格子 QCD 数値計算を用いた

QGP 相におけるクォークの探求

北沢正清

RCNP, 30, Oct., 2007

F. Karsch and M.K., Phys. Lett. B, in press.(arXiv:0708.0299)

Phase Diagram of QCD Phase Diagram of QCD Phase Diagram of QCD Phase Diagram of QCD

Tc

0

hadron phase(confined phase) color superconductivity

T Lattice QCD Monte Carlo simulation= first principle calculation of QCD

property of quarks in this region

•in the deconfined phase•as the basic degrees of freedom of QCD•will have many informations of the matter

Phase Diagram of QCD Phase Diagram of QCD Phase Diagram of QCD Phase Diagram of QCD

Tc

0

hadron phase(confined phase) color superconductivity

T

•Boyd, Gupta, Karsch, NPB 385,481(’92).•Petreczky, et al., NPPS106,513(’02).•Hamada, et al., hep-ph/0610010.

Lattice QCD Monte Carlo simulation= first principle calculation of QCD

property of quarks in this region

Quarks at Extremely High Quarks at Extremely High TT Quarks at Extremely High Quarks at Extremely High TT

•Hard Thermal Loop approx. ( p, , mq<<T )•1-loop (g<<1)

Klimov ’82, Weldon ’83Braaten, Pisarski ’89

( , ) p

“plasmino”

p / mT

/

mT

6T

gTm

0

1( , )

( , )S

p

p γ p

•Gauge independent spectrum

•2 collective excitations having a “thermal mass”

•The plasmino mode has a minimum at finite p.

p / m

/

m

Decomposition of Quark Propagator Decomposition of Quark Propagator Decomposition of Quark Propagator Decomposition of Quark Propagator

0

free

0

( ,)

)( ) (

SE E

p p

pp

p

0 ((

))

2

E m

E

p

p

pp

0

0

( )( , ) ( , )

(( ) ),

S S

S

p p

p

p

p

Free quark with mass mHTL ( high T limit )0

HTL

0

( , )( ) ( )

Sp p

p pp

p / mT

/

mT

0

0

( )

( , )

(

( )

, )

( , )

p

p

p

p

p

Quark Spectrum as a function of Quark Spectrum as a function of mm00 Quark Spectrum as a function of Quark Spectrum as a function of mm00

Quark propagator in hot medium at T >>Tc

- as a function of bare scalar mass m0

•How is the interpolating behavior?•How does the plasmino excitation emerge as m00 ?

m0 << gT

m0 >> gT

We know two gauge-independent limits:

m0mT-mT

+(,p=0) +(,p=0)

Fermion Spectrum in QED & Yukawa Model Fermion Spectrum in QED & Yukawa Model Fermion Spectrum in QED & Yukawa Model Fermion Spectrum in QED & Yukawa Model Baym, Blaizot, Svetisky, ‘92

0

1( )

2L i i m g

Yukawa model:

1-loop approx.:

m0/T=0.01

0.80.450.3

0.1

+(

,p=

0)

Spectral Function for g =1 , T =1

0 / 1m T thermal mass mT=gT/4

0 / 1m T single peak at m0

Plasmino peak disappearsas m0 /T becomes larger.

cf.) massless fermion + massive bosonM.K., Kunihiro, Nemoto,’06

Simulation Setup Simulation Setup Simulation Setup Simulation Setup

•quenched approximation•clover improved Wilson•Landau gauge fixing

T Lattice size

1.5Tc 6.64 483x12

6.87 643x16, 483x16

3Tc 7.19 483x12

7.45 643x16, 483x16

•vary bare quark mass m0 •for zero momentum p=0

•2-pole approx. for +(,p=0)

•wall source

Simulation Setup Simulation Setup Simulation Setup Simulation Setup

•quenched approximation•clover improved Wilson•Landau gauge fixing

T Lattice size

1.5Tc 6.64 483x12

6.87 643x16, 483x16

3Tc 7.19 483x12

7.45 643x16, 483x16

•vary bare quark mass m0 •for zero momentum p=0

•2-pole approx. for +(,p=0)

•wall source

Simulation Setup Simulation Setup Simulation Setup Simulation Setup

1 21 2( ) ( ) ( )E EZ Z

4-parameter fit E1, E2, Z1, Z2

•2-pole approx. for +(,p=0)

