The QCD Equation of State with almost Physical Quark Masses

0.5 MB·English

by  M. Cheng

BNL-NT-07/38, BI-TP 2007/20, CU-TP-1181
The QCD Equation of State with almost Physical Quark Masses
M. Chenga, N. H. Christa, S. Dattab, J. van der Heidec, C. Jungd, F. Karschc,d, O. Kaczmarekc,
E.Laermannc,R.D. Mawhinneya,C.Miaoc,P.Petreczkyd,e, K.Petrovf,C.Schmidtd, W.Soeldnerd andT.Umedag
a Physics Department,Columbia University,
New York, NY 10027, USA
b Department of Theoretical Physics,
Tata Institute of Fundamental Research,
Homi Bhabha Road, Mumbai 400005, India
cFakult¨at fu¨r Physik, Universit¨at Bielefeld,
D-33615 Bielefeld, Germany
dPhysics Department, Brookhaven National Laboratory,
Upton, NY 11973, USA
e RIKEN-BNL Research Center,
8 Brookhaven National Laboratory, Upton, NY 11973, USA
0 fNiels Bohr Institute, University of Copenhagen,
0
Blegdamsvej 17, DK-2100 Copenhagen, Denmark
2 gGraduate School of Pure and Applied Sciences,

n University of Tsukuba, Tsukuba, Ibaraki 305-8571, Japan
a
J (Dated: February 2, 2008)

5
We present results on the equation of state in QCD with two light quark flavors and a heavier
1
strange quark. Calculations with improved staggered fermions have been performed on lattices

] with temporal extent Nτ =4 and 6 on a line of constant physics with almost physical quark mass
t values; the pion mass is about 220 MeV, and the strange quark mass is adjusted to its physical
a
l value. High statistics results on large lattices are obtained for bulk thermodynamic observables,
- i.e. pressure, energy and entropy density, at vanishing quark chemical potential for a wide range
p
of temperatures, 140 MeV T 800 MeV. We present a detailed discussion of finite cut-off
e ≤ ≤
h effectswhichbecomeparticularlysignificantfortemperatureslargerthanabouttwicethetransition
[ temperature. At these high temperatures we also performed calculations of the trace anomaly on
latticeswithtemporalextentNτ =8. Furthermore,wehaveperformedanextensiveanalysisofzero
2 temperature observables including the light and strange quark condensates and the static quark
v potential at zero temperature. These are used to set the temperature scale for thermodynamic
4
observablesandtocalculaterenormalizedobservablesthataresensitivetodeconfinementandchiral
5
symmetry restoration and become order parameters in the infinite and zero quark mass limits,
3
respectively.
0
.
0 PACSnumbers: 11.15.Ha,11.10.Wx,12.38Gc, 12.38.Mh
1
7
0 I. INTRODUCTION
:
v
i Reaching a detailed understanding of bulk thermodynamics of QCD, e.g. the temperature dependence of pressure
X
andenergydensityaswellasthe equationofstate,p(ǫ)vs. ǫ,isoneofthe centralgoalsofnon-perturbativestudiesof
r
a QCDonthelattice. Theequationofstateclearlyisofcentralimportancefortheunderstandingofthermalproperties
of any thermodynamic system. It provides direct insight into the relevant degrees of freedom and their correlationin
differentphasesofstronglyinteractingmatter. We havesome understandingofthe equationofstate inlimiting cases
of high and low temperatures from perturbation theory [1–5] and hadrongas phenomenology [6], respectively. In the
transitionregionfromthelowtemperaturehadronicregimetothehightemperaturequarkgluonplasma,however,one
hastorelyonagenuinenon-perturbativeapproach,latticeregularizedQCD,tostudythenon-perturbativeproperties
of strongly interacting matter.
Lattice studies of bulk thermodynamics are particularly demanding as the most interesting observables, pressure
and energy density, are given in terms of differences of dimension 4 operators. These differences are particularly
difficult to evaluate because both terms being subtracted contain the pressure or energy density of the vacuum, an
unphysical quantity that is approximately 1/(aT)4 larger than the sought-after difference. Numerical signals thus
rapidly decrease with the fourth power of the lattice spacing, a, when one tries to approach the continuum limit at
fixed temperature (T). For this reason improved actions, which allow one to perform calculations on rather coarse
lattices with relatively small lattice discretization errors, are quite useful in thermodynamic calculations. Indeed,
the early calculations of bulk thermodynamics with standard staggered [7] and Wilson [8] fermion discretization
schemes have shown that at high temperature bulk thermodynamic observables are particularly sensitive to lattice
2
discretizationerrors. ThiscloselyfollowsobservationsmadeinstudiesofthethermodynamicsofSU(3)gaugetheories
[9]. Inordertominimizediscretizationerrorsathightemperature,improvedstaggeredfermionactions-thep4-action
[10] and the asqtad action [11] - have been used to study the QCD equation of state. Recent studies, performed with
the asqtad action with almost physical quark mass values on lattices with two different values of the lattice cut-off
[11], indeed show much smaller discretization errors than similar studies performed with the 1-link, stout smeared
staggered fermion action [12]. Another source for cut-off errors arises, however, from the explicit breaking of flavor
symmetry in the staggered fermion formulation. While this is not of much concern in the chirally symmetric high
temperaturephaseofQCD,itleadstocut-offdependentmodificationsofthehadronspectrumandthusmayinfluence
the calculation of thermodynamic observables in the low temperature hadronic phase of QCD. Techniques to reduce
flavor symmetry breaking through the introduction of so-called ’fat links’ are thus generally exploited in numerical
calculations with staggeredfermions [14].
