Non-perturbative study of the viscosity in SU ( 2 ) lattice - PowerPoint PPT Presentation

Non-perturbative study of the viscosity in SU ( 2 ) lattice gluodynamics. V. V. Braguta, A. Yu. Kotov Семинар "Теория адронного вещества при экстремальных условиях" Oct 30, 2013

Outline Introduction Transport coefficients in lattice calculations Improving statistical accuracy of the results Analytical continuation problem Numerical setup Results and discussion V. V. Braguta, A. Yu. Kotov Non-perturbative study of the viscosity... Oct 30, 2013 2 / 22

Introduction. Hydrodynamical description. Hydrodinamical description of the distribution of final particles One heavy ion collision produces a huge number of final particles Large number of particles ⇒ hydrodynamical description can be used In hydrodinamics transport coefficients control flow of energy, momentum, electrical charge and other quantities V. V. Braguta, A. Yu. Kotov Non-perturbative study of the viscosity... Oct 30, 2013 3 / 22

Shear viscosity. Value and bounds. Teaney D., Viscous Hydrodynamics and the Quark Gluon Plasma , arXiv:0905.2433 s ∼ ( 1 ↔ 3 ) 1 η • Experimentally preferred value: 4 π s < 5 1 • Experimental bound: η 4 π 1 η • KSS-bound: s ≥ 4 π V. V. Braguta, A. Yu. Kotov Non-perturbative study of the viscosity... Oct 30, 2013 4 / 22

Shear viscosity. Value and bounds. Cremonini S., Gursoy U. and Szepietowski P., On the Temperature Dependence of the Shear Viscosity and Holography , arXiv:1206.3581 Comparison of different liquids QGP the most superfluid liquid The aim: first principle calculation of transport coefficients V. V. Braguta, A. Yu. Kotov Non-perturbative study of the viscosity... Oct 30, 2013 5 / 22

Lattice simulations of QCD. Allows to study strongly interacting systems Based on the first principles of quantum field theory Acknowledged approach to study QCD Very powerful due to the development of computer systems V. V. Braguta, A. Yu. Kotov Non-perturbative study of the viscosity... Oct 30, 2013 6 / 22

Previous lattice calculations ( SU ( 3 ) gluodynamics). F. Karsch, H. W. Wyld. Phys. Rev. D35 (1987) A. Nakamura, S. Sakai Phys. Rev. Lett. 94, 072305 (2005) H. B. Meyer, Phys.Rev.Lett. 100 (2008) 162001 V. V. Braguta, A. Yu. Kotov Non-perturbative study of the viscosity... Oct 30, 2013 7 / 22

Viscosity in lattice calculations. Green-Kubo relation: ρ 12 , 12 ( ω, q = 0 ) η = π lim ω ω → 0 Green function measured on the lattice(Eucledian): � C 12 , 12 ( x 0 , p ) = β 5 d 3 x e i p x � T 12 ( 0 ) T 12 ( x 0 , x ) � Spectral function and correlator of stress-energy tensor: � ∞ ρ 12 , 12 ( ω, p ) cosh ω ( 1 2 L 0 − x 0 ) C 12 , 12 ( x 0 , p ) = β 5 d ω sinh ω L 0 0 2 Stress-energy tensor for gluodynamics: T µν = 2 tr ( F µσ F νσ − 1 4 δ µν F ρσ F ρσ ) Asymptotic behaviour - perturbation theory: ρ ( ω ) = 1 ( 4 π ) 2 ω 4 , ω → ∞ 3 10 V. V. Braguta, A. Yu. Kotov Non-perturbative study of the viscosity... Oct 30, 2013 8 / 22

Main difficulties. Large statistical errors in measuring correlator C 12 , 12 ( x 0 , 0 ) Improved action Multilevel algorithm Extracting spectral function ρ 12 , 12 from � ∞ ρ 12 , 12 ( ω, p ) cosh ω ( 1 2 L 0 − x 0 ) C 12 , 12 ( x 0 , p ) = β 5 d ω sinh ω L 0 0 2 Fit by model function Maximum entropy method Linear method ... V. V. Braguta, A. Yu. Kotov Non-perturbative study of the viscosity... Oct 30, 2013 9 / 22

Statistical error of the correlator C 12 , 12 . Improved action. � S unimpr = β S pl pl S pl − β impr � � S impr = β impr S rt 20 u 2 0 rt pl S pl , rt = 1 2 tr ( 1 − U pl , rt ) Increases accuracy but is not enough. V. V. Braguta, A. Yu. Kotov Non-perturbative study of the viscosity... Oct 30, 2013 10 / 22

Statistical error of the correlator C 12 , 12 . Multilevel algorithm. For t 1 and t 2 in different areas 1 � � O ( t 1 ) O ( t 2 ) � = � O ( t 1 ) � b . c � O ( t 2 ) � b . c N bc bc V. V. Braguta, A. Yu. Kotov Non-perturbative study of the viscosity... Oct 30, 2013 11 / 22

Analytical continuation problem. Fit by model function Maximum entropy method Linear method ... V. V. Braguta, A. Yu. Kotov Non-perturbative study of the viscosity... Oct 30, 2013 12 / 22

