ADP-04-023/T605 Quark propagator in Landau and Laplacian gauges with overlap fermions J.B.Zhanga,PatrickO.Bowmana,b,RyanJ.Coada,UrsM.Hellerc,DerekB.Leinwebera,andAnthonyG.Williamsa aCSSM Lattice Collaboration, Special Research Center for the Subatomic Structure of Matter (CSSM) and Department of Physics, University of Adelaide 5005, Australia bNuclear Theory Center, Indiana University, Bloomington IN 47405, USA cAmerican Physical Society, One Research Road, Box 9000, Ridge, NY 11961-9000, USA 5 (Dated: February 1, 2008) 0 0 The propertiesof themomentumspace quarkpropagator in Landau gauge and Gribov copy free 2 Laplacian gauge are studied for the overlap quark action in quenched lattice QCD. Numerical cal- culations are done on two lattices with different lattice spacing a and the same physical volume. n Wehavecalculated thenonperturbativewavefunction renormalization function Z(q) and thenon- a J perturbative mass function M(p) for a variety of bare quark masses and perform a simple linear extrapolation tothe chiral limit. Wefocus on thecomparison of thebehavior of Z(q) and M(p) in 4 thechirallimitinthetwogaugefixingschemesaswellasthebehaviorontwolatticeswithdifferent 1 latticespacinga. WefindthatthemassfunctionsM(p)areverysimilarforthetwogaugeswhilethe 2 wave-function renormalization function Z(q) is more strongly infrared suppressed in the Laplacian v gauge than in the Landau gauge on the finer lattice. For Laplacian gauge, it seems that the finite 5 a error is large on thecoarse lattice which has a lattice spacing a of about 0.124 fm. 4 0 PACSnumbers: 12.38.Gc,11.15.Ha,12.38.Aw,14.65.-q 0 1 4 I. INTRODUCTION Gribov problem in lattice gauge theory, see Ref. [15]. 0 An alternative approach is to work with the so-called t/ QuantumChromodynamics (QCD) is widely accepted Laplacian gauge [16]. This gauge is “Landau like” in a as the correct theory of the strong interaction and the the sense that it has similar smoothness and Lorentz l - quarkpropagatorisoneofitsfundamentalquantities. By invariance properties [17], but it involves a non-local p studying the momentum-dependentquarkmassfunction gaugefixing procedure that avoidslattice Gribov copies. e h intheinfraredregionwecangainvaluableinsightintothe ThegluonpropagatorhasalreadybeenstudiedinLapla- : mechanism of dynamical chiral symmetry breaking and cian gauge in Refs. [18, 19] and the improved staggered v the associated dynamical generation of mass. At high quark propagator in Laplacian gauge in Ref. [6]. It has i X momenta, one can use the quark propagator to extract been shown [20] that Landau and Laplacian gauges be- r the running quark mass [1]. come equivalent in the perturbative (high-momentum) a Lattice QCD provides a way to study the quark prop- regime and this has been confirmed by numerical stud- agator nonperturbatively. There have been several lat- ies [6, 18, 19]. tice studies of the momentum space quark propaga- In this paper we study the overlap quark propagator tor [2, 3, 4, 5, 6, 7, 8, 9, 10, 11] using different fermion in the Laplaciangauge andcomparethe results with the actions. The usual gauge for these studies has been Landau gauge to explore the effects of selecting a gauge Landau gauge, because it is a (lattice) Lorentz covari- conditionfree of Gribov copies. Unlike Asqtad fermions, ant gauge that is easy to implement on the lattice, and the overlap formalism provides a fermion action which the results from the lattice Landau gauge can be easily is free of doublers and preserves an exact form of chiral compared to studies that use