Scaling behavior of chiral phase transition in two-flavor QCD with improved Wilson quarks at finite density

0.11 MB·English

by  S. Ejiri

Scaling behavior of chiral phase transition in
two-flavor QCD with improved Wilson quarks at
finite density
1
1
0
2

n
a ShinjiEjiri∗, Y.Nakagawa
J
GraduateSchoolofScienceandTechnology,NiigataUniversity,Niigata950-2181,Japan
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E-mail: [email protected]
2

] S. Aoki,K.Kanaya,H.Ohno,H.Saito
t
a GraduateSchoolofPureandAppliedSciences,UniversityofTsukuba,Tsukuba,Ibaraki
l
- 305-8571,Japan
p
e T.Hatsuda
h
[
DepartmentofPhysics,TheUniversityofTokyo,Tokyo113-0033,Japan

1
v Y. Maezawa
2
8 MathematicalPhysicsLaboratory,RIKENNishinaCenter,Saitama351-0198,Japan
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5 T.Umeda
.
1
GraduateSchoolofEducation,HiroshimaUniversity,Hiroshima739-8524,Japan
0
1
(WHOT-QCDcollaboration)
1
:
v
Westudyscalingbehaviorofachiralorderparameterperformingasimulationoftwo-flavorQCD
i
X
with improved Wilson quarks. It has been shown that the scaling behavior of the chiral order
r
a parameter defined by a Ward-Takahashi identity agrees with the scaling function of the three-
dimensionalO(4) spin model at zero chemical potential. We extend the scaling study to finite
densityQCD. Calculatingderivativesofthe chiralorderparameterwith respecttothe chemical
potential in two-flavor QCD, the scaling property of chiral phase transition is discussed in the
lowdensityregion.Wemoreovercalculatethecurvatureofthephaseboundaryofthechirlphase
transitioninthetemperatureandchemicalpotentialplaneassumingtheO(4)scalingrelation.
TheXXVIIIInternationalSymposiumonLatticeFiledTheory
June14-19,2010
Villasimius,SardiniaItaly
∗Speaker.
(cid:13)c Copyrightownedbytheauthor(s)underthetermsoftheCreativeCommonsAttribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/
ScalingbehaviorofchiralphasetransitionwithimprovedWilsonquarks ShinjiEjiri
1. Introduction
ItisimportanttostudythenatureofQCDphasetransitionathightemperatureandlowdensity
for understanding theevolution ofthe early universe andanalyzing experimental data obtained by
the heavy ion corrosions. In order to extract unambiguous signals for the QCD phase transition
from the heavy ion collisions, quantitative calculations directly from the first principles of QCD
areindispensable. Atpresent, thelatticeQCDsimulation istheonlysystematic methodtodoso.
Many important properties of finite temperature quark matter have been uncovered by lattice
simulations. Recent developments of computational techniques enabled us to extend the study to
smallchemicalpotentials. MostlatticeQCDstudies atfinitetemperature (T)andchemicalpoten-
tial (m ) have been performed using staggered-type quark actions with the fourth-root trick of the
q
quark determinant. Because the theoretical base for the fourth-root trick is not confirmed, it isin-
dispensable tocarryoutsimulationsadoptingdifferentlatticequarkactionstocontrolandestimate
systematic errors due to lattice discretization. Another unsatisfactory point of the staggered-type
quarks is that, it was very difficult until recently [1] to confirm the scaling properties around the
critical point, which is universal to the three-dimensional O(4) spin model for 2-flavor QCD, as
expected from the effective sigma model analysis 1. This maysuggest large lattice artifacts in the
results withstaggered-type quarks.
Several years ago, the CP-PACSCollaboration has studied finite-temperature QCDusing the
clover-improved Wilson quark action coupled with the RG-improved Iwasaki glue [2, 3]. With
2-flavors of dynamical quarks, the phase structure, the transition temperature and the equation of
statehavebeeninvestigated. Incontrasttothecaseofstaggered-typequarks,bothwiththestandard
Wilsonquarkaction[4]andwiththeclover-improvedWilsonquarkaction[2],thesubtractedchiral
condensate showsthescalingbehaviorwiththecriticalexponentsandscalingfunctionoftheO(4)
spinmodelinaratherwiderangeoftheparameter spacenearthechiralphasetransition.
