Jet transport coefficient q ^ in (2+1)-flavor lattice QCD
HHU-EP-20/26-RTG
Jet transport coefficient ˆ q in (2+1)-flavor lattice QCD Amit Kumar, Abhijit Majumder, and Johannes Heinrich Weber
2, 3 Department of Physics and Astronomy, Wayne State University, Detroit, MI 48201, USA Department of Computational Mathematics, Science and Engineering & Department of Physics and Astronomy,Michigan State University, East Lansing, MI 48824, USA Institut f¨ur Physik, Humboldt-Universit¨at zu Berlin & IRIS Adlershof, D-12489 Berlin, Germany (Dated: November 13, 2020)We present the first calculation of the jet transport coefficient ˆ q in (2+1)-flavor QCD on a 4-dimensional Euclidean lattice. Carried out in a factorized approach, the light-like propagation of asingle hard parton within a jet is treated separately from the soft gluon field, off which it scatters.The mean square gain in momentum transverse to the direction of propagation is expressed in termsof the field-strength field-strength correlation function, which is then calculated on the lattice.Dispersion relations and the operator product expansion are used to relate the ˆ q defined on thelight-cone with the expectation of a diminishing series of local operators. Calculations of the localoperator products are carried out at finite temperature, over a range of lattice sizes, and comparedwith the results from quenched calculations, perturbative and phenomenological extractions of ˆ q . The study of dense QCD matter, produced in rela-tivistic heavy-ion collisions, using high transverse mo-mentum ( p T ) jets, currently boasts an almost establishedphenomenology [1–3]. The experimental data on variousaspects of jet modification is also extensive, with highstatistics available for a wide range of observables [4–14].Almost all of the evidence points to the formation of aQuark-Gluon Plasma (QGP), a state of matter where theQCD color charge is deconfined over distances larger thanthe size of a proton [15, 16]. Jets are expected to undergoconsiderable modification within the QGP compared toconfined conventional QCD matter [17].While a lot of the theoretical development of jetquenching has been focused on modifications to the par-ton shower, considerably less work has been carried outon the study of the interaction between a parton in thejet with the QGP itself. Most current calculations eithermodel the QGP as a set of slowly moving (or static) heavyscattering centers [17–20], or in terms of Hard-ThermalLoop (HTL) effective theory [21–24]. A third method as-sumes multiple exchanges and an effective Gaussian dis-tribution in the 2-D momentum (transverse to the jetparton) exchange between the parton and the medium.The second moment of this distribution, averaged overevents and scatterings, is referred to as ˆ q [25]:ˆ q = (cid:80) N events i =1 (cid:80) N i ( L ) j =0 [ k i,j ⊥ ] N events × L . (1)The meaning of the above equation is that given a paththrough a medium with a pre-determined density profile,a single parton may scatter N i ( L ) times while travers-ing a distance L < vτ i in event i ( τ i is the lifetime ofthe parton which travels at a speed v ). In each scatter-ing ( j ), it exchanges transverse momentum k i,j ⊥ . In thisLetter, we will only focus on momentum exchanges trans-verse to the direction of the jet parton, as these tend tohave a dominant effect on the amount of energy lost viabremsstrahlung from the parton [20, 26]. In realistic media, created in heavy-ion collisions, thedensity will vary over the length L , and thus, one nec-essarily averages over a non-uniform profile, which couldfluctuate from event to event. In most current simula-tions of the expanding QGP, one assumes that the densitygradients are small enough over a considerable portion ofthe evolution that one can apply relativistic viscous fluiddynamics, which assumes local thermal equilibrium. Sev-eral successful fluid dynamical simulations, which com-pare to RHIC and LHC data [27, 28], have used the equa-tion of state calculated in lattice QCD [29] as an input.Unlike the dynamical medium in a heavy-ion