In this paper we report on a calculation of bulk thermodynamics in QCD with almost physical light quark masses
and a physical value of the strange quark mass. Our calculations have been performed with a tree level Symanzik-
improvedgaugeactionand animprovedstaggeredfermionaction, the p4-actionwith 3-link smearing(p4fat3), which
removes (a2) cut-off effects at tree-level and also leads to small cut-off effects in (g2) perturbation theory [13].
O O
At each temperature, we perform simulations with two degenerate light quark masses and a heavier strange quark
mass for two different values of the lattice cut-off, corresponding to lattices with temporal extent N = 4 and 6.
τ
In these calculations we explore a wide range of temperatures varying from about 140 MeV to about 800 MeV.
This corresponds to the temperature interval relevant for current experimental studies of dense matter in heavy ion
collisions at RHIC as well as the forthcoming experiments at the LHC. Bare quark masses have been adjusted to
keep physical masses approximately constant when the lattice-cut off is varied. At high temperatures, T>350 MeV,
we also performed calculations on lattices with temporal extent N =8 to get control over cut-off effects∼in the high
τ
temperature limit.
We will startinthe next sectionby reviewing basic thermodynamic relationsin the continuumvalid for thermody-
namiccalculationsonsuchlines of constant physics (LCP).InsectionIIIweoutlinedetailsofourcalculationalset-up
with improved staggered fermions. In section IV we present our zero temperature calculations needed to define the
line of constant physics and the temperature scale deduced from properties of the static quark-antiquark potential.
SectionVis devotedto thepresentationofourbasicresult,thedifference betweenenergydensity andthreetimes the
pressure from which we obtain all other thermodynamic observables, e.g. the pressure, energy and entropy densities
aswellasthevelocityofsound. SectionVIisdevotedtoadiscussionofthetemperaturedependenceofPolyakovloop
expectationvalues andchiralcondensateswhichprovidesa comparisonbetweenthe deconfining andchiralsymmetry
restoring features of the QCD transition. We finally present a discussion of our results and a comparison with other
improved staggered fermion calculations of bulk thermodynamics in Section VII.
II. THERMODYNAMICS ON LINES OF CONSTANT PHYSICS
TostartourdiscussionofQCDthermodynamicsonthe latticewerecallsomebasicthermodynamicrelationsinthe
continuum. For large, homogeneous media the basic bulk thermodynamic observables we will consider here can be
obtainedfromthegrandcanonicalpartitionfunctionwithvanishingquarkchemicalpotentials,Z(T,V). Weintroduce
the grand canonical potential, Ω(T,V), normalized such that it vanishes at vanishing temperature,
Ω(T,V)=T lnZ(T,V) Ω , (1)
0
−
with Ω = lim T lnZ(T,V). With this we obtain the thermal part of the pressure (p) and energy density (ǫ)
0
T→0
1 T2∂Ω(T,V)/T
p= Ω(T,V) , ǫ= , (2)
V V ∂T
which by construction both vanish at vanishing temperature. Using these relations one can express the difference
between ǫ and 3p, i.e the thermal contribution to the trace of the energy-momentum tensor Θµµ(T), in terms of a
derivative of the pressure with respect to temperature, i.e.
Θµµ(T) ǫ 3p ∂
− =T (p/T4) , (3)
T4 ≡ T4 ∂T
In fact, it is Θµµ(T) which is the basic thermodynamic quantity conveniently calculated on the lattice. All other
bulk thermodynamic observables, e.g. p/T4, ǫ/T4 as well as the entropy density, s/T3 (ǫ+p)/T4, can be deduced
≡
3
from this using the above thermodynamic relations. In particular, we obtain the pressure from Θµµ(T) through
integration of Eq. 3,
p(T) p(T ) T 1
0 = dT′ Θµµ(T′) . (4)
T4 − T4 T′5
0 ZT0
Usually, the temperature for the lower integration limit, T , is chosen to be a temperature sufficiently deep in the
0
hadronic phase of QCD where the pressure p(T ), receives contributions only from massive hadronic states and is
0
already exponentially small. We will discuss this in more detail in Section V. Eq. 4 then directly gives the pressure
at temperature T. Using p/T4 determined from Eq. 4 and combining it with Eq. 3, we obtain ǫ/T4 as well as s/T3.
This makes it evident, that there is indeed only one independent bulk thermodynamic observable calculated in the
thermodynamic (largevolume)limit. Allother observablesarederivedthroughstandardthermodynamic relationsso
that thermodynamic consistency of all bulk thermodynamic observables is insured by construction!
We stress that the normalization introduced here for the grand canonical potential, Eq. 1, forces the pressure and
energy density to vanish at T = 0. As a consequence of this normalization, any non-perturbative structure of the
QCD vacuum, e.g. quark and gluon condensates, that contribute to the trace anomaly Θµµ(0), and would lead to
a non-vanishing vacuum pressure and/or energy density, eventually will show up as non-perturbative contributions
to the high temperature part of these thermodynamic observables. This is similar to the normalization used, e.g. in
the bag model and the hadron resonance gas, but differs from the normalization used e.g. in resummed perturbative
calculationathightemperature[15,16]orphenomenological(quasi-particle)modelsforthehightemperaturephaseof
QCD[17]. ThisshouldbekeptinmindwhencomparingresultsfortheEoSwithperturbativeandmodelcalculations.
