scaling properties of the chiral phase transition in the low density region of two-flavor qcd with...
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Scaling properties of the chiral phase transition in the low density region of two-flavor QCD with improved Wilson fermions
WHOT-QCD Collaboration:S. Aoki1, S. Ejiri2, T. Hatsuda3, K. Kanaya4, Y. Maezawa5, Y. Nakagawa2, H. Ohno6, H. Saito7, T. Umeda8, and S. Yoshida3
1Kyoto Univ., 2Niigata Univ., 3RIKEN, 4Univ. of Tsukuba, 5BNL, 6Bielefeld Univ., 7DESY, 8Hiroshima Univ., Abstract:We study scaling behavior of a chiral order parameter in the low density region, performing a simulation of two-flavor QCD with improved Wilson quarks. The scaling behavior of the chiral order parameter defined by a Ward-Takahashi identity agrees with the scaling function of the three-dimensional O(4) spin model at zero chemical potential. We extend the scaling study to finite density QCD. Applying the reweighting method and calculating derivatives of the chiral order parameter with respect to the chemical potential, the scaling properties of the chiral phase transition are discussed in the low density region. We moreover calculate the curvature of the phase boundary of the chiral phase transition in the temperature and chemical potential plane assuming the O(4) scaling relation.
1. Introduction and Motivation Quark mass dependence of QCD phase transition Expected nature of phase transitions (2-flavor QCD and O(4) spin model: same universality class?)
mud0
0
Physical point
1storder
2ndorder
Crossover
1storder
ms
QuenchedNf=2
Nf=3
Nf=2=0
Interesting properties in 2+1-flavor QCD• O(4) scaling for ms>>mtc
• Tri-critical scaling at mtc
• First order transition at ms < mtc
• O(4) scaring at small This study• Tri-critical point at finite
mud
ms
0
=0
Nf=2
mud0
mud=0
Nf=2
O(4)Tri-criticalpoint mtc
O(4)
Tri-criticalpoint
Z(2)
2. Wilson quark simulations Iwasaki improved gauge + Clover improved Wilson quark action
lattice 4163
,
211
,
110 )()(
xxg xWcxWcS
,81 ,331.0 101 ccc
: nxm wilson loop ,6 2g
4/31)8412.01( SWc
FKcUeUeK
UUKM
SWyxyx
a
yx
a
iyixiiyixiiyxyx
,,4̂44,4̂44
3
1,ˆ,ˆ,,
11
11
Clover term
q: quark chemical potential
,e det )( f
,
gSN
x
MxdUZ
mnW
Chiral limit at T=0
T0 pseudo-critical line(crossover)
determined by Polyakov loop
0m
4tN mPS/mV=0.65 mPS/mV=0.80 K T/Tpc K T/Tpc
1.50 0.150290 0.82(3) 1.50 0.143480 0.76(4) 1.60 0.150030 0.86(3) 1.60 0.143749 0.80(4) 1.70 0.148086 0.94(3) 1.70 0.142871 0.84(4) 1.75 0.146763 1.00(4) 1.80 0.141139 0.93(5) 1.80 0.145127 1.07(4) 1.85 0.140070 0.99(5) 1.85 0.143502 1.18(4) 1.90 0.138817 1.08(5) 1.90 0.141849 1.32(5) 1.95 0.137716 1.20(6) 1.95 0.140472 1.48(5) 2.00 0.136931 1.35(7) 2.00 0.139411 1.67(6) 2.10 0.135860 1.69(8) 2.10 0.137833 2.09(7)
#conf=500-2600
#noise=100-150 for each color and spin indices
2-flavor QCD
3. Scaling behavior of chiral order parameter at finite 3-1. O(4) scaling test for 2-flavor QCD 3-D O(4) spin model: magnetization (M), external field (h), reduced temperature (t) 2-flavor QCD: chiral order parameter , ud-quark mass (mq), critical (ct), (c)
)( 11 htfhM
,ψψM ,2 amh q
Scaling function:
ψψ2
2
T
cct
qt
t=0
q/T
mq=0
2
2
T
c q
ctqct
1
11htxdx
xdf
h
dtdM ,
00
2
2
q
q
dt
dMc
Td
Md
q
Note: c is the curvature of the critical line in the (, qT) plane.