collision,lattice simulations assume static media in thermal equi-librium. The use of lattice QCD input in fluid-dynamicalsimulations is predicated on the ability to reliably coarsegrain the system into space-time unit cells, over which in-trinsic quantities, e.g., temperature ( T ), entropy density( s ), pressure ( P ), remain approximately constant.The calculations in this Letter are an extension of theabove principle: Calculations of ˆ q in the static mediumof lattice QCD will be compared with phenomenolog-ical estimations, where jets are propagated through aQGP fluid dynamical simulation. These QGP simula-tions yield the space-time profiles for intrinsic quantitiesof the medium, e.g. T ( x, y, z, t ), s ( x, y, z, t ) etc. As jetsare propagated through the fluid simulation, the localvalue of ˆ q is obtained as ˆ q T / T , or ˆ q s / s . The fit pa-rameter ˆ q represents the value at T = T , the maximumtemperature at thermalization ( s is the entropy densityat T ). It is dialed to fit high p T hadron data. In this Let-ter, we will compare the dimensionless ratio ˆ q / T betweenthese different approaches [3].Following the framework in Ref. [30], we consider ajet parton with high energy E and virtuality Q suchthat E (cid:29) Q (cid:29) µ D , the Debye mass in the medium.The choice of a large E, Q leads to a diminished cou-pling α S ( Q ) with the medium, due to asymptotic free-dom [31, 32]. As a result, interactions between the jet a r X i v : . [ h e p - l a t ] N ov parton and a medium of limited extent will be dominatedby a single gluon exchange. In this limit, the expressionfor ˆ q simplifies to the form that will be used henceforth:ˆ q = (cid:80) N events i =1 [ k i ⊥ ] N events × L . (2)In light cone coordinates, the incoming quark, travel-ing in the − z direction, has two non-zero components, q + = ( q + q ) / √ (cid:28) q − = ( q − q ) / √ . The quark under-goes a single scattering off the gluon field in the mediumand gains transverse momentum k ⊥ (Fig. 1). In thisframe, the momentum of the quark changes from, q i ≡ [ q + , q − , , → q f ≡ [ q + + k ⊥ / (2 q − ) , q − , (cid:126)k ⊥ ] . (3)The matrix element for this process is given as M = (cid:104) q f |⊗(cid:104) X | (cid:82) T I dtd xg ¯ ψ q ( x ) γ µ t a A aµ ( x ) ψ ( x ) | n (cid:105)⊗| q i (cid:105) , where | n (cid:105) and | X (cid:105) represent the initial and final state of themedium. The factors ψ ( x ), ¯ ψ ( x )[= ψ † γ ] and A a ( x ) rep-resent the quark and gluon wave functions (and complexconjugate), with coupling g . The spatial integrations arelimited within a volume V = L and the time of interac-tion ranges from 0 to T I = L / c (we use particle physicsunits with (cid:126) , c = 1). Replacing the average over events,with an average over all initial states | n (cid:105) (energy E n )of the medium, weighted by a Boltzmann factor, with β = 1 /T the inverse temperature and Z the partitionfunction of the thermal medium, we obtain,ˆ q = (cid:88) n,X e − βE n ZT I (cid:90) d k ⊥ k ⊥ M ∗ n + q i → q f + X M n + q i → q f + X , (4) FIG. 1. A forward scattering diagram for the hard quarkundergoing a single scattering off the gluon field in the plasma.The vertical dashed line represents the cut-line.
Following standard methods outlined in Ref. [30],where factors of k ⊥ are turned into partial derivativesin y ⊥ and y − , we obtain the following well known ex-pression for ˆ q (assuming (cid:80) X | X (cid:105)(cid:104) X | = 1),ˆ q = c (cid:90) dy − d y ⊥ (2 π ) d k ⊥ e − i (cid:126)k ⊥ q − y − + i(cid:126)k ⊥ .(cid:126)y ⊥ × (cid:88) n (cid:104) n | e − βE n Z g Tr[ F + ⊥ µ (0) F + ⊥ µ ( y − , y ⊥ )] | n (cid:105) , (5)where c = 4 √ C R / ( N c − , C R (= C F for a quark) isthe representation specific Casimir, N c is the number ofcolors. In order to connect with non-perturbative calcu-lations, we replace g with the bare coupling constant g at the vertex between the hard quark and the glue field, F µν = t a F aµν is the bare gauge field-strength tensor.Computing the thermal and vacuum expectation valueof the non-perturbative operator g F + ⊥ µ (0) F + ⊥ µ ( y − , y ⊥ )is challenging due to the near light-cone separation be-tween the two field-strength tensors. The separation isslightly space-like y = − y ⊥ <