Wealsonotethatambiguitiesinnormalizingpressureandenergydensityatzerotemperaturedropoutinacalculation
of the entropy density which thus is the preferred observable for such comparisons.
III. LATTICE FORMULATION
In a lattice calculation, temperature and volume are given in terms of the temporal (N ) and spatial (N ) lattice
τ σ
extent as well as the lattice spacing, a, which is controlled through the lattice gauge coupling β 6/g2,
≡
1
T = , V =(N a(β))3 . (5)
σ
N a(β)
τ
As all observables that are calculated on the lattice, are functions of the coupling, β, we may rewrite Eq. 3 in terms
of a derivative taken with respect to β rather than T. Furthermore we adopt the normalization of the pressure as
introducedinEq.1. Thisinsuresaproperrenormalizationofthermodynamicquantitiesand,asaconsequence,forces
the pressure to vanish in the vacuum, i.e. at T =0.
Let us write the QCD partition function on a lattice of size N3N as
σ τ
Z (β,N ,N )= dU e−S(U) , (6)
LCP σ τ x,µ
Z x,µ
Y
whereU SU(3)denotesthegaugelinkvariablesandS(U)=βS (U) S (U,β)istheEuclideanaction,whichis
x,µ G F
∈ −
composedoutofapurelygluoniccontribution,S (U),andthefermionicpart,S (U,β),whicharisesafterintegration
G F
over the fermion fields. We will specify this action in more detail in the next section but note here that we will use
treelevelimprovedgaugeandfermionactions. Althoughitwouldbestraightforwardtointroduceone-looportadpole
improvement factors in the action the setup used here greatly simplifies the analysis of thermodynamic observables
and in some cases also gives a more direct relation to corresponding observables in the continuum.
When using only tree level improvement the gluonic action does not depend on the gauge coupling, β, and the
fermion action depends on β only through the bare light (mˆ ) and strange (mˆ ) quark masses. The subscript LCP
l s
in Eq. 6 indicates that we have defined the partition function Z on a line of constant physics (LCP), i.e. when
LCP
approaching the continuum limit by increasing the gauge coupling (β ) the bare quark masses (mˆ (β), mˆ (β))
l s
→ ∞
in the QCD Lagrangian are tuned towards zero such that the vacuum properties of QCD remain unchanged. The
quark masses thus are not independent parameters but are functions of β which are determined through constraints
imposed on zero temperature observables; e.g. one demands that a set of hadron masses remains unchanged when
the continuum limit is approached on a LCP.
WenowmayrewriteEq.3intermsofobservablescalculableinlatticecalculationsatzeroandnon-zerotemperature,
Θµµ(T) 1 dS 1 dS
= R (β)N4 . (7)
T4 − β τ N3N dβ − N3N dβ
(cid:18) σ τ (cid:28) (cid:29)τ σ 0 (cid:28) (cid:29)0(cid:19)
4
Here ... , with x = τ, 0 denote expectation values evaluated on finite temperature lattices of size N3N , with
N hNix, and zero temperature lattices, i.e. on lattices with large temporal extent, N3N with N σ Nτ >N ,
τ ≪ σ σ τ τ ≡ 0 σ
respectively. Furthermore, R denotes the lattice version of the QCD β-function which arises as a multiplic∼ative
β
factor in the definition of Θµµ(T) because derivatives with respect to T have been converted to derivatives with
respect to the lattice spacing a on lattices with fixed temporal extent N ,
τ
dβ dβ
R (β) T = a . (8)
β
≡ dT − da
We note that in the weak coupling, large β limit, R approaches the universal form of the 2-loop β-function of
β
3-flavor QCD,
R (β)=12b +72b /β+ (β−2) , (9)
β 0 1
O
with b =9/16π2 and b =1/4π4.
0 1
We analyze the thermodynamics of QCD with two degenerate light quarks (mˆ mˆ =mˆ ) and a heavier strange
l u d
≡
quark (mˆ ) described by the QCD partition function given in Eq. 6. For our studies of bulk thermodynamics we use
s
the same discretization scheme which has been used recently by us in the study of the QCD transition temperature
[22], i.e. we use an (a2) tree level improved gauge action constructed from a 4-link plaquette term and a planar
O
6-link Wilson loop as well as a staggered fermion action that contains a smeared 1-link term and bent 3-link terms.
We call this action the p4fat3-action; further details are given in Ref. [10] where the p4fat3 action was first used in
studies of the QCD equation of state on lattices with temporal extent N = 4 and larger quark masses. With this
τ
action, bulk thermodynamic quantities like pressure and energy density are (a2) tree level improved; corrections to
O
thehightemperatureidealgaslimitonlystartat (1/N4)andaresignificantlysmallerthanfortheNaikactionorthe
standardstaggeredactionwhichsuffersfromlargOe (1/Nτ 2)cut-offeffectsathightemperature. Ananalysisofcut-off
O τ
effects in the ideal gas limit and in (g2) lattice perturbation theory [13] shows that deviations from perturbative
O
results are already only a few percent for lattices with temporal extent as small as N =6.
τ
Following the notation used in Ref. [22] the Euclidean action is given as
S(U)=βS (U) S (U,β) , (10)
G F
−
with a gluonic contribution, S (U), and a fermionic part, S (U,β). The latter can be expressed in terms of the
G F
staggered fermion matrices, D , for two light (mˆ ) and a heavier strange quark (mˆ ),
mˆl(mˆs) l s
1 1
S (U,β) = TrlnD(mˆ (β))+ TrlnD(mˆ (β)) . (11)
F l s
2 4
Here we took the fourth root of the staggered fermion determinant to represent the contribution of a single fermion
flavor to the QCD partition function1.
