The Landau-gauge gluon propagator is computed for the SU(3) gauge theory on
lattices up to a size of $32^3 \times 200$. We use the standard Wilson action
at $\beta = 6.0$ and compare our results with previous computations using large
asymmetric and symmetric lattices. In particular, we focus on the impact of the
lattice geometry and momentum cuts to achieve compatibility between data from
symmetric and asymmetric lattices for a large range of momenta.Comment: Poster presented at Lattice2007, Regensburg, July 30 - August 4, 200

Ilgenfritz, E. -M.

Oliveira, O.

Silva, P. J.

Sternbeck, A.

English

arXiv

Also cited as: Lattice 2007, the XXV International Symposium on Lattice Field Theory, July 30 - August 4 2007, Regensburg, Germany / Gunnar Bali, Vladimir Braun, Christof Gattringer, Meinulf Göckeler, Andreas Schäfer, Peter Weisz, Tilo Wettig (eds.): 323/1-323/7.The Landau-gauge gluon propagator is computed for the SU(3) gauge theory on lattices up to a size of 323 ×200. We use the standard Wilson action at β=6.0 and compare our results with previous computations using large asymmetric and symmetric lattices. In particular, we focus on the impact of the lattice geometry and momentum cuts to achieve compatibility between data from symmetric and asymmetric lattices for a large range of momenta.Orlando Oliveira, P. J. Silva, E.-M. Ilgenfritz and A. Sternbec

Silva, P.

Ilgenfritz, E.

