SU(2) meets SU(3) in lattice-Landau-gauge gluon and ghost propagators
aa r X i v : . [ h e p - l a t ] O c t SU(2) meets SU(3) in lattice-Landau-gauge gluonand ghost propagators
A. Cucchieri
Instituto de Física de São Carlos, Universidade de São Paulo,Caixa Postal 369, 13560-970 São Carlos, SP, Brazil E-mail: [email protected]
T. Mendes
Instituto de Física de São Carlos, Universidade de São Paulo,Caixa Postal 369, 13560-970 São Carlos, SP, Brazil E-mail: [email protected]
Orlando Oliveira ∗ Department of Physics, University of Coimbra, 3004 516 Coimbra, PortugalE-mail: [email protected]
P. J. Silva
Department of Physics, University of Coimbra, 3004 516 Coimbra, PortugalE-mail: [email protected]
A comparative study of the lattice Landau gauge gluon and ghost propagators for SU(2) and SU(3)pure Yang-Mills theories is carried out. The data were specially produced with equivalent latticeparameters to allow for a careful comparison of the two cases. We find very good agreement be-tween the two theories. Our results seem to confirm the predicton of Schwinger-Dyson equationsthat the infrared exponents are independent of the gauge group SU(N).
The XXV International Symposium on Lattice Field TheoryJuly 30 - August 4 2007Regensburg, Germany ∗ Speaker. c (cid:13) Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/
U(2) meets SU(3) in lattice-Landau-gauge gluon and ghost propagators
Orlando Oliveira
1. Introduction and Motivation
The investigation of the infrared limit of QCD is of central importance for the comprehensionof the mechanisms of quark and gluon confinement and of chiral-symmetry breaking. However,despite the recent progress, we still do not have the full picture of the infrared structure of Yang-Mills theories.In what concerns gluon confinement, in Landau gauge, the infrared behavior of gluon andghost propagators is linked with the Gribov-Zwanziger [1, 2] and the Kugo-Ojima [3] confine-ment scenarios. These confinement mechanisms predict, at small momenta, an enhanced ghostpropagator and a suppression of the gluon propagator. Analytic studies of gluon and ghost propa-gators using Schwinger-Dyson equations (SDE) [4, 5, 6] seem to agree with the above scenarios.Schwinger-Dyson equations are an infinite tower of nonlinear equations. Typically, the computa-tion of a solution requires the definition of a truncation scheme and the parametrization of vertices.The above mentioned solutions are not the only known solutions. Indeed, in [7, 8] the authors founda set of solutions which do not comply with the above mechanisms. In what concerns the latticeresults for the gluon and ghost propagators, in Landau gauge, one side they seem to support theanalytical studies [9, 10, 11], on the other side they do not confirm the precise predictions obtainedwith SDE [12]. The solution of this apparent puzzle requires further studies.In the Schwinger-Dyson equations, when dynamic quarks are neglected, assuming that g ∼ / N c — as suggested by analysis of the large N c limit [13] — the SDE predict that gluon and ghostpropagators are independent of the number of colors (in the nonperturbative regime). In particular,they predict for the gluon and for the ghost propagators an infrared exponent that is independentof the gauge group SU ( N c ) . In this paper, we carry out a comparative study of lattice Landaugauge propagators for these two gauge groups. Our data were especially produced by consideringequivalent lattice parameters in order to allow a careful comparison of the two cases. For detailson the simulation see [14]. For another study comparing SU ( ) and SU ( ) propagators see [15].In the following the effect of Gribov copies is not taken into account.
2. Numerical Simulations
We consider four different sets of lattice parameters, with the same lattice size N and thesame physical lattice spacing a for the two gauge groups (see Table 1). The first three cases arechosen to yield approximately the same physical lattice volume V ≈ ( . ) . This allows acomparison of discretization effects. The fourth case corresponds to a significantly larger physicalvolume, V ≈ ( . ) , in order to study finite-size effects. For all four cases, 50 configurationswere generated using the Wilson action.The gluon and the ghost propagators D ab mn ( k ) = d ab (cid:18) d mn − k m k n k (cid:19) D ( k ) , (2.1) G ab ( k ) = − d ab G ( k ) (2.2)were computed for four different types of momenta: ( k , , , ) , ( k , k , , ) , ( k , k , k , ) and ( k , k , k , k ) .In the computation of D ( k ) and G ( k ) , an average over equivalent momenta and color componentswas always performed. 2 U(2) meets SU(3) in lattice-Landau-gauge gluon and ghost propagators
Orlando Oliveira N a (fm) Na (fm) b SU ( ) b SU ( ) .
102 1 .
632 2 . . .
073 1 .
752 2 . . .
054 1 .
728 2 . . .
102 3 .