Wealsointroducethelightandstrangequarkcondensatescalculatedatfinite(x=τ)andzerotemperature(x=0),
respectively,
1 1
ψ¯ψ TrD−1(mˆ ) , q =l, s , x=0, τ , (12)
h iq,x ≡ 4N3N q x
σ x
(cid:10) (cid:11)
as well as expectation values of the gluonic action density,
1
s S . (13)
h Gix ≡ N3N h Gix
σ x
All numericalcalculationshavebeen performedusing the RationalHybridMonte Carlo(RHMC) algorithm[21] with
parameters that have been optimized [22] to reach acceptance rates of about 70%. Some details on our tuning of
parameters of the RHMC algorithm have been given in [23].
For the discussion of the thermodynamics on a line of constant physics (LCP) it sometimes is convenient to
parametrize the quark mass dependence of S in terms of the light quark mass mˆ and the ratio h mˆ /mˆ rather
F l s l
≡
than mˆ and mˆ separately. We thus write the β-dependence of the strange quark mass as, mˆ (β) = mˆ (β)h(β). In
l s s l
1 There is a controversy regarding the validity of the rooting approach in numerical calculations with staggered fermions. For further
detailswerefertorecentreviewspresentedatLatticeconference[18–20]andreferencestherein.
5
the evaluation of (ǫ 3p)/T4 we then will need to know the derivatives of these parametrizations with respect to β.
−
We define
1 dmˆ (β) 1 dh(β)
l
R (β)= , R (β)= . (14)
m h
mˆ (β) dβ h(β) dβ
l
With these definitions we may rewrite Eq. 7 as
ǫ 3p d p ∂p/T4
− = T =R (β)
T4 dT T4 β ∂β
Θµµ(T(cid:16)) (cid:17)Θµµ(T) Θµµ(T)
= G + F + h , (15)
T4 T4 T4
with
Θµµ(T)
G = R [ s s ]N4 , (16)
T4 β h Gi0−h Giτ τ
Θµµ(T)
F = R R 2mˆ ψ¯ψ ψ¯ψ +mˆ ψ¯ψ ψ¯ψ N4 , (17)
T4 − β m l h il,0−h il,τ s h is,0−h is,τ τ
Θµµ(T) (cid:2) (cid:0) (cid:1) (cid:0) (cid:1)(cid:3)
h = R R mˆ ψ¯ψ ψ¯ψ N4 . (18)
T4 − β h s h is,0−h is,τ τ
(cid:2) (cid:3)
We will show in the next section that to a good approximation h(β) stays constant on a LCP. R thus vanishes on
h
the LCP andconsequently the lastterm in Eq.15, Θµµ, will notcontribute to the thermalpartof the traceanomaly,
h
Θµµ(T). The other two terms stay finite in the continuum limit and correspond to the contribution of the thermal
parts of gluon and quark condensates to the trace anomaly. We note that the latter contribution vanishes in the
chiral limit of three flavor QCD (mˆ , mˆ 0). The trace anomaly would then entirely be given by Θµµ(T) and
l s → G
the observables entering the calculation of bulk thermodynamic quantities in the chiral limit of QCD would reduce
to those needed also in a pure SU(3) gauge theory [9]. In fact, we also find that for physical values of the quark
masses the trace anomaly is dominated by the gluonic contribution, Θµµ(T). As will become clear in Section V
G
Θµµ(T) contributes less than 10% to the total trace anomaly for temperatures large than about twice the transition
F
temperature.
We also note that the prefactor in Eq. 17 will approach unity in the continuum limit as R attains a universal
m
form up to 2-loop level which is similar to that of R−1 and is only modified through the anomalous dimension of the
β
quark mass renormalization [24]. For the relevant combination of β-functions that enters the fermionic part of the
trace anomaly, one has
16b
(R (β)R (β))1−loop =1+ 0 . (19)
β m
− 3β
IV. STATIC QUARK POTENTIAL AND THE LINE OF CONSTANT PHYSICS
A. Construction of the line of constant physics
We will calculate thermodynamic observables on a line of constant physics (LCP) that is defined at T = 0 as
a line in the space of light and strange bare quark masses parametrized by the gauge coupling β. Each point on
this line corresponds to identical physical conditions at different values of the lattice cut-off which is tuned towards
the continuum limit by increasing the gauge coupling β. We define the line of constant physics by demanding (i)
that the ratio of masses for the strange pseudo-scalar and the kaon mass, m /m , stays constant and (ii) that m
s¯s K s¯s
expressedinunitsofthescaleparameterr staysconstant. Thelattergivesthedistanceatwhichtheslopeofthezero
0
temperature, static quark potential, V (r), attains a certain value. We also introduce the scale r , which frequently
q¯q 1
is used on finer lattices to convert lattice results expressed in units of the cut-off to physical scales,
dV (r) dV (r)
r2 q¯q =1.65 , r2 q¯q =1.0 . (20)
dr dr
(cid:18) (cid:19)r=r0 (cid:18) (cid:19)r=r1
We checked that (i) and (ii) also hold true, if we replace m by the mass of the light quark pseudo-scalar meson,
s¯s
m . However, errors on m r and (m /m ) are generally larger which, in particular at large values of β makes the
π π 0 π K
parametrization of the LCP less stringent.