different methods. Finite symmetry on the lattice. The latter feature makes over- volume effects anddiscretizationerrorshavebeen exten- lap fermions the action of choice for studying dynamical sivelyexploredinlatticeLandaugauge[10,11]. Unfortu- chiral symmetry breaking near the chiral limit. nately, lattice Landaugaugesuffers fromthe well-known We also compare Laplacian gauge results on two lat- problem of Gribov copies. Although the ambiguity orig- tices with the same physical volume and different lattice inally noticed by Gribov [12] is not present on the lat- spacings a to explore the finite a error. In this work, tice, since in practice one never samples from the same the (a2) mean-field improved gauge action is used to O gauge orbit twice, the maximization procedure used for generate the quenched gauge configurations. gauge fixing does not uniquely fix the gauge. In gen- eral, there are many local maxima for the algorithm to choose from, each one corresponding to a Gribov copy, II. GAUGE FIXING and no local algorithm can choose the global maximum from among them. While various remedies have been We consider the quark propagator in Landau and proposed [13, 14], they are either unsatisfactory or com- Laplacian gauges. Landau gauge fixing is performed by putationallyveryintensive. Forarecentdiscussionofthe enforcing the Lorentz gauge condition, ∂ A (x) = 0 µ µ µ P 2 onaconfigurationbyconfigurationbasis. Forthetadpole where ζ is the renormalization point. The renormaliza- improved plaquette plus rectangle (Lu¨scher-Weisz [21]) tion point boundary conditions are chosen to be gauge action which we use in the current work, we use the (a2)improvedgaugefixingscheme,this isachieved Z(ζ2;ζ2) 1 M(ζ2) m(ζ2). (3) O ≡ ≡ by maximizing the functional [22], wherem(ζ2)istherenormalizedquarkmassattherenor- 4 1 malization point. The functions A(ζ2;p2) and B(ζ2;p2), = , (1) F 3F1− 12u F2 or alternatively Z(ζ2;p2) and M(p2), contain all of the 0 nonperturbative information of the quark propagator. where 1 and 2 are Note that M(p2) is renormalization point independent, F F allofthe renormalization-pointdependence is carriedby 1 = Tr U (x)+U†(x) Z(ζ2;p2). F1 2 µ µ x,µ When all interactions for the quarks are turned off, X (cid:8) (cid:9) i.e., when the gluonfield vanishes (or the links are setto and one), the quark propagator has its tree-level form 1 F2 = 2 Tr Uµ(x)Uµ(x+µ)+Uµ†(x+µ)Uµ†(x) S(0)(p)= 1 , (4) x,µ i/p+m0 X (cid:8) (cid:9) respectively,andu0 istheusualplaquettemeasureofthe wherem0 is the barequarkmass. Whentheinteractions mean link. In this case, a Fourier accelerated, steepest- with the gluon field are turned on we have descents algorithm [23] is used to find a local maximum. There are, in general, many local maxima and these are S(0)(p) Sbare(a;p)=Z (ζ2;a)S(ζ2;p), (5) 2 → called lattice Gribov copies. This ambiguity in principle will remain a source of uncontrolled systematic error. where a is the regularization parameter - in this case, Laplacian gauge fixing is a nonlinear gauge fixing the lattice spacing - and Z (ζ2;a) is the quark wave- 2 that respects rotational invariance, has been seen to be function renormalizationconstantchosenso as to ensure smooth, yet is free of Gribov ambiguity. It is also com- Z(ζ2;p2)p2=ζ2 = 1. For simplicity of notation we sup- | putationallycheaperthenLandaugaugefixing. Thereis, press the a-dependence of the bare quantities. however, more than one way of obtaining such a gauge On the lattice we expect the bare quark propagators, fixinginSU(N)latticegaugetheory. Therearethreeim- in momentum space, to have a similar form as in the plementations of Laplaciangauge fixing employedin the continuum, except that the O(4) invariance is replaced literature: bya 4-dimensionalhypercubic symmetryonanisotropic lattice. Hence, the inverse lattice bare quark propagator 1. ∂2(I) gauge (QR decomposition), used by Alexan- takes the general form drou et al. [18]. 