PPT

Report
格子QCDによる有限密度系
シミュレーション
S. Muroya
Tokuyama Women’s College
in collabolation with
A. Nakamura, C. Nonaka and T. Takaishi
最近のレヴューです
Muroya, Nakamura, Nonaka
and Takaishi : PTP 110(03)615,
hep-lat/0306031
物理学最前線 “クォークマター” 宮村修 1986
物理学最前線 “クォークマター” 宮村修
物理学最前線 “クォークマター” 宮村修
高密度QCD 複雑な相構造
RHIC
JPARC
Thomas Schafer,
hep-ph/0304281
Ferro-Mgn.?
Q-Hall st ?
流体モデルのインプットに使っている
状態方程式の例( Nonaka, Honda, Muroya )
化学ポテンシャル
Lagrange未定定数
• 統計力学
  Tr{e
  ( H  N )
• 場の理論
 0   0  i
constant gauge field
P.A.M. Dirac (‘56)
Y. Nambu (‘68)
}
保存量 (保存電荷)
Chemical Potential on a Lattice
• Introducing the chemical potential on a lattice
(Wilson fermion) :hopping parameter
  Tr{e
 H
}

DUD  D e
4
S F    ,
 (SG  S F )
quark mass
U  ( x)  e

iA ( x )
  1    (1    )U  ( x ) x ゥ , x  ˆ  (1    )U  ( x ) x ゥ , x  ˆ

 1
  Tr{e
  ( H  N )
}
3


DUD  D e
 ( S G  S F (  ))
,
S F (  )    (  ) ,
 (  )  1    (1   i )U i ( x ) x ', x  iˆ  (1   i )U i ( x ) x ', x  iˆ


i 1
 e
P4  P4  i 

a
(1   4 )U 4 ( x ) x ', x  4ˆ  e

   5 5
 a

4
(1   4 )U ( x ' ) x ', x  4ˆ
det  : complex


Phase (sign) problem

   5 5
O 
1
Z

1
Z


det W : complex
DUD  D O e
DU O det  e
 SG  S F
SG
det   det  e
i
e
i
e
V
quench 計算では、化学ポテンシャル
の影響がいつから見え出すか?
•プロットはシミュレーション
•実線は πによるμc評価
•点線はバリオンによるμc評価
•破線は平均場近似
Dynamical Quark is
indispensable
I. Barbour et al, NP275 (’86
M.A. Stephanov, PRL(‘96)
chiral limit では μc = 0 か?
Wilson Fermion の固有値分布 β= 5.7, κ=0.16 , 4x4x4x4 Lattice
μ=0.0
μ=0.3
μ=0.2
μ=0.4
K-S Fermion の固有値分布 ( m =0.1, beta = 5.7)
μ=0
μ=0.3
μ=0.2
μ=0.4
Approach to high density state of
the Lattice QCD
• Reweighting method
– Fodor & Katz
– Grasgow
•
•
•
•
•
Taylor expansion
Imaginary Chemical Potential
Nishimura’s talk
Density of the state
Positive Measure model
Susceptibility against chemical potential
Irina’s talk
Susceptibility against chemical potential
クォーク数密度
MILC Collabolation
擬スカラーmeson mass の応答
second derivative for chemical potential
S  u  d
V   u    d
QCD-TARO
Collaboration
高次の微係数を計算する⇔物理量をで展開
/T
Gavai and Gupta,
quenched QCD, 4th order of 
Fodor-Katz, JHEP03(2002)014
T E  160  3 . 5 MeV ,  E  725  35 MeV
Standard gauge
+ Staggered fermion
NF  2 1
m u , d  0 . 025 ,
m s  0 .2
N s  4, N s  4, 6, 8
3
Reweighting
O 
1
Z

1
Z
 DU
det  ( ) e
 DU
Sg (0 )
e
Fodor and Katz
Multi-reweighting
method
Sg ( )
O
det  ( 0 ) e
S g (  0 ) S g (  )
det  ( )
det  ( 0 )
O
( 0 )
( )
  ( , m,  )
( 0 )
( )
Glasgow approach
Taylor expansion at high T and low 
• Allton et al. (Bielefeld-Swansea) hep-lat/0204010
Improved action
+ Improved staggered fermion
N F  2 , m q  0 . 1, 0.2
170 MeV
16  4
3
a=0.29
微分の4次まで
p
 p 
 4   4
T  T

