A quantitative comparison between the finite temperature behaviour of the
staggered and Wilson fermion formulations are performed. The comparison is
based on a physical quantity that is expected to be quite sensitive to the
fermionic features of the action. For that purpose we use the height of the
peak for $d\chi_s/dT$, where $\chi_s$ is the quark number susceptibility.Comment: 6 pages. Talk presented at Lattice 200

Bálint C. Tóth

Kálmán K. Szabó

Sándor D. Katz

Yasumichi Aoki

Zoltán Fodor

English

arXiv

Aoki, Yasumichi

Fodor, Zoltan

Katz, Sandor D

Szabo, Kalman K

Toth, Balint C

ELTE Digital Institutional Repository (EDIT)

arXiv:0711.0024v1  [hep-lat]  31 Oct 2007QCD finite T transitionComparison between Wilson and staggered resultsYasumichi AokiBrookhaven National Laboratory, Upton NY, USAE-mail: yaoki@bnl.govZoltán FodorDepartment of Physics, University of Wuppertal, GermanyInstitute for Theoretical Physics, Eötvös University, Budapest, HungaryE-mail: fodor@bodri.elte.huSándor D. KatzDepartment of Physics, University of Wuppertal, GermanyInstitute for Theoretical Physics, Eötvös University, Budapest, HungaryE-mail: katz@bodri.elte.huKálmán K. SzabóDepartment of Physics, University of Wuppertal, GermanyE-mail: szaboka@general.elte.huBálint C. Tóth ∗Institute for Theoretical Physics, Eötvös University, Budapest, HungaryE-mail: tothbalint@szofi.elte.huA quantitative comparison between the finite temperature behaviour of the staggered and Wilsonfermion formulations are performed. The comparison is based on a physical quantity that isexpected to be quite sensitive to the fermionic features of the action. For that purpose we use theheight of the peak fordχs/dT, whereχs is the quark number susceptibility.The XXV International Symposium on Lattice Field TheoryJuly 30 - August 4 2007Regensburg, Germany∗Speaker.c© Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike Licen. http://pos.sissa.it/QCD finite T transition – Comparison between Wilson and staggered results Bálint C. Tóth1. IntroductionStaggered fermions are computationally faster than Wilsonfermions, the discretization errorsscale withO(a2) and due to the well behaving spectrum of the Dirac operator, light quark masses –nowadays even the physical ones – can be reached. Note, however, that a straightforward definitionfor staggered fermions exists only for 4,8,... quark flavours. All works using 2 or 2+1 flavours ofstaggered quarks use the fourth root (or square root) trick to have only one (or two) flavour(s). Theaction is defined by taking the fourth root of the fermion determinant. Since such a prescription isnon-local at fixed lattice spacings, it is debated whether itis equivalent to a local continuum fieldtheory or not (see e.g. [1–3] and references therein).Large scale staggered studies are based on the expectation that i the continuum limit staggeredand Wilson results agree. Clearly, at fixed, non-vanishing lattice spacings deviations are possibleeven if the continuum results are the same. Therefore, any anal sis looking for the equivalence ornon-equivalence between the staggered and Wilson formalisshould fulfill two conditions. Firstof all, the analysis should be based on several lattice spacings and a controlled continuum ex-trapolation should be carried out. At least three differentlattice spacings are needed, for which theasymptotic scaling behaviour can be already observed (bothfor staggered and for nonperturbativelyclover-improved Wilson fermions one expects ana2 scaling). Secondly, the physical quantity cho-sen as a basis of such comparison not only has to be well definedand relatively easily measurable,but also has to be sensitive to the dynamical fermion sector.It is not so obvious how to find such a sensitive quantity. E.g.the masses of different hadronsprovide well defined physical quantities, however, these are not sensitive enough to the dynamicalfermion sector. Even the quenched calculations, where the dynamical fermions are completelyomitted, provide the physical hadron masses with an error abut or less then 10% [4]. Findingmeasurable differences in the continuum limit of such quantities would require extremely highprecision, thus, prohibitively large scale calculations.Thermodynamic observables can be much more sensitive to thefermionic content of the the-ory. This fact is related to the singular/non-singular behaviour of the finite temperature QCD tran-sition. E.g. on the one hand the quenched theory undergoes [5, 6] a first order phase transition atnon-vanishing temperatures (T). For first order phase transi io different observables behave in asingular way (the latent heat is infinite or the temperature derivative of the renormalized Polyakovloop has a discontinuity). On the other hand the finite temperature transition of QCD with staggeredfermions and physical quark masses has turned out to be a crossover [7]. As a consequence, noneof the physical quantities as the function of the temperature are infinite or discontinuous, at themost they only undergo a rapid change within a narrow