EEquation of State from Lattice QCD Calculations
Rajan Gupta
Theoretical Division, Los Alamos National Lab, Los Alamos, N.M. 87545, USA
Abstract
We provide a status report on the calculation of the Equation of State (EoS) of QCD at finitetemperature using lattice QCD. Most of the discussion will focus on comparison of recent resultsobtained by the HotQCD and Wuppertal-Budapest (W-B) collaborations. We will show that verysignificant progress has been made towards obtaining high precision results over the temperaturerange of T = 150 −
700 MeV. The various sources of systematic uncertainties will be discussedand the differences between the two calculations highlighted. Our final conclusion is that thelattice results of EoS are getting precise enough to justify being used in the phenomenologicalanalysis of heavy ion experiments at RHIC and LHC.
Key words:
Lattice QCD, Equation of State
PACS:
1. The Road to Precision Lattice QCD Calculations
One of the goals of simulations of lattice QCD is to provide a precise non-perturbativedetermination of the EoS of QCD over the temperature range 150 −
700 MeV that isbeing probed in experiments at RHIC and the LHC. The EoS, along with the transitiontemperature T c and transport coefficients such as shear viscosity, are crucial inputs intophenomenological hydrodynamical models used to describe the evolution of the quarkgluon plasma (QGP). In this talk I will mainly review the two recent and most completecalculations of the EoS by the HotQCD [1] and Wuppertal-Budapest [2] collaborations.Simulations of lattice QCD at finite temperature are carried out on a 4-D hypercubeof size aN τ × aN S × aN S × aN S where a is the lattice spacing usually denoted in unitsof GeV − or fermi. The spatial size aN S is taken large enough so that finite volumecorrections are under control and small. Past calculations show that for finite temperaturesimulations with N S /N τ = 4 these corrections are smaller than statistical errors for T < ∼ T c . For higher temperatures larger N S may be required. In QCD, the gauge coupling β ≡ /g is related to a by dimensional transmutation and the continuum theory isrecovered in the limit a → g → β → ∞ . To provide a perspective Preprint submitted to Elsevier 1 November 2018 a r X i v : . [ h e p - l a t ] A p r n how fine the current lattice simulations are note that N τ = 10 corresponds to a ≈ . ∼ − at the transition temperature T ∼
200 MeV.The second set of control parameters (inputs) in the simulations are the quark masses.The results discussed in this review are for 2 + 1 flavors, i.e. , two flavors of degenerate u and d quarks and a heavier strange quark. In nature 2 m s / ( m u + m d ) ≈ . m = ( m u + m d ) / ∼ . m s ∼
90 MeV. The small value of m provides thekey computational challenge because the most time consuming part of the simulationsis the inversion of the Dirac operator, a very large sparse matrix. This inversion is doneusing interative Krilov solvers that have critical slowing down in the limit m →
0. Thecomputational cost increases as m − / or faster and calculations become very expensivewith decreasing quark mass. For this reason 2 + 1 flavor simulations are typically donefixing the strange quark mass to its physical value and simulating at a number of valuesof m/m s from which extrapolations to the physical m are made. Recent calculationsby the W-B and HotQCD collaborations show that the computer power has reached astage where simulations can be done close to, or directly at, physical m . Thus, in thestate-of-the-art simulations, this source of systematic error (extrapolation in m ) is nowunder good control.The tuning of the set of parameters { g, m s , m } is done as follows. One first fixes theratio m s /m , ideally to m s /m = 27 .
5. Then for a judiciously chosen value of g , zerotemperature simulations are done to measure two independent physical quantities whosevalues are experimentally measured or well determined, and one of which is sensitive tostrange quark mass, for example the K-meson mass M K and the pion decay constant f π (or f K and M π ). The value of m s is tuned until the lattice results for these two quantitiesmatch their physical values. This fixes a and m s . Now depending on how finely one wantsto scan in T (or a ) a new value of g is chosen and the value of m s is again tuned toreproduce the observables, thus determining the new a keeping m s /m fixed. This processgenerates a set of { g, m s , m } values for which, by construction, the physics (defined bymatching lattice M K and f π to physical values) is fixed. This line in the { g, m s , m } space,since m s is tuned to the physical value and m s /m is fixed, is called a line of constantphysics (LoCP). The utility of simulating along LoCP is to reduce the three dimensionalspace of input parameters to a line along which only the lattice spacing is changing.This procedure provides better control over taking the continuum limit. The extent towhich { g, m s } would have varied had one chosen two different physical quantities, say M ss and M N , is a measure of variations in discretization errors along different LoCP. Theemphasis of the lattice community is to present results extrapolated to a → a , by simulatingat a number of values of a . In case of finite temperature calculations we extrapolateresults at fixed T to N τ → ∞ by simulating at a number of values of N τ .The lattice size in the Euclidean time direction defines the temperature of the system bythe relation T = 1 /aN τ . The scale a for fixed { g, m s , m } is the same for zero-temperatureand finite temperature lattices. Thus, knowing a corresponding to a given g and quarkmasses uniquely determines T for a given N τ . An important consequence of the fact that g or equivalently a or T is the single parameter that controls lattice simulations is thatonly one thermodynamic quantity can be determined, which for the extraction of EoS isthe trace anomaly I/T ≡ ( ε − p ) /T , where I is called the integration measure.The above approach for scanning in T is called the fixed N τ approach. A secondapproach that I will not cover in detail and which is being pursued by the WHOT2ollaboration [3] with improved Wilson fermions is called the fixed a approach. In thisapproach, for a given a (the same process is used for fixing { g, m s , m } ) one simulates ona number of different N τ lattices to scan in T . The advantage of this approach is thatonly a single zero-temperature matching calculation, needed to carry out subtractions oflattice artifacts in finite T data, is required for each a . The weakness of this approachis that the scan in T is limited by the coarseness and range of N τ values possible, i.e.N τ = 6 , , , ,
14, before one runs out of computational power. The recent results usingthe fixed a approach by the WHOT collaboration [3] are very encouraging and I referinterested readers to their paper for details.