0, similar to the case of aparton distribution function (PDF), as discussed in thework of Ji [33]. In that effort, one evaluates the PDF in aframe where the nucleon moves with a large momentumin the z -direction, yielding a series of covariant deriva-tives in y z ≡ y . The current Letter will reach a similarconclusion, though using analytic techniques.To recast ˆ q in terms of a series of local operators, weapply a method of dispersion as outlined in Ref. [30]. Inthis approach, a generalized coefficient is defined as,ˆ Q ( q + ) = c (cid:90) d yd ke iky (2 π ) q − (cid:104) g Tr[ F + ⊥ µ (0) F + ⊥ µ ( y )] (cid:105) ( q + k ) + i(cid:15) , (6)where (cid:104) . . . (cid:105) ≡ (cid:80) n (cid:104) n | . . . | n (cid:105) e − βE n /Z . The object ˆ Q ( q + )has a branch cut in a region where q + ∼ T (cid:28) q − corre-sponding to the quark propagator with momentum q + k going on mass shell (Fig. 1). In this region, the incom-ing hard quark is light-like, i.e. q = 2 q + q − ∼
0, and thediscontinuity of ˆ Q ( q + ) is related to the physical ˆ q as Disc [ ˆ Q ( q + )]2 πi (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) at q + ∼ T = ˆ q. (7)In addition to the thermal discontinuity, ˆ Q ( q + ) alsohas an additional vacuum discontinuity in the region q + ∈ (0 , ∞ ) due to real hard gluon emission processes.In this region, the incoming hard quark is time-like. Ifinstead, one takes q + (cid:28)
0, e.g. q + = − q − , the incomingquark becomes space-like and there is no discontinuityon the real axis of q + . In this deep space-like region, thequark propagator can be expanded as follows:1( q + k ) (cid:39) − q − ( q − − ( k + − k − )) = − q − ) ∞ (cid:88) n =0 (cid:32) √ k z q − (cid:33) n . (8)Using integration by parts, the factor of exchangedgluon momentum k z [Eq. (8)] is replaced with the regularspatial derivative ∂ z acting on the field-strength F + ⊥ µ ( y )[Eq. (6)]. A set of higher order contributions from gluonscattering diagrams can be added to promote the reg-ular derivative to a covariant derivative (in the adjointrepresentation). With all factors of k removed from theintegrand [Eq. (6)], except for the phase factor, k can beintegrated out ( (cid:82) d ke iky ) to yield δ ( y ), setting y to theorigin. This yields ˆ Q ( q + = − q − ) [ ≡ ˆ Q | q + = − q − ] as,ˆ Q (cid:12)(cid:12)(cid:12) q += − q − = c (cid:104) g Tr[ F + ⊥ µ (0) ∞ (cid:88) n =0 (cid:32) i √ D z q − (cid:33) n F + ⊥ µ (0)] (cid:105) /q − . (9)In the above equation, each term in the series is a lo-cal gauge-invariant operator, and hence, one can directlycompute their expectation value on the lattice.To relate ˆ Q ( q + = − q − ) to the physical ˆ q , consider thefollowing contour integral in the q + complex-plane: I = (cid:73) dq + πi ˆ Q ( q + )( q + + q − ) = ˆ Q ( q + = − q − ) , (10)where the contour is taken as a small anti-clockwise cir-cle centered around point q + = − q − , with a radius smallenough to exclude regions where ˆ Q ( q + ) may have discon-tinuities. Alternatively, the integral can be evaluated byanalytically deforming the contour over the branch cutof ˆ Q ( q + ) for q + ∈ ( − T , ∞ ) and obtaining Eq. (10) as:ˆ Q ( q + = − q − ) = T (cid:90) − T dq + πi Disc [ ˆ Q ( q + )]( q + + q − ) + ∞ (cid:90) dq + πi Disc [ ˆ Q ( q + )]( q + + q − ) . (11)The limits − T and T in the first integral represent lowerand upper bounds of q + [= k + + k ⊥ / (2 q − + 2 k − ) ], beyondwhich the thermal discontinuity in ˆ Q ( q + ) on the real axisof q + is zero. In this region, the hard incoming quark isclose to on-shell, i.e. q = 2 q + q − ≈ q + ∈ (0 , ∞ ) undergoes vacuum-like splitting. Hence, thesecond integral is temperature independent.Using Eqs. (7-11), we obtain (suppressing y − = y ⊥ = 0),ˆ qT = c ∞ (cid:80) n =0 (cid:16) Tq − (cid:17) n (cid:28) g T Tr (cid:20) F + ⊥ µ (cid:16) i √ D z T (cid:17) n F + ⊥ µ (cid:21)(cid:29) (T − V)( T + T ) / T , (12)where the subscript T-V