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Scaling study of the chiral phase transitionin two-flavor QCD for the improved Wilson
quarks at finite density
H. Ohnofor WHOT-QCD Collaboration
The 3rd A01 group workshopCCS, University of Tsukuba, July 7, 2010
H. Ohnofor WHOT-QCD Collaboration
Plan of this talk
• Introduction– QCD phase diagram– Scaling behavior of QCD
• Scaling study with Wilson quarks– Scaling behavior at finite μ– Derivative of chiral order parameter with respect to μ/T
• Numerical results• Conclusion and future plan
2Scaling study of the chiral phase transition in two-flavor QCD for the improved Wilson quarks at finite density
H. Ohnofor WHOT-QCD Collaboration
QCD phase diagram
3Scaling study of the chiral phase transition in two-flavor QCD for the improved Wilson quarks at finite density
Hadron gas
Quark-gluon-plasma
critical end pointCrossover
χSB
• Early universe• Heavy ion collisions• Neutron star
Important to understand
H. Ohnofor WHOT-QCD Collaboration
Critical behavior of QCD at zero chemical potential
4Scaling study of the chiral phase transition in two-flavor QCD for the improved Wilson quarks at finite density
mud
00
Physical point
1storder
2ndorder
Crossover
1storder
ms
QuenchedNf=2
N f=3
O(4)
Tri-criticalpoint mtc
Z(2)
Z(2)
Interesting subject:• O(4) scaling at ms >> mtc
• Tri-critical point• 1st order phase transition at ms << mtc
H. Ohnofor WHOT-QCD Collaboration
Scaling behavior of chiral order parameter in 2-flavor QCD
5Scaling study of the chiral phase transition in two-flavor QCD for the improved Wilson quarks at finite density
)4(~)2()2( OSUSU Chiral symmetry of 2-flavor QCD
3-d O(4) Heisenberg model
[Wilczek, Intn.J.Mod.Phys. A7('92)3911; Rajagopal, Wilczek, NP B399('93)395]
Recent lattice studies
Wilson quarks• Local for any Nf
• Conserved flavor symmetry• Explicit chiral symmetry breaking
• Higher computational cost
KS quarks• Non-local except for Nf = 4• Explicit flavor symmetry breaking
• Partly conserved chiral symmetry• Lower computational cost
Consistent with O(4) scaling
O(2)
Simulation at physical point
Subtraction and renormalization
[CP-PACS, PRD63, 034502(2000)]O(N) scaling?
[BNL-Bielefeld-GSI, arXiv:0909.5122]
H. Ohnofor WHOT-QCD Collaboration
Recent study with Wilson quarks
6/20Scaling study of the chiral phase transition in two-flavor QCD for the improved Wilson quarks at finite density
[CP-PACS, PRD63, 034502(2000)]
• Iwasaki improved gauge• Clover improved Wilson quark• Nf = 2• 163×4 lattice
O(4) scaling function
H. Ohnofor WHOT-QCD Collaboration
Critical behavior of 2-flavor QCD at finite density
7Scaling study of the chiral phase transition in two-flavor QCD for the improved Wilson quarks at finite density
mud0
mud=0
O(4)
mud
Quenched
00
1storder
2ndorder
Crossover
1storder
mud
Interesting subject:• O(4) scaling at small μ• Curvature of critical or crossover line •Tri-critical point at finite μ
Nf = 2
H. Ohnofor WHOT-QCD Collaboration
Our study
8Scaling study of the chiral phase transition in two-flavor QCD for the improved Wilson quarks at finite density
We investigate scaling behavior
• of 2-flavor QCD• at finite chemical potential• with Wilson quarks
• using derivative of chiral order parameter• calculating curvature of 2nd order critical line in β-μ plane
H. Ohnofor WHOT-QCD Collaboration
Comparison with 3-d O(4) spin model
9Scaling study of the chiral phase transition in two-flavor QCD for the improved Wilson quarks at finite density
3-d O(4) spin model
magnetization: Mexternal field: hreduced temperature: t
2-flavor QCD
chiral order parameter:ud-quark mass: mq
critical value of β: βct
ψψM
amh q2
ctt
)( 11 htfhM Scaling function:
H. Ohnofor WHOT-QCD Collaboration
Chiral order parameter with Ward-Takahashi identity
10Scaling study of the chiral phase transition in two-flavor QCD for the improved Wilson quarks at finite density
[Bochichio et al., NPB262, 331 (1985)]
yxts
q yPxPNN
aZmψψ
,3
WI)()(
2
xyxyPxPamyPxA aaq
aa 2
,
0
02 4
PtP
PtAmamq
22KZ
aaaa TATP 55 ,
Ward-Takahashi identities in the continuum limit
Wilson quarks : explicit chiral symmetry breaking
Ward-Takahashi identities
: Tree level renormalization factor
A proper subtraction and renormalization are required.