Adelaide Research &amp; Scholarship

PUBLISHED VERSION  http://hdl.handle.net/2440/58357   Orlando Oliveira, P. J. Silva, E.-M. Ilgenfritz and A. Sternbeck The gluon propagator from large asymmetric lattices Proceedings of Science, 2007; PoS(LATTICE 2007): 323/1-323/7. Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike Licence.                    PERMISSIONS  http://pos.sissa.it/POSlicense.html  Given the Open Access nature of this proceedings collection, PoS has chosen to apply the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License. http://creativecommons.org/licenses/by-nc-nd/4.0/  21 October 2015      PoS(LATTICE 2007)323The gluon propagator from large asymmetric latticesOrlando Oliveira∗Department of Physics, University of Coimbra, 3004 516 Coimbra, PortugalE-mail: orlando@teor.fis.uc.ptP. J. SilvaDepartment of Physics, University of Coimbra, 3004 516 Coimbra, PortugalE-mail: psilva@teor.fis.uc.ptE.-M. IlgenfritzHumboldt-Universität zu Berlin, Institut für Physik, Newtonstr. 15, 12489 Berlin, Germany,E-mail: ilgenfri@physik.hu-berlin.deA. SternbeckCSSM, School of Chemistry & Physics, The University of Adelaide, SA 5005, AustraliaE-mail: andre.sternbeck@adelaide.edu.auThe Landau-gauge gluon propagator is computed for the SU(3) gauge theory on lattices up to asize of 323× 200. We use the standard Wilson action at β = 6.0 and compare our results withprevious computations using large asymmetric and symmetric lattices. In particular, we focuson the impact of the lattice geometry and momentum cuts to achieve compatibility between datafrom symmetric and asymmetric lattices for a large range of momenta.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 Licence. http://pos.sissa.it/PoS(LATTICE 2007)323The gluon propagator from large asymmetric lattices Orlando Oliveira1. Introduction and motivationThe investigation of the Landau-gauge gluon propagatorDabµν(q) = δ ab(δµν −qµqνq2)D(q2) (1.1)in lattice QCD dates back to more than 20 years. Despite the efforts made by a number of authors,there are questions which remain to be answered. For large momenta, let us say q > 2 GeV, theresults from different groups provide a consistent picture of the propagator and agree well withcalculations performed using other non-perturbative techniques. On the other hand, the infraredlimit is still an open issue and has been a field of intense research in the last years (see the con-tributions to this conference, e.g. [1], and references therein). From the point of view of latticesimulations, the questions to be answered yet are numerous and sometimes are even not easy tohandle with. For example, it is still under debate how to deal with the Gribov ambiguity in lat-tice simulations, and how close present lattice data come to Landau-gauge gluodynamics in thecontinuum and infinite-volume limit.In order to access the infrared limit of the gluon propagator, two of us (O.O. and P.J.S.) recentlyproposed and explored the use of large asymmetric lattices [2, 3, 4], i.e. L3×T with T ≫ L. Theprice of relying on such kind of lattices are the control, or the lack of it, of additional finite-volume effects coming from a breaking of the Z4 symmetry, a remnant of the O(4) continuumsymmetry on a symmetric hypercubic lattice. When previous studies on symmetric lattices haveshown strong finite-volume effects in the infrared region [5], the situation is more dramatic forasymmetric lattices. For example, there the gluon dressing function Z(q2) = q2D(q2) computed atequal time-like and spatial momenta are not necessarily compatible within pure statistical errors inthe low-momentum region.On the other hand, the access to very low momenta is much more computationally intensive insimulations on symmetric lattices compared to those on asymmetric ones. Therefore, if somehowthe asymmetry-induced finite-volume effects were brought under control, data at much lower mo-menta than currently available could be obtained. Indeed, then still the infinite volume limit has tobe taken, but in a situation where more data were available in the infrared momentum region.Having now access to a considerably larger spatial volume for the asymmetric case, in thisstudy we report on some first results obtained comparing data on symmetric and asymmetric lat-tices, namely 163× 256, 183× 256, 323× 200 and 324. In particular, we look for regions in thelattice momentum space where the differences between time-like and spatial momenta disappearand where not. For our simulations we use the standard Wilson gauge action with β = 6.0 fixed.This value corresponds to an inverse lattice spacing of about a−1 = 1.94 GeV. To relate our data atthe different lattice momenta to their continuum counterparts we useqµ =2asin(pinµLµ), nµ = 0, 1, . . . ,Lµ −1, (1.2)where Lµ is the lattice extent in direction µ . Definitions and details on the gauge fixing are givenin [7] and for the 323× 200 data in [8]. In the following, whenever possible, a Z3 average overequivalent momenta is performed.2PoS(LATTICE 2007)323The gluon propagator from large asymmetric lattices Orlando Oliveira0 1 2 3 4 5 6 7 8q (GeV)051015q2 D(q2)Temporal MomentaSpatial On-axis MomentaCylindrical CutConical CutBare Gluon Dressing Function323x200,      β = 6.00 2 4 6 8q (GeV)051015q2 D(q2)Temporal MomentaCylindrical Cut(111X) Cut(1116)  CutBare Gluon Dressing Function323x200,      β = 6.0Figure 1: The gluon dressing function on a 323× 200 lattice. The different momentum cuts are discussedin the text.2. Asymmetric lattice: 323×200The gluon propagator and dressing function were computed for 39 configurations using a323 × 200 lattice. The data shows discretization effects similar to those seen in the symmetriclattices. Indeed, it is well-known that for momenta above q > 1 GeV the propagator is not a simplefunction of q2 = q2µ alone. The traditional approach is to apply momenta cuts [6] which reduce thedependence of the propagator and the dressing function on other Z4 (in our case Z3) invariants to aunique curve. This is better seen in the gluon dressing function. In order to illustrate this effect, inFig. 