represents the vacuum sub-tracted expectation value and T + T (cid:39) √ T representsa width of the thermal discontinuity of ˆ Q ( q + ). The aboveexpression of transport coefficient ˆ q contains several fea-tures. First, each term in the series is local, allowing fortheir computation on the lattice. Second, the successiveterms in the series are suppressed by the hard scale q − ,and hence, computing only first few terms will be suffi-cient [34]. We also note that the expression of ˆ q given inEq. (12) is applicable for the hard quark traversing eitherthe pure glue plasma or a quark-gluon plasma.To compute the operators on the lattice, we performthe Wick rotation x → − ix , A → iA = ⇒ F i → iF i .We study the first three non-zero operators in the ˆ q se-ries [35], ˆ O n = g / T (cid:80) i =1 Tr( F i [ i √ D z / T ] n F i − [3 → g F µν via tadpole-improved clover-leaf op-erators projected to anti-Hermitian traceless matrices, ig F µν ( x ) = P Q µν ( x ) a L u = ig F µν + O ( a L ) (13)with Q µν = [ U µ,ν + U ν, − µ + U − µ, − ν + U − ν,µ ] / U µ,ν beingthe plaquette) and P Q = ( Q − Q † − Tr[ Q − Q † ] /N c ) /
2. We define u = (cid:112) (cid:104) Tr[ U µ,ν ] (cid:105) /N c and use symmetric, tadpole-improved differences in covariant derivatives. Note thateach operator contains temperature-independent, addi-tive contributions that diverge in the continuum limit,but cancel exactly in the vacuum-subtracted operators.We generate T > n σ / n τ = 4for n τ = 4 , T = / ( a L n τ ) ], and also corresponding T = 0 lattices with n τ = n σ using the MILC code [36]and the USQCD software stack [37]. We use the Ra-tional Hybrid Monte Carlo (RHMC) algorithm [38] withhighly improved staggered quark (HISQ) action [39] andtree-level Symanzik gauge action [40, 41] for (2+1)-flavorQCD. The leading cutoff effects are O ( a ) and O ( g a ).We employ tuned input parameters (bare lattice cou-pling β = 10 /g , and bare quark masses), and use the r lattice scale following Refs. [40–43] by the HotQCDand TUMQCD collaborations. This setup has a physi-cal strange and two degenerate light quarks with m l = m s /
20 corresponding to a pion mass of about 160 MeVin the continuum limit. We generate pure gauge ensem-bles via the heat-bath algorithm using Wilson gauge ac-tion [44] with β = 6 /g and leading cutoff effects O ( a ).See Supplemental Material for details about the gaugeensembles [45]. T − − − − − −
10 110 > i O < =4 τ n =6 τ n =8 τ n > O > < O > < O < ppp ) × ( ) × ( ) × ( ) × ( ) × ( ) × ( FIG. 2. T and n τ dependence of the expectation values ofvacuum-subtracted operators ˆ O n in (2+1)-flavor QCD. Thefeatures are qualitatively similar in SU(3) pure gauge theory.The bands represent continuum estimates. Our data set does not show unambiguous linear scal-ing towards the continuum due to lack of fine enoughlattices (three ensembles with n τ ≥ n τ = 8, especially inˆ O , ˆ O . In order to provide a continuum estimate [46],we first interpolate for each n τ the expectation value ofˆ O n in the temperature using splines and propagate theerrors via bootstrap. Then we extrapolate linearly to thecontinuum at fixed temperature by varying the subset of n τ values and use either zero, one, or two monomials in1 /n τ . If we cannot propagate the error directly, we resortto bootstrap for the error propagation. Finally, we esti-mate the continuum limit as the average of all fits anda corrected error from the spread of the fits if the latterexceeds the naively propagated error.The vacuum-subtracted expectation values of the firstthree non-zero operators ˆ O n , n = 0 , ,