temperature range. Clearly, the height ofsuch a peak is expected to be quite sensitive to the details ofthe fermionic properties of the action.As we increase or decrease the quark masses the peak turns outto be more and more singular andafter a while a second order then a first order phase transitioregion is reached. Thus, the fermioniccontent (quenched or unquenched with physical or non-physical quark masses) manifests itself ina very pronounced way.Choosing a physical quantity that undergoes a rapid change ad finding the maximum of itsderivative with respect to the temperature may provide a quantity sensitive to the finite temperaturebehaviour of the system.2QCD finite T transition – Comparison between Wilson and staggered results Bálint C. TóthThe aim of this study is to provide a quantitative comparisonbetween the behaviour of thestaggered and Wilson fermion formulations. We attempt to perform an analysis at two differentlattice spacings (thus no conclusive continuum extrapolation can be done yet). The observable wehave chosen is related to the transition at non-vanishing temperatures and expected to be sensitiveto the fermionic properties of the theory.One physical quantity that undergoes a rapid change around the transition temperature is thequark number susceptibility, which is defined via [8]χsT2=1TV∂ 2 logZ∂ µ2∣∣∣∣µ=0, (1.1)whereµ is the quark chemical potential. The quark number susceptibility can be directly mea-sured, and it automatically has the correct continuum limitwithout the need for renormalization.In addition, the maximum of its derivative with respect to the emperature, that is, the rate at whichthe susceptibility changes during the transition, is sensitive to the dynamical fermion sector. Theseproperties make the quark number susceptibility a good candid te for the quantity that a comparisonof the Wilson and staggered fermion formulations should be based on. Since the rooting procedureof the determinant is less transparent (might be more problematic) for odd number of flavours weuse three flavours. (The one flavour theory, where there is no chiral symmetry breaking [9], is notsuitable for our purposes.)2. Action parametersThe three flavours were degenerate for both Wilson and staggered calculations, and the gaugeaction used was the Symanzik tree-level improved gauge action [10]. In both cases the latticessizes were 323×8 and 323×10, and the configurations were generated using the RationalHybridMonte-Carlo algorithm [11].Wilson calculations: Three steps of stout smearing [12] with smearing parameterρ = 0.1 wereused. The gauge coupling constant was in the rangeβ = 3.2−3.7. In addition the femionic sectorwas clover improved [13] with a tree level clover coefficientc = 1.0. Note, that for this type ofsmeared fermions the tree level clover coefficient essentially le ds to anO(a) improved action [14].Staggered calculations: Two steps of stout smearing with smearing parameterρ = 0.15 wasused [15], and the gauge coupling constant was in the rangeβ = 3.5−4.0. This staggered actionwithin this lattice spacing range was shown to be in the scaling regime [16].3. Setting the scaleTo be able to make sure that the staggered and Wilson calculations are performed at the sameset of physical parameters, the line of constant physics wasdefined by fixing the ratio of the pseu-doscalar and the vector meson massesmPS/mV .If the finite temperature behaviour of the Wilson and the staggered formulations are different,the difference is likely to be most apparent at small quark mass parameters. For small quark masseswe are closer and closer to the first order phase transition region, thus all the differences are easier3QCD finite T transition – Comparison between Wilson and staggered results Bálint C. TóthFigure 1: The scale for the staggered calculations: the bare staggered quark mass (left panel) and the vectormeson mass (right panel) as a function of the gauge coupling co stant.Figure 2: The scale for the Wilson calculations: the bare Wilson quarkmass (left panel), the PCAC quarkmass (middle panel) and the vector meson mass (right panel) as a function of the gauge coupling constant.to see in the height of the peak of our observable. Decreasingthe quark mass, however, causesthe computational costs to rise. Therefore, one needs to finda compromise between making thecomparison more sensitive by lowering the quark mass (or in other words loweringmPS/mV) andkeeping the computational costs reasonable.As a compromise we have chosenmPS/mV = 0.5, which sets the quark mass aboutms/3,wherems is the physical strange quark mass. This relationship completely defines the line ofconstant physics. In three flavour lattice QCD we have two parameters. One of them is the quarkmass which is essentially set by the relationship between thpseudoscalar and vector mass ratio.The other one is the lattice spacing, which is dominantly given by the gauge coupling.The bare quark mass and the vector meson mass corresponding to the different gauge coupligvalues for the staggered case are shown in Figure 1. For