1 we plot the gluon dressing function for different choices of momenta (purely time-like anddifferent cuts of spatial momenta).The plot shows that, within our limited statistics, there is very good agreement between thedressing functions computed using purely spatial on-axis and purely temporal momenta. The figuredoes not include the dressing function for all purely spatial momenta. However, in what concernsthe purely spatial momenta, the gluon dressing function for on-axis momenta evolves typicallyalong the lower edge of the spatial (including off-axis) momentum data. The diagonal choice ofmomenta, i.e. the cylindrical cut [6] where nµ ≈ ±nν (see the left plot in Fig. 1) picks up anunique propagator which is slightly above the propagator for the on-axis choice for momenta at qlarger than ∼ 1 GeV. Note that this “democratic” choice of momenta has been successfully used tosuppress discretization effects such that data from different volumes and lattice spacings match atlarger momenta.In the right plot of Fig. 1 we also show two naive generalizations to the asymmetric case ofthe on-axis case and of a “democratic” choice of momenta (the cylinder cut), labelled as (111X)and (1116) cuts. The former includes momenta only of type nµ = (±1,±1,±1,±nt) with nt =0,1, . . . ,T/2 (softening of on-axis momenta), while the latter includes momenta defined around thedirection nµ = (±n,±n,±n,±6nt ) with n = 0,1, . . . ,L−1 and nt = n,n±1,n±2. This direction isclose to the diagonal in the elongated volume (remember that T/L = 6.25). Note that the (111X)cut reproduces the results of the symmetric lattice for the on-axis choice of momenta, while the(1116) cut follows the symmetric lattice data for the cylindrical and conical cuts. Given that thecylindrical and conical cuts seem to reduce the finite-volume effects for momenta above 1 GeV,3PoS(LATTICE 2007)323The gluon propagator from large asymmetric lattices Orlando Oliveira0 1 2 3 4 5 6 7 8q (GeV)051015q2 D(q2)On-axis MomentaCylindrical CutConical CutBare Gluon Dressing Function324,      β = 6.00 1 2 3 4 5 6 7 8q (GeV)051015q2 D(q2)Cylindrical Cut        323x200Temporal Momenta 323x200On-axis Momenta    324Cylindrical Cut        324Bare Gluon Dressing Function324 vs.  323x200,      β = 6.0Figure 2: The gluon dressing function for a 324 lattice is shown on the left hand side. The right figurecompares the same data with those on a 323×200 lattice. Different cuts have been applied to the data.Fig. 1 suggests that for the asymmetric lattices one should use the (1116) cut, or a variant of it, forthat momentum region.3. Symmetric lattice: 324The gluon dressing function Z(q2) = q2D(q2), computed for an ensemble of 50 gauge config-urations on a 324 lattice, is given in the left plot of Fig. 2 applying various cuts. The lattice datashows similar discretization effects as for the asymmetric lattice. This is illustrated in the right plotof the same figure. There, data for the two lattices 324 and 323× 200 are shown for two differentmomentum cuts at larger momenta and good agreement is found. Moreover, within our limitedstatistics, the cuts produce similar effects for both lattices where the on-axis data lie in both casessystematically below cylinder-cut data for q > 2 GeV.In Fig. 3 the lattice gluon dressing function for a 323×200 and a 324 lattice are compared formomenta below 2 GeV. Again, two different cuts (on-axis and cylindrical) are considered. We findthat in this momentum range the dressing functions Z(q2) for the two lattice geometries are in goodagreement, even though we cannot compare at the low-lying momenta. A comparison with datafrom larger symmetric lattices is necessary to become more confident in this. At least, the goodmatching between 0.5 GeV and 2 GeV is encouraging in what concerns the use of asymmetriclattices to extract reliable infrared properties in future lattice simulations.4. The impact of the spatial volumeNow we discuss the volume dependence of the propagator D(q 6= 0) as function of the physicalmomentum q. In Fig. 4 we compare the gluon propagator on one hand and the dressing function onthe other hand for various asymmetric lattices, for 324 (all at β = 6.0) and for the continuum Dyson-Schwinger solution of Ref. [10]. All propagators were renormalized according to the conditionD(q2)∣∣q2=µ2 =1µ2 , (4.1)4PoS(LATTICE 2007)323The gluon propagator from large asymmetric lattices Orlando Oliveira0 1 2q (GeV)051015q2 D(q2)Cylindrical Cut        323x200Temporal Momenta 323x200On-axis Momenta    324Cylindrical Cut        324Bare Gluon Dressing Function324 vs. 323x200,      β = 6.0Figure 3: The gluon dressing function at low momenta for the 324 and the 323×200 lattices for two differentmomenta cuts.0 1 2q (GeV)0123456789101112D(q2)DSE323x 200324 On-axis183x 256163x 256Gluon PropagatorRenormalized at 3 GeV0 1 2 3q (GeV)00.511.52Z(q2 )DSE323x 200324 On-axis183x 256163x 256Gluon Dressing FunctionRenormalized at 3 GeVFigure 4: The gluon propagator (left) and the gluon dressing function (right) for various asymmetric latticesand for 324 at β = 6.0 . compared with the DSE solution.with the choice µ = 3 GeV. For asymmetric lattices, the data is for time-like (on-axis) momenta.For the symmetric lattice, the plot includes only on-axis momenta. Fig. 4 shows that the two resultsbecome closer as the lattice volume increases.5. Fitting the IR gluon dressing functionIn previous investigations it was verified that the lattice gluon dressing function and the contin-uum Dyson-Schwinger solution are well described not by a pure power law but by both functionsZI(q2) = ω(q2q2 + Λ2)2κ, ZII(q2) = ω(q2)2κ(q2)2κ +(Λ2)2κ(5.1)5PoS(LATTICE 2007)323The gluon propagator from large asymmetric lattices Orlando OliveiraLattice qmax κ Λ χ2/d.o. f . # conf163×256 664 0.5090+19−20 409+4−4 0.71 155183×256 711 0.5320+28−30 389+5−6 1.14 150323×200 728 0.532+12−12 465+25−23 1.28 39Table 1: Parameters corresponding to a fit of the lattice gluon dressing function according to ZI(q2). qmaxand Λ are given in MeV.Lattice qmax κ Λ χ2/d.o. f . # conf163×256 664 0.5077+16−17 409+4−3 0.69 155183×256 711 0.5266+29−21 391+3−7 1.09 150323×200 728 0.528+10−8 464+22−23 1.16 39Table 2: Parameters corresponding to a fit of the lattice gluon dressing function according to ZII(q2). qmaxand Λ are given in MeV.for momenta below ∼ 700 MeV.The results of fitting the lattice dressing function for purely temporal momenta with ZI and ZIIare reported in tables 1 and 2, respectively; qmax is the highest momentum included in the fits. Notethat, the exponent κ agrees within one standard deviation for the two largest volumes. Furthermore,for these lattices, κ agrees with the estimate of O.O. and P.J.S. [3], κ ∼ 0.53, from using ratios ofpropagators to suppress the volume dependence. The ratio method discussed in [3], if applied tothe 323×200 data, estimates κ = 0.565±0.040 for the infrared exponent.The results give κ consistently above 0.5. If this really represents the infrared asymptotics, itsupports a vanishing q→ 0 limit of the gluon propagator D(q 6= 0). Moreover, one should keep inmind that the fits to a pure power law provide always a κ < 0.5, with κ increasing with the latticevolume. It should be noted, however, that so far lattice simulations have always reported a finiteand not vanishing gluon propagator at zero momentum [12].6. Results and ConclusionsIn this work the gluon propagator and dressing function has been analysed for various asym-metric lattices and a comparison to 324 data has been done. Despite the observed finite-volumeeffects, for volumes as large as 323 × 200 the dressing function Z(q2) for purely temporal mo-menta agrees well, within the available statistics, with the corresponding function for purely spatialon-axis momenta.In what concerns the 323× 200 and 324 data, the momentum cuts produce similar results forthe full range of momenta, i.e. the gluon propagator/dressing function for on-axis momenta aresystematically below cone-cut or cylinder-cut data for q > 1 GeV.6PoS(LATTICE 2007)323The gluon propagator from large asymmetric lattices Orlando OliveiraThe behaviour of the lattice infrared gluon dressing function Z(q2) is well described by the twoansätze ZI and ZII for q < 700MeV . The fits to the data provide κ values which support a vanishingzero momentum limit of the gluon propagator for all the lattices reported here. The measurementof the infrared exponent κ for the two larger lattices suggests a value κ ∼ 0.53, in agreement withthe estimate discussed in [3].AcknowlegdementsP.J.S. acknowledges F.C.T. financial support via grant SFRH/BD/10740/2002. This work wassupported in part by F.C.T. under contracts POCI/FP/63436/2005 and POCI/FP/63923/2005. E.-M. I. thanks the DFG for support through the Forschergruppe FOR 465. A. S. is supported by theAustralian Research Council.References[1] C. S. Fischer et al., arXiv:0709.3205 [hep-lat] (2007); A. Sternbeck, et al., PoS(LATTICE2007)340,(2007); I. L. Bogolubsky et al., PoS(LATTICE2007)290, (2007); PoS(LATTICE2007)318;O. Oliveira, et al., PoS(LATTICE2007)332, (2007); A. Cucchieri, et al, PoS(LATTICE2007)297,(2007).[2] O. Oliveira, P. J. Silva, Braz. J. Phys. 37, 201 (2007).[3] O. Oliveira, P. J. Silva, arXiv:0705.0964[hep-lat] (2007).[4] O. Oliveira, P. J. Silva, AIP Conf. Proc. 756, 290 (2005).[5] See, for example, A. Cucchieri, T. Mendes, O. Oliveira, P. J. Silva, arXiv:0705.3367[hep-lat] (2007).[6] D. B. Leinweber, J. I. Skullerud, A. G. Williams, C. Parrinello, Phys. Rev. D58 (1998) 031501; Phys.Rev. D60 (1999) 094507.[7] P. J. Silva, O. Oliveira, Phys. Rev. D74, 034513 (2006).[8] A. Sternbeck, E.-M. Ilgenfritz, M. Müller-Preussker, A. Schiller, Phys. Rev. D72, 014507 (2005).[9] O. Oliveira, P. J. Silva, PoS(LAT2005)287, (2005).[10] C. S. Fischer, M. R. Pennington, Phys. Rev. D73, 034029 (2006).[11] O. Oliveira, P. J. Silva, EPJA31, 790 (2007)[12] See, for example, E.-M. Ilgenfritz, M. Müller-Preussker, A. Sternbeck, A. Schiller, I. L. Bogolubsky,Braz. J. Phys. 37, 193 (2007).7

The gluon propagator from large asymmetric lattices

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The gluon propagator from large asymmetric lattices

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The gluon propagator from large asymmetric lattices

Abstract

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