2. ∂2(II) gauge, where the Laplacian gauge transfor- (Sbare)−1(p) i C (p)γ +B(p). (6) µ µ mation is projected onto SU(3) by maximising its ≡ ! µ trace [19]. X With the periodic boundary conditions in the spatial di- 3. ∂2(III) gauge (Polar decomposition), the original rections and anti-periodic in the time direction, the dis- prescription described in Ref. [16] and tested in crete lattice momenta will be Ref. [17]. 2π N 2π 1 N i t All three versions reduce to the same gauge in SU(2). pi = ni , and pt = Nt ,(7) N a − 2 N a − 2 − 2 For a more detailed discussion, see Ref. [19]. For SU(3) i (cid:18) (cid:19) t (cid:18) (cid:19) staggeredquarks,thestudyinRef.[6]indicatethat∂2(I) where n =1,..,N and n =1,..,N , N and N are the and ∂2(II) gauge give very similar results, and ∂2(III) i i t t i t lattice extent in spatial and temporal direction respec- gauge is very noisy. In this work we will only use the tively. ∂2(II) gauge. The overlap fermion formalism [24, 25] realizes an ex- act chiral symmetry on the lattice and is automatically (a) improved. The massive overlap operator can be III. QUARK PROPAGATOR ON THE LATTICE O written as [26] In a covariant gauge in the continuum, the renormal- 1 D(η)= [1+η+(1 η)γ ǫ(H )] , (8) ized Euclidean space quark propagator has the form 2 − 5 w S(ζ2;p)= 1 = Z(ζ2;p2) , (2) where Hw(x,y) = γ5Dw(x,y) is the Hermitian Wilson- i/pA(ζ2;p2)+B(ζ2;p2) i/p+M(p2) Dirac operator, ǫ(Hw) = Hw/ Hw2 is the matrix sign p 3 function, and the dimensionless quark mass parameter η IV. NUMERICAL RESULTS is A. Simulation parameters m0 η , (9) ≡ 2m w In this paper we work on two lattices with different where m0 is the bare quark mass and m is the Wilson lattice spacing, a, and very similar physical volumes. w quark mass which, in the free case, must be in the range The gauge configurations are created using a tadpole 0 < m < 2. The bare quark propagator in coordinate improved plaquette plus rectangle (Lu¨scher-Weisz [21]) w space is given by the equation gauge action through the pseudo-heat-bath algorithm. For each lattice size, 50 configurations are used. Lat- Sbare(m0) D˜−1(η), (10) tice parameters are summarized in Table I. The lattice ≡ c spacing a is determined from the static quark potential where with a string tension √σ =440 MeV [30]. Landau gauge fixing to the gauge configuration 1 D˜−1(η) D˜−1(η) and was done using a Conjugate Gradient Fourier Ac- c ≡ 2m w celeration [31] algorithm with an accuracy of θ ≡ ∂ A (x)2 < 10−12. The improved gauge-fixing µ µ | | 1 scheme was used to minimize gauge-fixing discretization D˜−1(η) D−1(η) 1 . (11) ePrrors [22]. For the Laplacian gauge fixing, we only use ≡ 1 η − − the∂2(II)gauge[19]. Inthiscaseweconstructthegauge (cid:2) (cid:3) When all the interactions are turned off, the inverse transformationbyprojectingM(x)constructedfromthe barelattice quarkpropagatorbecomesthe tree-levelver- three lowest lying eigenmodes, onto SU(3) by means of sion of Eq. (6) trace maximisation. Effectively, we maximise the trace ofG(x)M(x)† by iterationoverCabibbo-MarinariSU(2) subgroups. (S(0))−1(p)≡i Cµ(0)(p)γµ!