Green s Functions and Topological Configurations

Transcription

1 Green s Functions and Toological Configurations Deartment of Theoretical Physics, Institute of Physics, Karl-Franzens University Graz, Universitätslatz, A-8 Graz, Austria and Deartment of Comlex Physical Systems, Institute of Physics, Slovak Academy of Sciences, Dúbravská cesta 9, SK-8 Bratislava, Slovakia Axel.Maas@Uni-Graz.at There are, among others, currently two imortant views on the non-erturbative structure of Yang- Mills theory. One is through toological configurations and one is through Green s functions, in articular their (asymtotic) infrared behavior. Based on both views, various scenarios for confinement, chiral symmetry breaking and other non-erturbative effects have been develoed. However, if both views are correct then they can only be different asects of the same underlying hysics, and it must be ossible to relate them. After discussing the current status of the understanding of this connection, smeared and cooled configurations in lattice gauge theory are used to determine the roerties of Green s functions in the low-momentum regime. It is found that the qualitative roerties are essentially unchanged comared to results on unsmeared configurations. This is also the case when the configurations are smeared sufficiently strongly to reach the almost (anti-)self-dual domain. PoS(Confinement8) 8th Conference Quark Confinement and the Hadron Sectrum Setember -, 8 Mainz, Germany Seaker. Suorted by the DFG under grant number MA 9/- and by the FWF under grant number P. I am grateful to the organizers for inviting me to give this talk, and for this interesting conference. c Coyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike Licence. htt://os.sissa.it/

2 Green s Functions and Toological Configurations. Connecting confinement scenarios There are two views on the long-distance dynamics of Yang-Mills theory (and QCD), which are currently emloyed rominently []. One view is based on toological configurations of various tyes []. Most investigated among them are vortices and monooles, but there are also other ossibilities like merons, calorons, instantons, and others. These configurations rovide direct access to the generation of (quark) confinement, chiral symmetry breaking, the finite temerature hase transition and other non-erturbative features of Yang-Mills theory and QCD. The other view is by means of Green s functions []. These rovide access to (gluon) confinement, chiral symmetry breaking, bound states and also other non-erturbative henomena. They also ermit directly the connection to erturbation theory. Both sets of scenarios are not yet comlete, and the discussion on various asects of them continues [,, ]. Nonetheless, the most develoed scenarios of both cases, the combined vortex and monoole scenario [, ] and the scenarios of Gribov and Zwanziger and of Kugo and Ojima (GZKO) [,, ], rovide already a quite encomassing view of the long-distance shae of QCD. Given the considerable successes of both views, it seems rather unlikely that either is fundamentally flawed. If therefore both are correct views of the same underlying hysics, they must be connected. Understanding this connection may also rovide insight into the yet missing ieces of both views, and may also hel to settle ersisting roblems of the individual views. This relation is to a large extent unexlored. However, there are a number of observations, mostly on the connection of the infrared roerties of Landau and Coulomb gauge roagators to vortices, monooles, and instantons. In this context it is imortant that most toological configurations are gauge-deendent quantities, and that their identification in various gauges is differing in difficulty []. On the other hand, the GZKO scenario can be alied to the class of linear-λ gauges, including the Landau and Coulomb gauge. This requirement of gauge-fixing is due to the concet of quarks and gluons as the basic building blocks of matter, as these are gauge-deendent. However, at least vortices, instantons [7], and monooles [8] can be transferred to Landau gauge, although they are not in one-to-one corresondence to their counterarts in other gauges [9]. Hence, in the following only Landau (and to some extent Coulomb) gauge will be discussed, desite its comlexities [,, ]. In this gauge it was ossible to show that vortices aear inside and on the boundary of the socalled first Gribov region [7], monooles [8] and instantons [7] on the boundary of the first Gribov region, and instantons also on the common boundary of the first Gribov region and the so-called fundamental modular region [7]. According to the Gribov-Zwanziger scenario [], these boundaries are resonsible for the infrared roerties of Green s functions. Hence, these toological configurations should be relevant field configurations for the Green s functions. An indirect indication for this is that the sectrum and eigenstates of the Faddeev-Poov oerator, which determines the ghost roagator, are affected by toological configurations [7, 8, ]. In articular, they give rise to additional zero-modes, which in rincile enhances the ghost roagator. This has also been demonstrated exlicitly by determining the ghost [] and gluon roagators [] in lattice gauge theory after removal of vortices. In this case, the low-momentum roerties are qualitatively altered. However, one of the most surrising results in this context is that instantons also lay a role [7, ]. PoS(Confinement8)