Wewouldliketohighlight scalingproperties oftheQCDphasetransitionbynumericalsimu-
lations withdynamical Wilson quarks. AsshowninFig.1(left)for m =0,thephasetransition is
q
expected tobefirstorder whentheupand downquark masses (m )andstrange quark mass(m )
ud s
aresufficientlylargeorsmall,andbecomescrossoverintheintermediateregionbetweenthem. We
expect a second order transition in the chiral limit of 2-flavor QCD, i.e. m ≡m =0,m =¥ ,
ud q s
andthenatureofthetransition changes asm decreases. Itbecomesfirstorderbelowthetrictitical
s
point (m ). When m 6=0, the phase diagram in the (T,m ) plane is expected as Fig. 1 (right) for
E q q
2-flavorQCDwithm =0andm 6=0. Atlowdensity,thephasetransitionisexpectedtobesecond
q q
orderform =0andcrossoverform 6=0. Toconfirmthisphasestructure, thescalingstudyinthe
q q
crossover region isimportant. Recently, thescaling studywiththephysical strange masshasbeen
doneusingastaggered-type quarkactioninRef.[1],andthescalingpropertyincludingachemical
potential has been also discussed [5]. Therefore, it is worth revisiting the scaling study using the
Wilson-type quarkaction. Inparticular, wewanttoclarifythescalingproperty atfinite m .
q
In this report, we study 2-flavor QCD, as a first step. Previous results for O(4) scaling at
m =0 are summarized in Sec. 2. We then extend the discussion to finite density in Sec. 3. The
q
scaling properties aretestedbynumericalsimulations inSec.4. Theconclusion isgiveninSec.5.
1Preciselyspeaking,thechiraltransitionof2-flavorQCDwithstaggered-typequarksisuniversaltotheO(2)spin
model,whosescalingpropertiesareclosetothoseoftheO(4)model.
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ScalingbehaviorofchiralphasetransitionwithimprovedWilsonquarks ShinjiEjiri
2 flavor QCD quenched QCD
1storder
quark-gluon plasma phase
2ndorder
crossover
crossover nd
ms T 2ndorder 2critoicrdael rpoint
m tricritical
triEcritical point point 1storder
physical point?
3 flavor QCD mq=0 mq>0
hadron phase
2ndorder
0 1storder m m
0 ud q
Figure 1: Left: Mass-dependenceof the natureof (2+1)-flavorQCD. Right: Expected phase structure of
2-flavorQCDatfinitetemperatureanddensityform =0andm 6=0.
q q
2. Scaling behaviorofthechiral order parameter for2-flavorQCD
Thechiralphasetransition in2-flavor QCDatfinitetemperature isexpected tohavethesame
criticalpropertiesasthe3-dimensionalO(4)spinmodel. Hence,thescalingtestsofthechiralorder
parameter hyy¯ i provide us with a useful way to study universality properties of the chiral phase
transition for2-flavorQCD.
The order parameter of the O(4) spin model is given by the magnetization M. In the vicinity
ofthesecondordercritical point,M satisfiesthefollowingscalingrelation:
M/h1/d = f(t/h1/y), (2.1)
where h is the external magnetic field, t is the reduced temperature which is defined by t =
(T −T | )/T | . Thecritical exponents havethe values 1/y≡1/(bd )=0.537(7) and 1/d =
c h=0 c h=0
0.2061(9), and f(x) is the scaling function. In 2-flavor QCD, we identify M ∼hyy¯ i, h∼2m a
q
and t ∼b −b . Here, b is the critical point in the chiral limit, and a is the lattice spacing. We
ct ct
comparethescalingfunctions of2-flavorQCDandO(4)spinmodel.
