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

• Experimentally preferred value:

η s ∼ (1 ↔ 3) 1 4π

• Experimental bound:

η s < 5 1 4π

• KSS-bound:

η s ≥ 1 4π

• 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: η = π lim

ω→0

ρ12,12(ω, q = 0) ω Green function measured on the lattice(Eucledian): C12,12(x0, p) = β5

• d3xeipxT12(0)T12(x0, x)

Spectral function and correlator of stress-energy tensor: C12,12(x0, p) =β5 ∞ ρ12,12(ω, p)cosh ω( 1

2L0 − x0)

sinh ωL0

2

dω Stress-energy tensor for gluodynamics: Tµν = 2 tr(FµσFνσ − 1 4δµνFρσFρσ) Asymptotic behaviour - perturbation theory: ρ(ω) = 1

10 3 (4π)2 ω4, ω → ∞

• 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 C12,12(x0, 0)

Improved action Multilevel algorithm

Extracting spectral function ρ12,12 from C12,12(x0, p) =β5 ∞ ρ12,12(ω, p)cosh ω( 1

2L0 − x0)

sinh ωL0

2

dω

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 C12,12. Improved action.

Sunimpr = β

• pl

Spl Simpr = βimpr

• pl

Spl − βimpr 20u2

• rt

Srt Spl,rt = 1 2 tr(1 − Upl,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 C12,12. Multilevel algorithm.

For t1 and t2 in different areas O(t1)O(t2) = 1 Nbc

• bc

O(t1)b.cO(t2)b.c

• 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.

C12,12(x0, p) =β5 ∞ ρ12,12(ω, p)cosh ω( 1

2L0 − x0)

sinh ωL0

2

dω Proposed in the first work on transport coefficients:

• F. Karsch, H. W. Wyld. Phys. Rev. D35 (1987)

ρ(ω)/ω = A

• γ

(m − ω)2 + γ2 + γ (m + ω)2 + γ2

• 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.

C12,12(x0, p) =β5 ∞ ρ12,12(ω, p)cosh ω( 1

2L0 − x0)

sinh ωL0

2

dω We discretize ω, Nω ∼ O(103) ρ(ω)

K(ω,xi)

− − − − → G(xi) = Gi χ2 =

• i,j

(Gi − G (0)

i

)(S−1)ij(Gj − G (0)

j

) min χ2 → ∼ 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 χ2 we minimize χ2 + αS, Entropy S determines how our function is close to some model function µ(ω) (which summarizes our prior knowledge about the spectral function). S =

Nw

• m=1
• ρm − µm − ρm log ρm

µm

• 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 +

• l

alul(ω)), where m(ω) is an initial approximation : m(ω) = Aω4 tanh2 ω

4T tanh ω 2T

, and ul(ω) are eigenmodes of H(ω, ω′) =

i

K(ti, ω)K(ti, ω′) with K(t, ω) = m(ω)

cosh(ω( 1

2T −t))

sinh ω

2T

al are selected to minimize χ2.

• 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. ˆ ρ(ω) = m(ω)(1 + ˆ a(ω))

K(ti,w)

− − − − → Gi

linear

− − − → ρ(ω) = m(ω)(1 + a(ω)) Resolution function: a(ω) =

• dωˆ

a(ω)δ(ω, ω′)

• 0.06
• 0.04
• 0.02

0.02 0.04 0.06 0.08 0.1 0.12 0.14 5 10 15 20 25 30 35 40 δ(0,ω) 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 C12,12(x0, 0)

Improved action Multilevel algorithm

Extracting spectral function ρ12,12 from C12,12(x0, p) =β5 ∞ ρ12,12(ω, p)cosh ω( 1

2L0 − x0)

sinh ωL0

2

dω

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 2

• pl

tr(1 − Upl) Lattice 8 × 323 β = 2.643 T/Tc ≈ 1.2 Clover-shaped discretization for Fµν 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 ρ12,12.

0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 5 10 15 20 25 30 ρ12,12/sh(ω/2T) 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 4π ≈ 0.08 Perturbative result: η/s ∼ 2 Experimental bound and preferred value: η/s < 5 1 4π ≈ 0.4 η/s ∼ (1 ↔ 3) 1 4π

• V. V. Braguta, A. Yu. Kotov

Non-perturbative study of the viscosity... Oct 30, 2013 21 / 22

Unsatisfactory attempts to increase lattice size

1e-06 1e-05 0.0001 0.001 0.01 0.1 2 4 6 8 10 12 14 C12,12 x0/a Data

• V. V. Braguta, A. Yu. Kotov

Non-perturbative study of the viscosity... Oct 30, 2013 22 / 22