Temperature dependence of shear viscosity of SU(3) gluodynamics within lattice simulation

Transcription

1 Temperature dependence of shear viscosity of SU(3) gluodynamics within lattice simulation V.V. Braguta ITEP 20 April, 2017

2 Outline: Introduction Details of the calculation Shear viscocity Fitting of the data Backus-Gilbert method Bulk viscosity (preliminary results) Conclusion

3 Relativistic Hydrodynamics T µν = (e + p)u µ u ν + pg µν + (η µ u ν + ζ µν α u α ) +... α = αν ν, µν = g µν u µ u ν µ u ν = µ u ν + ν u µ 2 3 µν α u α EOM µ T µν = 0 Non-relativistic limit (u µ = (1, v)) Continuity equation: t ρ + ρ( v) + v ρ = 0 v Navier Stokes equation: i t + v k v i = 1 x k ρ p x i Viscous stress tensor: Π ik = η ( v i + v k 2 l ) v x k x i 3δik x ζδ ik v l l x l η shear viscosity, ζ bulk viscosity 1 Π ki ρ x k

4 Calculation of transport coefficients: Shear viscosity η = lim ω 0 1 ω 0 dt d 3 x e iωt [T 12 (x), T 12 (0)] ρ 12,12 (ω) = 1 12,12 π ImGR (ω, k = 0) 1 η = π lim ω 0 ω ρ 12,12(ω) Bulk viscosity ζ = 1 9 lim ω 0 1 ω 0 dt d 3 x e iωt [T µ µ (x), Tν ν (0)] ρ µµ,νν (ω) = 1 µµ,νν π ImG R (ω, k = 0) ζ = π 9 lim ω 0 1 ω ρ µµ,νν(ω) Analytic continuation G E (ω, p) = G R (iω, p), ω > 0 C E (t) = ( 0 dωρ(ω) ch ω 2T ωt ) sh( ω 2T )

5 Elliptic flow from STAR experiment (Nucl. Phys. A 757, 102 (2005)) dn d φ (1 + 2v 1cos(φ) + 2v 2 cos 2 (φ)) Quark-gluon plasma is close to ideal liquid ( η s M. Luzum and P. Romatschke, Phys. Rev. C 78, (2008) = (1 3) 1 4π )

6 S.Cremonini, U.Gursoy, P.Szepietowski, JHEP 1208 (2012) 167 Comparison of different liquids Quark-gluon plasma is the most ideal liquid

7 Study of shear viscosity (perturbative calculation + pion gas)

8 Study of shear viscosity (effective models) R. Marty, E. Bratkovskaya, W. Cassing, J. Aichelin and H. Berrehrah, Phys. Rev. C 88, (2013)

9 Other works (SU(3) gluodynamics): Karsch, F. et al. Phys.Rev. D35 (1987) A. Nakamura, S. Sakai Phys. Rev. Lett. 94, (2005) H. B. Meyer, Phys.Rev. D76 (2007) H. B. Meyer, Nucl.Phys. A830 (2009) 641C-648C Results: η s = ± (T /T c = 1.65, ) η s = ± (T /T c = 1.24, ) η s = 0.20 ± 0.03 (T /T c = 1.58, ) η s = 0.26 ± 0.03 (T /T c = 2.32, ) SU(2) gluodynamics: η s = ± (T /T c = 1.2, ) N.Yu. Astrakhantsev, V.V. Braguta, A.Yu. Kotov, JHEP 1509 (2015) 082

10 Lattice calculation of shear viscocity The first step: Measurement of the correlation function: The second step: C E (t) = T 12 (t)t 12 (0) Calculation of the spectral function ρ(ω): C E (t) = ( 0 dωρ(ω) ch ω η = π lim ω 0 ρ(ω) ω 2T ωt ) sh( ω 2T )

11 Details of the calculation SU(3) gluodynamics Two-level algorithm Lattice size Temperatures T /T c = 0.9, 0.925, 0.95, 1.0, 1.1, 1.2, 1.35, 1.5 Accuracy 2 3% at t = 1 2T T 12 (x)t 12 (y) ( T 11 (x)t 11 (y) T 11 (x)t 22 (y) ) Clover discretization for the ˆF µν Renormalization of EMT: F. Karsch, Nucl.Phys. B205 (1982)

12 Correlation functions

13 Spectral function C(t) = ( dωρ(ω) ch ω ( 0 Properties of the spectral function: ρ(ω) 0, ρ( ω) = ρ(ω) Asymptotic freedom: ρ(ω) NLO 90% of the total contribution t = 1/2/T Hydrodynamics: ρ(ω) ω 0 = η π ω sh 2T ωt ) ω 2T ) ( ω = 1 d A 10 (4π) ω Nc αs 9π )