Analytical continuation problem. Fit by model function. � ∞ ρ 12 , 12 ( ω, p ) cosh ω ( 1 2 L 0 − x 0 ) C 12 , 12 ( x 0 , p ) = β 5 d ω sinh ω L 0 0 2 Proposed in the first work on transport coefficients: F. Karsch, H. W. Wyld. Phys. Rev. D35 (1987) � � γ γ ρ ( ω ) /ω = A ( m − ω ) 2 + γ 2 + ( m + ω ) 2 + γ 2 A , m , γ - parameters Clearly ignores asymptotic behaviour V. V. Braguta, A. Yu. Kotov Non-perturbative study of the viscosity... Oct 30, 2013 13 / 22

Analytical continuation problem. Maximum entropy method. � ∞ ρ 12 , 12 ( ω, p ) cosh ω ( 1 2 L 0 − x 0 ) C 12 , 12 ( x 0 , p ) = β 5 d ω sinh ω L 0 0 2 We discretize ω , N ω ∼ O ( 10 3 ) K ( ω, x i ) ρ ( ω ) − − − − → G ( x i ) = G i χ 2 = ( G i − G ( 0 ) )( S − 1 ) ij ( G j − G ( 0 ) � ) i j i , j min χ 2 → ∼ O ( 10 ) equations V. V. Braguta, A. Yu. Kotov Non-perturbative study of the viscosity... Oct 30, 2013 14 / 22

Analytical continuation problem. Maximum entropy method. Instead of χ 2 we minimize χ 2 + α S , Entropy S determines how our function is close to some model function µ ( ω ) (which summarizes our prior knowledge about the spectral function). N w � � ρ m − µ m − ρ m log ρ m � S = µ m m = 1 Doesn’t work for small lattice sizes. V. V. Braguta, A. Yu. Kotov Non-perturbative study of the viscosity... Oct 30, 2013 15 / 22

Analytical continuation problem. Linear method: � ρ ( ω ) = m ( ω )( 1 + a ( ω )) = m ( ω )( 1 + a l u l ( ω )) , l where m ( ω ) is an initial approximation : A ω 4 m ( ω ) = , tanh 2 ω 4 T tanh ω 2 T and u l ( ω ) are eigenmodes of H ( ω, ω ′ ) = � K ( t i , ω ) K ( t i , ω ′ ) with i cosh ( ω ( 1 2 T − t )) K ( t , ω ) = m ( ω ) sinh ω 2 T a l are selected to minimize χ 2 . V. V. Braguta, A. Yu. Kotov Non-perturbative study of the viscosity... Oct 30, 2013 16 / 22

Analytical continuation problem. Resolution function. ρ ( ω ) = m ( ω )( 1 + a ( ω )) Let ˆ ρ ( ω ) be a true spectral function. K ( t i , w ) linear ρ ( ω ) = m ( ω )( 1 + ˆ ˆ a ( ω )) − − − − → G i − − − → ρ ( ω ) = m ( ω )( 1 + a ( ω )) Resolution function: � a ( ω ) δ ( ω, ω ′ ) a ( ω ) = d ω ˆ 0.14 0.12 0.1 0.08 0.06 δ (0, ω ) 0.04 0.02 0 -0.02 -0.04 -0.06 0 5 10 15 20 25 30 35 40 ω /T V. V. Braguta, A. Yu. Kotov Non-perturbative study of the viscosity... Oct 30, 2013 17 / 22

Main difficulties. Large statistical errors in measuring correlator C 12 , 12 ( x 0 , 0 ) Improved action Multilevel algorithm Extracting spectral function ρ 12 , 12 from � ∞ ρ 12 , 12 ( ω, p ) cosh ω ( 1 2 L 0 − x 0 ) C 12 , 12 ( x 0 , p ) = β 5 d ω sinh ω L 0 0 2 Fit by model function Maximum entropy method Linear method ... V. V. Braguta, A. Yu. Kotov Non-perturbative study of the viscosity... Oct 30, 2013 18 / 22

Numerical setup. SU ( 2 ) -gluodynamics with Wilson action: S = β � tr ( 1 − U pl ) 2 pl Lattice 8 × 32 3 β = 2 . 643 T / T c ≈ 1 . 2 Clover-shaped discretization for F µν Two-level algorithm for measuring stress-energy tensor correlator. V. V. Braguta, A. Yu. Kotov Non-perturbative study of the viscosity... Oct 30, 2013 19 / 22

Spectral function ρ 12 , 12 . 0.18 0.16 0.14 0.12 ρ 12,12 /sh( ω /2T) 0.1 0.08 0.06 0.04 0.02 0 0 5 10 15 20 25 30 ω /T V. V. Braguta, A. Yu. Kotov Non-perturbative study of the viscosity... Oct 30, 2013 20 / 22

Numerical results. η s = 0 . 111 ± 0 . 032 KSS-bound: η/ s ≥ 1 4 π ≈ 0 . 08 Perturbative result: η/ s ∼ 2 Experimental bound and preferred value: η/ s < 5 1 4 π ≈ 0 . 4 η/ s ∼ ( 1 ↔ 3 ) 1 4 π V. V. Braguta, A. Yu. Kotov Non-perturbative study of the viscosity... Oct 30, 2013 21 / 22

Unsatisfactory attempts to increase lattice size 0.1 Data 0.01 0.001 C 12,12 0.0001 1e-05 1e-06 2 4 6 8 10 12 14 x 0 /a V. V. Braguta, A. Yu. Kotov Non-perturbative study of the viscosity... Oct 30, 2013 22 / 22