Infrared Exponents and Running Coupling of SU(N) Yang-Mills Theories

We present approximate solutions for the gluon and ghost propagators as well as the running coupling in Landau gauge Yang-Mills theories. We solve the corresponding Dyson-Schwinger equations in flat Euclidean space-time without any angular approximation. This supplements recently obtained results employing a four-torus, i.e. a compact space-time manifold, as infrared regulator. We confirm previous findings deduced from an extrapolation with tori of different volumes: the gluon propagator is weakly vanishing in the infrared and the ghost propagator is highly singular. For non-vanishing momenta our propagators are in remarkable agreement with recent lattice calculations.Comment: 11 pages, 4 figure

Truncating first-order Dyson-Schwinger equations in Coulomb-Gauge Yang-Mills theory

The non-perturbative domain of QCD contains confinement, chiral symmetry breaking, and the bound state spectrum. For the calculation of the latter, the Coulomb gauge is particularly well-suited. Access to these non-perturbative properties should be possible by means of the Green's functions. However, Coulomb gauge is also very involved, and thus hard to tackle. We introduce a novel BRST-type operator r, and show that the left-hand side of Gauss' law is r-exact. We investigate a possible truncation scheme of the Dyson-Schwinger equations in first-order formalism for the propagators based on an instantaneous approximation. We demonstrate that this is insufficient to obtain solutions with the expected property of a linear-rising Coulomb potential. We also show systematically that a class of possible vertex dressings does not change this result.Comment: 22 pages, 4 figures, 1 tabl

Vertex functions and infrared fixed point in Landau gauge SU(N) Yang-Mills theory

The infrared behaviour of vertex functions in an SU(N) Yang-Mills theory in Landau gauge is investigated employing a skeleton expansion of the Dyson-Schwinger equations. The three- and four-gluon vertices become singular if and only if all external momenta vanish while the dressing of the ghost-gluon vertex remains finite in this limit. The running coupling as extracted from either of these vertex functions possesses an infrared fixed point. In general, diagrams including ghost-loops dominate in the infrared over purely gluonic ones.Comment: 14 pages, 8 figures, v2: typos corrected, version to be published in PL

Numerical Study of the Ghost-Gluon Vertex in Landau gauge

We present a numerical study of the ghost-gluon vertex and of the corresponding renormalization function \widetilde{Z}_1(p^2) in minimal Landau gauge for SU(2) lattice gauge theory. Data were obtained for three different lattice volumes (V = 4^4, 8^4, 16^4) and for three lattice couplings \beta = 2.2, 2.3, 2.4. Gribov-copy effects have been analyzed using the so-called smeared gauge fixing. We also consider two different sets of momenta (orbits) in order to check for possible effects due to the breaking of rotational symmetry. The vertex has been evaluated at the asymmetric point (0;p,-p) in momentum-subtraction scheme. We find that \widetilde{Z}_1(p^2) is approximately constant and equal to 1, at least for momenta p > ~ 1 GeV. This constitutes a nonperturbative verification of the so-called nonrenormalization of the Landau ghost-gluon vertex. Finally, we use our data to evaluate the running coupling constant \alpha_s(p^2).Comment: 19 pages, 6 figures, 9 tables, using axodraw.sty; minor modifications in the abstract, introduction and conclusion

Renormalization flow of Yang-Mills propagators

We study Landau-gauge Yang-Mills theory by means of a nonperturbative vertex expansion of the quantum effective action. Using an exact renormalization group equation, we compute the fully dressed gluon and ghost propagators to lowest nontrivial order in the vertex expansion. In the mid-momentum regime, $p^2\sim\mathcal{O}(1)\text{GeV}^2$, we probe the propagator flow with various {\em ans\"atze} for the three- and four-point correlations. We analyze the potential of these truncation schemes to generate a nonperturbative scale. We find universal infrared behavior of the propagators, if the gluon dressing function has developed a mass-like structure at mid-momentum. The resulting power laws in the infrared support the Kugo-Ojima confinement scenario.Comment: 28 pages, 5 figures. V2: Typos corrected and reference adde

Rota-Baxter algebras and new combinatorial identities

The word problem for an arbitrary associative Rota-Baxter algebra is solved. This leads to a noncommutative generalization of the classical Spitzer identities. Links to other combinatorial aspects, particularly of interest in physics, are indicated.Comment: 8 pages, improved versio

Analytic structure of the gluon and quark propagators in Landau gauge QCD

In Landau gauge QCD the infrared behavior of the propagator of transverse gluons can be analytically determined to be a power law from Dyson-Schwinger equations. This propagator clearly shows positivity violation, indicating the absence of the transverse gluons from the physical spectrum, i.e. gluon confinement. A simple analytic structure for the gluon propagator is proposed capturing all important features. We provide arguments that the Landau gauge quark propagator possesses a singularity on the real timelike axis. For this propagator we find a positive definite Schwinger function.Comment: 6 pages, 3 figures; summary of a talk given at several occasions; to be published in the proceedings of the international conference QCD DOWN UNDER, March 10 - 19, Adelaide, Australi

Nonperturbative structure of the quark-gluon vertex

The complete tensor structure of the quark--gluon vertex in Landau gauge is determined at two kinematical points (`asymmetric' and `symmetric') from lattice QCD in the quenched approximation. The simulations are carried out at beta=6.0, using a mean-field improved Sheikholeslami-Wohlert fermion action, with two quark masses ~ 60 and 115 MeV. We find substantial deviations from the abelian form at the asymmetric point. The mass dependence is found to be negligible. At the symmetric point, the form factor related to the chromomagnetic moment is determined and found to contribute significantly to the infrared interaction strength.Comment: 16 pages, 11 figures, JHEP3.cl

Non-perturbative Propagators, Running Coupling and Dynamical Quark Mass of Landau gauge QCD

The coupled system of renormalized Dyson-Schwinger equations for the quark, gluon and ghost propagators of Landau gauge QCD is solved within truncation schemes. These employ bare as well as non-perturbative ansaetze for the vertices such that the running coupling as well as the quark mass function are independent of the renormalization point. The one-loop anomalous dimensions of all propagators are reproduced. Dynamical chiral symmetry breaking is found, the dynamically generated quark mass agrees well with phenomenological values and corresponding results from lattice calculations. The effects of unquenching the system are small. In particular the infrared behavior of the ghost and gluon dressing functions found in previous studies is almost unchanged as long as the number of light flavors is smaller than four.Comment: 34 pages, 10 figures, version to be published by Phys. Rev.

Light's Bending Angle due to Black Holes: From the Photon Sphere to Infinity

The bending angle of light is a central quantity in the theory of gravitational lensing. We develop an analytical perturbation framework for calculating the bending angle of light rays lensed by a Schwarzschild black hole. Using a perturbation parameter given in terms of the gravitational radius of the black hole and the light ray's impact parameter, we determine an invariant series for the strong-deflection bending angle that extends beyond the standard logarithmic deflection term used in the literature. In the process, we discovered an improvement to the standard logarithmic deflection term. Our perturbation framework is also used to derive as a consistency check, the recently found weak deflection bending angle series. We also reformulate the latter series in terms of a more natural invariant perturbation parameter, one that smoothly transitions between the weak and strong deflection series. We then compare our invariant strong deflection bending-angle series with the numerically integrated exact formal bending angle expression, and find less than 1% discrepancy for light rays as far out as twice the critical impact parameter. The paper concludes by showing that the strong and weak deflection bending angle series together provide an approximation that is within 1% of the exact bending angle value for light rays traversing anywhere between the photon sphere and infinity.Comment: 22 pages, 5 figure