14 Spectral function C(t) = ( dωρ(ω) ch ω ( 0 Properties of the spectral function: ρ(ω) 0, ρ( ω) = ρ(ω) Asymptotic freedom: ρ(ω) NLO 90% of the total contribution t = 1/2/T Hydrodynamics: ρ(ω) ω 0 = η π ω sh 2T ωt ) ω 2T ) ( ω = 1 d A 10 (4π) ω Nc αs 9π ) Ansatz for the spectral function (QCD sum rules motivation) ρ(ω) = η π ωθ(ω 0 ω) + Aρ lat (ω)θ(ω ω 0 )

15 Lattice spectral function ρ lat Takes into account discretization errors in temporal direction

16 Spectral function ρ 1 (ω) = η π ωθ(ω 0 ω) + Aρ lat (ω)θ(ω ω 0 ) χ 2 /dof 1, A 1, ω 0 /T 7 8 Two additional ansatzs: ρ 2 (ω) = 1 2 B ω( 1 + tanh [ γ(w 0 w) ]) Aρ lat (ω)( 1 + tanh [ γ(w w 0 ) ]) ρ 3 (ω) = B ω ( 1 + Cω 2) θ(ω 0 ω) + Aρ lat (ω)θ(ω ω 0 )

17 Properties of the spectral function Hydrodynamical approximation works well up to ω < πt 1GeV (H.B. Meyer, arxiv: ) Asymptotic freedom works well from ω > 3 GeV Poor knowledge of the spectral function in the region ω (1, 3) GeV Main source of uncertainty in the fitting procedure

18 Backus-Gilbert method for the spectral function Problem: find f (ω) from the integral equation C(x i ) = ( dωf (ω)k(x 0 i, ω), K(x i, ω) = ch ω Define an estimator f ( ω) (δ( ω, ω) - resolution function): Let us expand δ( ω, ω) as f ( ω) = 0 dωˆδ( ω, ω)f (ω) ) 2T ωx ) i sh( ω 2T δ( ω, ω) = i b i ( ω)k(x i, ω) f ( ω) = i b i ( ω)c(x i ) Goal: minimize the width of the resolution function b i ( ω) = j W 1 R ij j, ij R i W 1 R ij j W ij = dωk(x i, ω)(ω ω) 2 K(x j, ω), R i = dωk(x i, ω) Regularization by the covariance matrix S ij : W ij λw ij + (1 λ)s ij, 0 < λ < 1

19 Resolution function δ(0, ω) (T /T c = 1.2, λ = 0.001) Width of the resolution function ω/t 4 Hydrodynamical approximation works up to ω/t < π Problem: large contribution from ultraviolet tail (more than 100%)

20 Model for the ultraviolet contibution ρ ultr = Aρ lat (ω)θ(ω ω 0 )

21 Solution: Take ultraviolet contribution in the form: ρ ultr = Aρ lat (ω)θ(ω ω 0 ) Determine the value of the constant A from the BG method Subtract ultraviolet contribution and obtain η/s as a function of ω 0

22 For T /T c = 1. ω 0 /T = 7.33 ± 0.47 η/s = 0.22 ± 0.04

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24 Our results

25 Estimation of the η/s for QCD with dynamical quarks (N f = 3): ( ) ( η ) s QCD = (η/s) QCD (η/s) ( η ) Gluodynamics s Gluodynamics pert

26 Our results

27 Bulk viscosity ζ = 1 9 lim ω 0 1 ω C E (t) = ( 0 dωρ µµ,νν (ω) ch ω ζ = π 9 lim ω 0 1 ω ρ µµ,νν(ω) 0 dt d 3 x e iωt [T µ µ (x), T ν ν (0)] 2T ωt ) sh( ω 2T Violation of conformal symmetry in QCD In conformal theory T µ µ = 0 Conformal symmetry is violate by quantum corrections Trace anomaly: T µ µ = β(g) 2g Tr[ G µν G µν ] + i m i qq )

28 Properties of spectral density: ρ µµ,νν (ω) 0, ρ µµ,νν ( ω) = ρ µµ,νν (ω) Asymptotic freedom: ρ(ω) NLO ω = d A ( 11αs (4π) 2 ) 2ω 4 Hydrodynamics: ρ(ω) ω 0 = 9 ζ π ω The contribution of asymptotic freedom is suppressed ( α 2 s ) Bulk viscosity from midpoint C E (β/2) = 1 0 dωρ hyd (ω) ) sh( ω 2T

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30 Conclusion: We calculated η/s for set of temperatures T /T c (0.9, 1.5) Applied fitting procedure and Backus-Gilbert method for the SF η/s is close to N=4 SYM and in agreement with experiment Large deviation from perturbative results