.) 北澤正清（大阪大学） for FlowQCD Collaboration Asakawa, Hatsuda, Iritani, Itou, MK, Suzuki FlowQCD, PRD90,011501R (2014) ATHIC2014, 7 Aug. 2014, Osaka U.

Lattice QCD First principle calculation of QCD Monte Carlo for path integral hadron spectra, chiral symmetry, phase transition, etc.

Gradient Flow Luscher, 2010 A powerful tool for various analyses on the lattice

energy momentum pressure stress Poincaresymmetry Einstein Equation Hydrodynamic Eq.

: nontrivial observable on the lattice Definition of the operator is nontrivial because of the explicit breaking of Lorentz symmetry ① ② Its measurement is extremely noisy due to high dimensionality and etc. the more severe on the finer lattices ex:

因小失大 smalllosebigstem from

因小失大  miss the wood for the trees  見小利則大事不成孔子（論語、子路 13 ） smalllosebigstem from

If we have

Thermodynamics direct measurement of expectation values If we have

Fluctuations and Correlations viscosity, specific heat,... Thermodynamics direct measurement of expectation values If we have

Fluctuations and Correlations viscosity, specific heat,... Thermodynamics direct measurement of expectation values Vacuum Structure  vacuum configuration  mixed state on 1 st transition Hadron Structure  confinement string  EM distribution in hadrons If we have

Themodynamics: Integral Method : energy density : pressure directly observable

Themodynamics: Integral Method : energy density : pressure directly observable  measurements of e-3p for many T  vacuum subtraction for each T  information on beta function Boyd+ 1996

Gradient Flow

YM Gradient Flow Luscher, 2010 t: “flow time” dim:[length 2 ]

YM Gradient Flow Luscher, 2010 t: “flow time” dim:[length 2 ]  transform gauge field like diffusion equation  diffusion length  This is NOT the standard cooling/smearing  All composite operators at t>0 are UV finite Luescher,Weisz,2011

失小得大 coarse graining  This is NOT the standard cooling/smearing

Small Flow Time Expansion of Operators and EMT

Small t Expansion Luescher, Weisz, 2011 remormalized operators of original theory an operator at t>0 t  0 limit original 4-dim theory

Constructing EMT Suzuki, 2013 DelDebbio,Patella,Rago,2013  gauge-invariant dimension 4 operators

Constructing EMT 2 Suzuki coeffs. Suzuki, 2013

Constructing EMT 2 Suzuki coeffs. Remormalized EMT Suzuki, 2013

Numerical Analysis: Thermodynamics Thermodynamics direct measurement of expectation values

Gradient Flow Method lattice regularized gauge theory gradient flow continuum theory (with dim. reg.) continuum theorycontinuum theory (with dim. reg.)(with dim. reg.) gradient flow analytic (perturbative) “ 失小得大 ”

Gradient Flow Method lattice regularized gauge theory gradient flow continuum theory (with dim. reg.) continuum theorycontinuum theory (with dim. reg.)(with dim. reg.) gradient flow analytic (perturbative) measurement on the lattice “ 失小得大 ”

lattice regularized gauge theory gradient flow continuum theory (with dim. reg.) continuum theorycontinuum theory (with dim. reg.)(with dim. reg.) gradient flow Caveats Gauge field has to be sufficiently smeared! measurement on the lattice analytic (perturbative) Perturbative relation has to be applicable!

lattice regularized gauge theory gradient flow continuum theory (with dim. reg.) continuum theorycontinuum theory (with dim. reg.)(with dim. reg.) gradient flow Caveats Gauge field has to be sufficiently smeared! analytic (perturbative) Perturbative relation has to be applicable! measurement on the lattice

O(t) effect classic al non-perturbative region in continuum

on the lattice O(t) effect classic al  t  0 limit with keeping t>>a 2 non-perturbative region

Numerical Simulation lattice size: 32 3 xN t Nt = 6, 8, 10  = 5.89 – 6.56 ~300 configurations  SU(3) YM theory  Wilson gauge action Simulation 1 (arXiv: ) RCNP KEK using lattice size: 64 3 xN t Nt = 10, 12, 14, 16  = 6.4 – 7.4 ~2000 configurations twice finer lattice! Simulation 2 (new, preliminary) using KEK efficiency ~40%

 -3p at T=1.65T c Emergent plateau! the range of t where the EMT formula is successfully used! Nt=6,8,10 ~300 confs.

 -3p at T=1.65T c Emergent plateau! the range of t where the EMT formula is successfully used! Nt=6,8,10 ~300 confs.

Entropy Density at T=1.65Tc Emergent plateau! Direct measurement of e+p on a given T! systematic error Nt=6,8,10 ~300 confs. NO integral / NO vacuum subtraction

Continuum Limit 0 Boyd xNt Nt = 6, 8, 10 T/Tc=0.99, 1.24, 1.65

Continuum Limit 0 Comparison with previous studies Boyd xNt Nt = 6, 8, 10 T/Tc=0.99, 1.24, 1.65

Numerical Simulation lattice size: 32 3 xN t Nt = 6, 8, 10  = 5.89 – 6.56 ~300 configurations lattice size: 64 3 xN t Nt = 10, 12, 14, 16  = 6.4 – 7.4 ~2000 configurations  SU(3) YM theory  Wilson gauge action Simulation 1 (arXiv: ) Simulation 2 (new, preliminary) RCNP KEK using using KEK efficiency ~40% twice finer lattice!

Entropy Density on Finer Lattices  The wider plateau on the finer lattices  Plateau may have a nonzero slope T = 2.31Tc 64 3 xNt Nt = 10, 12, 14, confs. FlowQCD,2013 T=1.65Tc Preliminary!!

Continuum Extrapolation 0 T=2.31Tc 2000 confs Nt = 10 ~ 16 Nt= Continuum extrapolation is stable a  0 limit with fixed t/a Preliminary!! Preliminary!!

Summary

EMT formula from gradient flow This formula can successfully define and calculate the EMT on the lattice. It’s direct and intuitive Gradient flow provides us novel methods of various analyses on the lattice avoiding 因小失大 problem

Other observables full QCD Makino,Suzuki,2014 non-pert. improvement Patella 7E(Thu) Many Future Studies!! O(a) improvement Nogradi, 7E(Thu); Sint, 7E(Thu) Monahan, 7E(Thu) and etc.

Numerical Analysis: EMT Correlators Fluctuations and Correlations viscosity, specific heat,...

EMT Correlator  Kubo Formula: T 12 correlator  shear viscosity  Energy fluctuation  specific heat  Hydrodynamics describes long range behavior of T 

EMT Correlator : Noisy… Nakamura, Sakai, PRL,2005 N t =8 improved action ~10 6 configurations … no signal Nt=16 standard action 5x10 4 configurations With naïve EMT operators

Energy Correlation Function T=2.31Tc b=7.2, Nt= confs p=0 correlator Preliminary!! smeared

Energy Correlation Function T=2.31Tc b=7.2, Nt= confs p=0 correlator   independent const.  energy conservation Preliminary!! smeared

Energy Correlation Function T=2.31Tc b=7.2, Nt= confs p=0 correlator  specific heat Preliminary!!  Novel approach to measure specific heat! smeared Gavai, Gupta, Mukherjee, 2005 differential method / cont lim.