+B(0)(p). (12) Ournumericalcalculationbeginswithanevaluationof µ theinverseofD(η)withtheunfixedgaugeconfigurations, X where D(η) is defined in Eq. (8). We approximate the WecalculateS(0)(p)directlybysettingthelinkstounity matrix sign function ǫ(H ) by the 14th order Zolotarev w in the coordinate space, doing the matrix inversion and approximation [32]. We then calculate Eq. (10) for each then taking its Fourier transform. It is then possible to configurationandrotateittoLandauorLaplaciangauge identify the appropriate kinematic lattice momentum q by using the corresponding gauge transformation matri- directly from the definition ces G (x) . Afterward we take the ensemble averageto i { } obtain Sbare(x,y). The discrete Fourier transformation qµ ≡Cµ(0)(p). (13) is then applied to Sbare(x,y) and the momentum-space bare quark propagator,Sbare(p) is obtained finally. The form of q (p ) is shown and its analytic form given µ µ We use the mean-field improved Wilson action in the inRef.[9]. Havingidentifiedtheappropriatekinematical overlap fermion kernel. The value κ = 0.19163 is used lattice momentum q, we can now define the bare lattice in the Wilson action, which provides m a = 1.391 for propagator as w the regulator mass in the interacting case [9]. We calcu- Z(p) late the overlap quark propagator for ten quark masses Sbare(p) . (14) on each ensemble by using a shifted Conjugate Gradient ≡ i/q+M(p) solver. For the two lattices considered here, the quark This ensures that the free lattice propagator is identical mass parameter η was adjusted to make the tree level to the free continuum propagator. Due to asymptotic bare quark mass in physical units, the same on the two freedom the lattice propagator will also take the contin- lattices. For example,we chooseµ=0.018,0.021,0.024, uum form at large momenta. In the gauge sector, this 0.030, 0.036, 0.045, 0.060, 0.075, 0.090, and 0.105 on type of analysis dramatically improves the gluon propa- ensemble 1, i.e., the 163 32 lattice with a = 0.093 × gator [27, 28, 29]. fm. This corresponds to bare masses in physical units The two Lorentz invariants can then obtained by of m0 =2µmw = 106, 124, 142, 177, 212, 266, 354, 442, 531, and 620 MeV respectively. Z−1(p)= 1 Tr /qS−1(p) (15) Theresultsoflattice2(123×24)inLandaugaugewere 12iq2 { } presentedin detailin Ref. [9], andthe results oflattice 1 Z(p) (163 32)inLandaugaugewerealsoreportedinRef.[11]. M(p)= Tr S−1(p) . (16) Here×wewillfocusonthecomparisonoftheresultsoftwo 12 { } lattice gauge fixing schemes, i.e., the Landau gauge and ThismeansthatZ(p)isdirectlydependentonourchoice the Gribov copy free Laplacian gauge, to probe the be- of momentum, q, while M(p) is not. havior of the overlap fermion propagator with different 4 TABLE I:Lattice parameters. Action Volume N N β a (fm) u Physical Volume (fm4) Therm Samp 0 Improved 163×32 5000 500 4.80 0.093 0.89650 1.53×3.00 Improved 123×24 5000 500 4.60 0.124 0.88888 1.53×3.00 B. Laplacian gauge In Fig. 1 we show the results for all ten masses for both the mass and wave-function renormalization func- tions, M(p) and Z(R)(q) Z(ζ;q) respectively. As was ≡ shown in Ref. [11], the continuum limit is more rapidly approached when the mass function is plotted against thediscretelatticemomentump,whilethewave-function renormalization function Z(R)(q) is plotted against the kinematic momentum q. The renormalization point in Fig. 1 for Z(R)(q) has been chosen to be ζ = 5.31 GeV in the q-scale. In the plots of M(p), the data is orderedas one would expect by the values for bare quark mass m0, i.e., the larger the bare quark mass m0, the higher the M(p) curve. For large momenta, the function