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理研スパコン研究会 2010/Sep./24
有限温度QCDにおける
励起モード
北沢正清
（阪大）
Lattice Study of QCD @T>0
static
macro
(bulk)
EoS
susceptibilities
(fluctuations)
order parameters
…
micro
screening mass
potential
wave func.
…
dynamical
Lattice Study of QCD @T>0
spectral function
static
dynamical
macro
(bulk)
EoS
susceptibilities
(fluctuations)
order parameters
…
transport coefficients
(viscosity)
dynamic critical
phenomena?
micro
screening mass
potential
wave func.
…
excitation modes
(quasi-particles;
hadrons, quarks,
gluons, etc.)
mass, decay rates
Spectral Functions at T>0
 ( , p)   Im FT O( x), O(0)   ( x0 )
T>0
 (,p)
 (,p=0)
T=0


Difficulties at T>0: continuous spectrum
fixed temporal extent: T=1/aNt

 D( )  
e(1/ 2T  )
d  / 2T
 ( )
  / 2T
e
e
Extracting Spectral Functions

D( )   dK (, )  ()

D( )  T O( )O(0)
lattice observable
discrete and noisy
e(  / 2 )
K ( , )   / 2   / 2
e
e
spectral function
continuous
Ill-posed problem
MEM analysis of  ()
most probable image estimated by
lattice data + prior knowledge
Asakawa, Hatsuda, Nakahara, 1999
qualitative structure of  ().
errors only for average for finite range
Dilepton (Photon) Production Rate
Direct probes in heavy ion collisions.
Emissions from all stages are superposed
PHENIX, PRC,2010
dRee

1
R


Im


4
4 2
q0 / T
d q
12 Q
e
1
 R   FT  j ( x), j (0)   ( x0 )
g
vector channel propagator
e+
e-
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
How do quarks and gluons disappear?
perturbatively, decay width ~g2T
•Is quasi-particle picture for quarks and gluons irrelevant in sQGP?
How do hadrons cease to exist?
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 s channels.
Hatsuda, Kunihiro, ’85
and other excitation modes?
glueballs, diquarks, …
Hydrodynamical Models at RHIC
Viscous hydrodynamics
contains transport coefficients:
•shear viscosity h
•bulk viscosity z
•relaxation times  , …
Is the ratio h/s close to the
conjectured lower bound 1/4 ?
Schenke, et al. 1009.3244
 (,p)
Spectral Functions at T>0

peak
slope at the origin
 transport coefficients
Kubo formulae h ~ lim
0
 ( )

•shear viscosity : T12
•bulk viscosity : T
•electric conductivity : Jii
quasi-particle excitation
width ~ decay rate
Approximation
So far, almost all studies on spectral functions are
performed in quenched approximation.
Because we need
larger Nt (finer a) to increase number of data points
larger L > 4/T
to eliminate finite volume effects
higher statistics
The largest lattice thus far: Nt ~ 48, Nx~128
Quenched QCD is not the QCD.
However, some nonperturbative features would be
revealed in the quenched analysis.
Charmonium Spectra
Asakawa, Hatsuda, 2004
Asakawa, Hatsuda, 2004
Datta, et al., 2004
Umeda, et al., 2002
Aarts, et al., 2006
Jakovac, et al., 2007
…
MEM analysis:
All calculation concludes
that J/y survives up to ~1.5Tc
Only a few studies on light quarks
Recent progress:
subtract the constant mode Umeda,2007
default model dependence Ding, et al.,2009
Transport Coefficients
shear:
Karsch, Wyld, 1987
Nakamura, Sakai, 2005
Meyer, 2007
bulk:
Meyer, 2008
electric conductivity:
Gupta, 2004
Aarts, et al., 2007
Several approaches:
fitting with Lorentz-type ansatz,
ansatz from hydrodynamics at low energy
Electric Conductivity:
Spectrum in vector channel
Aarts, et al., 2007
1
6  0
s  lim 
i
ii ( )
 0.4(1)

Cut-off Dependence in Vector Channel
Kaczmarek, et al., xQCD2010
Correlator for vector channel
for T=1.5Tc with different a
GV ( )   yg y ( , x)yg y (0,0)
x,
G ( ) : free & continuum
free
V
strong a dependence
Spatial Volume Dependence of mT
mT/T
643x16
483x16
T=3Tc
1283x16
N3 / Ns3 ~ 1/ V
•Strong spatial volume
dependence of mT.
mT/T=0.725(14)
Spatial Volume Dependence of mT
mT/T
643x16
483x16
T=3Tc
1283x16
•Strong spatial volume
dependence of mT.
mT/T=0.725(14)
N3 / Ns3 ~ 1/ V
pmin
N
2

 2 T
Lx
Nx
•Nx/N=4  pmin~1.57T
•Nx/N=8  pmin~0.79T
k2nB(Ek)
Discretization of p
m=0
m=T
Nx/N=4
k/T
Exploiting Sum Rules
Kharzeev, Tuchin, 2008
Romatschke, Son, 2009
Ellis, Kapusta, Tang, 1998
Spectral function can be constrained by sum-rules.
For bulk sector:
2



0
d
 ( )  T 0 ( )  

  3s  4  (  3 p)

 s

QCD thermodynamics serves as
an additional information for the spectrum!
Some difficulties:
()T=0() is not positive semi-definite. MEM is not applicable.
We must know T=0 spectrum.
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
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。
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
Spectral Function at T>0
nonzero T
1
 ( )   e En / T (1  e / T ) (  En  Em ) n Oˆ m
Z m,n
T=0
 ( )    (  Em ) 0 Oˆ m
m
2
2

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