H. Ohnofor WHOT-QCD Collaboration
Scaling behavior for finite density
11Scaling study of the chiral phase transition in two-flavor QCD for the improved Wilson quarks at finite density
The critical line runs on mq = 0 axis in (mq,μ) plane
t=0
μq/T
mq=0
β
2
2
T
c q
ctqct
c is the curvature of the critical ine in the (β, μq/T) plane.
To confirm this scaling property, we calculate the second derivative of
WI with respect to μq/T.
1
11htxdx
xdf
h
dtdM
,
00
2
2
q
q
dt
dMc
Td
Md
q
,WI
M
,2 amh q2
2
T
c
ct
qt
H. Ohnofor WHOT-QCD Collaboration
Derivative of chiral order parameter
12Scaling study of the chiral phase transition in two-flavor QCD for the improved Wilson quarks at finite density
M
NMMMM
Fdetlntr
2tr
f5
15
1
25
15
12
2
202332
WI2
03
WI 2 ,
2AFF
NN
maZ
TF
NN
maZ
tsqts
tNTa qq
,trtrdetln
,trdetln 11
2
21
2
21
M
MM
MM
MMM
MM
,tr 51
51
0 MMF
,detlndetln
2
f2
2
f2
M
NM
N
,2Tr4tr2tr
tr5
115
115
15
1115
15
12
21
25
15
12
M
MMM
MMMM
MM
MMMM
MM
MM
,tr2
tr5
15
1151
51
MM
MM
MM
,detln
trdetln
tr2
2f5
15
12
2
f51
51
M
NMMM
NMM
These operators can be calculated by random noise method
H. Ohnofor WHOT-QCD Collaboration
Simulation setup
• Action– Iwasaki improved gauge action– Clover improved Wilson fermion action– Nf =2
• Lattice size– 163×4
• # conf.– 500-600
• # noise vectors– 50 for each color and spin indices
13Scaling study of the chiral phase transition in two-flavor QCD for the improved Wilson quarks at finite density
β Κ T/Tpc
1.80 0.141139 0.93
1.85 0.140070 0.99
1.90 0.138817 1.08
1.95 0.137716 1.20
β Κ T/Tpc
1.80 0.145127 1.07
1.85 0.143502 1.18
1.90 0.141849 1.32
1.95 0.140472 1.48
mps/mv = 0.65 mps/mv = 0.80
H. Ohnofor WHOT-QCD Collaboration
Numerical result at zero μ (1)
14Scaling study of the chiral phase transition in two-flavor QCD for the improved Wilson quarks at finite density
Filled symbols: CP-PACS, PRD63, 034502(2000)Open symbols: New data in this study
mq : measured in T=0 simulations.
IW amq2as a function of
H. Ohnofor WHOT-QCD Collaboration
Numerical result at zero μ (2)
15Scaling study of the chiral phase transition in two-flavor QCD for the improved Wilson quarks at finite density
β,δ: critical exponents of O(4) spin model.
Filled symbols: CP-PACS, PRD63, 034502(2000)Open symbols: New data in this studyDashed line:O(4) scaling function [Toussaint,’97]
Comparison with O(4) scaling function
Scaling behavior is consistent with 3-d O(4) spin model.
H. Ohnofor WHOT-QCD Collaboration
Numerical result at finite density (1)
16Scaling study of the chiral phase transition in two-flavor QCD for the improved Wilson quarks at finite density
c =0.05
c =0.04
c =0.03
c =0.02
Second derivative ofIW
/1/
/1/1
0
2
2 )(
htxq dxxdf
chTd
Md
q
Solid line: O(4) scaling function with
Scaling behavior is roughly consistent with our expectation.
H. Ohnofor WHOT-QCD Collaboration
Numerical result at finite density (2)
17Scaling study of the chiral phase transition in two-flavor QCD for the improved Wilson quarks at finite density
350328.02
2
cTd
d
q
ct
/1/
/1/1
0
2
2 )(
htxq dxxdf
hTd
Mdc
q
dada
Td
d
Td
Td
T q
ct
q
c
c
2
2
2
21To calculate the curvature of Tc(μq),
we need a(dβ/da)in the chiral limit.
Curvature of βc(μq)
7df ,77.3df2 NN
H. Ohnofor WHOT-QCD Collaboration
Conclusion
• Scaling behavior of chiral order parameter is investigated in two-flavor QCD.
• At zero chemical potential, – it is consistent with 3-d O(4) spin model.
• At finite density,– chemical potential dependence for scaling variables is almost
consistent with our expectation– and the curvature of second order critical line in β-μ plane is
roughly calculated.
18Scaling study of the chiral phase transition in two-flavor QCD for the improved Wilson quarks at finite density
)35(0328.0c
H. Ohnofor WHOT-QCD Collaboration
Future plan
• More precise scaling study– increasing the statistics and the number of noise vectors
• Lighter mud
• 2+1-flavor simulation
19Scaling study of the chiral phase transition in two-flavor QCD for the improved Wilson quarks at finite density