A careful treatment must be required because the chiral symmetry is explicitly broken for
Wilson quarks at finite a. In Ref. [2, 4], it have been shown that the O(4) scaling of Eq. (2.1)
is well satisfied when one defines the quark mass m a and the chiral order parameter hyy¯ i by
q
Ward-Takahashi identities [6]. Thequarkmassisdefinedbythefollowingcorrelation function for
temporaldirection:
2m a=−m hA¯ (t)P¯(0)i hP¯(t)P¯(0)i, (2.2)
q PS 4
(cid:14)
wherePandAm arethepseudo-scalar andvectormesonoperators, respectively, mPS isthepseudo-
scalar meson mass, and the bar means the spatial average. The chiral order parameter is given by
thefollowingequation:
hyy¯ i= 2mqa(cid:229) hP(x)P(x′)i= 2mqa(2K)2 tr D−1g D−1g . (2.3)
N3N N3N 5 5
s t x,x′ s t (cid:10) (cid:0) (cid:1)(cid:11)
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ScalingbehaviorofchiralphasetransitionwithimprovedWilsonquarks ShinjiEjiri
1.5
1.5 <yy > <yy >
1/d b =1.80
h b =1.85
b =1.90
1.0 b =1.95
1.0
b =1.80
b =1.85
b =1.90
0.5 b =1.95 0.5
b =2.00
b =2.10
b =2.20
0.0 0.0
0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.0 0.5 1.0 1.5 2.0
2m a 1/y
q t / h
Figure2:Left:Theciralorderparameterin2-flavorQCDasafunctionoftheWard-identityquarkmassfor
eachb . Right:O(4)scalingplotofthechiralorderparameter.
Here,Disthequarkmatrix. Thequarkmassandthechiralcondensate satisfytheWard-Takahashi
identity inthecontinuum limit:
h¶ m Am (x)P(x′)i−2mqahP(x)P(x′)i=d (x−x′)hyy¯ i. (2.4)
We plot the data of the chiral condensate as a function of the quark mass in the left panel
of Fig. 2. The closed symbols are obtained in the study of Ref. [2] by CP-PACS Collaboration,
and the open symbols are new data in this study. (See Sec. 4.) We then reconstruct the data into
M/h1/d = f(t/h1/y)forb =1.80−1.95 and2m a=0.0−0.9, which areshowninFig.2(right).
q
The critical exponents of the O(4) spin model are used. The dashed line is the scaling function
obtained bytheO(4)spinmodelinRef.[7]. Weadjustthreefitparameters inthisanalysis. Oneis
the critical value of b , which is b =1.462(66) in Ref. [2], and the others are used for adjusting
ct
the scales of the horizontal and vertical-axes to the scaling function of the O(4) spin model. This
scaling plot indicates that the scaling function of 2-flavor QCDisconsistent with that ofthe O(4)
spinmodel.
3. Scaling property atfinite density
Next,wediscusstheuniversalitypropertyof2-flavorQCDatfinitedensity. Itisexpectedthat,
when m is increased from 0 in the chiral limit, the second order phase transition changes to first
q
order at the tricritical point, as shown in the right panel of Fig. 1. The O(4) scaling behavior is
expected around the second order transition line evenat finitedensities because theaction has the
chiral symmetryinthechiral limit. Atsmall m ,wemayidentify thescaling variables, t andh,by
q
thefollowingwayforfinitedensityQCD:
c m 2
M=hyy¯ i, t =b −b + q , h=2m a. (3.1)
ct q
2 T
(cid:16) (cid:17)
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ScalingbehaviorofchiralphasetransitionwithimprovedWilsonquarks ShinjiEjiri
Table1: Simulationparametersform /m =0.65(left)andm /m =0.80(right)[8].
