Gluon Propagators and Confinement

We present SU(3) gluon propagators calculated on 48*48*48*N_t lattices at beta=6.8 where N_t=64 (corresponding the confinement phase) and N_t=16 (deconfinement) with the bare gauge parameter,alpha, set to be 0.1. In order to avoid Gribov copies, we employ the stochastic gauge fixing algorithm. Gluon propagators show quite different behavior from those of massless gauge fields: (1) In the confinement phase, G(t) shows massless behavior at small and large t, while around 5<t<15 it behaves as massive particle, and (2) effective mass observed in G(z) becomes larger as z increases. (3) In the deconfinement phase, G(z) shows also massive behavior but effective mass is less than in the confinement case. In all cases, slope masses are increasing functions of t or z, which can not be understood as addtional physical poles.Comment: 6 pages in Postscrip

Large volume behavior of Yang-Mills propagators

We summarize results on finite-volume effects in the propagators of Landau gauge Yang-Mills theory using Dyson-Schwinger equations on a 4-dimensional torus. We demonstrate explicitly how the solutions for the gluon and the ghost propagator tend towards their respective infinite volume forms in the corresponding limit. We discuss the relation of our solutions with results from lattice Monte-Carlo simulations.Comment: 7 pages, 2 figures, Presented by CF at the XXV International Symposium on Lattice Field Theory, July 30 - August 4 2007, Regensburg, German

Non-perturbative Landau gauge and infrared critical exponents in QCD

We discuss Faddeev-Popov quantization at the non-perturbative level and show that Gribov's prescription of cutting off the functional integral at the Gribov horizon does not change the Schwinger-Dyson equations, but rather resolves an ambiguity in the solution of these equations. We note that Gribov's prescription is not exact, and we therefore turn to the method of stochastic quantization in its time-independent formulation, and recall the proof that it is correct at the non-perturbative level. The non-perturbative Landau gauge is derived as a limiting case, and it is found that it yields the Faddeev-Popov method in Landau gauge with a cut-off at the Gribov horizon, plus a novel term that corrects for over-counting of Gribov copies inside the Gribov horizon. Non-perturbative but truncated coupled Schwinger-Dyson equations for the gluon and ghost propagators $D(k)$ and $G(k)$ in Landau gauge are solved asymptotically in the infrared region. The infrared critical exponents or anomalous dimensions, defined by $D(k) \sim 1/(k^2)^{1 + a_D}$ and $G(k) \sim 1/(k^2)^{1 + a_G}$ are obtained in space-time dimensions $d = 2, 3, 4$. Two possible solutions are obtained with the values, in $d = 4$ dimensions, $a_G = 1, a_D = -2$, or $a_G = [93 - (1201)^{1/2}]/98 \approx 0.595353, a_D = - 2a_G$.Comment: 26 pages. Modified 2.25.02 to update references and to clarify Introduction and Conclusio

The Extremely High Energy Cosmic Rays

Experimental results from Haverah Park, Yakutsk, AGASA and Fly's Eye are reviewed. All these experiments work in the energy range above 0.1 EeV. The 'dip' structure around 3 EeV in the energy spectrum is well established by all the experiments, though the exact position differs slightly. Fly's Eye and Yakutsk results on the chemical composition indicate that the cosmic rays are getting lighter over the energy range from 0.1 EeV to 10 EeV, but the exact fraction is hadronic interaction model dependent, as indicated by the AGASA analysis. The arrival directions of cosmic rays are largely isotropic, but interesting features may be starting to emerge. Most of the experimental results can best be explained with the scenario that an extragalactic component gradually takes over a galactic population as energy increases and cosmic rays at the highest energies are dominated by particles coming from extragalactic space. However, identification of the extragalactic sources has not yet been successful because of limited statistics and the resolution of the data.Comment: The review paper including 21 figures. 39 pages: To be published in Journal of Physics