2, computed in(2+1)-flavor QCD over a range of temperatures covering150MeV 220 MeVin (2+1)-flavor QCD or at T ≈ 300 MeV in the SU(3)pure gauge theory, their impact on ˆ q is negligible at theachievable accuracy. This can be understood from thefact that the ˆ O n are suppressed in Eq. (12) by powers ofthe ratio ( T / q − ) n compared to the leading term ˆ O . T / T q Lattice [(2+1)-flavor]Lattice [Pure SU(3)] =0] f N HTL [ =3] f N HTL [JETJETSCAPE FIG. 3. Lattice QCD determination of ˆ q for a 100 GeV quarktraversing a pure glue and (2+1)-flavor QCD plasma. Contin-uum results are estimated using n τ = 8 , α S ( µ ) evaluated at 2 πT ≤ µ ≤ πT . Given the one gluon exchange approximation [Eq. (9)],the q − dependence of ˆ q / T is indeed small. In the fol-lowing, the light-cone momentum q − of the hard quarktraversing the plasma was taken to be 100 GeV. Thetemperature dependence of the resulting ˆ q / T is shownin Fig. 3 for the continuum estimates using both thequenched SU(3) or unquenched (2+1)-flavor lattices.The red band represents ˆ q / T for (2+1)-flavor QCD(HISQ and Symanzik gauge action) and constrains ˆ q / T =2 . T = 800 MeV. The blue band represents ˆ q / T as a function of T for SU(3) pure gluon plasma (Wilsongauge action) and constrains ˆ q / T = 1 . T = 800 MeV. Errors in both results are large due to the chal-lenges in the continuum extrapolation.The transport coefficient ˆ q / T exhibits a rapid rise inmagnitude in the transition region and slightly above,i.e. in the temperature range 150 MeV (cid:46) T (cid:46) 250 MeVfor (2+1)-flavor QCD or 250 MeV (cid:46) T (cid:46) 350 MeV forthe SU(3) pure gauge theory, and is flat within errorsabove 400 MeV. This is similar to the T dependence ofthe scaled entropy density s / T [50]. The reduced mag-nitude of the upward shift of the (2+1)-flavor QCD ˆ q / T calculations can be understood by scaling the additionalquark degrees of freedom with the ratio of quark-to-gluonCasimir [ C F / C A = ( N c − / (2 N c ) = / ]. Expectedly, thelattice results do not show any log-like rise at lower T , asone observes in any leading order (LO) perturbative HTLbased calculation [51] (Note the HTL bands in Fig. 3).In Fig. 3, we also present a comparison with phe-nomenological extractions of ˆ q / T extracted by theJET [3] and JETSCAPE [48] collaborations. The JETcollaboration applied several disparate models of energyloss with either a sole T dependence of the ratio ˆ q / T ,or one obtained from HTL effective theory [52]. TheJETSCAPE extraction applied an amalgam of theoriesfor different epochs of the jet shower, with a data-driven(Bayesian) determination of ˆ q / T , allowed to depend on T , the energy and scale of a given parton in the shower.In this Letter, we carried out the first semi-realistic(2+1)-flavor lattice QCD calculation of the jet quenchingparameter ˆ q , which is the leading coefficient affecting jetmodification in the QGP. This represents a rigorous first-principles framework for calculations of ˆ q . Continuum es-timates were presented for the SU(3) pure gluon plasmaand (2+1)-flavor QCD. While the proximity of the lat-tice calculations with phenomenological extractions is en-couraging, several caveats need to be considered: Theperturbative portions of the current calculation will haveto be extended to higher-order, at which point ˆ q willhave to be renormalized (including possible operator mix-ing). The ˆ q used in the JET and JETSCAPE extraction(carried out in the single gluon approximation) are simi-larly not renormalized (the JETSCAPE extractions allowfor E, µ dependence but no operator mixing). Improvedcomparisons in the future will require rigorous control ofthe continuum extrapolation through the use of finer lat-tices and higher statistics. The current lattice calculationincluded a single parton undergoing a single scattering offthe medium simulated on the lattice. Future efforts willneed to consider the effect of gluon emission and multipleinteractions within the lattice, and the matching of therenormalized lattice results to a continuum scheme, e.g., M S [53]. Phenomenological extractions will also need tobe performed with a renormalized ˆ q in the same scheme. Acknowledgements. This work was supported in partby the National Science Foundation (NSF) under grantnumber ACI-1550300 (SSI-JETSCAPE), by the U.S. De-partment of Energy (DOE) under grant number DE-SC0013460. 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Majumder, Phys.Rev.Lett. , 262301(2010), arXiv:0910.3016 [hep-ph]. [52] S. Caron-Huot, Phys. Rev. D , 065039 (2009),arXiv:0811.1603 [hep-ph].