PS V PS V
b K T/T Traj. b K T/T Traj.
pc pc
1.80 0.145127 1.07(4) 5000 1.80 0.141139 0.93(5) 6000
1.85 0.143502 1.18(4) 5000 1.85 0.140070 0.99(5) 6000
1.90 0.141849 1.32(5) 5000 1.90 0.138817 1.08(5) 6000
1.95 0.140472 1.48(5) 5000 1.95 0.137716 1.20(6) 6000
Here, c is the curvature of the critical line in the (b ,m /T) plane. On the other hand, h does not
q
have a m -dependent term at low density. Because the critical line is expected to run along the
q
m =0axisinthelowdensityregionofthe(m ,m /T)plane, hiszeroatm =0.
q q q q
Itisworthdiscussing thephasestructure investigating thescalingbehavior ofthechiralorder
parameter. One of the easiest ways for confirming this scaling property is to calculate the second
derivative ofthechiralorderparameter. Weexpectthefollowingscalingproperties:
d2M dM dM/dt df(x)
=c , = . (3.2)
d(m /T)2(cid:12) dt (cid:12) h1/d −1/y(cid:12) dx (cid:12)
q (cid:12)m q=0 (cid:12)m q=0 (cid:12)m q=0 (cid:12)x=t/h1/y
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
Westudythesecondderivativeperformingnumericalsimulationsof2-flavorQCD(N =2)at
f
m =0. Thederivativeofthechiralorderparameteriscomputedbythefollowingway. Wedefine
q
¶ ntr D−1g D−1g ¶ nlndetD
C =(2K)2 5 5 , Q =N , (3.3)
n (cid:0)¶ (m a)n (cid:1) n f ¶ (m a)n
q q
A =hQ i, A =hQ i+ Q2 , F =hC i,
1 1 2 2 1 0 0
F =hC i+hC Q i, F (cid:10)=hC(cid:11)i+2hC Q i+hC Q i+ C Q2 . (3.4)
1 1 0 1 2 2 1 1 0 2 0 1
(cid:10) (cid:11)
Then,thederivativesofthechiralcondensate aregivenby
2m a
hyy¯ i = q F , (3.5)
(cid:12) N3N 0
(cid:12)m q=0 s t
¶ hyy¯ i (cid:12)(cid:12) = 2mqa (F −F A )=0, (3.6)
¶ (m /T)(cid:12) N3N2 1 0 1
q (cid:12)m q=0 s t
(cid:12)
¶ 2hyy¯ i(cid:12) 2m a 2m a
= q F −2F A −F A +2F A2 = q (F −F A ), (3.7)
¶ (m /T)2(cid:12) N3N3 2 1 1 0 2 0 1 N3N3 2 0 2
q (cid:12)m =0 s t (cid:0) (cid:1) s t
(cid:12)
whereweusedth(cid:12)epropertiesthatA andF arezeroforoddn’satm =0. Theoperators,C ,Q ,
n n q n n
canbecalculated byarandomnoisemethod.
4. Numerical simulationsforthe calculationofthe chiral condensate
We calculate the second derivative of the chiral order parameter for 2-flavor QCD at m =0
q
usingtheconfigurationsobtainedinthesimulationsofRefs.[8,9]. TheRG-improvedgaugeaction
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ScalingbehaviorofchiralphasetransitionwithimprovedWilsonquarks ShinjiEjiri
0.1
0.1 - d2M/d(m /T)2 b =1.80 d2M/d(m /T)2 b =1.80
h(1/d -1/y) b =1.85 h(1/d -1/y)df/dx b =1.85
b =1.90 0.08 b =1.90
0.08 c=0.05 b =1.95 b =1.95
c=0.04 0.06
0.06
c=0.03
0.04
0.04
c=0.02
0.02 0.02
0 0
0.5 1 1.5 0.4 0.6 0.8 1 1.2 1.4 1.6
1/y 1/y
t/h t/h
Figure3:Left:O(4)scalingplotofthesecondderivativeofthechiralorderparameter.Right:Thecurvature
ofthecriticallineinthe(b ,m /T)planedeterminedbythechiralorderparameter.
q
and the N =2 clover-improved Wilson quark action are adopted. The simulation parameters are
f
summarized inTable1. Thedetailsofthesimulation aregiveninRef.[8,9].
