QCD thermodynamics with 2+1 flavors at nonzero chemical potential

0.33 MB·English

by  C. Bernard

QCD thermodynamics with 2+1 flavors at nonzero chemical potential
C. Bernard
Department of Physics, Washington University, St. Louis, MO 63130, USA
C. DeTar and L. Levkova
Physics Department, University of Utah, Salt Lake City, UT 84112, USA
8
0 Steven Gottlieb
0
2 Department of Physics, Indiana University, Bloomington, IN 47405, USA

n
a
J U.M. Heller

9
American Physical Society, One Research Road,

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a Box 9000, Ridge, NY 11961-9000, USA
l
-
p
e J.E. Hetrick
h
[
Physics Department, University of the Pacific, Stockton, CA 95211, USA

4
v
0 R. Sugar
3
3 Department of Physics, University of California, Santa Barbara, CA 93106, USA
1
.
0
1 D. Toussaint
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0 Department of Physics, University of Arizona, Tucson, AZ 85721, USA
:
v
i (Dated: February2,2008)
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r
Abstract
a
WepresentresultsfortheQCDequationofstate,quarkdensitiesandsusceptibilities atnonzerochemical
potential, using 2+1 flavor asqtad ensembles with N =4. The ensembles lie on a trajectory of constant
t
physics for which m ≈ 0.1m . The calculation is performed using the Taylor expansion method with
ud s
termsuptosixthorderinµ/T.
PACSnumbers:12.38.Gc,12.38.Mh,25.75.Nq
1
I. INTRODUCTION
The equation of state (EOS) of QCD is of special interest to the interpretation of data from
heavy-ion collision experiments and to the development of nuclear theory and cosmology. The
EOS at zero chemical potential (µ = 0) has been extensively studied on the lattice. However, to
approximate most closely the conditions of heavy ion collision experiments (for example RHIC
has µ∼15 MeV [1]) or of the interior of dense stars, the inclusion of nonzero chemical potential
is necessary. Unfortunately, as is well known, inclusion of a nonzero chemical potential makes
thefermion determinantin numerical simulationscomplexand straightforward MonteCarlo sim-
ulation not applicable. Several methods have been developed to overcome or circumvent this
problem. Theyincludethereweightingtechniques[2, 3], simulationswithanimaginarychemical
potential combined with analytical continuation [4, 5] or canonical ensemble treatment [6], and
lastly,theTaylorexpansionmethod[7, 8], whichisemployedhere. In thismethodoneTaylorex-
pandsthequantitiesneededforthecomputationoftheEOSaroundthepointµ=0wherestandard
Monte Carlo simulations are possible. The expansion parameter is the ratio µ/T, where T is the
temperature. To ensure fast convergence of the Taylor series, the expansion parameter should be
sufficientlysmall. Numericalcalculationsshowsatisfactoryconvergenceforµ/T <∼1(seereviews
[9, 10]).
In our simulations we use 2+1 flavors of improved staggered fermions. In such simulations
where the number of flavors is not equal to a multiple of four, the so-called “fourth root trick”
is employed to reduce the number of “tastes”. While this trick is still somewhat controversial,
there is a growing body of numerical [11] and analytic [12] evidence that it leads to the correct
continuumlimit. For simulationsat nonzero chemical potentialthe problemsof rooting are much
moresevere[13]. However,theTaylorexpansionmethodisnotdirectlyaffectedbythisadditional
problem with rooting since the coefficients in the Taylor series are calculated in the theory with
zero chemical potential. The Taylor expansion method is generally considered reliable in regions
wherethestudiedphysicsquantitiesare analytic.