6
Leadingorderchiralperturbationtheorysuggeststhattheratio(m /m )2 is proportionalto mˆ /(mˆ +mˆ ). One
s¯s K s l s
thusexpectsthis ratiotostayconstantforfixedh=mˆ /mˆ . Thisis,indeedfulfilledinthe entireregimeofcouplings,
s l
β, explored in our calculations (see Table I). The first condition for fixing the LCP parameters thus, in practice, has
been replaced by choosing h = mˆ /mˆ to be constant. As a consequence we find R (β) = 0, which simplifies the
l s h
calculation of thermodynamic quantities.
In order to define a line of constant strange quark mass, as a second condition for the LCP we demand that the
product m r stays constant. For our LCP we chose 1.59 as the value for the product. Here one should note
s¯s 0
that m determined in our calculations only receives contributions from connected diagrams and does not include
s¯s
disconnected loops. In order to compare our value (1.59, see discussion below) to a physical one, we therefore follow
the argumentation of Ref. [25] and adopt m = 2m2 m2 = 686 MeV as the physical mass of our strange
s¯s K − π
pseudoscalar. Together with the scale r = 0.469(7) fm as determined in Ref. [26] through a comparison of r with
0 0
p
level splittings of the charmonium system [27], this yields m r 1.63. Of course, there is some ambiguity in this
s¯s 0
≃
choice as current determinations of r differ by about 10% [26, 28]. This introduces some systematic error in the
0
definition of the physical LCP. The main reason for deviation from the physical LCP in the present calculation,
however, is due to the choice of the light quark masses which are about a factor two too large.
Fixing the light and strange pseudo-scalar masses in units of r required some trial runs for several β values. We
0
then usedthe leadingorderchiralperturbationtheoryansatz m2 mˆ (or m2 mˆ )to choosemˆ andmˆ mˆ /10
s¯s ∼ s π ∼ l s l ≡ s
at several values of the gauge coupling and used a renormalization group inspired interpolation to determine quark
mass values at several other β values at which high statistics simulations have been performed. It turned out that
these values are best fitted by m r = 1.59. We thus use this value rather than the value 1.63 mentioned above, to
s¯s 0
defineourLCP.Forallothersimulationswethenusedtheresultsofthesezerotemperaturecalculationstodetermine
the quark mass values that belong to a line of constant physics characterized by:
LCP: (i)m r =1.59 , (ii)h mˆ /mˆ =10
s¯s 0 s l
≡
In general our calculations are thus performed at parameter values close to the LCP which is defined by the above
condition. The parameters of all our zero temperature calculations performed to determine the LCP, results for
meson masses and parameters of the static quark potential are summarized in Table I. As can be seen, at our
actual simulation points the results for m r fluctuate around the mean value by a few percent. We also checked
s¯s 0
the sensitivity of the meson masses used to determine the LCP to finite volume effects. At β = 3.49 and 3.54 we
performedcalculationson324 lattices inadditionto the 163 32lattices. As canbee seenfromTable I resultsform
s¯s
·
and m agree within statistical errors and volume effects are at most on the level of 2% for the light pseudo-scalar.
K
The LCP is furthermore characterized by m /m = 0.435(2) and m /m = 1.33(1). Using r = 0.469(7) fm to
π K s¯s K 0
convertto physicalscales we find that on the LCP the light and strangepseudo-scalarmasses are m 220(4)MeV,
π
≃
m 669(10) MeV and the kaon mass is given by m 503(6) MeV.
s¯s K
≃ ≃
B. The static quark potential and the scale r0
On the LCP we determine several parameters, e.g. the short distance scale r and the linear slope parameter, the
0
string tension σ, that characterize the shape of the static quark potential calculated at T = 0 in a fixed range of
physicaldistances. Thedistancer ,definedinEq.20,isusedtodefinethe temperaturescaleforthe thermodynamics
0
calculations.
The static quark potential, V (r), has been calculated from smeared Wilson loops as described in [22] for all
q¯q
parameter sets listed in Table I. We checked that the the smeared Wilson loops project well onto the ground state
at all values of the cut-off by verifying the independence of the extracted potential parameters on the number of
smearing levels used in the analysis. The set of gauge couplings, β [3.15,4.08], used in this analysis covers a large
∈
interval in which the lattice cut-off changes by a factor 6 from a 0.3 fm down to a 0.05 fm. When analyzing
≃ ≃
the static potential over such a wide range of cut-off values one should make sure that the potential is analyzed in
approximately the same range of physical distances. The fit interval [(r/a) ,(r/a) ] for fits with a Cornell type
min max
ansatz for the static potential thus has been adjusted for the different values of gauge couplings such that it covers
approximately the same range of physical distances, r /2 < r < 2r , or 0.25fm<r<1fm. We confirmed our analysis
0 0
of the static quark potential and the determination of r a∼lso in∼dependently by∼usi∼ng spline interpolations which are
0
not biased by a particular ansatz for the form of the potential.