Asymptotic Scaling and Infrared Behavior of the Gluon Propagator

The Landau gauge gluon propagator for the pure gauge theory is evaluated on a 32^3x64 lattice with a physical volume of (3.35^3x6.7)fm^4. Comparison with two smaller lattices at different lattice spacings allows an assessment of finite volume and finite lattice spacing errors. Cuts on the data are imposed to minimize these errors. Scaling of the gluon propagator is verified between beta=6.0 and beta=6.2. The tensor structure is evaluated and found to be in good agreement with the Landau gauge form, except at very small momentum values, where some small finite volume errors persist. A number of functional forms for the momentum dependence of the propagator are investigated. The form D(q^2)=D_ir+D_uv, where D_ir(q^2) ~ (q^2+M^2)^-\eta and D_uv is an infrared regulated one-loop asymptotic form, is found to provide an adequate description of the data over the entire momentum region studied - thereby bridging the gap between the infrared confinement region and the ultraviolet asymptotic region. The best estimate for the exponent \eta is 3.2(+0.1/-0.2)(+0.2/-0.3), where the first set of errors represents the uncertainty associated with varying the fitting range, while the second set of errors reflects the variation arising from different choices of infrared regulator in D_uv. Fixing the form of D_uv, we find that the mass parameter M is (1020+/-100)MeV.Comment: 37 pages, RevTeX, 16 postscript figures, 7 gif figures. Revised version accepted for publication in Phys. Rev. D. Model functions and discussion of asymptotic behaviour modified; all model fits have been redone. This paper, including postscript version of all figures, can be found at http://www.physics.adelaide.edu.au/~jskuller/papers

Infrared behavior of the gluon propagator in lattice Landau gauge: the three-dimensional case

We evaluate numerically the three-momentum-space gluon propagator in the lattice Landau gauge, for three-dimensional pure-SU(2) lattice gauge theory with periodic boundary conditions. Simulations are done for nine different values of the coupling $\beta$, from $\beta = 0$ (strong coupling) to $\beta = 6.0$ (in the scaling region), and for lattice sizes up to $V = 64^3$. In the limit of large lattice volume we observe, in all cases, a gluon propagator decreasing for momenta smaller than a constant value $p_{dec}$. From our data we estimate $p_{dec} \approx 350$ MeV. The result of a gluon propagator decreasing in the infrared limit has a straightforward interpretation as resulting from the proximity of the so-called first Gribov horizon in the infrared directions.Comment: 14 pages, BI-TP 99/03 preprint, correction in the Acknowledgments section. To appear in Phys.Rev.

On the Infrared Exponent for Gluon and Ghost Propagation in Landau Gauge QCD

In the covariant description of confinement, one expects the ghost correlations to be infrared enhanced. Assuming ghost dominance, the long-range behavior of gluon and ghost correlations in Landau gauge QCD is determined by one exponent kappa. The gluon propagator is infrared finite (vanishing) for kappa =1/2 (kappa > 1/2) which is still under debate. Here, we study critical exponent and coupling for the infrared conformal behavior from the asymptotic form of the solutions to the Dyson-Schwinger equations in an ultraviolet finite expansion scheme. The value for kappa is directly related to the ghost-gluon vertex. Assuming that it is regular in the infrared, one obtains kappa = 0.595. This value maximizes the critical coupling alpha_c(kappa), yielding alpha_c^max = (4 Pi/Nc) 0.709 approx. 2.97 for Nc=3. For larger kappa the vertex acquires an infrared singularity in the gluon momentum, smaller ones imply infrared singular ghost legs. Variations in alpha_c remain within 5% from kappa = 0.5 to 0.7. Above this range, alpha_c decreases more rapidly with alpha_c -> 0 as kappa -> 1 which sets the upper bound on kappa.Comment: 22 Pages, 10 Figures, LaTeX2e, revtex4, some notes and references added in response to communication

Lattice Landau gauge with stochastic quantisation

We calculate Landau gauge ghost and gluon propagators in pure SU(2) lattice gauge theory in two, three and four dimensions. The gauge fixing method we use, sc. stochastic quantisation, serves as a viable alternative approach to standard gauge fixing algorithms. We also investigate the spectrum of the Faddeev-Popov operator. At insufficiently accurate gauge fixing, we find evidence that stochastic quantisation samples configurations close to the Gribov horizon. Standard gauge fixing does so only at specific parameters; otherwise, there is a clear difference. However, this difference disappears if the gauge is fixed to sufficient accuracy. In this case, we confirm previous lattice results for the gluon and ghost propagator in two, three and four dimensions.Comment: 14 pages, 15 figure