The Taylor expansion method has been used to study the phase structure and the EOS of two
flavor QCD [7, 14, 15, 16, 17]. Our work improveson the previous studies by the addition of the
strange quark to the sea. Our calculations are performed on 2+1 flavor ensembles generated with
the R algorithm [18] and using the asqtad quark action [19] and a one-loop Symanzik improved
gauge action [20]. These improved actions have small discretization errors of O(a a2,a4) and
s
2
O(a 2a2,a4), respectively. Thisisvery importantsincewestudytheN =4case, wherethelattice
s t
spacing (a =1/(TN )) is quite large, especially at low temperatures. Our ensembles lie along a
t
trajectoryofconstantphysicsforwhichtheratiooftheheavyquarkmassandthelightquarkmass
is m /m ≈ 0.1, and the heavy quark mass itself is tuned approximately to the physical value
ud s
of the strange quark mass. The determination of the Taylor expansion coefficients, other than
the zeroth order ones computed already previously, is necessary only on the finite temperature
ensembles(forourstudyN =4). Nozero-temperaturesubtractionsareneededforthem. Wehave
t
determined the contributions to the energy density, pressure and interaction measure due to the
presence of a nonzero chemical potential. We also present results for the quark susceptibilities
and densities. In addition, we have calculated the isentropic EOS, which is highly relevant for
the heavy-ion collision experiments, where, after thermalization, the created matter is supposed
to expand without further increase in entropy or change in the baryon number. All the results are
obtained with the strange quark density fixed to n =0 regardless of temperature, appropriate for
s
theexperimentalconditions. Thisrequiresthetuningofthestrangequarkchemicalpotentialalong
thetrajectoryofconstantphysics.
II. THETAYLOREXPANSIONMETHOD
In this section we give a brief description of the Taylor expansion method for the thermody-
namicquantitieswestudyandas appliedto theasqtadfermionformulation.
A. Calculatingthepressure
Theasqtad quark matrixforagivenflavorwithnonzero chemical potentialis:
1
M = Mspatial+ h (x) U(F)(x)eµl,hd −U(F)†(x−0ˆ)e−µl,hd (1)
l,h l,h 2 0 0 x+0ˆ,y 0 x,y+0ˆ
+U(L)(x)e3µl,hd h −U(L)†(x−30ˆ)e−3µl,hd ,
0 x+30ˆ,y 0 x,y+30ˆ
i
whereµ =µ andµ =µ arethequarkchemicalpotentialsinlatticeunitsforthelight(uandd)
l ud h s
quarks andtheheavy (stranges)quark, respectively. In theabove
Mspatial = am d + (cid:229) 3 1h (x) U(F)(x)d −U(F)†(x−kˆ)d (2)
l,h l,h x,y 2 k k x+kˆ,y k x,y+kˆ
k=1
h
+U(L)(x)d −U(L)†(x−3kˆ)d ,
k x+3kˆ,y k x,y+3kˆ
i
3
with m the light and strange quark masses. The superscripts F and L on the links U denote
l,h µ
the type of links, “fat” and “long”; appropriate weights and factors of the tadpole strength u are
0
(F) (L)
includedinU andU . Thepartitionfunctionbasedon theasqtadquark matrixis
µ µ
Z = DUen4l lndetMlen4hlndetMhe−Sg, (3)
Z
wheren =2isthenumberoflightquarksandn =1isthenumberofheavyquarks. Thepressure
l h
p can beobtainedfrom theidentity
p lnZ
= , (4)
T4 T3V
where T is the temperature andV the spatial volume. It can be Taylor expanded in the following
manner
¥ n m
p (cid:229) µ¯l µ¯h
= c (T) , (5)
T4 nm T T
n,m=0 (cid:18) (cid:19) (cid:18) (cid:19)
where µ¯ is the nonzero chemical potential in physical units. Due to the CP symmetry of the
l,h
partition function, only the terms with n+m even are nonzero. The expansion coefficients are
defined by
1 1 N3 ¶ n+mlnZ
c (T)= t , (6)
nm n!m!Ns3¶ (µlNt)n¶ (µhNt)m µl,h=0
(cid:12)
with µ = aµ¯ and N and N the spatial and temporal exte(cid:12)nts of the lattice. All coefficients
l,h l,h s t (cid:12)
need to be calculated on the finite-temperature ensembles only, except for c (T). The latter is
00
the pressure divided by T4 at µ =0, which needs a zero-temperature subtraction. It should be
l,h
calculated by other means, such as the integral method, which we have already done in [21]. The
c (T) coefficientsarelinearcombinationsofobservablesA andaregiveninAppendixB. The
nm nm
A observablesare obtainableas linearcombinationsofvariousproductsoftheoperators
nm
n ¶ nlndetM
l l
L = (7)
n 4 ¶ µn
l
n ¶ mlndetM
h h
H = , (8)
m 4 ¶ µm
h
evaluatedat µ =0. Forthedefinitionsand explicitformsoftheA seeAppendixB.