Z(R)(q) demon- strateslittlemassdependence. DeviationofZ(R)(q)from itsasymptoticvalueof1isasignofdynamicalsymmetry breaking, so we expect the infrared suppression to van- ishinthelimitofaninfinitelyheavyquark. Inthefigure for Z(R)(q), the smaller the bare mass, the more pro- nounced is the dip at low momenta. Similarly, at small bare masses M(q) falls off more rapidly with increasing momenta,whichisunderstoodfromthefactthatalarger proportionoftheinfraredmassisduetodynamicalchiral symmetry breaking at small bare quark masses. These results are much the same as in Ref. [11], which display the data on the same lattices in Landau gauge. This qualitative behavior is also consistent with what is seen in Dyson-Schwinger based QCD models [33, 34]. In Fig. 2 we plot the data after a linear chiral extrap- FIG. 1: (Color online). The functions M(p) and Z(R)(q) ≡ olation for both functions M(p) and Z(R)(q) Z(ζ;q) Z(ζ2;q)renormalizedatζ =5.31GeV(intheqscale)forall ≡ in Laplacian gauge. The mass function M(p) is shown ten quark masses in Laplacian gauge on the163×32 lattice. against p and while the wave function renormalization The mass function M(p) is plotted versus the discrete mo- function Z(R)(q) is shown against q with the renormal- mentum p defined in Eq. (7), p = p2, over the interval µ ization point chosen as at 5.31 GeV in the q scale. We [0,5] GeV, and Z(R)(q) is plotted against the kinematic mo- pP seethatbothM(p)andZ(R)(q)deviatestronglyfromthe mentum q defined in Eq. (13), q = q2, over the interval µ tree-levelbehavior,whichareM(p)=m0 andZ(R)(q)= [0,12] GeV.Thedata correspond tobarequarkmasses (from bottom to top) µ = 0.018, 0.021, 0p.02P4, 0.030, 0.036, 0.045, 1. Inparticular,asinearlierstudiesoftheLandaugauge 0.060, 0.075, 0.090, and 0.105, which in physical units corre- quark propagator[4, 5, 6, 9, 11], we find a clear signal spond to m0 = 2ηmw ≃ 106, 124, 142, 177, 212, 266, 354, of dynamical mass generation and a significant infrared 442, 531, and 620 MeV respectively suppression of the Z(ζ;q) function. gauge fixings, and the effect of Gribov copies. Before we C. Gauge fixing comparison make the comparison, we first briefly present some data inLaplaciangaugeonthe163 32lattice,ourfinelattice. Next we present the results in the two gauge fixing × Alldatahasbeencylindercut[27]. Statisticaluncertain- schemes for comparison. All data have been cylinder ties are estimated via a second-order, single-elimination cut [27]. First we give the results on the 163 32 lat- × jackknife. tice. 5 FIG.2: TheplotofthefunctionsM(p)andZ(R)(q)≡Z(ζ;q) FIG. 3: (Color online). The comparison of two gauge fixing afteralinearextrapolationtothechirallimit. Themassfunc- results at finite bare quark mass (m0 = 106 MeV) on the tionsM(p)isplottedagainstthediscretemomentumpinthe fine 163 ×32 lattice with a = 0.093 fm. Z(R)(q) is renor- upper part of the figure and Z(R)(q) with the renormaliza- malized to one at the renormalization point ζ =5.31 GeV tionpointζ =5.31GeV(intheq-scale) isplottedagainst the (in the q-scale). For the mass function M(p), Landau gauge kinematic momentum q in the lower part of the figure. and Laplacian gauge are very similar, while the wave func- tion renormalization functions Z(R)(q) differ in the infrared region. Fig.3reportsresultsforthemassandrenormalization functions at our lightest bare quark mass of m0 = 106 MeV. These results may be compared with the Asqtad Landau gauge or the Gribov-copy free Laplacian gauge. results of Ref. [6], where Fig. 9 compares results of var- Wenowproceedtocomparethedatainthechirallimit. ious