the Wilson calculations, the bare quarkmass, the current algebra quark mass and the vector meson mass are shown in Figure 2.4QCD finite T transition – Comparison between Wilson and staggered results Bálint C. TóthFigure 3: The quark number susceptibility (left panel) and its derivative (right panel) as a function of thetemperature normalized with the vector meson mass.4. ResultsTo be able to compare the staggered and Wilson results the temperature was made dimension-less by dividing by the vector meson mass. The quark number susceptibilities as a function ofT/mV are shown in the left panel of Figure 3. The derivative of the susceptibility was obtained byfitting cubic polynomials to the susceptibility points, then taking the derivative of the polynomial.The slight change due to the variation of the fitting range is taken as a systematic error. Thesederivatives are shown in the right panel of Figure 3.5. ConclusionsCalculating the maximum of the derivative of the quark number susceptibility with respect tothe temperature yields a sensitive quantity, based on whicht e finite temperature behaviour of theWilson and the staggered fermion formulations can be compared. We have performed calculationsusing lattices with temporal extensionsNt = 8 and 10. AtNt = 8 the maximum of the dervative inthe Wilson case is a factor of 2 higher than the staggered result, whereas atNt = 10 the heights ofthe peaks get closer to one another. In order to be conclusiveone needs to get one or more stepscloser to the continuum limit by performing calculations using Nt = 12 or even finer lattices.AcknowledgementsThis research was partially supported by OTKA Hungarian Science Grants No. T34980, T37615,M37071, T032501, AT049652, EU Grant I3HP and by DFG German Research Grant No. FO502/1-1. The computations were carried out on the 370 process r PC cluster of Eötvös University,5QCD finite T transition – Comparison between Wilson and staggered results Bálint C. Tóthon the 1024 processor PC cluster of Wuppertal University andon the Blue Gene/L in Research Cen-tre Jülich. We used a modified version of the publicly available MILC code [17] with next-neighborcommunication architecture for PC-clusters [18].References[1] C. Bernard, M. Golterman and Y. Shamir,Observations on staggered fermions at non-zero latticespacing, Phys. Rev.D73 (2006) 114511 [hep-lat/0604017].[2] M. Creutz,Why rooting fails, PoS(LATTICE 2007)007 (2007) [arXiv:0708.1295 [hep-lat]].[3] A. Kronfeld, Lattice QCD with staggered quarks: Why, where, and how (not),PoS(LATTICE 2007)016 (2007).[4] CP-PACS Collaboration, S. Aoki et al.,Light hadron spectrum and quark masses from quenchedlattice QCD, Phys. Rev.D67 (2003) 034503 [hep-lat/0206009].[5] T. Celik, J. Engels and H. Satz,The order of the deconfinement transition in SU(3) Yang-Mills theory,Phys. Lett.B125 (1983) 411.[6] M. Fukugita, M. Okawa and A. Ukawa,Order of the deconfining phase transition in SU(3) latticegauge theory, Phys. Rev. Lett.63 (1989) 1768.[7] Y. Aoki, G. Endrodi, Z. Fodor, S. D. Katz and K. K. Szabo,The order of the quantumchromodynamics transition predicted by the standard modelf particle physics, Nature443 (2006)675–678 [hep-lat/0611014].[8] MILC Collaboration, C. Bernardet. al., QCD thermodynamics with three flavors of improvedstaggered quarks, Phys. Rev.D71 (2005) 034504 [hep-lat/0405029].[9] M. Creutz,One flavor QCD, Annals Phys.322 (2007) 1518–1540 [hep-th/0609187].[10] K. Symanzik,Continuum limit and improved action in lattice theories. 1.Principles andφ4 theory,Nucl. Phys.B226 (1983) 187.[11] M. A. Clark and A. D. Kennedy,The RHMC algorithm for 2 flavors of dynamical staggered fermions,Nucl. Phys. Proc. Suppl.129 (2004) 850–852 [hep-lat/0309084].[12] C. Morningstar and M. J. Peardon,Analytic smearing of SU(3) link variables in lattice QCD, Phys.Rev.D69 (2004) 054501 [hep-lat/0311018].[13] B. Sheikholeslami and R. Wohlert,Improved continuum limit lattice action for QCD with Wilsonfermions, Nucl. Phys.B259 (1985) 572.[14] R. Hoffmann, A. Hasenfratz and S. Schaefer,Non-perturbative improvement of nHYP smeared Wilsonfermions, PoS(LATTICE 2007)104 (2007) [arXiv:0710.0471 [hep-lat]].[15] Y. Aoki, Z. Fodor, S. D. Katz and K. K. Szabo,The equation of state in lattice QCD: With physicalquark masses towards the continuum limit, JHEP01 (2006) 089 [hep-lat/0510084].[16] Y. Aoki, Z. Fodor, S. D. Katz and K. K. Szabo,The QCD transition temperature: Results withphysical masses in the continuum limit, Phys. Lett.B643 (2006) 46–54 [hep-lat/0609068].[17] MILC Collaboration,Public lattice gauge theory code, online athttp://physics.indiana.edu/˜sg/milc.html.[18] Z. Fodor, S. D. Katz and G. Papp,Better than $1/Mflops sustained: A scalable PC-based parallelcomputer for lattice QCD, Comput. Phys. Commun.152 (2003) 121–134 [hep-lat/0202030].6

QCD finite T transition: Comparison between Wilson and staggered results

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QCD finite T transition -- Comparison between Wilson and staggered results

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