2. Taste Symmetry breaking with Staggerd fermions
In the naive discretization of the Dirac action there automatically is a 2 doublingof flavors. In the staggered fermion approach, using a lattice symmetry called “spindiagonalization”, the degeneracy is reduced from 16 to 4 by placing a single degree offreedom at each lattice site. Under this construction a 2 hypercube is the basic unit cellthat reduces to a point in the continuum limit. The 16 degrees of freedom in the cellrepresent, in the continuum limit, four identical copies (called taste) of Dirac fermions. Onthe lattice this four-fold degeneracy gives rise to a proliferation of particles propagatingin the QCD vacuum, for example, there are 16 pions distinguished by their taste ratherthan just one. If taste symmetry was unbroken at finite a , the 4-fold degeneracy couldbe handled by just dividing results by the appropriate degeneracy factor. Problems arisebecause this degeneracy is broken at finite a and one does not know a priori how large thissystematic error is and how it effects simulations of QCD thermodynamics in particular.The most common approach to quantify this effect is to study the difference in themasses of the 16 pions, and determine how these differences vanish when lattice resultsare extrapolated to the continuum limit.The HotQCD collaboration has studied the consequences of taste symmetry breakingutilizing three versions of improved staggered fermions − the asqtad, p4 and HISQ/treeformulations [4]. In Fig. 1 we show preliminary HotQCD HISQ/tree results for M π − M G where M G is the Goldstone pion mass versus a , and also compare with results from thestout and asqtad actions. The large spread in masses that increases on coarser latticesshows that taste symmetry is indeed badly broken [5]. Based on such studies the conclu-sion is that at any given a , the taste breaking is least in HISQ/tree followed by stout,asqtad, and p4 actions. One consequence of this discretization error for thermodynamicsis that the contribution of any state, for example the pion, is not just from the lowesttaste state (the Goldstone pion) but is some weighted average of the 16 pions (or theappropriate multiplets for other states). Thus, at any given a the effective masses of allhadrons are larger than the desired ground state value. The magnitude of the effect isshown in the right panel of Fig. 1 which plots the root mean squared mass of the 16 pionstates corresponding to a Goldstone pion mass of 140 MeV. Results at low temperatures, T <
150 MeV, and on small N τ lattices are most susceptible to this discretization error.Taste breaking also puts a caveat on the above described tuning of quark masses: setting m s /m = 27 . m s /m values to control the chiral extrapolationfollowed by taking the continuum limit using a number of N τ lattices provides the best3 (cid:47) -m G2 )/(200 MeV) a [fm ] (cid:97) i (cid:97) (cid:97) (cid:97) (cid:97) i (cid:97) j (cid:97) i (cid:97) (cid:97) i (cid:97) (cid:97) i (cid:97) stout, (cid:97) i (cid:97) j (cid:47) [MeV] a [fm]RMS pion 306 MeVRMS pion 200 MeV HISQ/treestoutasqtad Fig. 1. The splitting between the 16 pions due to taste symmetry breaking for HISQ/tree, stout andasqtad actions (left panel). The right panel shows the root mean squared mass corresponding to aGoldstone pion mass of 140 MeV. Note that the HISQ action used so far for thermodynamics does notinclude tadpole improvement of the gauge action and is therefore called HISQ/tree in [4]. understanding of systematic errors and for obtaining physical results.A second issue with staggered simulations that include the strange quark (or in futurethe charm quark) is the need to take the fourth root of the determinant to compensatefor the 4-fold degeneracy. Creutz [6] claims that this “rooting” is a fundamental flawof the staggered formulation, however, its effect in current simulations may be smallsince the strange quark mass is large, whereas the review by Sharpe [7] (covering a largebody of work) shows that while the staggered formulation may be ugly, it gives physicalresults in the continuum limit. (For degenerate u and d quarks, this rooting problemis overcome because of a lattice symmetry whereby the square root of the determinantcan be written as the determinant on even (or odd) sites.) In a perfect world one wouldlike to use a Wilson-like action that maintains the continuum flavor structure and chiralsymmetry, i.e. , domain wall or overlap fermions, however, to date most thermodynamicalsimulations use staggered fermions for two reason: they are much faster (10 − × ) tosimulate than even simple Wilson fermions and because of a residual chiral symmetrythat protects the Goldstone pion.