gauge fixing schemes for the renormalization func- Fig.4 showsthe comparisonofthe massfunction, M(p), tion. There, Landau gauge results are seen to lie sig- andthewavefunctionrenormalizationfunction,Z(R)(q), nificantly higher than the Laplacian gauge results in the in Landaugaugeand Laplaciangauge. We see that they infrared. However,withoverlapfermions,Fig.3indicates give similar performance in terms of rotational symme- the Landau gauge results lie much closer to the Lapla- tryandstatisticalnoise. Lookingmoreclosely,wecansee cian gauge results. Given the improved chiral properties that Landau gauge gives a slightly cleaner signal at this of the overlap operator, the present results should pro- latticespacing. Wealsonotethatatverylargemomenta, vide a better indicationofthe continuumlimit behavior. the two gauge fixing schemes give similar results as ex- In either case, the same qualitative behavior of Landau pected. Although Laplacian gauge is a non-local gauge Gauge sitting above Laplacian gauge in the infrared is fixing scheme and difficult to understand perturbatively, observed. it is equivalent to Landau gauge in the asymptotic re- ThemassfunctionofFig.3revealsanapproximatein- gion[20]. Intheinfraredregion,themassfunction,M(p), varianceonthe selectionofLandauorLaplaciangauges. in the two gauges are very similar. For the mass func- Figs. 12 and 13 of Ref. [6] indicate that the mass func- tionsthere isahintthatthe datainLaplaciangaugeare tionofAsqtadfermionsisalsoinsensitivetothechoiceof a little higher than for the Landau gauge, although they 6 FIG. 4: (Color online). The comparison of two gauge fixing FIG. 5: (Color online). The comparison of two gauge fixing results after a linear extrapolation to the chiral limit on the results in the chiral limit on the coarse lattice, i.e., 123×24 fine lattice, i.e., 163×32 at a = 0.093 fm. Z(R)(q) is renor- with a = 0.124 fm. The small gauge dependence of the in- malized to one at the renormalization point ζ =5.31 GeV fraredbehavioroftheZ-functionissimilartothatonthefine (in the q-scale). For the mass function M(p), Landau gauge lattice in Fig. 4. The infrared mass functions, M(p), appear andLaplaciangaugeareverysimilar,whilethewavefunction different on this coarse lattice whereas they were similar on renormalizationfunctionsZ(R)(q)aresimilarinthelargemo- thefinelattice. ThissuggestslargerO(a2)errorsinLaplacian mentum region but differ in the infrared region. gauge. agreewithinstatisticalerrors. With greaterstatistics we thetwogaugefixingschemesgivesimilarresults,whilein may resolve a small difference. For the renormalization the infrared region, Z(R)(q) is more strongly suppressed function Z(R)(q), there are systematic differences in the in Laplacian gauge than in Landau gauge. For the mass infrared region. The Z(R)(q) is more strongly infrared function, M(p), the situation is different to that seen on suppressed in the Laplacian gauge than in the Landau thefinelattice. Intheinfraredregion,thedatainLapla- gauge. Thatis consistentwithwhatwasseeninthe case cian gauge sit higher than in Landau gauge. While this oftheAsqtadquarkaction[6]whencomparingthesetwo is consistent with the infrared behavior of the renormal- gauges. ization function, Z(R)(q), which is more suppressed in Now we give the results fromthe coarselattice. Fig. 5 Laplacian gauge than in Landau gauge, there is no sim- shows the comparison of the mass function M(p) and ilar signal for our fine lattice 163 32. In that case (see the wave function renormalization function Z(R)(q) in Fig. 