Lattice discretisation errors in the Landau gauge condition are examined. An
improved gauge fixing algorithm in which order a^2 errors are removed is
presented. Order a^2 improvement of the gauge fixing condition displays the
secondary benefit of reducing the size of higher-order errors. These results
emphasise the importance of implementing an improved gauge fixing condition.Comment: LATTICE99 (Improvement and Renormalization), 3 pages, 1 figur

A.G. Williams

Cucchieri

D.B. Leinweber

D.G. Richards

Davies

F.D.R. Bonnet

Leinweber

Lepage

P.O. Bowman

Nuclear Physics B - Proceedings Supplements

English

arXiv

Crossref

Improved Landau gauge fixing and discretisation errors

Copyright © 1999. Submitted to Cornell University’s online archive www.arXiv.org in 1999 by Patrick O. Bowman. Post-print sourced from www.arxiv.org.Lattice discretisation errors in the Landau gauge condition are examined. An improved gauge fixing algorithm in which order a^2 errors are removed is presented. Order a^2 improvement of the gauge fixing condition displays the secondary benefit of reducing the size of higher-order errors. These results emphasise the importance of implementing an improved gauge fixing condition.F. D. R. Bonnet, P. O. Bowman, D. B. Leinweber, D. G. Richards and A. G. William

Bonnet, F.

Bowman, P.

Leinweber, D.

Richards, D.

Williams, A.

