aa r X i v : . [ h e p - l a t ] S e p Chiral transition and deconﬁnement transition inQCD with the highly improved staggered quark(HISQ) action
Alexei Bazavov a and Peter Petreczky b (for HotQCD collaboration) a Department of Physics, University of Arizona, Tucson, AZ 85721, USA b Physics Department, Brookhaven National Laboratory, Upton, NY 11973, USA
We report preliminary results on the chiral and deconﬁnement aspects of theQCD transition at ﬁnite temperature using the Highly Improved Staggered Quark(HISQ) action on lattices with temporal extent of N τ = 6 and 8. The chiral aspectsof the transition are studied in terms of quark condensates and the disconnectedchiral susceptibility. We study the deconﬁnement transition in terms of the strangequark number susceptibility and the renormalized Polyakov loop. We made con-tinuum estimates for some quantities and ﬁnd reasonably good agreement betweenour results and the recent continuum extrapolated results obtained with the stoutstaggered quark action. Introduction
Improved staggered fermion formulations are widely used to study QCD at non-zerotemperatures and densities, see e.g. Ref. [1, 2] for recent reviews, for, at least, tworeasons: they preserve a part of the chiral symmetry of continuum QCD which allowsone to study the chiral aspects of the ﬁnite temperature transition, and are relativelyinexpensive to simulate numerically because due to the absence of an additive massrenormalization the Dirac operator is bounded from below. However, lattice artifactsrelated to taste symmetry breaking turned out to be numerically large. To reduce thetaste violations smeared links, i.e. weighted averages of diﬀerent paths on the latticethat connect neighboring points, are used in the staggered Dirac operator and severalimproved staggered formulations, like p4, asqtad, stout and HISQ diﬀer in the choice ofthe smeared gauge links. The ones in the p4 and asqtad actions are linear combinationsof single links and diﬀerent staples [3, 4] and therefore are not elements of the SU(3)group.It is known that projecting the smeared gauge ﬁelds onto the SU(3) group greatlyimproves the taste symmetry . The stout action  and the HISQ action implementthe projection of the smeared gauge ﬁeld onto the SU(3) (or simply U(3)) group and Present address: Physics Department, Brookhaven National Laboratory, Upton, NY 11973, USA The HotQCD Collaboration members are: A. Bazavov, T. Bhattacharya, M. Cheng, N.H. Christ,C. DeTar, S. Gottlieb, R. Gupta, U.M. Heller, C. Jung, F. Karsch, E. Laermann, L. Levkova, C. Miao,R.D. Mawhinney, S. Mukherjee, P. Petreczky, D. Renfrew, C. Schmidt, R.A. Soltz, W. Soeldner,R. Sugar, D. Toussaint, W. Unger and P. Vranas O ( a ) lattice artifacts in thermodynamicquantities. The p4 and asqtad actions implement this improvement by introducing 3-linkterms in the staggered Dirac operator.In this paper we report preliminary results on the chiral and deconﬁnement transitionin QCD at non-zero temperature obtained with the HISQ action which combines theremoval of tree level O ( a ) lattice artifacts with the addition of projected smeared linksthat greatly improve the taste symmetry. We also compare our results to the continuumextrapolated results obtained with the stout action . The Highly Improved Staggered Quark (HISQ) action developed by the HPQCD/UKQCDcollaboration  reduces taste symmetry breaking and decreases the splitting betweendiﬀerent pion tastes by a factor of about three compared to the asqtad action. The netresult, as recent scaling studies show [9, 10], is that a HISQ ensemble at lattice spacing a has scaling violations comparable to ones in an asqtad ensemble at lattice spacing 2 a/ m s was set to its physi-cal value adjusting the quantity q m K − m π = m η s ¯ s ≃ √ Bm s to the physical value686.57 MeV. We used two values of the light quark mass: m l = 0 . m s and m l = 0 . m s .These correspond to the lightest pion mass of about 160 MeV and 320 MeV respectively. Calculations have been performed on 24 × × × lattices for m l = 0 . m s and on 16 ×
32 lattices for m l = 0 . m s . The molecular dynamics (MD) tra-jectories have length of 1 time unit (TU) and the measurements were performed every 5TUs at zero and 10 TUs at ﬁnite temperature. However, for the few smallest beta valuesthe trajectories had length of 1 / / V ( r ). We have calculated the Sommer scale r deﬁned as r dVdr (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) r = r = 1 . . (1)The potential has been normalized to the string potential V string = − π r + σr, (2)2 ∆ l,s N τ =8N τ =6stout cont.cont. ms/20 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 120 140 160 180 200 220T [MeV] ∆ lR N τ =8N τ =6stout, cont.cont. ms/20 Figure 1: The subtracted chiral condensate and ∆ Rl compared to the continuum estimatefor the stout action . The solid black line is our continuum estimate for the HISQaction.at r = 1 . r or equivalently to the value 0 . /r at r = r . The additive constantdetermined by this normalization is used to calculate the renormalization constant forthe Polyakov loop as will be discussed later. To convert from lattice units to physicalunits we use the value r = 0 .
