Lorenz von Smekal

The infrared behaviour of QCD Green's functions: Confinement, dynamical symmetry breaking, and hadrons as relativistic bound states

Abstract Recent studies of QCD Greens functions and their applications in hadronic physics are reviewed. We discuss the definition of the generating functional in gauge theories, in particular, the role of redundant degrees of freedom, possibilities of a complete gauge fixing versus gauge fixing in presence of Gribov copies, BRS invariance and positivity. The apparent contradiction between positivity and colour antiscreening in combination with BRS invariance in QCD is considered. Evidence for the violation of positivity by quarks and transverse gluons in the covariant gauge is collected, and it is argued that this is one manifestation of confinement. We summarise the derivation of the Dyson–Schwinger equations (DSEs) of QED and QCD. For the latter, the implications of BRS invariance on the Greens functions are explored. The possible influence of instantons on DSEs is discussed in a two-dimensional model. In QED in (2+1) and (3+1) dimensions, the solutions for Greens functions provide tests of truncation schemes which can under certain circumstances be extended to the DSEs of QCD. We discuss some limitations of such extensions and assess the validity of assumptions for QCD as motivated from studies in QED. Truncation schemes for DSEs are discussed in axial and related gauges, as well as in the Landau gauge. Furthermore, we review the available results from a systematic non-perturbative expansion scheme established for Landau gauge QCD. Comparisons to related lattice results, where available, are presented. The applications of QCD Greens functions to hadron physics are summarised. Properties of ground state mesons are discussed on the basis of the ladder Bethe-Salpeter equation for quarks and antiquarks. The Goldstone nature of pseudoscalar mesons and a mechanism for diquark confinement beyond the ladder approximation are reviewed. We discuss some properties of ground state baryons based on their description as Bethe-Salpeter/Faddeev bound states of quark–diquark correlations in the quantum field theory of confined quarks and gluons.

A Solution to Coupled Dyson-Schwinger Equations for Gluons and Ghosts in Landau Gauge

A truncation scheme for the Dyson{endash}Schwinger equations of QCD in Landau gauge is presented which implements the Slavnov{endash}Taylor identities for the 3-point vertex functions. Neglecting contributions from 4-point correlations such as the 4-gluon vertex function and irreducible scattering kernels, a closed system of equations for the propagators is obtained. For the pure gauge theory without quarks this system of equations for the propagators of gluons and ghosts is solved in an approximation which allows for an analytic discussion of its solutions in the infrared: The gluon propagator is shown to vanish for small spacelike momenta whereas the ghost propagator is found to be infrared enhanced. The running coupling of the non-perturbative subtraction scheme approaches an infrared stable fixed point at a critical value of the coupling, {alpha}{sub c}{approx_equal}9.5. The gluon propagator is shown to have no Lehmann representation. The results for the propagators obtained here compare favorably with recent lattice calculations. {copyright} 1998 Academic Press, Inc.

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

In gauge theories without Higgs mechanism, particles carrying the global charges of the gauge group cannot strictly be localized. Localized physical states are necessarily neutral in QED and colorless in QCD. The extension to all gauge invariant and thus physical states is possible only with a mass gap in the physical world. Then, color-electric charge superselection sectors do not arise in QCD and one concludes confinement. The necessary conditions for this were formulated more than twenty years ago. In the next subsection we briefly recall these conditions, and how they constrain the infrared behavior of ghost and gluon propagators in Landau gauge QCD. Based on linear-covariant gauges, their derivation may not fully be divorced from perturbation theory. Their essence is quite generic and summarized in the Kugo-Ojima criterion which should apply in one way or another, whenever some form of BRS cohomology construction does for gauge theories. One way towards a non-perturbative definition of the Landau gauge is provided via stochastic quantization for which the full 5-dimensional BRS machinery is in the garage. The time-independent diffusion equation of this formulation is closely related to the Dyson-Schwinger equations (DSEs) in 4 dimensions as we describe next. Some of the necessary extensions, which have already been implemented in previous DSE studies of infrared exponents for other reasons, imply the Kugo-Ojima criterion. We summarize these studies, and how they are confirmed qualitatively in this way, at the end of the introduction. In Sec. II, we set up the DSE structures relevant for our present study. We summarize general properties

Goldstone theorem and diquark confinement beyond rainbow ladder approximation

Abstract The quark Dyson-Schwinger equation and meson Bethe-Salpeter equation are studied in a truncation scheme that extends the rainbow-ladder approximation such that, in the chiral limit, the isovector, pseudoscalar meson remains massless. Repulsive contributions, which appear at higher order in the Bethe-Salpeter kernel, eliminate the bound state pole in the quark-quark (diquark) channel that is present in rainbow-ladder approximation. The net effect of higher order terms on the meson bound state masses is small.