Adelaide Research & Scholarship

arXiv:hep-lat/9909110v2  17 Dec 19991Improved Landau Gauge Fixing and Discretisation Errors ∗F. D. R. Bonneta, P. O. Bowmana, D. B. Leinwebera, D. G. Richardsb and A. G. Williamsac †aCSSM and The Department of Physics and Mathematical Physics, University of Adelaide, AdelaideSA 5005, AustraliabDept. of Physics and Astronomy, University of Edinburgh, Edinburgh EH9 3JZ, Scotlandand TJNAF, 12000 Jefferson Avenue, Newport News, VA 23606, USA.cDepartment of Physics and SCRI, Florida State University, Tallahassee, Florida, USALattice discretisation errors in the Landau gauge condition are examined. An improved gauge fixing algorithmin which O(a2) errors are removed is presented. O(a2) improvement of the gauge fixing condition displaysthe secondary benefit of reducing the size of higher-order errors. These results emphasise the importance ofimplementing an improved gauge fixing condition.1. IntroductionGauge fixing in lattice gauge theory simulationsis crucial for many calculations e.g. the studyof gauge dependent quantities such as the gluonpropagator [1]. However, the standard latticeLandau gauge condition [2] is the same as the con-tinuum condition,∑µ ∂µAµ = 0, only to leadingorder in the lattice spacing, a.The focus of this talk is to use mean-field-improved perturbation theory [3] to compare dif-ferent lattice definitions of the Landau gauge,and quantify the sizes of the discretisation errors.In particular, we derive a new O(a2) improvedLandau-gauge-fixing functional, and a method ofgeneralising this to O(an).2. Lattice Landau GaugeGauge fixing on the lattice is achieved by max-imising a functional whose extremum implies thegauge fixing condition. The usual Landau gaugefixing functional is [2]FG1 [{U}] =∑µ,x12Tr{UGµ (x) + UGµ (x)†}, (1)where UGµ (x) = G(x)Uµ(x)G(x + µˆ)†. By takingthe functional derivative of (1), it can be shown∗Supported in part by the Australian Research Council†Talk presented by AGWthat a maximum of that functional implies thecontinuum Landau gauge, with O(a2) errors. Itcan further be shown that the gauge fixing con-dition implies that∑µ∂µAµ(x) =∑µ{−a212∂3µAµ(x)−H1}, (2)where H1 represents O(a4) and higher-orderterms. Na¨ıvely one might hope that higher-orderderivatives in the brackets are small, but it willbe shown that the terms on the R.H.S. of (2) arelarge compared to the numerical accuracy possi-ble in gauge fixing algorithms.This “one-link” functional can be generalisedto functionals using “n-link” terms:FGn [U ] ≡∑x,µ12n2Tr{UGnµ(x) + h.c.}(3)whereUGnµ(x) ≡ UGµ (x)UGµ (x + µˆ)...UGµ (x + (n − 1)µˆ).ThenδFGnδωa(x)=12n2i∑µTr{[UGnµ(x− µˆ)− UGnµ(x)−h.c.]T a} (4)= ga2(∑µTr{[∂µAµ(x) +2(na)2{(na)44!∂3µAµ(x)2+(na)66!∂5µAµ(x) + ...}]T a}+O(g2a2))O(a2) errors can be removed from the gaugefixing condition by taking a linear combination ofthe one-link and two-link functionals:FGImp =43FG1 −112u0FG2 (5)where we have included the plaquette-based,mean-field (tadpole) improvement parameter,u0 [3].To perform the gauge fixing we adopt a “steep-est descents” approach [2]. The gauge transfor-mation is G(x) = exp{−iα∑µ ∂µAµ(x)}. Tomaximise, for example, FG1 , we use (4) to derivethe gauge transformationG1(x) = exp{α2∆1(x)}, (6)where∆1(x) =∑µ{UGµ (x − µˆ)− UGµ (x)− h.c.}traceless.Similarly, ∆2 and ∆Imp are obtained from thefunctional derivatives of F2 and FImp respec-tively. For a given functional, FGi , the gaugefixing algorithm proceeds by calculating the rel-evant ∆i, applying the associated gauge trans-formation to the gauge field, and iterating untilthe lattice Landau gauge condition is satisfied, towithin some numerical accuracy. The approachto Landau gauge is measured byθi =1V Nc∑xTr{∆i(x)∆i(x)†}. (7)A configuration fixed using ∆1(x) will satisfy(2). It will also satisfy∆2(x) = −2iga2∑µ{a24∂3µAµ(x) −H1 +H2}and similarly,∆Imp(x) = −2iga2∑µ{−a212∂3µAµ(x)−H1 +HImp} . (8)Since the improved measure has no O(a2) error ofits own, (8) provides an estimate of the absolutesize of these discretisation errors.Table 1Values of the gauge-fixing measures obtained us-ing the improved gluon action on 64 lattices atthree values of the lattice spacing, fixed to Lan-dau gauge with the one-link, two-link and im-proved functionals respectively.β u0 F θImp θ2θImpθ23.92 0.837 1 0.102 0.921 0.1114.38 0.880 1 0.0585 0.526 0.1115.00 0.904 1 0.0410 0.369 0.111β u0 F θImp θ1θ1θImp3.92 0.837 2 57.5 32.3 0.5634.38 0.880 2 53.4 30.0 0.5635.00 0.904 2 52.2 29.4 0.563β u0 F θ1 θ2θ1θ23.92 0.837 Imp 0.0638 1.02 0.06254.38 0.880 Imp 0.0366 0.586 0.06255.00 0.904 Imp 0.0261 0.417 0.06253. Calculations on the LatticeWe use an O(a2) tadpole-improved action. Forthe exploration of gauge fixing errors we con-sider 64 lattices at β = 3.92, 4.38, and 5.00, cor-responding to lattice spacings of approximately0.35, 0.17, and 0.1 fm respectively.The configurations are gauge fixed, using Con-jugate Gradient Fourier Acceleration [4] untilθ1 < 10−12. θImp and θ2 are then measured, to seethe size of the residual higher order terms. Theevolution of the gauge fixing measures is shownfor one of the lattices in Fig. 1. This procedure isthen repeated, fixing with each of the other twofunctionals, and the results are shown in Table 1.Results from additional lattices, as well as a moredetailed discussion, are in [5].If we fix a configuration to Landau gauge byusing the basic, one-link functional, then the im-proved measure, θImp, will consist entirely of thediscretisation errors. Looking at Table 1, we seethat at β = 4.38, θImp = 0.059, a substantial de-viation from the continuum Landau gauge com-pared to the tolerance of the gauge fixing. Wenote that the relationship between the function-als in (5) provides a constraint on the gauge fixing3Figure 1. The gauge fixing measures for a 64 lat-tice with Wilson action at β = 6.0. This lat-tice was gauge fixed with ∆1, so θ1 drops steadilywhilst θ2 and θImp plateau at much higher values.measures. For example, when fixing with ∆1θImpθ2=(− 112)2(− 112+ 13)2=19≃ 0.111. (9)A configuration fixed using ∆Imp(x) will satisfy∑µ∂µAµ(x) =∑µ{−HImp} . (10)With the help of (2) we see that∆1(x) = −2iga2∑µ{a212∂3µAµ(x) +H1 −HImp}and (8) are identical to within a sign. If the threedifferent methods presented all fixed in exactlythe same way, then the θImp of a configurationfixed with ∆1, would be equal to θ1 when theconfiguration is fixed with ∆Imp. It is clear fromthe table that they are not, signaling the higher-order derivative terms ∂nµAµ(x), contained in thethe Hi, take different values depending on thegauge fixing functional used.Examining the values in Table 1 reveals that inevery case θ1 is smaller when we have fixed withFImp than θImp under the F1. This suggests thatthe additional long range information used by theimproved functional is producing a gauge fixedconfiguration with smaller, higher-order deriva-tives; a secondary effect of improvement.Equally, one can compare the value of θ2 whenfixed using F1, and θ1 when fixed using the F2.In this case, their differences are rather large andare once again attributed to differences in the sizeof higher-order derivatives of the gauge field. F2is coarser, knows little about short range fluctu-ations, and fails to constrain higher-order deriva-tives. Similar conclusions are drawn from a com-parison of θ2 fixed with the FImp and θImp fixedwith F2.We also find that in terms of the absolute er-rors, the Wilson action at β = 6.0 is comparableto the improved lattice at β = 4.38, where thelattice spacing is three times larger.4. ConclusionsWe have fixed gluon field configurations to Lan-dau gauge by three different functionals: one-linkand two-link functionals, both with O(a2) errors,and an improved functional, with O(a4) errors.Using these functionals we have devised a methodfor estimating the discretisation errors involved.Lattice Landau gauge, in its standard implemen-tation, deviates from its continuum counterpartby one part in 20, despite fixing the Lattice gaugecondition to one part in 1012. Our results indicatethat order O(a2) improvement of the gauge fix-ing condition improves comparison with the con-tinuum Landau gauge through: 1) the elimina-tion of O(a2) errors and 2) reducing the size ofhigher-order errors.REFERENCES1. D.B. Leinweber et al. Phys. Rev. D 58,031501 (1998), hep-lat/9803015; ibid., Phys.Rev. D 60, 094507 (1999), hep-lat/9811027,and references therein.2. C.T.H. Davies et al, Phys. Rev. D 37, 1581(1988).3. G.P. Lepage & P.B. Mackenzie, Phys. Rev. D48, 2250 (1993).4. A. Cucchieri & T. Mendes, Phys. Rev. D 57,3822 (1998).5. F.D.R. Bonnet et al., Austral.J.Phys. 52(1999) 939-948, hep-lat/9905006

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Improved Landau Gauge Fixing and Discretisation Errors

Improved Landau Gauge Fixing and Discretisation Errors

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