ThelefthandpartofFig.1showsthestaticquarkpotentialforseveralofourparametersets. Wehaverenormalized
7
β 100mˆl Nσ3·Nτ mπa ms¯sa mKa r0/a √σa c(g2)r0
3.150 1.100 163 32 0.3410( 2) 1.0474( 1) 0.7854( 2) 1.467(72) 0.75(18) 0.97(12)
·
3.210 1.000 163 32 0.3262( 1) 0.9988( 1) 0.7496( 1) 1.583(36) 0.685(75) 1.118(68)
·
3.240 0.900 163 32 0.3099( 2) 0.9485( 2) 0.7125( 3) 1.669(31) 0.658(36) 1.243(67)
·
3.277 0.765 163 32 0.2881( 7) 0.8769( 5) 0.6599( 6) 1.797(19) 0.612(53) 1.362(53)
·
3.290 0.650 163 32 0.2667( 8) 0.8104( 7) 0.6101( 8) 1.823(16) 0.623(32) 1.362(29)
·
3.335 0.620 163 32 0.2594( 3) 0.7884( 2) 0.5941( 5) 1.995(11) 0.5668(73) 1.504(22)
·
3.351 0.591 163 32 0.2541( 7) 0.7692( 5) 0.5800( 7) 2.069(12) 0.551(11) 1.594(24)
·
3.382 0.520 163 32 0.2370( 6) 0.7194( 5) 0.5422( 5) 2.230(14) 0.5100(82) 1.718(57)
·
3.410 0.412 163 32 0.2098( 4) 0.6371( 6) 0.4796( 8) 2.503(18) 0.440(10) 2.073(49)
·
3.420 0.390 243 32 0.2029( 8) 0.6177( 5) 0.4675( 5) 2.577(11) 0.4313(56) 2.124(33)
·
3.430 0.370 243 32 0.1986( 6) 0.6000( 3) 0.4529( 5) 2.6467(81) 0.4225(53) 2.178(17)
·
3.445 0.344 243 32 0.1909( 7) 0.5749( 4) 0.4335( 5) 2.813(15) 0.3951(68) 2.388(35)
·
3.455 0.329 243 32 0.1833(10) 0.5580( 6) 0.4204( 8) 2.856(20) 0.3895(68) 2.375(42)
·
3.460 0.313 163 32 0.1808(16) 0.5443(11) 0.4102(11) 2.890(16) 0.3831(84) 2.391(55)
·
3.470 0.295 243 32 0.1686(19) 0.5233( 8) 0.3940(12) 3.065(18) 0.3592(75) 2.617(41)
·
3.490 0.290 163 32 0.1689(14) 0.5115(11) 0.3842(11) 3.223(31) 0.3423(66) 2.757(59)
·
3.490 0.290 324 0.1679(8) 0.5113(8) 0.3840(7)
3.510 0.259 163 32 0.1525(40) 0.4740(20) 0.3554(22) 3.423(61) 0.322(14) 2.934(92)
·
3.540 0.240 163 32 0.1495(24) 0.4458(20) 0.3358(19) 3.687(34) 0.3011(46) 3.128(51)
·
3.540 0.240 324 0.1469(11) 0.4451(6) 0.3339(11)
3.570 0.212 243 32 0.1347(53) 0.4053(18) 0.3028(23) 4.009(26) 0.2743(38) 3.414(47)
·
3.630 0.170 243 32 0.1126(20) 0.3386( 7) 0.2537( 8) 4.651(41) 0.2352(44) 3.939(59)
·
3.690 0.150 243 32 0.1020(90) 0.2960(20) 0.2230(30) 5.201(48) 0.2116(36) 4.320(63)
·
3.760 0.130 242 32 48 0.0857(32) 0.2530(16) 0.1894(16) 6.050(61) 0.1810(29) 4.984(73)
· ·
3.820 0.125 323 32 0.0830(40) 0.2310(38) 0.1744(50) 6.835(44) 0.1701(21) 5.541(106)
·
3.920 0.110 323 32 0.0750(70) 0.2020(10) 0.1550(20) 7.814(83) 0.1423(24) 6.037(72)
·
4.080 0.081 323 32 0.0700(70) 0.1567(36) 0.1220(50) 10.39(23) 0.1060(35) 7.710(183)
·
TABLEI:Lightquarkandstrangepseudo-scalarmesonmassesandparametersofthestaticquarkpotentialcalculatedonzero
temperature lattices of size Nσ3Nτ. The last column gives the renormalization constants times r0 needed to renormalize the
heavy quark potential to thestring potential at distance r/r0 =1.5.
these potentials by matching2 all potentials at a large distance, r/r = 1.5, to a common value that is taken to be
0
identical to the large distance string potential, V (r)= π/12r+σr. The result of this matching is shown in the
string
−
lower part of Fig. 1(left) and the constant shifts needed to obtain these renormalized potentials are listed in Table I.
The goodmatchingofallthe potentialdataobtainedatdifferentvaluesofthe cut-offalreadygivesagoodideaofthe
smallness of finite cut-off effects in this observable. We note that this matching procedure provides renormalization
constants for the static quark potential, which we also will use later to renormalize the Polyakov loop expectation
value.
To further analyze the shape of the static quark potential we determined the scale parameter r /a as well as the
0
square root of the string tension in lattice units, √σa. These parameters have been obtained from three and four
parameter fits. As described in [22] the latter fit ansatz has been used to estimate systematic errors in our analysis
of the scale parameters.
Results for r /a and √σa are given in Table I. We note that the product r √σ stays constant on the LCP and
0 0
changes by less than 2% in the entire range of couplings β in which the lattice cut-off changes by a factor 6. For
a 0.15 fm we used a quadratic fit ansatz, (r √σ) =r √σ+c(a/r )2, to fit 10 data points. The asymptotic value
≤ 0 a 0 0
for r √σ coincides within errors with a simple averageover all values of (r √σ) in this interval. This confirms that
0 0 a
(a2) corrections indeed are small for this product. Similarly we determined the scale parameter r frequently used
1
O
2 Furtherdetailsonthematchingofthezerotemperatureheavyquarkpotentials,itsapplicationtotherenormalizationofthePolyakov
loopandfinitetemperaturefreeenergieswillbepublishedelsewhere.
