We present lattice results for the gluon propagator for SU(2) and SU(3) in the Laplacian gauge which avoids lattice Gribov copies. In SU(3) we compare with the most recent lattice calculation in Landau gauge and with various approximate solutions of the Dyson Schwinger equations (DSE)

Alexandrou, C

De Forcrand, Philippe

Follana, E

We present lattice results for the gluon propagator for SU(2) and SU(3) in the Laplacian gauge which avoids lattice Gribov copies. In SU(3) we compare with the most recent lattice calculation in Landau gauge and with various approximate solutions of the Dyson Schwinger equations (DSE).We present lattice results for the gluon propagator for SU(2) and SU(3) in the Laplacian gauge which avoids lattice Gribov copies. In SU(3) we compare with the most recent lattice calculation in Landau gauge and with various approximate solutions of the Dyson Schwinger equations (DSE)

Alexandrou, C.

de Forcrand, Ph.

Follana, E.

CERN Document Server

The Infrared behaviour of the gluon propagator in SU(2) andSU(3) without lattice Gribov copiesC. Alexandroua, Ph. de Forcrandb,c, E. FollanaaaDepartment of Physics, University of Cyprus, CY-1678 Nicosia, CyprusbInst. fu¨r Theoretische Physik, ETH Ho¨nggerberg, CH-8093 Zu¨rich,Switzerlandc CERN, Theory Division, CH-1211 Geneva 23, SwitzerlandWe present lattice results for the gluon propagator for SU(2) and SU(3) in theLaplacian gauge which avoids lattice Gribov copies. In SU(3) we compare withthe most recent lattice calculation in Landau gauge and with various approximatesolutions of the Dyson Schwinger equations (DSE).IntroductionWe rst summarize the results obtained within the Landau gauge1: Bysolving approximately the DSE, Mandelstam found an infrared enhanced gluonpropagator of the form D(q2)q→0 1q4 . Avoiding gauge copies, Gribov obtainedD(q2)  q2q4+m4 . Using the \pinch technique", Cornwall 2 obtained a solutionwhich fullls the Ward identities, allows a dynamical mass generation, and alsopredicts a nite value for D(0)  D(q2 = 0) consistent with our data.Early results for the gluon propagator obtained directly from Lattice QCDon small lattices 4 were interpreted in terms of a massive scalar propagator.Results on larger lattices were accounted for by assuming a positive anoma-lous dimension 5: D(q2)  1q2(1+α)+m2. A recent, detailed study of the gluonpropagator uses very large lattices 6. Since we want to compare our resultswith these, we follow closely their analysis and refer to Refs. 6,7 for details.In the Laplacian gauge, the longitudinal part of the gluon propagator doesnot vanish; the transverse scalar function D(q2) can be extracted from Dabµν(q)as D(q2) = 13{∑µ18∑aDaaµµ(q)}− 13 F (q2)q2 , where F (q2) is determined byprojecting the longitudinal part of Daaµν(q) using the symmetric tensor qµqν .Gauge Fixing ProcedurePrevious lattice studies all xed to Landau gauge by using a local iterativemaximization algorithm, which converges to any one of many local maxima(lattice Gribov copy), but fails to determine the global one. To overcome thisproblem, we use a dierent gauge condition, the Laplacian gauge 3, which∗Talk given by C. Alexandrou1is Lorentz-symmetric and gives a smooth gauge eld like the Landau gauge,but which species the gauge unambiguously. We consider the maximizationof Q = Re∑x,µ Tr[g(x)Uµ(x)gy(x + µ^)− g(x)g(x)y]. If one relaxes the re-quirement that g 2 SU(N), maximizing Q is equivalent to minimizing thequadratic form∑xy fxxyfy, with (U) the covariant Laplacian. Using the(N −1) lowest-lying eigenvectors of (U), one can construct Ω(x) 2 SU(N) 7.ResultsIn Fig.1 we show the transverse gluon propagator for SU(2) Yang-Millstheory in two dierent volumes; m0 √D(0)−1 for the 164 lattice. Changingthe volume has little eect, in particular on D(0). We observe similarly smallvolume eects in SU(3). This is strikingly dierent from Landau gauge, whereZwanziger has argued that D(0) should vanish in the innite lattice volumelimit 8. This prediction is indeed consistent with recent lattice results in SU(2)at nite temperature 9. In contrast, in the Laplacian gauge, we nd that D(0)is nite and independent of the volume V for V larger than about 1/2fm4 D(0)2. We nd D(0) = 58(2) in lattice units at β = 6.0, i.e. D(0)−1/2 = 248(5)MeV (using a−1 = 1.885 GeV), corresponding to a length scale of about 0.8 fm.In Fig.2 we compare results for the gluon propagator in SU(3) quenchedQCD in Laplacian and Landau gauges. (m0 √D(0)−1 in the Laplaciangauge). Scaling is checked on the 163 32 lattice for β = 5.8 and 6.0. Makinga cylindrical cut in the momenta 6 to minimize lattice artifacts, we nd thatscaling is very well satised for the Laplacian gauge, with both sets of datafalling on a universal curve 7.We t to our data the same models as considered by Leinweber et al. 6in Landau gauge. Since we have observed scaling, we use our results at theFigure 1: The SU(2) gluon propagator intwo different volumes.Figure 2: The SU(3) gluon propagator inLaplacian and Landau gauges.2Model Z m λ or α A D(0) χ2/d.o.fGribov 2.63(2) 0.203(7) 0 5.7Stingl 2.63(2) 0.203(13) 0.002 (1.1) 0 5.7Marenzoni 2.47(3) 0.199(6) 0.237(5) 62 4Cornwall 7.08(9) 0.281(4) 0.265(8) 59 2.5Model A 1.96(1) 0.654(17) 2.181(67) 8.91(41) 43 1.2Table 1: best fit of parameter values to our β = 6.0 data on the 163 × 32 lattice.ner lattice spacing (β = 6.0) for the ts. Table 1 and Fig. 3 summarize theresults of the ts to the various models. We nd that Gribov{type models areexcluded, whereas Cornwall’s model is clearly favored among all analyticallymotivated models. Model \A" 6, which gives a better t, is phenomenological,contains one more parameter, and misses D(0) by 25%. One can then use thet to Cornwall’s model to analytically continue to negative q2 and determinethe gluon pole mass. This is carried out in Ref. 7.In conclusion, we see signicantmodications from Landau gauge inthe infrared. In particular, we ndthat D(0) obeys scaling, is nite,and volume independent for largeenough volumes. We nd supportfor Cornwall’s model which ts themomentum dependence of the prop-agator rather well, whereas modelswith infrared enhancement of thetype 1/(q2)2 or Gribov{type sup-pression are excluded. Figure 3: Fits to various modelsReferences1. J. E. Mandula, Phys. Rep. 315 (1999) 273.2. J. M. Cornwall, Phys. Rev. D26 (1982) 1453.3. J. C. Vink and U. Wiese, Phys. Lett. B 289 (1992) 122.4. J. E. Mandula and M. Ogilvie, Phys. Lett. B185 (1987) 127;R. Guptaet al., Phys. Rev. D36 (1987) 2813.5. P. Marenzoni, G. Martinelli and N. Stella, Nucl. Phys. B455 (1995) 339.6. D. Leinweber et al., Phys. Rev. D60 (1998) 094507; T. G. Williams, thisvolume.7. C. Alexandrou, Ph. de Forcrand and E. Follana, hep-lat/0008012.38. D. Zwanziger, Phys. Lett. B257 (1991) 168.9. A. Cucchieri, hep-lat/9908050.4

The Infrared behaviour of the gluon propagator in SU(2) and SU(3) without lattice Gribov copies

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