l,h nm
Figure1comparesthecut-offeffectsduetothefinitetemporalextentN inthefreetheorycase
t
forthecoefficientsc ,c ,c andc forthreedifferentstaggeredfermionactions: thestandard,
00 20 40 60
the Naik (asqtad) and the p4 action. The results for the first three coefficients are normalized to
their respective Stefan-Boltzmann (SB) values. The SB value for c is zero (and the same holds
60
for c ) . In the SB limit, the c coefficients are, of course, equal to half of the SB values of c
06 0n n0
4
for0<n≤4. Allothercoefficientswithn,m6=0arezero intheSBlimit. In theinteractingcase,
the coefficients which are zero in the SB limit can aquire non-zero values. Figure 1 shows that
the asqtad action has better scaling properties than the standard (unimproved)staggered action at
N =4,butitisclearthatastudyatlargerN isimportantforfurtherreductionofthediscretization
t t
errors.
FIG.1: Theexpansion coefficientsc ,c ,c andc forthepressureinthefreetheorycaseasafunction
00 20 40 60
ofN.
t
B. Calculatingtheinteraction measureandenergydensity
TheinteractionmeasureI can beTaylorexpandedin amannersimilarto thepressure
I =−Nt3dlnZ = (cid:229)¥ b (T) µ¯l n µ¯h m, (9)
T4 N3 dlna nm T T
s n,m (cid:18) (cid:19) (cid:18) (cid:19)
whereagainonly termseveninn+m are nonzeroand
1 N3 ¶ n+m dlnZ
b (T)= − t . (10)
nm n!m!N3¶ (µ N )n¶ (µ N )m dlna
s l t h t (cid:12)µl,h=0(cid:18) (cid:19)
(cid:12)
The derivative with respect to lna is taken along a trajector(cid:12)y of constant physics. The fermionic
(cid:12)
part of dlnZ, consideringtheform oftheasqtad action,is
dlna
dSf = (cid:229) nf d(mfa)trhM−1i+ du0 trhM−1dMf i . (11)
dlna 4 dlna f dlna f du
(cid:28) (cid:29) f=h,l (cid:20) 0 (cid:21)
No volume normalization of the various traces is assumed in the above. The gauge part, taking
intoaccount theexplicitformoftheSymanzikgaugeaction, is
−dS db db db
g rt pg
=hGi=h6 P+12 R+16 Ci, (12)
dlna dlna dlna dlna
(cid:28) (cid:29)
5
where P, R and C are the appropriate sums of the plaquette, rectangle and parallelogram terms,
respectively(here theyarenot normalizedtothevolume). Thustheb (T) coefficients become
nm
b (T) = − 1 Nt3 (cid:229) nf d(mfa) tr ¶ n+mhM−f 1i
nm n!m!N3 4  dlna ¶ (µ N )n¶ (µ N )m
s f=l,h (cid:12)(cid:12)µl,h=0 l t h t (cid:12)(cid:12)µl,h=0
(cid:12)
 (cid:12)
du ¶ n+mhM−1(cid:12)dMfi (cid:12)(cid:12)
+ 0 tr f du0
dlna ¶ (µ N )n¶ (µ N )m(cid:12) 
(cid:12)µl,h=0 l t h t (cid:12)
(cid:12) (cid:12)µl,h=0
(cid:12) (cid:12)
1 N(cid:12)3 ¶ n+mhGi (cid:12) 
− t (cid:12) (13)
n!m!N3 ¶ (µ N )n¶ (µ N )m
s l t h t (cid:12)µl,h=0
(cid:12)
(cid:12)
Theexplicitformsoftheb (T)coefficientsarem(cid:12)orecomplexthanthoseforc (T)andwesave
nm nm
them for Appendix C. The SB limit of all b coefficients is zero. In the presence of interactions
nm
their values can become different from zero. For the computation of the b (T) coefficients, in
nm
additiontothederivativesofthefermionmatrixandthegaugeactionwithrespecttothechemical
potentials, we have to know the derivativesof the action parameters with respect to lna along the
trajectory of constant physics. The latter have been determined in our previous work on the EOS
atzerochemicalpotential[21],alongwiththecoefficientb (T),whichistheinteractionmeasure
00
divided by T4 in that case. The coefficients c (T) can be obtained from b (T) by integration
nm nm
alongthetrajectory ofconstantphysics. Thiscan serveas aconsistencycheck ofthecalculation.
Theenergy densitye issimplyobtainedfrom thelinearcombination
e I+3p
= . (14)
T4 T4
C. Quarknumberdensitiesandsusceptibilities
TheTaylorexpansionforthequarknumberdensitiescanbeobtainedfromthatforthepressure.
