格格 QCD 格格格格格格格格 格格格格格格格格 S. Muroya Tokuyama Women’s College in collabolation with A. Nakamura, C. Nonaka and T. Takais hi
Jan 03, 2016
格子 QCD による有限密度系シミュレーション
S. MuroyaTokuyama Women’s College
in collabolation withA. 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
Thomas Schafer,hep-ph/0304281
RHIC
JPARC
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• 統計力学
• 場の理論
}Tr{e )( NH
i 00
constant gauge field P.A.M. Dirac (‘56) Y. Nambu (‘68)
保存量 (保存電荷)
Lagrange 未定定数
3
1ˆ,'ˆ,'
)()1()()1(1)(i
ixxiiixxii xUxU
4̂,'444̂,'44 )'()1(e)()1(e
xx
a
xx
a xUxU
• Introducing the chemical potential on a lattice (Wilson fermion)
4
1ˆ,ゥˆ,ゥ )()1()()1(1
xxxx xUxU
iPP 44
: hopping parameter quark mass
,FS
Chemical Potential on a Lattice
)(e)( xiAxU )(e}Tr{e FG SSH DDUD
55 complex :det
,e}Tr{e ))(()( FG SSNH DDUD ,)()( FS
GSedet1
e1
ODU
Z
ODDUDZ
O FG SS
Phase (sign) problem
iedetdet Vi ee
55 complex:Wdet
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?
chiral limit では μc = 0 か ?
•プロットはシミュレーション•実線は π による μ c評価•点線はバリオンによる μ c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)
μ=0.0 μ=0.2
μ=0.4μ=0.3
Wilson Fermion の固有値分布 β= 5.7, κ=0.16 , 4x4x4x4 Lattice
K-S Fermion の固有値分布 ( m =0.1, beta = 5.7)
μ=0.2μ=0
μ=0.3 μ=0.4
Approach to high density state of the Lattice QCD
• Reweighting method– Fodor & Katz– Grasgow
• Taylor expansion
• Imaginary Chemical Potential
• Density of the state
• Positive Measure model
• Susceptibility against chemical potential
Nishimura’s talk
Irina’s talk
Susceptibility against chemical potential
クォーク数密度MILC Collabolation
second derivative for chemical potential
擬スカラー meson mass の応答
duV
duS
QCD-TAROCollaboration
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
MeV MeV 35725,5.3160 EET
12 FN2.0 ,025.0, sdu mm
8 ,6 ,4 ,43 ss NN
Standard gauge + Staggered fermion
Reweighting
ODUZ
O gS )(e)(det1
ODUZ
ggg SSS
)(det
)(dete)(dete
1
0
)()(0
)( 00
Fodor and KatzMulti-reweighting
method)( 0
)( 0
)(
)(
),,( m
Glasgow approach
• Allton et al. (Bielefeld-Swansea) hep-lat/0204010
Improved action + Improved staggered fermion
,2FN4163
0.2 ,1.0qmMeV
a=0.29
Taylor expansion at high T and low
40,
4,
4 T
p
T
p
T
p
TT
微分の4次まで
Imaginary Chemical Potential
deForcrand and Philipsen NPB642(02)290; hep-lat/0307020
Im
)()( 210
I
IIC acca
D’Elia and Lombardo Phys.Rev. D67 (2003) 014505
At small )()(log 64
42
20 OaaaZ )()(log64
42
20 IIII OaaaZ ImI
complex:det M real:det M
ReIm i
Standard gauge + Staggered fermion
,2FN 250.0qm46 ,48 33
3
I Z(3) symmetry
Fodor-Katz
Allton et al.
deForcrand-Philipsen D’Elia and Lombardo
Consistent !? YES
• Effective theory• Finite Isospin
• Two-color QCD Pseudo-Real
du )*,(det)(det
2)(det)(det)(det
U U * 2 2
2*
2
24̂,'44*
4̂,'44*
3
1ˆ,'
*ˆ,'
*',2
*
);',(
'1e1e
)'()1()()1();',(
xx
xUxU
xUxUxx
xx
a
xx
a
iixxiiixxiixx
)'det);',(det)};',({det ** ;x'(x,xxxx
real:det Monte Carlo Calculation Works Well !
Models free from Sing Problem
Color SU(2) at Finite Density
0.01
0.1
1
10
100
0 1 2 3 4 5 6 7 8
RHO K160(periodic)
mu=0.0mu=0.1mu=0.2mu=0.3mu=0.4mu=0.5mu=0.6mu=0.7mu=0.8mu=0.9
G(n
t)
nt
0.001
0.01
0.1
1
10
100
0 1 2 3 4 5 6 7 8
Pi Kapp=160 (periodic)
mu = 0.0mu=0.1mu=0.2mu=0.3mu=0.4mu=0.5mu=0.6mu=0.7mu=0.8mu=0.9
G(n
t)
nt
4 X83
Clear evidence of meson mass decrease at finite chemical potential !
Color SU(2) at Finite Chemical Potential
Color SU(2) at Finite Chemical Potential
Peculiar behavior of a vector meson at finite density
Mass of becomes small ! Remind us of the CERES Experiment
a a=0.160 =0.175
• Nf = 2, 4
Thermodynamical Quantities4
a a
a
L
GE BNGluon energy density
Polyakov line
Baryon number density
Polyakov Line Susceptibility
• Anti periodic (spatial direction) periodic (spatial direction)
0
0.0002
4 X83
0.0001
0 0.4 0.8a
nnLLL
L
L
Polyakov Line Susceptibility
4 periodic4
a
nnLLL
L
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 andPetersson, 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
Nakamura, Sasai, Takaishi, 基研研究会(2003)
Bielfelt-Swansea,PRD68(03)
Thomas Schafer,hep-ph/0304281
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?
RHIC
JPARC
Ferro-Mgn.? Q-Hall st ?
Muroya, Nakamura, Nonaka and Takaishi : PTP 110 ( 03 ) 615, hep-lat/0306031