5), the data in Laplacian ga×uge agree with that in the chirallimit inLandaugaugeandLaplaciangaugeon Landaugaugewithin errorbars,althoughthere is a hint the 123 24 lattice with a = 0.124fm. As in the case of that the data in Laplaciangauge are a little higher than the163 ×32lattice,atverylargemomenta,thetwogauge inLandaugaugeintheinfraredregion. Thisinfraredbe- × fixing schemes give similar results. For the renormaliza- havior of the mass function in Laplacian gauge is likely tionfunction Z(R)(q), the situationis verysimilar to the caused by the finite lattice spacing errors, i.e., on fine case of the 163 32 lattice, i.e., at very large momenta, enough lattices we expect Landau and Laplacian gauge × 7 well at large momenta, but there is a substantial differ- ence in the infrared region. A similar comparison was made between these two lattices in Landau gauge [11]. Inthatcase,boththe renormalizationfunction, Z(R)(q), and the mass function, M(p), agree well on the two lat- tices. This indicates that in Landau gauge, the finite a errors are small even for our coarse lattice with lattice spacing a = 0.124fm. But in Laplaciangauge,the finite a errorsarenot negligible for ourcoarselattice 123 24. × A likely explanation for this is the fact that our imple- mentationoftheLaplaciangaugeisnot (a2)improved. O V. SUMMARY AND OUTLOOK The momentum-space quark propagator has been studied in Landau gauge as well as the Gribov copy free Laplacian gauge on two lattices with the same physical volume but with different lattice spacings a . We calcu- lated the nonperturbative momentum-dependent wave- function renormalization,Z(q), and the nonperturbative mass function, M(p), for a variety of bare quark masses. We also performed a simple linear extrapolation to the chiral limit. At very large momenta the two gauge-fixing schemes givesimilarresultsasexpected. Laplaciangaugeisequiv- alent to the Landau gauge in the asymptotic region. In the infrared region, the mass function, M(p), in the two gauges are very similar on the fine lattice, but differ on the coarse lattice. The present Laplacian gauge fixing is not (a2) improved. For our fine lattice in the infrared FIG. 6: (Color online). The comparison of Laplacian gauge O region,themassfunction,M(p),agreeswithinstatistical resultsinthechirallimitontwolattices, i.e.,123×24with a errors in the two gauge fixings. However, there is a hint =0.124fmand163×32witha=0.093fm. Therenormaliza- tion point for Z(R)(q)≡Z(ζ;q) is chosen to be ζ =5.31 GeV that the data in Laplacian gauge may be a little higher than in Landau gauge, which would be consistent with (in the q-scale). For the mass functions M(p), the value on 123×24 lattice is higher than that on the163×32 lattice in the behavior of the renormalization function, Z(R)(q). the infrared region. while the wave function renormalization For the renormalizationfunction, Z(R)(q), there are sys- functionZ(R)(q)ontwolatticesagreewithinerrorbars. This tematic differences inthe infraredregion. The renormal- suggests that the finite a errors are not small on the coarse ization function is more strongly infrared suppressed in lattice in Laplacian gauge. the Laplacian gauge than in the Landau gauge. results for M(p) to be very similar. VI. ACKNOWLEDGMENTS Finally, we present the data for the two lattices in Laplacian gauge to further explore possible finite a er- rors. Fig. 6 shows the comparison of the mass function, We thank the Australian National Computing Facil- M(p), and the wave function renormalization function ity for Lattice Gauge Theory and