8
8 0 0.05 0.1 0.15 0.2 0.25 0.3
r V- (r)
0 qq
a [fm]
6 1.5
4 1.4
r /r
2 1.3 0 1
3.382 r σ1/2
3.43 0
3.51
1.2
0 3.57
3.63
3.69
3.76 1.1
-2 3.82
3.92 a/r
r/r0 Vstring 1.0 0
0 0.5 1 1.5 2 2.5 3 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
FIG. 1: The static quark potential in units of the scale r0 versus distance r/r0 (left) and dimensionless combinations of the
potentialshapeparametersr0/r1andr0√σextractedfromfitstothesepotentials(right). Thelefthandfigureshowspotentials
for several values of β taken from our entire simulation interval, β [3.15:4.08]. The lowest curve in this figure combines all
∈
potentials by matching them to the string potential (solid line) as explained in the text. Curves in the right hand figureshow
quadratic fits and a fit to a constant with a 1% error band. The lattice spacing has been converted to physical units using
r0 =0.469 fm.
to set the scale in calculations performed on finer lattices. Both fits for r √σ and r /r yield χ2/dof 0.7. From
0 0 1
≃
this analysisweobtainthe parameterscharacterizingthe shape ofthe heavyquarkpotentialatmassesinthe vicinity
of the LCP,
r √σ = 1.1034(40),
0
r /r = 1.4636(60). (21)
0 1
We note that the result obtainedhere for r /r is in goodagreementwith the correspondingcontinuumextrapolated
0 1
value,r /r =1.474(7)(18),determinedwiththe asqtadactionfromananalysisofthe quarkmassdependence ofthis
0 1
ratio at two different values of the lattice spacing, a 0.12 fm and a 0.09 fm, respectively [36]. We show results
≃ ≃
for r /r and r √σ calculated at parameter sets close to the LCP in Fig. 1 (right).
0 1 0
Despite the good scaling behavior of dimensionless combinations of scale parameters deduced from the static
potential,oneexpects,ofcourse,tostillfindsubstantialdeviationsfromasymptoticscalingrelationsthatarecontrolled
byuniversal2-loopβ-functions. Forthescaleparameterr /aweparametrizedeviationsfromasymptoticscalingusing
0
a rational function ansatz,
r 1+e aˆ2(β)+f aˆ4(β)
0 r r
rˆ = , (22)
0 ≡ a a R (β)(1+b aˆ2(β)+c aˆ4(β)+d aˆ6(β))
r 2 r r r
where
β 6b −b1/(2b20)
0
R (β) = exp (23)
2
−12b β
(cid:18) 0(cid:19)(cid:18) (cid:19)
denotes the 2-loop β-function of QCD for three massless quark flavors and aˆ(β) = R (β)/R (3.4). With this
2 2
parametrization it is straightforwardto calculate the β-function R entering all basic thermodynamic observables,
β
r dr /a −1
0 0
R (β)= . (24)
β
a dβ
(cid:18) (cid:19)
Furthermore, we need a parametrization of the β-dependence of the bare quark masses to determine the second β-
function entering the thermodynamic relations,i.e. R (β) defined in Eq.14. For this purpose we use a parametriza-
m
tion of the product of the bare light quark mass, mˆ and rˆ that takes into account the anomalous scaling dimension
l 0
of quark masses [24],
12b 4/9
0
mˆ rˆ =a P(β), (25)
l 0 m
β
(cid:18) (cid:19)
9
bm cm dm em fm gm
-2.149(121) 1.676(178) -0.365(144) -2.290(162) 1.829(425) -0.356(335)
ar br cr dr er fr
13.250(363) -1.201(91) 0.054(196) 0.406(109) -1.682(103) 0.823(76)
TABLEII:Parameters ofthefitofthescaleparameter r0 inlattice unitsbased ontheansatz giveninEq.22(lowerhalf) and
thefitoftherenormalization groupinvariantcombination oflightquarkmassesandr0 (Eq.25)onthelineofconstantphysics
(upperhalf). Theχ2/d.o.f forthesefitsare1.5for19degreesoffreedom(lowerhalf)and0.84for18degreesoffreedom(upper
half).
0.040
10.0 m r (β/12b )4/9
l 0 0
0.035
8.0
r /a 0.030
0
6.0
0.025
4.0
0.020
2.0 3-interv. fit 0.015
RG fit
β
β
0.0 0.010
3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4.0 4.1 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4.0 4.1
FIG. 2: The scale parameter rˆ0 r0/a versus β = 6/g2 (left) and its product with the bare light quark mass on the LCP
≡
(right). The two curves shown in the left hand part of this figure correspond to two different fit ans¨atze. As explained in the
text in addition to the renormalization group motivated ansatz given in Eq. 22 the result from a 3-interval fit is shown. The
curvein theright hand part of thefigureshows a fit based on theansatz given in Eqs. 25 and 26.
with a being related to the renormalizationgroupinvariant quark mass in units of r and P(β) being a sixth order
m 0
rational function that parametrizes deviations from the leading order scaling relation for the bare quark mass,
1+b aˆ2(β)+c aˆ4(β)+d aˆ6(β)
m m m
P(β)= . (26)
1+e aˆ2(β)+f aˆ4(β)+g aˆ6(β)
m m m
This ansatz insures that the parametrization for the two β-functions as well as the parametrizationof their product,
R (β)R (β), reproduces the universal 2-loop results given in Eqs. 9 and 19.
