有限温度 QCD における 励起モード 理研スパコン研究会 2010/Sep./24 北沢正清 （阪大）

有限温度 QCD における

励起モード

理研スパコン研究会 2010/Sep./24

北沢正清（阪大）

Lattice Study of QCD @Lattice Study of QCD @TT>0 >0 Lattice Study of QCD @Lattice Study of QCD @TT>0 >0

EoSsusceptibilities (fluctuations)order parameters…

macro(bulk)

micro

static dynamical

screening masspotentialwave func.…

Lattice Study of QCD @Lattice Study of QCD @TT>0 >0 Lattice Study of QCD @Lattice Study of QCD @TT>0 >0

EoSsusceptibilities (fluctuations)order parameters…

macro(bulk)

micro

static dynamical

screening masspotentialwave func.…

transport coefficients (viscosity)dynamic critical phenomena?

excitation modes (quasi-particles; hadrons, quarks, gluons, etc.)mass, decay rates

spectral function

Spectral Functions at Spectral Functions at TT>0 >0 Spectral Functions at Spectral Functions at TT>0 >0

0( , ) Im FT ( ), (0) ( )O x O x p

(

,p=

0)

(

,p)

T = 0 T > 0

Difficulties at T>0: continuous spectrumfixed temporal extent: T=1/aNt

(1/ 2 )

/ 2 / 2( ) ( )

T

T T

eD d

e e

Extracting Spectral Functions Extracting Spectral Functions Extracting Spectral Functions Extracting Spectral Functions

( ) ( , ) ( )D d K

(( )) ) (0OD T O

( / 2 )

/ 2 / 2( , )

eK

e e

Ill-posed problem

lattice observablediscrete and noisy

spectral functioncontinuous

MEM analysis of ()most probable image estimated bylattice data + prior knowledge Asakawa, Hatsuda, Nakahara, 1999

qualitative structure of ().errors only for average for finite range

Dilepton (Photon) Production Rate Dilepton (Photon) Production Rate Dilepton (Photon) Production Rate Dilepton (Photon) Production Rate

e+

e-

04 4 2 /

1Im

12 1Ree

q T

dR

d q Q e

Direct probes in heavy ion collisions.Emissions from all stages are superposed

PHENIX, PRC,2010

0FT ( ), (0) ( )R j x j x

vector channel propagator

Quark Number Scaling at RHIC Quark Number Scaling at RHIC Quark Number Scaling at RHIC Quark Number Scaling at RHIC

v2 >0

Quark number scaling indicates the existence of quasi-particles having quark quantum number at early stage.

Excitation Modes in Hot Medium Excitation Modes in Hot Medium Excitation Modes in Hot Medium Excitation Modes in Hot Medium

Are there hadronic modes above Tc?

•fate of charmonia: signal of realization of QGP phase Matsui, Satz, ’86

•soft modes of chiral transition in and channels.

and other excitation modes?

glueballs, diquarks, …

How do hadrons cease to exist?

Hatsuda, Kunihiro, ’85

perturbatively, decay width ~g2T

How do quarks and gluons disappear?

•Is quasi-particle picture for quarks and gluons irrelevant in sQGP?

Hydrodynamical Models at RHIC Hydrodynamical Models at RHIC Hydrodynamical Models at RHIC Hydrodynamical Models at RHIC

Schenke, et al. 1009.3244

Viscous hydrodynamicscontains transport coefficients:

•shear viscosity •bulk viscosity •relaxation times , …

Is the ratio /s close to the conjectured lower bound 1/4?

Spectral Functions at Spectral Functions at TT>0 >0 Spectral Functions at Spectral Functions at TT>0 >0

quasi-particle excitationwidth ~ decay rate transport coefficients

(

,p)

peak

Kubo formulae

slope at the origin

0

( )~ lim

•shear viscosity : T12

•bulk viscosity : T•electric conductivity : Jii

Approximation Approximation Approximation Approximation

So far, almost all studies on spectral functions are performed in quenched approximation.

to increase number of data points larger Nt (finer a)

to eliminate finite volume effectslarger L > 4/T

The largest lattice thus far: Nt ~ 48, Nx~128

Because we need

higher statistics

Quenched QCD is not the QCD.However, some nonperturbative features would be revealed in the quenched analysis.