•wall source

T Lattice size

1.5Tc 6.64 483x12

6.87 643x16, 483x16

3Tc 7.19 483x12

7.45 643x16, 483x16

•quenched approximation•clover improved Wilson•Landau gauge fixing

•vary bare quark mass m0 •for zero momentum p=0

Correlation Function Correlation Function Correlation Function Correlation Function 0

0

( , ) ( ) ( )

( ) ( )S

C C C

C C

0

( )C

•We neglect 4 points near the source from the fit.•2-pole ansatz works quite well!! ( 2/dof.~2 in corr. fit )

643x16, = 7.459, = 0.1337, 51confs.

/T

1 2 ( )1 2( ) e eE EC z z

Fitting result

mm00 Dependence of Dependence of CC++(() ) mm00 Dependence of Dependence of CC++(() )

( )C

= 0.134

= 0.132

= 0.130

/T

•Shape of C+() changes from chiral symmetric to single pole structures.

c=0.13390

m0: small

m0: large

0

1 1 1

2 c

m

Spectral Function Spectral Function Spectral Function Spectral Function

E1E2

Z1

Z2

E1E2

Z1Z2

T = 3Tc 643x16 (= 7.459)

E2

E1

2

1 2

Z

Z Z

= m0 pole of free quark

m0 / T

E /

TZ

2 / (

Z1+

Z 2)

T=3Tc

1

2

1

2

( ) ( )

( )

E

E

Z

Z

0

1 1 1

2 c

m

Spectral Function Spectral Function Spectral Function Spectral Function T = 3Tc 643x16 (= 7.459)

E2

E1

2

1 2

Z

Z Z

= m0 pole of free quark

m0 / T

E /

TZ

2 / (

Z1+

Z 2)

1

2

1

2

( ) ( )

( )

E

E

Z

Z

•Limiting behaviors for are as expected.•Chiral symmetry of quark propagator restores around m0=0.•Quarks in the chiral limit have a thermal mass!•E2>E1 : qualitatively different from the 1-loop result.

0 00,m m

T=3Tc

Temperature Dependence Temperature Dependence Temperature Dependence Temperature Dependence

•mT /T is insensitive to T.•The slope of E2 and minimum of E1 is much clearer at lower T.

T = 3Tc

T =1.5Tc

minimum of E1

E2

E1

2

1 2

Z

Z Z

m0 / T

E /

TZ

2 / (

Z1+

Z 2)

643x16

Lattice Spacing Dependence Lattice Spacing Dependence Lattice Spacing Dependence Lattice Spacing Dependence

643x16 (= 7.459)

483x12 (= 7.192)

E /

T

E2

E1

m0 / T

same physical volumewith different a.

•No lattice spacing dependence within statistical error.

T=3Tc

Spatial Volume Dependence Spatial Volume Dependence Spatial Volume Dependence Spatial Volume Dependence

E2

E1

m0 / T

E /

T

T=3Tc

643x16 (= 7.459)

483x16(= 7.459)

same lattice spacingwith different aspect ratio.

•Excitation spectra have clear volume dependence even for N/N=4.

Extrapolation of Thermal Mass Extrapolation of Thermal Mass Extrapolation of Thermal Mass Extrapolation of Thermal Mass

Extrapolation of thermal mass to infinite spatial volume limit:

•Small T dependence of mT/T, •while it decreases slightly with increasing T.•Simulation with much larger volume is desireble.

3Tc

1.5TcmT /T

3 3/N N

643x16 483x16

T=1.5Tc

T=3Tc

mT /T = 0.800(15)mT = 322(6)MeV

mT /T = 0.771(18)mT = 625(15)MeV

Charm Quark & J/ Charm Quark & J/ charm quark

•Z2/(Z1+Z2) は十分小さい　　　 c-quark は、 free quark に近い粒子描像を持つ。•J/ 粒子は閾値 2mc より高いエネルギーを持つ？

threshold 2mc

PreliminaryPreliminary

T = 1.5Tc

Finite Momentum Finite Momentum

E2

E1

p / T

E /

T

•E2<E1 for finite momentum.

00

0 0

( , ) ( , )

( , )

( , )

( ) ( )

V

S

S S

S p

S

S S

p p

p

p

p p

In the chiral limit,

01(

2)

pp

Preliminary!!!