Forexample,thelightquark numberdensity,n ,is
ud
nud ¶ lnZ (cid:229)¥ µ¯l n−1 µ¯h m
= = nc (T) , (15)
T3 ¶ µ¯ /T T3V nm T T
l (cid:18) (cid:19) n=1,m=0 (cid:18) (cid:19) (cid:18) (cid:19)
and theheavy one, n , is
s
ns ¶ lnZ (cid:229)¥ µ¯l n µ¯h m−1
= = mc (T) . (16)
T3 ¶ µ¯ /T T3V nm T T
h (cid:18) (cid:19) n=0,m=1 (cid:18) (cid:19) (cid:18) (cid:19)
6
Similarly, the quark number susceptibilities are derivatives of the quark number densities with
respect tothechemical potentials. Thus,thediagonallight-lightquark susceptibilitybecomes
c uu ¶ nud (cid:229)¥ µ¯l n−2 µ¯h m
= = n(n−1)c (T) , (17)
T2 ¶ µ¯ /T T3 nm T T
l n=2,m=0 (cid:18) (cid:19) (cid:18) (cid:19)
(cid:16) (cid:17)
and theheavy-heavy diagonaloneis
c ss ¶ ns (cid:229)¥ µ¯l n µ¯h m−2
= = m(m−1)c (T) . (18)
T2 ¶ µ¯ /T T3 nm T T
h n=0,m=2 (cid:18) (cid:19) (cid:18) (cid:19)
(cid:16) (cid:17)
Lastly,themixedquarksusceptibilityhastheform
c us ¶ nud (cid:229)¥ µ¯l n−1 µ¯h m−1
= = nmc (T) . (19)
T2 ¶ µ¯ /T T3 nm T T
h n=1,m=1 (cid:18) (cid:19) (cid:18) (cid:19)
(cid:16) (cid:17)
III. SIMULATIONS
Theasqtad-Symanzikgaugeensemblesweuseinthisstudyhavespatialvolumesof123 or163
and N =4, and are generated using the R algorithm. They are a subset of the ensembles in our
t
EOS calculation at zero chemical potential [21]. The ensembles lie on an approximate trajectory
of constant physics for which m ≈ 0.1m , and m is tuned to the physical strange quark mass
ud s s
within 20%. Along the trajectory, the p to r mass ratio is mp /mr ≈0.3. Table I in [21] contains
the run parameters and trajectory numbers of the ensembles used here. They are the ones that
have the gauge coupling values of b =6.0, 6.075, 6.1, 6.125, 6.175, 6.2, 6.225, 6.25, 6.275, 6.3,
6.35, 6.6 and 7.08. The last column of that table shows the lattice scale. For explanation of the
scalesettingandothersimulationdetailswereferthereadertosectionIIIof[21]. Theobservables
that need to be measured along the trajectory of constant physics in order to construct the Taylor
coefficients in the expansion for the pressure are L and H defined by Eqs. (7) and (8). For the
n m
interactionmeasuredeterminationthefollowingobservableshavetobecalculated in addition:
¶ ntrM−1 ¶ mtrM−1
l = l , h = h , (20)
n ¶ µn m ¶ µm
l h
¶ ntr(M−1dMh) ¶ mtr(M−1dMh)
l = l du0 , c = h du0 (21)
n ¶ µn m ¶ µm
l h
andthegluonicobservablesP,RandC. InAppendixCweshowhowtheyenterinthecoefficients
b (T). To sixth order in the Taylor expansion, the number of fermionic observables (L , H ,
nm n m
7
l , h , l , c ) that need to be determined is 40. We calculate them stochastically employing
n m n m
random Gaussian sources. In the region outside the phase transition or crossover we use 100
sources and double that number insidethe transition/crossoverregion. This ensures that we work
with statistical errors dominated by the gauge fluctuations and not by the ones coming from the
stochasticestimators.
The ensembles we are working with have been generated using the inexact R algorithm which
introduces finite step-size errors. In our previous study of these ensembles [21] we measured the
step-sizeerrorinbothgluonicandfermionicobservables. Theerrorwasconsiderablylessthan1%
in the relevant gluonic and fermionic observables, measured on the high temperature ensembles.