WeusedtherandomnoisemethodofRef.[9]forthecalculationoftheinverseofthematrices
in Eq. (3.3). As weemphasized in Ref. [9], it is important to apply the noise method only for the
space index and to solve the inverse exactly for the spin and color indices without applying the
noise method to obtain reliable results. We choose the number of noise vectors 50 for each color
and spin indices. The Ward-identity quark mass obtained by the zero temperature simulation in
Ref.[2]isusedforthescaling plot. IthasbeenshowninRef.[2]thattherange b >1.95isoutof
the scaling region, i.e. thedataofhyy¯ idonotagree withtheO(4)scaling function inthat range.
We thus plot the results of the second derivative of the chiral condensate at b ≤1.95 in the left
panel of Fig. 3. The colored line is the derivative of the scaling function c(df(x)/dx), which is
obtained byperforminganumericaldifferentiation of f(x)intherightpanelofFig.2numerically.
The additional parameter c is chosen as c=0.02,0.03,0.04 and 0.05 from bottom to top. These
results areroughlyconsistent withtheexpectedscaling behavior. Becausetheerrorisstilllarge, it
istooearlyfordrawingadefiniteconclusion. Inparticular, theerrorbecomeslargeasb decreases.
But,theresultsuggests thatthechiralorderparametersatisfiesthescaling relationinEq.(3.2).
We then calculate the curvature of the critical line assuming the scaling relation. The ratio
of (d2M/d(m /T)2)h−1/d +1/y and df(x)/dx| is plotted in Fig. 3 (right). This gives c ≡
q x=t/h1/y
d2b /d(m /T)2, i.e. the curvature of the critical line in the chiral limit. Taking the average, we
ct q
obtaind2b /d(m /T)2=−0.0328(35). Moreover,thecurvatureofthecriticaltemperatureT (m )
ct q c q
at m =0canbecalculated by
q
1 d2T d2b db
c ct
=− a . (4.1)
T d(m /T)2 d(m /T)2(cid:30) da
c q q
In this calculation, we need a(db /da) at b in the chiral limit. The value of a(db /da) for the
ct
lattice action which we used have been calculated in Ref. [3] for the pseudo-scalar-vector mass
ratiom /m ≥0.65. Ifwetrytoestimateadopting thevaluenearthepseudo-critical temperature
PS V
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ScalingbehaviorofchiralphasetransitionwithimprovedWilsonquarks ShinjiEjiri
at m /m = 0.65, which is a(db /da) ≈ −0.5, the value of (1/T )(d2T /d(m /T)2) ≈ −0.07.
PS V c c q
This value is smaller than that we expect from the data of the chemical freeze out, but this result
is similar to the result in Ref. [5]. Further studies are, of course, necessary. However, it is found
that this scaling analysis provides asystematic way toinvestigate the phase boundary ofthe finite
density QCDinthelowdensityregion.
5. Summary
Westudiedthescalingpropertyofthechiralphasetransitioninthelowdensityregionoffinite
temperature and density QCD. Assuming the phase structure shown in Fig. 1 (right), the scaling
function in the low density region is discussed. To confirm the scaling property, we calculated
the second derivative of the chiral order parameter performing a simulation of 2-flavor QCD with
improved Wilson quarks andcompared theresults withanexpected scaling function. Because the
chiralsymmetryisexplicitlybrokenfortheWilsonquarkatfinitelatticespacing,weusedthechiral
orderparameterdefinedbyaWard-Takahashiidentityforthescalinganalysis. Thescalingbehavior
of the chiral order parameter for 2-flavor QCD is roughly consistent with the scaling behavior of
thecorresponding spinmodelevenatfinitedensity.
Moreover, we calculated the curvature of the critical line in the chiral limit assuming the
scaling relation. It is important to reduce the error by increasing the statistics and the number
of noise vectors for more precise study. However, we found that this scaling analysis provides a
systematic waytoinvestigate thephaseboundary offinitedensityQCDinthelowdensityregion.
This work is in part supported by Grants-in-Aid of the Japanese Ministry of Education, Cul-
ture, Sports, Science and Technology, (Nos. 21340049, 22740168, 22840020, 20340047) and by
theGrant-in-Aid forScientificResearchonInnovativeAreas(Nos. 20105001, 20105003).
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