both the Australian Z(R)(q) in the chiral limit in Laplacian gauge on the Partnership for Advanced Computing (APAC) and the 123 24 and the 163 32 lattices. For the renormal- South Australian Partnership for Advanced Computing izati×on function, Z(R)(×q), results from the two lattices (SAPAC) for generous grants of supercomputer time have small differences, but agree with each other within which have enabled this project. This work is supported errors. For the mass function, M(p), the results agree by the Australian Research Council. [1] P. O. Bowman, U. M. Heller, D. B. Leinweber, 128, 23 (2004). A.G.WilliamsandJ.B.Zhang,Nucl.Phys.Proc.Suppl. 8 [2] C.W.Bernard,D.Murphy,A.Soni,K.Yee,Nucl.Phys. ibid, D 65, 117502 (2002). B (Proc. Suppl.), 17, 593 (1990) [19] P.O.Bowman, U.M.Heller,D.B.LeinweberandA.G. [3] C. W. Bernard, A. Soni, K. Yee, Nucl. Phys. B (Proc. Williams, Phys. Rev.D 66, 074505 (2002). Suppl.),20, 410 (1991). [20] P.vanBaal,Nucl.Phys.B(Proc.Suppl.)42,843(1995). [4] J. I. Skullerud and A. G. Williams, Phys. Rev. D 63, [21] M. Lu¨scher and P. Weisz, Commun. Math. Phys. 97, 59 054508 (2001); Nucl. Phys. B (Proc. Suppl.), 83, 209 (1985). (2000). [22] F.D.R.Bonnet,P.O.Bowman,D.B.Leinweber,A.G. [5] J.Skullerud,D.B.LeinweberandA.G.Williams,Phys. Williams and D. G. Richards, Austral J. Phys. 52, 939 Rev.D 64, 074508 (2001). (1999). [6] P.O.Bowman, U.M.Heller, andA.G. Williams, Phys. [23] C.T.H. Davieset al.,Phys.Rev. D 37, 1581 (1998). Rev.D 66 014505 (2002). [24] R. Narayanan and H. Neuberger Nucl. Phys.B443, 305 [7] P.O.Bowman,U.M.Heller,D.B.Leinweber,andA.G. (1995). Williams, Nucl.Phys. (Proc. Suppl.) 119, 323 (2003). [25] H. Neuberger, Phys.Lett. B427, 353 (1998). [8] T. Blum et al., Phys.Rev.D 66, 014504 (2002). [26] R. G. Edwards, U. M. Heller, and R. Narayanan, Phys. [9] F. D. R. Bonnet, P. O. Bowman, D. B. Leinweber, A. Rev. D 59, 094510 (1999). G. Williams and J. B. Zhang, Phys. Rev. D 65, 114503 [27] D. B. Leinweber, J. I. Skullerud, A. G. Williams and (2002). C. Parrinello, Phys. Rev. D 58, 031501(R) (1998); 60, [10] J. B. Zhang, F. D. R. Bonnet, P. O. Bowman, D. B. 094507 (1999); 61, 079901(E) (1999). Leinweber, A. G. Williams, Nucl. Phys. (Proc. Suppl.) [28] F. D. R. Bonnet, P. O. Bowman, D. B. Leinweber and 129, 495 (2004). A. G. Williams, Phys. Rev.D 62, 051501(R) (2000). [11] J. B. Zhang, P. O. Bowman, D. B. Leinweber, A. G. [29] F. D. R. Bonnet, P. O. Bowman, D. B. Leinweber, A. Williams,andF.D.R.Bonnet,Phys.Rev.D70,034505 G.WilliamsandJ.M.Zanotti,Phys.Rev.D64,034501 (2004). (2001). [12] V.N. Gribov, Nucl.Phys. B 139, 1 (1978). [30] F. D. R. Bonnet, D. B. Leinweber, A. G. Williams and [13] J. E. Hetrick & Ph. de Forcrand, Nucl. Phys. B (Proc. J.M.Zanotti,Symanzik improvement inthe static quark Suppl.)63, 838 (1998). potential, hep-lat/9912044. [14] J. F. Markham & T. D. Kieu, Nucl. Phys. B (Proc. [31] A. Cucchieri and T. Mendes, Phys. Rev. D 57, R3822 Suppl.) 73, 868 (1999); O. Oliveira and P.J. Silva, Nucl. (1998). Phys.B (Proc. Suppl.) 106, 1088 (2002). [32] J. van den Eshof et al., Comput. Phys. Commun. 146, [15] A. G. Williams, Prog. Theor. Phys. Suppl. 151, 154 203 (2002); Nucl. Phys. B (Proc. Suppl.), 106, 1070 (2003). (2002). [16] J. C. Vink and U-J. Wiese, Phys. Lett. B 289, 122 [33] C. D. Roberts and A. G. Williams, Prog. Part. Nucl. (1992). Phys. 33, 477 (1994). [17] J. C. Vink,Phys. Rev.D 51, 1292 (1995). [34] R. Alkofer and L. von Smekal, Phys. Rept. 353, 281 [18] C. Alexandrou, Ph. de Forcrand and E. Follana, Phys. (2001). Rev. D 63, 094504 (2001). ibid, D 65, 114508 (2002),