For the EOS at zero chemical potential it is necessary to subtract the high temperature and zero
temperature values. In the difference the effect of the step-size error becomes somewhat more
pronounced. The contributionsto the EOS due to nonzero chemical potential, computed here, do
notrequirezerotemperaturesubtractions. Thus,basedontheobservationsnotedabove,weexpect
anystep-sizeerrors inthesecontributionstobeconsiderablysmallerthan ourstatisticalerrors.
IV. NUMERICALRESULTS
Figure 2 shows our results for the temperature dependence of the c (T) and the c (T) coef-
n0 0m
ficients. They all show rapid changes in the phase transition region and relatively quickly reach
the Stefan-Boltzmann (SB) ideal gas values around 1.5T - 2T . Unsurprisingly, the errors of the
c c
higher order coefficients are larger than the ones for the lowest order coefficients. They are worst
for the sixth order coefficients c (T) and c (T). Although the magnitude of the coefficients
60 06
decreases with each order in the Taylor expansion, for µ¯/T ∼1 the sixth order terms contributea
great deal of noise in the thermodynamic quantities at the present level of statistics. Very similar
conclusionscanbemadeaboutthegeneralbehavioroftherestofthepressurecoefficients,c (T)
nm
withbothn,m6=0,showninFig. 3. By comparisonwiththec (T)and c (T) coefficients,they
n0 0m
are smallerand soare theircontributionstothevariousthermodynamicquantities.
Figures4and5showthecoefficientsintheTaylorexpansionoftheinteractionmeasure. Here
again we see the rapid changes/large fluctuations around the transition region, the fast approach
to the SB limit at high temperatures and the increase in magnitude of the errors and the decrease
in magnitude of the coefficients with each successive order. In principle, each c (T) coefficient
nm
could be obtained from b (T) by integrating the latter along the trajectory of constant physics.
nm
8
FIG.2: Taylorexpansion coefficients c (T)andc (T)for p/T4.
n0 0m
For example, in Fig. 6 the c (T) coefficient obtained directly using Eq. (5) is compared to its
20
valuecalculatedbyintegratingb (T). Thecomparisonshowsthatwithinthestatisticalerrors the
20
two results are the same. Similar calculations were done for the rest of the coefficients and the
consistencybetweentheresultsfromthetwomethodswassatisfactoryconsideringthelargeerrors
on thevaluesobtainedby integration.
Having determined the c (T) and b (T) coefficients we can now calculate the EOS to sixth
nm nm
orderinthechemicalpotentials. Wealsodeterminethequarkdensitiesandvarioussusceptibilities
to fifth and fourth order, respectively. Since we want to work at strange quark density n = 0 to
s
approximate the experimental conditions, we tuned µ¯ /T along the trajectory of constant physics
h
in order to achieve that condition within the statistical error. Figure 7 (left) shows, for several
values of µ¯ /T, that with µ¯ /T =0 a slightly negative n is generated due to the nonzero c (T)
l h s n1
terms. After the introduction of an appropriate nonzero µ¯ /T for each studied temperature and
h
9
FIG.3: Taylorexpansion coefficients c (T)withn,m6=0for p/T4.
nm
µ¯ /T, Fig. 7 (right) shows our approximation of the condition n = 0. The effect of the tuning
l s
on thermodynamic quantities, other than n /T3 itself, is small, because of the smallness of the
s
“mixed expansion coefficients” c (T) and b (T) for n,m 6= 0. For our level of statistics the
nm nm
typicaleffect is withinthestatisticalerrors on thestudiedquantities.
Figures 8 and 9 show the corrections to the pressure, interaction measure and energy density
due to the presence of a nonzero µ¯ /T. The correction to the pressure, for example, is the dif-
l
ference D p/T4 = p(µ 6=0)/T4−p(µ =0)/T4, which is Eq. (5) minus the zeroth order term
l,h l,h
c (T) = p(µ = 0)/T4. Similarly for the interaction measure and energy density, the correc-
00 l,h
tionsareD I/T4 =I(µ 6=0)/T4−I(µ =0)/T4 andDe /T4=e (µ 6=0)/T4−e (µ =0)/T4,
l,h l,h l,h l,h
whichmeansagainthatthezerothordertermsaresubtractedfromtheTaylorexpansionsforthese
quantities. Qualitatively,ourEOS results are similarto the previoustwo-flavor studies [16]. The
corrections to the thermodynamic quantities grow with increasing µ¯ /T and so do the statistical
l
errors. Thelatteris duetotheincreasing contributionsfrom higherorderterms, whichare noisier
10