β m
In Fig. 2 we show our results for rˆ = r /a and mˆ rˆ together with the fits described above. The fit parameters
0 0 l 0
defining the quark masses on the LCP have been obtained from χ2-fits in the interval β [3.1,4.08]. Results for the
∈
fit parameters are given in Table II. In addition we find a = 0.0190(9) which turns into a value of 8.0(4) MeV in
m
physicalunits. Fitresultsforr /adifferfromtheactuallycalculatedvaluesgiveninTableIIbylessthanonepercent.
0
Likeinthepuregaugetheorycalculationsoftheequationofstate,wealsofindforQCDwithlightdynamicalquarks
that, in the parameter range of interest for finite temperature calculations, β-functions deviate significantly from the
asymptotic scaling form. In particular, we find a dip in R at β 3.43. For small values of N , the interesting
β τ
≃
parameter range thus includes the crossoverregion from the strong to weak coupling regime.
Weusetheinterpolatingfitsforrˆ andmˆ rˆ ,todeterminethetwoβ-functionsR andR thatenterthecalculations
0 l 0 β m
ofthermodynamicquantities. AsallbasicthermodynamicobservablesaredirectlyproportionaltoR ,weshouldcheck
β
thesensitivityofR ontheparticularinterpolationformused. Wethushaveusedacompletelydifferentinterpolation
β
that restricts the renormalization group motivated ansatz to the small coupling regime, β 3.52, and uses purely
≥
rational functions to piecewise fit 2 intervals at smaller β. We find that results for R are sensitive to the fit ansatz
β
onlyforsmallβ-values,i.eβ<3.25,wherethedependenceofrˆ onβ becomesweak. Asdiscussedlatertheuncertainty
0
onR atsmallvaluesofthe∼couplingonlyaffectsthethreesmallesttemperaturesusedfortheanalysisoftheequation
β
of state on the N =4 lattices.
τ
10
3.0
1.4
Rβ 2.5 -Rβ Rm
1.2
1.0 2.0
0.8
1.5
0.6
1.0
0.4
0.5
0.2
β β
0.0 0.0
3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4.0 4.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4.0 4.1
FIG.3: Theβ-functionontheLCP(Eq.24)(left)andtheproductRβRm (right). Thehorizontallinesshowtheweakcoupling
behavior given in Eqs. 9 and 19. Thetwo curves result from two different fits of rˆ0 as discussed in thetext.
Using the parametrizations of rˆ and mˆ rˆ given in Eq. 22 and Eq. 25 as well as the above discussed piecewise
0 l 0
interpolation of rˆ we now can derive the two β-functions R (β) and R (β). In Fig. 3 we show R as well as the
0 β m β
combination R R which enter the calculation of the gluonic and fermionic contributions to (ǫ 3p)/T4. For rˆ
β m 0
− −
as well as for the two β-functions, we show result obtained with our two different fit ansa¨tze. As can be seen, the
differentfitformsleadtodifferencesinthefitresultattheedgesoftheparameterrangeanalyzed. Wetakecareofthis
in our analysis of the equation of state by averaging over the results obtained with the two different fit ansa¨tze and
by including the difference of both fit results as a systematic error. We note that the β-function R has a minimum
β
at β 3.43. This characterizes the transition from strong to weak coupling regions and is similar to what is known
≃
from β-functions determined in pure gauge theory [9] as well as in QCD with heavier quark masses [10]. The details
of this region will differ in different discretization schemes as the QCD β-functions are universal only up to 2-loop
order in perturbation theory. In order to understand the origin of cut-off effects in thermodynamic observables it is,
however,important to havegood controlover R in this non-universalregime as well, as R enters the calculationof
β β
all relevant lattice observables as an overall multiplicative factor3.
V. BULK THERMODYNAMICS
A. The trace anomaly: (ǫ−3p)/T4
ThebasiclatticeobservablesneededtodeterminetheQCDequationofstatewithourtreelevelimprovedgaugeand
fermion actions are expectation values of the gauge action as well as the light and strange quark chiral condensates
calculated on the LCP on finite (N N ) and zero (N >N ) temperature lattices. We have performed finite
τ σ τ σ
≪
temperature calculations on lattices with temporal extent N∼ = 4, 6 and 8. In all cases the spatial extent of the
τ
lattices (N ) was at least four times larger than the temporal extent (N ), i.e. most finite temperature calculations
σ τ
have been performed on lattices of size 1634 and 2436, respectively. In particular at high temperature, we found it
important to increase the spatial volume in our calculations on N = 6 lattices to check for possible finite volume
τ
effects and also to add a few calculations on N = 8 lattices to get control over the cut-off dependence seen in the
τ
trace anomaly. In these cases, calculations on 3236 and 3238 lattices have been performed. For all parameter sets,
corresponding zero temperature calculations have been performed on lattices of size 16332 and 24332. In a few cases
we used lattices of size 242 32 48 as well as 324. The length of individual calculations on the finite temperature
· ·
latticesvariedbetween6500and35000trajectoriesontheN =4latticesand5000to17600iterationsontheN =6
τ τ
lattices, where Metropolis updates were done after hybrid Monte Carlo evolutions of trajctory length τ = 0.5.
MD
At all values of the gauge couplings the length of runs on zero temperature lattices has been adjusted such that the
3 We note that in order to insure thermodynamic consistency the β-function used in the definition of thermodynamic quantities has to
bedeterminedfromthecut-offdependence oftheobservableusedtosetthetemperaturescale,i.e. r0/ainourstudy.