3. The Trace Anomaly
The results from the HotQCD [1] [4] and W-B [2] collaborations for
I/T ≡ ( ε − p ) /T ,the single thermodynamic quantity calculated on the lattice, are shown in Fig. 2.Before making detailed comparisons it is important to stress here, and applicable to alldiscussion that follows, that the HotQCD results do not yet incorporate extrapolationto the physical quark mass or the continuum limit. The most extensive data are for N τ = 6 , m s /m = 10 with new ongoing calculations at m s /m = 20 and N τ = 12.The W-B results at N τ = 6 , m s /m includingat m s /m = 28 .
15 where the final physical value is quoted. (Recall, however, the caveatabout the uncertainty in locating the physical value of m due to taste breaking.) TheW-B data at N τ = 10 and 12 are more limited in T values and are at m s /m = 28 . N τ = 8 results to represent the continuum value. Data with4 (cid:161) -3p)/T HISQ/tree: N (cid:111) =8N (cid:111) =6p4: N (cid:111) =8asqtad: N (cid:111) =12N (cid:111) =8stouts95p-v1
Fig. 2. Integration measure
I/T from W-B (left) and preliminary HotQCD (right) comparing differentactions. The curve (right panel) is a a parametrization of ε − p/T that is based on HRG and thep4/asqtad lattice data at high temperatures [8]; and the W-B data are shown in purple for comparison. N τ = 10 at T ≤
365 MeV (red points in left panel of Fig. 2) and the three points (greenpoints in Fig. 2) with N τ = 12 provide a consistency check since they show no significantdiscretization effects relative to N τ = 8 data.The overall form of the results by the two collaborations is similar. There are, however,two significant differences between the HotQCD and W-B data. The first is the value of I at the peak, ∼ . .
1, and the peak in the W-B data is shifted to lower T byabout 20 MeV. W-B collaboration attribute these differences to the lack of extrapolationof HotQCD data in quark mass and a , i.e. , residual discretization errors. PreliminaryHotQCD results with N τ = 8 lattices using HISQ/tree fermions (also shown in Fig. 2)give a similar peak height and position as the asqtad action (the p4 results are higherbut decreasing with N τ ). The agreement between asqtad and HISQ/tree and since theHISQ/tree action is the more improved than stout, has smaller discretization errors andless taste breaking, it is not clear if, today, we have a simple resolution of the difference.The forthcoming results with HISQ/tree action on N τ = 8 and 12 lattices being simulatedby the HotQCD collaboration should help clarify these issues.
4. Pressure, Energy Density, Entropy and Speed of Sound
The pressure p can be determined from the trace anomaly using the following relations: IT ≡ Θ µµ ( T ) T ≡ ε − pT = T ∂∂T (cid:16) pT (cid:17) (1) p ( T ) T − p ( T ) T = T (cid:90) T dt Θ µµ ( t ) t (2)The results for pressure and energy density are summarized in Figs. 3 and 4. To obtainpressure p there are two issues that need to be addressed when carrying out the inte-gration in Eq. 2. The first is to construct a smooth function that represents the latticedata for ( ε − p ) /T over the whole range of T since I/T has been calculated only at a5 ig. 3. HotQCD collaboration [1] results for ε/T and 3 p/T on N τ = 8 lattices with m s /m = 10. finite number of values of T . The second is the choice of T above which Θ µµ ( T ) /T iswell-determined and at which point p ( T ) can be estimated reliably.The HotQCD collaboration has investigated a number of ansatz for parameterizingΘ µµ ( T ) /T and find that the results for p do not vary significantly. The uncertainty dueto the ansatz is shown by the error bars on p/T at T = 275 ,
540 MeV in Fig. 3. TheW-B collaboration uses a variant of the method − they parameterize the pressure itselfand then evaluate its derivatives to match to I . The band in Fig 4 show the uncertainty.The second issue is more significant. The systematic errors in lattice data grow as T islowered and are expected to be large below T = 150 MeV. At the same time p ( T = 150)is not negligible and a priori unknown. One approach is to use the hadron resonance gas(HRG) model for p ( T = 150 MeV). This requires that there be reasonable agreementbetween the HRG and lattice values at T = 150 MeV. The HotQCD data approaches theHGR from below and at T = 150 MeV there is a significant difference. Another approachis to use T = 100 MeV where there is more confident in the HRG value but then onehas to confront the uncertainty in matching and parameterizing Θ µµ ( T ) /T between T = 100 −