[53] S. Capitani, Phys. Rept. , 113 (2003), arXiv:hep-lat/0211036 [hep-lat].[54] A. Hasenfratz and P. Hasenfratz, , 241 (1980).[55] S. Booth et al. (QCDSF-UKQCD), Phys. Lett. B ,229 (2001), arXiv:hep-lat/0103023.[56] A. Deur, S. J. Brodsky, and G. F. de T´eramond, Progressin Particle and Nuclear Physics , 1–74 (2016). Supplemental Material for “Jet transportcoefficient ˆ q in (2+1)-flavor lattice QCD” In this document, we list the parameters used in gen-erating gauge ensembles for pure SU(3) and (2+1)-flavorQCD lattices. In the presented lattice calculations, theunquenched lattices were generated at the physical valueof the strange quark mass m s and the light sea quarkmasses of m l = m s / 20 using the HISQ and tree-levelSymanzik improved gauge action. We employed the Ra-tional Hybrid Monte Carlo algorithm (RHMC). In TableI, II and III, we present the strange quark mass ( am s ) inunits of lattice spacing a , temperature ( T ) and time units(TU) for n τ = 4 , , T = 0.The temperatures for different β = 10 /g ’s have beenfixed using the r scale and taken from Refs. [41]. β = 10 /g am s T (MeV) T (cid:54) =0) T =0)5.9 0.132 201 10000 100006.0 0.1138 221 10000 100006.285 0.079 291 10000 100006.515 0.0603 364 10000 100006.664 0.0514 421 20000 100006.95 0.0386 554 10000 100007.15 0.032 669 10000 100007.373 0.025 819 10000 10000TABLE I. The parameters to generate (2+1)-flavor QCDgauge ensembles with m l = m s / 20 for lattice size n τ = 4with aspect ratio n s /n τ = 4 . β = 10 /g am s T (MeV) T (cid:54) =0) T =0)6.0 0.1138 147 10000 100006.215 0.0862 181 10000 100006.285 0.079 194 10000 100006.423 0.067 222 7600 100006.664 0.0514 281 10000 70006.95 0.0386 370 10000 80007.15 0.032 446 10000 86007.373 0.025 547 10000 100007.596 0.0202 667 8600 100007.825 0.0164 815 9140 10000TABLE II. The parameters to generate (2+1)-flavor QCDgauge ensembles with m l = m s / 20 for lattice size n τ = 6with aspect ratio n s /n τ = 4 . In Table IV, V and VI, we provide β , temperature andthe collected statistics for pure SU(3) lattices. The scalesetting was done using the two-loop perturbative renor-malization group (RG) equation with non-perturbativecorrection factor [ f ( β )] given as a = f ( β )Λ L (cid:20) g π (cid:21) − exp (cid:20) − π g (cid:21) (1) where Λ L is a lattice parameter. We estimated the non-perturbative factor by adjusting the function f ( β ) suchthat T c / Λ L is independent of bare coupling constant g .In this calculation, the Λ L was set 5 . T c ≈ 265 MeV [50]. β = 10 /g am s T (MeV) T (cid:54) =0) T =0)6.515 0.0604 182 7300 64006.575 0.0564 193 8650 68006.664 0.0514 211 10000 50006.95 0.0386 277 10000 59507.28 0.0284 377 10000 65507.5 0.0222 459 10000 50007.596 0.0202 500 10000 94007.825 0.0164 611 10000 79008.2 0.01167 843 10000 5000TABLE III. The parameters to generate (2+1)-flavor QCDgauge ensembles with m l = m s / 20 for lattice size n τ = 8with aspect ratio n s /n τ = 4 . β = 6 /g T (MeV) T (cid:54) =0) T =0)5.6 209 10000 100005.7 271 10000 100005.8 336 10000 100005.9 406 10000 100006.0 482 10000 100006.2 658 10000 100006.35 816 10000 100006.5 1003 10000 100006.6 1146 10000 10000TABLE IV. The parameters to generate pure SU(3) gaugeensembles using Wilson’s pure SU(3) gauge action for latticesize n τ = 4 with aspect ratio n s /n τ = 4 . β = 6 /g T (MeV) T (cid:54) =0) T =0)5.60 139 10000 100005.85 247 10000 100005.90 271 10000 100006.00 321 10000 100006.10 377 10000 100006.25 472 10000 100006.45 625 10000 100006.60 764 10000 100006.75 929 10000 100006.85 1056 10000 10000TABLE V. The parameters to generate pure SU(3) gauge en-sembles using Wilson’s pure SU(3) gauge action for latticesize n τ = 6 with aspect ratio n s /n τ = 4 . β = 6 /g T (MeV) T (cid:54) =0) T =0)5.70 135 10000 100005.95 221 10000 100006.00 241 10000 100006.10 283 10000 100006.20 329 10000 100006.35 408 10000 100006.55 536 10000 100006.70 653 10000 100006.85 792 10000 100006.95 899 10000 10000TABLE VI. The parameters to generate pure SU(3) gaugeensembles using Wilson’s pure SU(3) gauge action for latticesize n τ = 8 with aspect ratio n s /n ττ