264 2 . . Table 1:
Lattice setup. The lattice spacing was computed from the string tension, assuming √ s = S U ( ) Renormalized Gluon Propagator renormalization scale = 3 GeV0 1 2 3010 S U ( ) Figure 1:
Gluon propagator as function of momenta given in GeV for lattices with volume V ≈ ( . ) . In order to compare the propagators from the different simulations, the gluon and ghost prop-agators were renormalized accordingly to D ( q ) (cid:12)(cid:12) q = m = m , G ( q ) (cid:12)(cid:12) q = m = m , (2.3)using m =
3. The Propagators
The gluon propagator for V ≈ ( . ) is reported in figure 1. In figure 2, the data for differentvolumes, same b value is displayed. The corresponding figures for the ghost propagator are fig. 3and fig. 4, respectively. 3 U(2) meets SU(3) in lattice-Landau-gauge gluon and ghost propagators
Orlando Oliveira S U ( ) Renormalized Gluon Propagator renormalization scale = 3 GeV0 1 2 3 4010 S U ( ) Figure 2:
Gluon propagator as function of momenta given in GeV for b SU ( ) = . b SU ( ) = . For the gluon propagator, figure 1 show some discretization effects which are stronger for theSU(3) data. On the other hand, figure 2 shows finite volume effects, specially for SU(2). In orderto try to understand such effects, in figure 5 one plots the ratios of SU(3) over SU(2) propagatorsfor all the simulations. Note that the plots include ratios of D ( ) , i.e. the most left point shouldbe taken with care. In what concerns the gluon propagator, given the relatively small statistics andgiven that there is no clear systematics in data, one can not conclude on the nature of observed smalldifferences. Anyway, the SU(3) and SU(2) propagators are, at least, qualitatively similar. Giventhe small differences one can also claim quantitative agreement between the two propagators.In what concerns the ghost propagator, the data seems more stable than the gluon points.Indeed, comparing figures 1-4 and the ratios of propagators in fig. 5, fig 6 the ghost data fluctuatesless. Moreover, for the full range of momenta the ratios of ghost propagators are compatible withone at the level of two standard deviations. Therefore, for the ghost propagator one can concludein favour of quantitative and qualitative agreement between SU(2) and SU(3).
4. Results and Conclusions
In summary, considering a careful choice of the lattice parameters, we were able to carry out anunambiguous comparison of the lattice Landau gluon and ghost propagators for SU ( ) and SU ( ) gauge theories. The data show that the two cases have very similar finite-size and discretizationeffects. Moreover, we find very good agreement between the two Yang-Mills theories (for ourvalues of momenta larger than 1 GeV), for all lattice parameters and for all types of momenta.Below 1 GeV, the results for the two gauge groups show some differences, especially for the gluon4 U(2) meets SU(3) in lattice-Landau-gauge gluon and ghost propagators
Orlando Oliveira S U ( ) Renormalized Ghost Propagator
Renormalization Scale 3 GeV1 2 3 401234 S U ( ) Figure 3:
Ghost propagator as function of momenta given in GeV for lattices with volume V ≈ ( . ) . S U ( ) Renormalized Ghost Propagator
Renormalization Scale = 3 GeV1 2 3 4051015 S U ( ) Figure 4:
Ghost propagator as function of momenta given in GeV for b SU ( ) = . b SU ( ) = . U(2) meets SU(3) in lattice-Landau-gauge gluon and ghost propagators
Orlando Oliveira k000kk00kkk0kkkk Ratios of Gluon SU(3)/SU(2) Propagator B = 6.0B = 6.0B = 6.2B = 6.4
Figure 5:
SU(3)/SU(2) gluon propagator as function of momenta given in GeV. k000kk00kkk0kkkk Ratios of Ghost SU(3)/SU(2) Propagators B = 6.0B= 6.2B = 6.0B = 6.4
Figure 6:
SU(3)/SU(2) ghost propagator as function of momenta given in GeV. U(2) meets SU(3) in lattice-Landau-gauge gluon and ghost propagators
Orlando Oliveira propagator. However, given the lattice volumes considered, further studies are required beforedrawing conclusions about the comparison between SU ( ) and SU ( ) propagators in the deep-IRregion. In this sense, we claim that our results support the prediction from the Schwinger-Dysonequations that the propagators are the same for all SU ( N c ) groups in the nonperturbative region. Acknowledgments
P.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.
References [1] V. N. Gribov,
Nucl. Phys. B , 1 (1978).[2] D. Zwanziger,
Phys. Lett. B , 168 (1991);
Nucl. Phys. B , 127 (1991);
Nucl. Phys. B , 657(1994).[3] T. Kugo, I. Ojima,
Prog. Theor. Phys. Suppl. , 1 (1979) [Erratum Prog. Theor. Phys. Suppl. , 1121(1984)].[4] L. von Smekal, A. Hauck, R. Alkofer, Ann. Phys. , 1 (1998).[5] C. Lerche, L. von Smekal,
Phys. Rev. D , 125006 (2002).[6] D. Zwanziger, Phys. Rev. D , 094039 (2002).[7] A. C. Aguilar, A. A. Natale, P. S. Rodrigues da Silva, Phys. Rev. Lett. , 152001 (2003).[8] A. C. Aguilar, A. A. Natale, JHEP , 57 (2004).[9] O. Oliveira, P. J. Silva, arXiv:0705.0964 [ hep-lat], (2007).[10] P. J. Silva, O. Oliveira, Phys. Rev.
D74 , 034513 (2006).[11] P. J. Silva, O. Oliveira,
PoS
LATTICE2007 , 323 (2007).[12] A. Cucchieri, T. Mendes,
PoS ( LATTICE2007 ), 297 (2007).[13] G. t’Hooft,
Nucl. Phys. B , 461 (1974).[14] A. Cucchieri, T. Mendes, O. Oliveira, P. J. Silva, arXiv:0705.3367 [hep-lat] , (2007).[15] A. Sternbeck, D.B. Leinweber, L. von Smekal, A.G. Williams, PoS ( LATTICE2007 ), 340 (2007).), 340 (2007).