3-2. Chiral order parameter defined by Ward-Takahashi identities Ward-Takahashi identities in the continuum limit
Because the chiral symmetry is explicitly broken for Wilson quarks, we define
and by the Ward-Takahashi identities [Bochichio et al., NPB262, 331 (1985)].
yxts
q yPxPNN
Kamψψ
,3
2WI
)()(22
xyxyPxPamyPxA q 2
0
02 4
PtP
PtAmamq
WIψψ
amq
55 , AP
4. Summary 1) We discussed the scaling property of the chiral phase transition in the low density region. 2) We calculated the chiral order parameter and its second derivative performing a simulation of 2-flavor QCD with Wilson quarks and compared the results with an expected scaling function. 3) The scaling behavior is roughly consistent with our expectation. 4) The curvature of the critical line in the () plane was estimated.
3-6. Curvature of the critical line in the ciral limit cTd
d
q
ct
2
2
These operators can be calculated by the random noise method.
,trtrdetln
,trdetln 11
2
21
2
21
M
MM
MM
MMM
MM
,2Tr4tr2tr
tr5
115
115
15
1115
15
12
21
25
15
12
M
MMM
MMMM
MM
MMMM
MM
MM
,tr2
tr5
15
112
51
51
MM
MM
MM
51
51
3
2
,3
2WI
tr2222
MMNN
KamyPxP
NN
Kam
ts
q
yxts
q
We estimate the curvature c by three method.1)Global fit in the scaling plot (Sec. 3-4). c = 0.0290 (magenta line)
2) From the ratio of the 2nd derivative and df/dx. (1-1) Method1: c = 0.0273 (42) (Blue line) (1-2) Method2: c = 0.0257 (43) (Black line)
dada
Td
d
Td
Td
T q
ct
q
c
c
2
2
2
21To calculate the curvature of Tc(q),
we need a(d/da) in the chiral limit.
3-3. Reweighting method for the chiral order parameter at finite
0,
0detlndetln
0,
0detlndetln5
15
1
51
51
,51
51
0
f
0
f
f
e
e tredet tr
1tr )(
MMN
MMN
SNMM
MMMDUZ
MM g
32
22
d
detlnd
2d
detlnd0detlndetln
OMM
MM
32
51
5122
51
51
51
51
51
51 tr
2
tr0trtr
O
d
MMd
d
MMdMMMM
We compare the scaling functions of 2-flavor QCD and O(4) spin model.
The second derivative:
3-4. Scaling plot at finite and O(4) scaling function
Consistent with O(4) scaling function.
029.0 ,510.1 cct
)( 11 htfhM 2
2
T
cct
qt
O(4) scaling function: J. Engels et al., Nucl. Phys. B 678 (2003) 533
mud
Quenched
00
1storder2ndorder
Crossover1storder
We neglect O().
3-5-2. method 2
2
2f5
15
12
2
f51
51
f5
15
1
25
15
12
2
detlntr
detlntr
detlntr2
tr
M
NMMM
NMMM
NMMMM
F
20233
2
2
WI2
03
2WI 22
,22
AFFNN
Kma
TF
NN
Kma
tsqts
tNTa qq
,tr 51
51
0 MMF ,detlndetln
2
f2
2
f2
M
NM
N
Second derivative of for 2-flavor QCD IW
O(4) scaling functionc=0.06
c=0.05
c=0.04
c=0.03
c=0.02
111
22
htx
q
dx
xdfc
h
Tdd
These results show the expected scaling behavior although the error is still large.
3-5. Derivatives of the chiral order parameter at =0 3-5-1. method 1 We fit the data of at finite by , where x and y are the fit parameters. (The first derivative is zero due to the symmetry: –.)
2WITyx
IW
0at 2
1 , 2
WI2
WI
Tyx
q
method 1 method 2
We calculate the expectation value of the following operators and obtain the derivatives.
The expectation value at finite is computed by the reweighting method with O(3) error.
We fit the data of by the O(4) scaling function adjusting the parameter ct and c in t.IW
Best values:
Symbols: results at =0 Lines on the right of the symbols: result at 0We omitted data having large error.
In method 1, we fit the data in /T<1.0
cf) Result by an improved staggered: Keczmarek et al., Phys. Rev. D 83 (2011) 014504
)4)(2(059.02
12
2
Td
Td
T q
c
c
mPS/mV=0.65 mPS/mV=0.80