3 Green s Functions and Toological Configurations Smearing rogress D() Gluon dressing function D() Smearing stes APE ste APE stes DG () Smearing stes Smearing rogress.... Smearing stes DG ().... Ghost dressing function Smearing stes APE ste APE stes.... Figure : The gluon (to) and ghost (bottom) roagators as a function of the smearing ste (left anel) after mild smearing and measured after each smearing ste. Some individual stes are shown in the right anel. Results are from a lattice at β =. (a =. fm, V = (. fm ) ).. Proagators in toological backgrounds To investigate this fact more closely, smeared and/or cooled lattice gauge theory configurations can be used. In articular, when smearing sufficiently long, the configurations will be reduced to essentially (anti-)self-dual ones, with an action being almost a multile of the instanton action. To investigate this in detail, SU() lattice configurations in four dimensions [7] were smeared using a standard APE-smearing [8]. This rocedure is essentially as effective as other methods of smearing for the identification of long-range (toological) structures [8]. During the smearing, the smeared configurations were fixed to minimal Landau gauge, and the gluon and ghost roagators were determined using standard methods [7]. By this it was ossible to monitor the roagators during the smearing rocess, which has been done dee into the region of urely toological configurations. Results for rather small lattices, and thus only useful for investigations of the lowmomentum roerties, are shown in figure for mild smearing. In figure, results dee in the self-dual regime are shown. A more extended study, for significantly larger volumes and different PoS(Confinement8)

4 Green s Functions and Toological Configurations D() DG () Smearing rogress Smearing rogress Smearing rogress Smearing rogress Gluon dressing function D() DG () Smearing stes APE stes APE stes.... Ghost roagator dressing function Smearing stes APE stes APE stes.... Figure : Same as in figure, but smearing is now extended to APE stes, and measurements are erformed every stes. This is dee in the self-dual regime.. discretizations, will be shown elsewhere [9]. The roagators lose their ultraviolet roerties under smearing, as exected. The gluon roagator is damed out comletely, while the ghost roagator becomes essentially tree-level-like. Their low-momentum roerties are, however, not qualitatively affected. Nonetheless, the quantitative effects are considerable, and there might be even stronger effects observable once larger volumes, and thus smaller momenta, can be reached. In fact, the further smearing roceeds, the lower the momentum becomes from which on non-trivial roerties of the roagators are observed. The residual configurations, i. e., the configurations which the smeared configurations have to be combined with on the grou level to obtain the original ones, yield essentially trivial Green s functions, as exected [9]. It is also ossible to determine the roagators in sectors with fixed toological charge. Although this requires large amounts of statistics, reliminary findings indicate that there is no ronounced deendence of the roagators on the toological charge. This will be discussed in detail elsewhere [9]. In rincile, such an analysis could also be erformed for the instanton number. PoS(Confinement8)

5 Green s Functions and Toological Configurations. Concluding remarks The fact that the low-momentum roerties of the roagators kee their qualitative roerties under smearing rovides two insights. On the one hand, this imlies that the long-range structure is shaing the low-momentum roerties of roagators. This has already been anticiated based on the findings in center-trivial grous [, ]. On the other hand, reroducing the low-momentum roerties of the Green s functions in functional or other continuum calculations indicates that at least art of the dynamics due to toological configurations has been catured. Note that these findings are reliminary, and also do not reach far enough into the infrared to make (yet) a statement on the asymtotic roerties of the roagators in smeared (toological) field configurations. Nonetheless, the fact that smearing does not qualitatively modify the lowmomentum roerties of Green s functions is in agreement with the exectations [7]. References [] R. Alkofer and J. Greensite, J. Phys. G (7) S [arxiv:he-h/]. [] J. Greensite, Prog. Part. Nucl. Phys. () [arxiv:he-lat/]. [] C. S. Fischer, J. Phys. G () R [arxiv:he-h/7]. [] C. S. Fischer, A. Maas and J. M. Pawlowski, arxiv:8.987 [he-h]. [] A. Cucchieri and T. Mendes, arxiv: [he-lat]. [] M. Faber, J. Greensite and Š. Olejník, JHEP () [arxiv:he-lat/7]. [7] A. Maas, Eur. Phys. J. C 8 () 79 [arxiv:he-th/7]. [8] A. Maas, Nucl. Phys. A 79 (7) [arxiv:he-th/]. [9] J. Greensite, Š. Olejník and D. Zwanziger, Phys. Rev. D 9 () 7 [arxiv:he-lat/]. [] A. Maas, arxiv:88.7 [he-lat]. [] A. Sternbeck and L. von Smekal, PoS LATTICE8 (8) 7 [arxiv:8.7 [he-lat]]. [] D. Zwanziger, Phys. Rev. D 9 () [arxiv:he-h/8] and references therein. [] J. Greensite, Š. Olejník and D. Zwanziger, JHEP () 7 [arxiv:he-lat/7]. [] J. Gattnar, K. Langfeld and H. Reinhardt, Phys. Rev. Lett. 9 () [arxiv:he-lat/]. [] K. Langfeld, et al. arxiv:he-th/97; J. Gattnar, et al. Nucl. Phys. B 7 () [arxiv:he-lat/]. [] P. Boucaud, F. De Soto, A. Le Yaouanc, J. P. Leroy, J. Micheli, O. Pene and J. Rodriguez-Quintero, Phys. Rev. D 7, () [arxiv:he-h/]. [7] A. Cucchieri, A. Maas and T. Mendes, Phys. Rev. D 7, () [arxiv:he-lat/]. [8] F. Bruckmann, et al., Eur. Phys. J. A (7) [arxiv:he-lat/]. [9] A. Maas, in rearation. [] A. Maas and Š. Olejník, JHEP 8 (8) 7 [arxiv:7. [he-lat]]. [] A. Maas, Mod. Phys. Lett. A () 797 [arxiv:he-h/]. PoS(Confinement8)