T ,
p
T
4
T , 0
Imaginary Chemical Potential
deForcrand and Philipsen NPB642(02)290; hep-lat/0307020
D’Elia and Lombardo Phys.Rev. D67 (2003) 014505
At small 
log Z (  )  a 0  a 2   a 4   O (  )
2
4
6
 I  Im 
det M : complex
I 

3
2
det M : real
Z(3) symmetry
Im    i Re 
 C ( a  I )  c 0  c1 ( a  I )
2
 I  Im 
Standard gauge
N F  2 , m q  0 . 0 25
3
3
+ Staggered fermion 8  4 , 6  4
6
log Z (  I )  a 0  a 2  I  a 4  I  O (  I )
4
Allton et al.
Fodor-Katz
Consistent !?
YES
deForcrand-Philipsen
D’Elia and Lombardo
Models free from Sing Problem
• Effective theory
• Finite Isospin
u  d
det  (  ) det  (   )  det  (  )
det  (   )  det  (  )*,
U    2U   2
• Two-color QCD
3

 ( x , x ' ;   )   2   x , x '    (1  
*
*
*
i
i 1
 1   U
 e
 a
*
4
4
)U i ( x ) x ', x  iˆ
Pseudo-Real

 (1   )U ( x ' )
*
i

i
x ', x  iˆ
 x  x ', x  4ˆ  e   a 1   * 4 U 4  x '  x ', x  4ˆ   2
  2  ( x , x ' ;   ) 2
*
{det  ( x , x ' ;   )}  det  ( x , x ' ;   )  det  (x, x' ;  '  )
*
det  : real
*
Monte Carlo Calculation Works Well !
2
Color SU(2) r at Finite Density
3
4 X8 ,   0.160
mu = 0.0
mu=0.1
mu=0.2
mu=0.3
100
mu=0.4
mu=0.5
mu=0.6
mu=0.7
10
mu=0.8
mu=0.9
Pi Kapp=160 ( periodi c)
100

10
G (n t)
G ( n t)
1
mu=0.0
mu=0.1
mu=0.2
mu=0.3
mu=0.4
RHO K160( periodic)
r
mu=0.5
mu=0.6
mu=0.7
mu=0.8
mu=0.9
1
0.1
0.1
0.01
0.001
0.01
0
1
2
3
4
nt
5
6
7
8
0
1
2
3
4
5
nt
Clear evidence of r meson mass decrease
at finite chemical potential !
6
7
8
Color SU(2)
r at Finite Chemical Potential
Peculiar behavior of a vector meson at finite density
=0.160
a
=0.175
Mass of r becomes small !
Remind us of the CERES Experiment
a
Thermodynamical Quantities
• Nf = 2,
 EG 
4 4   0.7
Baryon number density
Gluon energy density
a

NB

L
Polyakov line

a
a
Polyakov Line Susceptibility
3
4 X8
 L

 L L
n n
• Anti periodic (spatial direction)
 L
periodic
(spatial direction)
0.160

0.0002
0.0001
 L

0
0
0.4 a
0.8
Polyakov Line Susceptibility
4
4 periodic
 L

 L L
n n
 L

a

粒子対凝縮 ?
Kogut-Toublan-SInclare
外場の入ったシミュレーション
Sinclare and Kogut,
 condensation with _I
diquark condensation in colorSU(2)
(see Nishida’s talk)
phase quenching
重みだと思う
2 flavor finite iso-spin model  phase quench model
Configulation の update は可能なはず
 の大きいところは揺らぎが小さい?
Bilic, Demeterfi and
Petersson, NPB337(‘92)

R-algorithm
Nakamura, Sasai, Takaishi, 基研研究会(2003)
Nakamura, Sasai, Takaishi, 基研研究会(2003)
位相の揺らぎ
Bielfelt-Swansea,PRD68(03)
Nakamura, Sasai, Takaishi, 基研研究会(2003)
高密度 Lattice QCD
•Lattice simulation for small  seems to work enough
•SU(3)の複雑な相構造まで届いてはいない
•カラーを持った凝縮を出せるか?
•高密度状態は計算可能か?
Muroya, Nakamura, Nonaka
and Takaishi : PTP 110(03)615,
hep-lat/0306031
RHIC
JPARC
Thomas Schafer,
hep-ph/0304281
Ferro-Mgn.?
Q-Hall st ?

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