Charmonium Spectra Charmonium Spectra Charmonium Spectra Charmonium Spectra

MEM analysis:All calculation concludes that J/ survives up to ~1.5Tc

Asakawa, Hatsuda, 2004Datta, et al., 2004Umeda, et al., 2002Aarts, et al., 2006Jakovac, et al., 2007…

Asakawa, Hatsuda, 2004

Recent progress:subtract the constant mode Umeda,2007default model dependence Ding, et al.,2009

Only a few studies on light quarks

Transport Coefficients Transport Coefficients Transport Coefficients Transport Coefficients shear:Karsch, Wyld, 1987Nakamura, Sakai, 2005Meyer, 2007bulk:Meyer, 2008electric conductivity:Gupta, 2004Aarts, et al., 2007

Electric Conductivity:Spectrum in vector channel Aarts, et al., 2007

0

( )1lim 0.4(1)

6ii

i

Several approaches:

fitting with Lorentz-type ansatz,ansatz from hydrodynamics at low energy

Cut-off Dependence in Vector Channel Cut-off Dependence in Vector Channel Cut-off Dependence in Vector Channel Cut-off Dependence in Vector Channel

Correlator for vector channelfor T=1.5Tc with different a

Kaczmarek, et al., xQCD2010

,

( ) ( , ) (0,0)Vx

G x

strong a dependence

free ( )VG : free & continuum

Spatial Volume Dependence of Spatial Volume Dependence of mmTT Spatial Volume Dependence of Spatial Volume Dependence of mmTT

mT/T 483x16643x16

1283x16mT/T=0.725(14)

T=3Tc

•Strong spatial volume dependence of mT.

3 3/ ~ 1/N N V

Spatial Volume Dependence of Spatial Volume Dependence of mmTT Spatial Volume Dependence of Spatial Volume Dependence of mmTT

mT/T 483x16643x16

1283x16mT/T=0.725(14)

T=3Tc

•Strong spatial volume dependence of mT.

k2 nB(E

k) m=0

m=T

k/T

Nx/N=4

Discretization of p

min

22

x x

Np T

L N

•Nx/N=4 pmin~1.57T•Nx/N=8 pmin~0.79T

3 3/ ~ 1/N N V

Exploiting Sum Rules Exploiting Sum Rules Exploiting Sum Rules Exploiting Sum Rules

0

0

( ) ( )23 4 ( 3 )Td s p

s

Spectral function can be constrained by sum-rules.

Kharzeev, Tuchin, 2008Romatschke, Son, 2009Ellis, Kapusta, Tang, 1998

For bulk sector:

Some difficulties:()T=0() is not positive semi-definite. MEM is not applicable.We must know T=0 spectrum.

QCD thermodynamics serves asan additional information for the spectrum!

Summary Summary Summary Summary

有限温度 QCD の動的な性質を格子 QCD の数値解析により理解することは、極めて重要かつ興味深い課題である。

スペクトル関数の評価には注意深い解析が必要である。

•相関関数の格子間隔および体積依存性•解析の結果得られたスペクトル関数の誤差評価

既存の解析の更新、および新しい物理量の解析のため、より大きな計算機資源が必要である。

•より大きな格子、高統計•フル QCD における解析

解析接続をより確実に行うための手法の開発や、信頼性の正しい評価等において、更なる理論的進展が望まれる。

Spectral Function at Spectral Function at TT>0 >0 Spectral Function at Spectral Function at TT>0 >0

/2

/

,

1 ˆ( ) (1 ) ( )n TE Tn

nm

m

e E O mEZ

e n

2ˆ( ) ( ) 0m

m

E O m

nonzero T

T = 0