469 fm as determined in .
In the limit of zero light quark masses QCD has a chiral symmetry and the ﬁnite tem-perature transition is a true phase transition. The order parameter for this transition isthe light chiral condensate h ¯ ψψ i l . However, even at ﬁnite values of the quark mass thechiral condensate will show a rapid change in the transition region indicating an eﬀectiverestoration of chiral symmetry. The ﬂuctuations of the chiral condensate, also called thedisconnected chiral susceptibility, will have a peak at the transition temperature. Recentstudies with the p4 action suggest that for the physical strange quark mass and twolight quarks the chiral transition is of the second order for vanishing light quark mass,belonging to the O ( N ) universality class . Thus, the universal properties of the chiraltransition in the limit of zero light quark masses govern the transition for suﬃcientlysmall but non-zero light quark masses . The corrections to scaling turned out tobe small for m l = m s /
20 and thus the temperature variation of the chiral condensateis to a large extent determined by the singular part of the free energy density which isuniversal . This allows to determine the chiral transition temperature for non-zeroquark masses. The temperature derivative of the chiral condensate and disconnectedchiral susceptibility will diverge in the limit m l →
0. Since the chiral condensate has an3 χ disc /T N τ =6, 0.05m s N τ =8, 0.05m s N τ =6, 0.20m s Figure 2: The disconnected chiral susceptibility calculated with the HISQ action on N τ = 6 and 8 lattices for m l = 0 . m s and m l = 0 . m s .additive ultraviolet renormalization we consider the subtracted chiral condensate ∆ l,s ( T ) = h ¯ ψψ i l,τ − m l m s h ¯ ψψ i s,τ h ¯ ψψ i l, − m l m s h ¯ ψψ i s, . (3)Here the subscript l and s refer to light and strange chiral condensates, while subscript0 and τ to the expectation value at zero and non-vanishing temperature. Another pos-sibility to get rid of the additive renormalization in the chiral condensate is to considerthe quantity ∆ Rl ( T ) = − m s r ( h ¯ ψψ i l,τ − h ¯ ψψ i l, ) (4)which up to a constant multiplicative factor is identical to the quantity introduced in Ref. and was called the renormalized chiral condensate. In Fig. 1 we show our numericalresults for ∆ l,s and ∆ Rl and compare them to the continuum extrapolated stout results. To facilitate this comparison for m l = 0 . m s we performed a combined polynomialﬁt of the N τ = 6 and N τ = 8 results allowing for lattice spacing dependent coeﬃcientsin this ﬁt. This allows us to give an estimate of these quantities in the continuum limitshown as the solid black lines. The continuum estimates for HISQ are diﬀerent fromthe continuum extrapolated stout results. This is presumably due to the fact that thequark masses used in the stout calculations are smaller, namely m l = m s / . N τ = 12lattices and smaller quark masses will be needed to clarify this issue quantitatively.We also calculated the disconnected chiral susceptibility. In Fig. 2 we show the lightquark disconnected chiral susceptibility. The peak position in this quantity also deﬁnesthe chiral transition temperature. To estimate the peak position in this quantity we haveperformed ﬁts of the lattice data using diﬀerent functional forms. We also varied the ﬁtintervals. Our analysis gives N τ = 6 : T c = 168(4)(5)MeV (0 . m s ); T c = 185(4)(5)MeV (0 . m s ) (5)4 τ = 8 : T c = 165(4)(5)MeV (0 . m s ) , (6)where the ﬁrst error reﬂects the uncertainty in the determination of the peak positionestimated using diﬀerent ﬁt forms and varying the ﬁt range. The second error is due tothe scale determination. The peak positions in the disconnected chiral susceptibilities aresigniﬁcantly lower than those obtained with p4 and asqtad actions for N τ ≤ N τ = 12 lattices. Unfortunately no published stout data are available for the disconnected chiralsusceptibility and direct comparison between HISQ and stout results is not possible here.Therefore, in Ref.  we compared HISQ and stout data for the renormalized chiralsusceptibility introduced in . Some small discrepancies between the stout and HISQresults have been found there which again are probably due to slightly diﬀerent quarkmasses. Note that within errors of the calculations the peak positions in the renormalizedchiral susceptibility are consistent between HISQ and stout . By the deconﬁnement transition we mean liberation of many degrees of freedom, whichalso could be understood as a transition from hadronic degrees of freedom to partonicones, and the onset of color screening. The aspects of the deconﬁnement transition relatedto color screening are studied in terms of the Polyakov loop and Polyakov loop correlators.The Polyakov loop needs to be renormalized and after proper renormalization it is relatedto the free energy of a static quark anti-quark pair at inﬁnite separation F ∞ ( T ) , L ren ( T ) = exp( − F ∞ ( T ) / (2 T )) . (7)The renormalized Polyakov loop can be obtained from the bare Polyakov loop as L ren ( T ) = z ( β ) N τ L bare ( β ) = z ( β ) N τ *