150 MeV. The HotQCD collaboration use p = 0 at T = 100 MeV for theircentral value and the HRG value to estimate the uncertainty whose magnitude is shownby the black square on ε/T at T = 550 MeV in Fig. 3. The W-B collaboration showthat a modified “lattice” HRG calculation, taking into account taste breaking in pionand kaon states, fits the lattice data between T = 100 −
140 MeV. Nevertheless, theychoose p ( T = 100 M eV, m s = m ) = 0 for the normalization.Once I and p are determined the energy density is given by ε/T = I/T + 3 p/T ,entropy by s = ( ε + p ) /T and the speed of sound c s by c s = dpdε = ε d ( p/ε ) dε + pε . (3)A comparison of results for ε , p and s is shown in Figures 3, 4, and 5.The W-B collaboration apply two corrections to the estimate for p ( T ). The first is toguarantee that the lattice results for each N τ match the continuum Stefan-Boltzmannvalue at T = ∞ . To do this they construct the ratio of the continuum Stefan-Boltzmannvalue for p to its free-field ( T = ∞ ) lattice value (the continuum integrals are replaced bylattice sums appropriate for each N τ ) and then correct the lattice data at all T by this6 ig. 4. Results for ε/T and p/T from the W-B collaboration [2]. To compare results for p note thatthe W-B data are for p/T whereas the HotQCD results in Fig 3 are for 3 p/T .
0 5 10 15 20 100 150 200 250 300 350 400 450 500 550 0.4 0.6 0.8 1 1.2
T [MeV] s/T Tr s SB /T p4: N τ =86 asqtad: N τ =86 Fig. 5. Comparison of entropy density obtained by W-B (left) and HotQCD (right) collaborations. TheHotQCD data are for m s /m = 10. ratio. This ratio is large, 1 .
517 and 1 .
283 for N τ = 6 and 8 respectively. Since N τ = 8data are used to define the continuum estimate, this correction is too large to justifyon the basis of a tree-level improvement of lattice observables (lattice operators used toprobe the physics). Furthermore, this correction is also applied to I , ε and s . The secondcorrection made by the W-B collaboration is to shift upward their results for p/T byhalf the difference, 0 .
06, between the lattice and HRG estimates at T = 100 MeV. Again,to me, this is not a well-motivated correction of data. Future simulations and betterunderstanding of the low T region will hopefully alleviate the need for such corrections.A comparison of results for the speed of sound are shown in Fig. 6. The fundamentalquantity needed to calculate it is p/ε as shown in Eq. 3. Two features in the data areworth commenting on. First, data in Fig. 4 show that, in the transition region fromhadronic matter to QGP, the energy density is changing more rapidly than the pressure.This implies that c s should show a dip in the transition region as is indeed observed.Second, c s rises quickly after the transition region and reaches close to the relativisticBoltzmann gas value of 1 / T ∼
400 MeV.
5. Prospects for improvement in the EoS of QCD in the near future
Significant progress has been made in determining the EoS using lattice QCD in thelast three years. The current lattice results for the EoS have already given the Heavy Ion7 ig. 6. Comparison of the speed of sound c s . The W-B data (left panel) are plotted versus T . TheHotQCD data for p/ε and c s versus the energy density ε using the fits to ε and p (right panel). community a much better understanding of the dynamics of the QGP. Lattice estimatesare now being used in hydrodynamic analysis of the evolution of the QGP in experimentsat the RHIC at BNL and at the LHC.There are a number of ways in which both HotQCD and W-B collaborations areimproving their estimates:– The HotQCD collaboration will present their estimates of the continuum values withHISQ/tree action on N τ = 6 , m s /m = 10 and 20.– Both collaborations will include the charm quark to provide results with (2 + 1 + 1)dynamical flavors. Preliminary partially-quenched estimates suggest that the charmquark contribution starts to become large at above T ∼
300 MeV.– To fully control finite volume effects at
T > N s /N τ = 4.– In addition to simulations with staggered fermions, simulations with improved Wilsonand domain wall fermions are maturing [3] [9] and will provide independent checks ofthe staggered results in, hopefully, the near future.With these improvements a number of unresolved issues such as the location and heightof the peak in I , control over systematic errors at low temperatures, and the impact ofcharm quark at high temperatures, should be addressed over the next couple of years.So stay tuned. Acknowledgements:
I thank T. Nayak, R. Verma and P. Ghosh for the invitationto a very informative conference and acknowledge the support of DOE grant KA140102.
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