Preliminary!!!

Effect of Dynamical Quarks Effect of Dynamical Quarks Effect of Dynamical Quarks Effect of Dynamical Quarks

Quark propagatorin quench approximation:

screen gluon field suppress mT?

meson loop will have strong effect if mesonic excitations exist

In full QCD,

massless fermion + massive boson 3 peaks in quark spectrum! M.K., Kunihiro, Nemoto, ‘06

まとまとめ め まとまとめ め

•light クォークは、ゲージ場の媒質効果により温度程度の熱質量を獲得する。•Heavy クォーク極限では plasmino の寄与は無視でき、自由粒子のそれへ漸近する。•比 mT/T の温度依存性は、今回調べた温度領域で非常に小さい。

•1-loop とは定性的に異なる振る舞い。•強い空間体積依存性。

Puzzles :

臨界温度付近の QGP 相におけるクォークは、熱質量およびplasmino を伴った崩壊幅の小さい準粒子として振る舞っている。

格子 QCD はクォークの解析に適している。

展望 展望

有限温度多体系としてのQGP の物性物理

基礎理論（ QCD ）基礎理論（ QCD ）

クォーク

重イオン衝突実験重イオン衝突実験

クォークの微視的理解cf. M.K., Kunihiro, Nemoto, Mitsutani

•QGP 中のメソン励起、熱力学量•有効模型の構成

観測量

•体積効果： Karsch et al., 1283x16,

　　 LQGP collaboration, in progress

•full QCD•有限運動量•ゲージ依存性、 T~Tc & T >>Tc

•グルオンの pole mass

数値解析の展望

Choice of Source Choice of Source Choice of Source Choice of Source

Wall source, instead of point source

( , ) ( , ) ( ,0)S x

p 0 x 0

,

1( , ) ( , ) ( ,0)S

V

x y

p 0 x y

point:

wall :

•same (or, less) numerical cost•quite effective to reduce noise!!

the larger spatial volume, the more effective!

t

0K mD

result sourceK

result o rce1

s uK

t

What’s the source?

point

wall

Elliptic Flow Elliptic Flow vv22Elliptic Flow Elliptic Flow vv22

v2

pT 空間v2 >0 v2 <0

•RHIC エネルギーでは v2>0 。•完全流体模型が低 pT でよく成り立つ。

•v2 が飽和する pT は粒子により異なる。

非常に小さい粘性係数

反応平面

短い平均自由行程

実空間

strongly coupled QGP (sQGP)

Elliptic Flow Elliptic Flow vv22Elliptic Flow Elliptic Flow vv22

v2

pT 空間v2 >0 v2 <0

•RHIC エネルギーでは v2>0 。•完全流体模型が低 pT でよく成り立つ。

•v2 が飽和する pT は粒子により異なる。

非常に小さい粘性係数

反応平面

短い平均自由行程

実空間

strongly coupled QGP (sQGP)

クォーク数による scaling が非常に良く成り立っている。　　　　　　　　　　　　　　　 Nonaka, et al.

Recombination Model の成功。

1exp( ( )

1det exp( )

, ) ( ,0) G F

G

DUD D S SZ

D KU SZ

x 0

Quark Propagator in Quenched Lattice Quark Propagator in Quenched Lattice Quark Propagator in Quenched Lattice Quark Propagator in Quenched Lattice

quenched approx.

1conf.

conf.

1

NK Configurations are distributed

with a weight exp(SG).

0K mD fermion matrix:

Wilson fermion:

in continuum

0mK D r 0

1

2 8m r

0

1 1 1

2 c

m

We can calculate quark propagator with various m0

for a given set of gauge(-fixed) configuration!

Dirac Structure of Quark Propagator Dirac Structure of Quark Propagator Dirac Structure of Quark Propagator Dirac Structure of Quark Propagator

00( , ) ( , ) ( , ) ( , )V SS S S p S p p p p

quark propagator

00

0 0

( , ) ( ) ( )

( ) ( )

SS S S

S S

0

0 0( ) ( )

( ) ( )S S

S S

S S

11

00

01

2

00

11

in stand. repr.p=0

evenodd

0 SS S S ( ) ( )S S

Chiral symmetric Ss=0 S+ is an even function.