13 Tr N τ − Y x =0 U ( x , ~x ) + . (8)Here the multiplicative renormalization constant z ( β ) is related to the additive normal-ization of the potential c ( β ) as z ( β ) = exp( − c ( β ) /
2) discussed in the previous section.The renormalized Polyakov loop has been calculated for pure gauge theory [18, 19], 3-ﬂavor QCD  as well as for 2-ﬂavor QCD . More recently it has been calculatedfor 2+1 ﬂavor QCD with physical strange quark mass and light quark masses close tothe physical values [7, 23, 24, 25, 15, 14]. In Fig. 3 we show our results for the renor-malized Polyakov loop calculated with the HISQ action using N τ = 6 and 8 lattices and m l = 0 . m s and compare them to the continuum extrapolated results obtained withthe stout action . The decrease of F ∞ ( T ), and thus the increase in the Polyakov loopcould be related to the onset of screening at high temperatures (e.g. see discussion inRef. ). On the other hand, in the low-temperature region the increase of L ren isrelated to the fact that there are many static-light meson states that can contribute tothe static quark free energy close to the transition temperature, while far away from thetransition temperature it is determined by the binding energy of the lowest static-light5 ren (T) N τ =6N τ =8stout cont. χ s /T T [MeV]HISQ: N τ =8 N τ =6stout: continuum Figure 3: The renormalized Polyakov loop (left) and the strange quark number suscep-tibility (right) calculated with the HISQ action for m l = 0 . m s and compared to thecontinuum extrapolated stout results. The lines correspond to continuum extrapolation.mesons. The strange quark number susceptibility is deﬁned as the second derivative ofthe pressure with respect to the quark chemical potential χ s = ∂ p ( T, µ s ) ∂µ s | µ s =0 . (9)It describes strangeness ﬂuctuations at zero strange quark chemical potential. Atlow temperatures strangeness is carried by strange hadrons, which are heavy comparedto the temperature. As a result strangeness ﬂuctuations are suppressed in the low-temperature region. At high temperatures, on the other hand, strangeness is carried bylight quarks and strangeness ﬂuctuations are close to the value given by an ideal quarkgas. Deconﬁnement will manifest itself as a rapid increase in the strangeness ﬂuctuationsin some temperature interval, reﬂecting the change in the relevant degrees of freedomfrom hadronic to partonic. Therefore, strangeness ﬂuctuations are used as a probe ofthe deconﬁnement transition [7, 23, 24, 25, 15, 14]. In Fig. 3 we show the strangequark number susceptibility calculated for the HISQ action on N τ = 6 and 8 lattices for m l = 0 . m s .We performed a combined polynomial ﬁt of our lattice results where the coeﬃcient ofthe polynomial had a correction. This allows to estimate strangeness ﬂuctuations in thecontinuum limit. We compared our results to those obtained with the stout action andextrapolated to the continuum. As one can see there is reasonable agreement betweenHISQ and stout results in the continuum limit. The main diﬀerence is the inﬂectionpoint in the stout calculations at temperatures of about 165 MeV which is not visiblein the HISQ results. Note, that strangeness ﬂuctuations approach the continuum limitfrom below. This is expected due to the lattice spacing dependence of the hadron masses. 6 Conclusions
We studied the deconﬁnement and chiral transition in QCD at non-zero temperatureusing the HISQ action and compared our results to recent continuum extrapolated resultsobtained with the stout action. We ﬁnd reasonable agreement in the continuum limitbetween the results obtained with diﬀerent actions for the renormalized Polyakov loopand the strange quark number susceptibility. We also calculated the chiral condensateand disconnected chiral susceptibility for the HISQ action. For these quantities a directcomparison with stout results is more diﬃcult due the slightly diﬀerent light quark massesused in the two calculations. Nonetheless, current HISQ results indicate a chiral transitiontemperature that is signiﬁcantly lower than the previous estimates obtained with theasqtad and p4 actions using lattices with the temporal extent N τ ≤
8. It would also beinteresting to compute the equation of state with the HISQ action and compare it to thevery recent results obtained with the stout action .
This work has been supported in part by contracts DE-AC02-98CH10886 and DE-FC02-06ER-41439 with the U.S. Department of Energy and contract 0555397 with the NationalScience Foundation. The numerical calculations have been performed using the USQCDresources at Fermilab as well as the BlueGene/L at the New York Center for Computa-tional Sciences (NYCCS).
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