Geometric misalignment and calibration in cone‐beam tomography

We present a new high-precision method for the geometric calibration in cone-beam computed tomography. It is based on a Fourier analysis of the projection-orbit data, recorded with a flat-panel area detector, of individual point-like objects. For circular scan trajectories the complete set of misalignment parameters which determine the deviation of the detector alignment from the ideal scan geometry are obtained from explicit analytic expressions. To derive these expressions we show how to disentangle the problems of calculating misalignment parameters and point coordinates. The calculation of the coordinates of the point objects inside the scanned volume, in units of the distance from the focal spot to the center of rotation, is then possible analytically likewise. We simulate point-projection data on a misaligned detector with various amounts of randomness added to mimic measurement uncertainties. This data is then employed in our calibration to validate the method by comparing the resulting misalignment parameters and point coordinates to the known true ones. We also present our implementation and results for the geometric calibration of micro-CT systems. The effectiveness of the corresponding misalignment correction in reducing image artifacts is exemplified by reconstructed micro-CT images.

The Strong coupling and its running to four loops in a minimal MOM scheme

Abstract We introduce the minimal momentum subtraction (MiniMOM) scheme for QCD. Its definition allows the strong coupling to be fixed solely through a determination of the gluon and ghost propagators. In Landau gauge this scheme has been implicit in the early studies of these propagators, especially in relation to their non-perturbative behaviour in the infrared and the associated infrared fixed-point. Here we concentrate on its perturbative use. We give the explicit perturbative definition of the scheme and the relation of its β-function and running coupling to the MS ¯ scheme up to 4-loop order in general covariant gauges. We also demonstrate, by considering a selection of N f = 3 examples, that the apparent convergence of the relevant perturbative series can in some (though not all) cases be significantly improved by re-expanding the MS ¯ coupling version of this series in terms of the MiniMOM coupling, making the MiniMOM coupling also of potential interest in certain phenomenological applications.

Scaling behavior and positivity violation of the gluon propagator in full QCD

The Landau-gauge gluon propagator is studied using the coarse and fine dynamical MILC configurations. The effects of dynamical quarks are clearly visible and lead to a reduction of the nonperturbative infrared enhancement relative to the quenched case. Lattice spacing effects are studied and found to be small. The gluon spectral function is shown to clearly violate positivity in both quenched and full QCD.

Large volume behaviour of Yang-Mills propagators

We investigate finite volume effects in the propagators of Landau gauge Yang-Mills theory using Dyson-Schwinger equations on a 4-dimensional torus. In particular, we demonstrate explicitly how the solutions for the gluon and the ghost propagator tend towards their respective infinite volume forms in the corresponding limit. This solves an important open problem of previous studies where the infinite volume limit led to an apparent mismatch, especially of the infrared behaviour, between torus extrapolations and the existing infinite volume solutions obtained in 4-dimensional Euclidean space-time. However, the correct infinite volume limit is approached rather slowly. The typical scales necessary to see the onset of the leading infrared behaviour emerging already imply volumes of at least 10 to 15 fm in lengths. To reliably extract the infrared exponents of the infinite volume solutions requires even much larger ones. While the volumes in the Monte-Carlo simulations available at present are far too small to facilitate that, we obtain a good qualitative agreement of our torus solutions with recent lattice data in comparable volumes.

Quark-meson-diquark model for two-color QCD

We introduce a two-flavor quark-meson-diquark model for two-color QCD and its extensions to include gauge-field dynamics as described by the Polyakov loop. Grand potential and phase structure are being studied both in mean-field approximation and with the functional renormalization group. The model provides an explicit example for the importance of baryonic degrees of freedom: When they are omitted, the phase diagram closely resembles that of the corresponding (Polyakov)-quark-meson models for QCD, in particular, including their critical endpoint. In order to reproduce the well-established main features based on the symmetries and breaking patterns of two-color QCD, however, they must be included and there is no critical endpoint. The competing dynamics of collective mesonic and baryonic fluctuations is well described by the functional renormalization group equation in lowest-order derivative expansion for the effective potential, which we solve numerically on a two-dimensional grid in field space.

On the influence of three-point functions on the propagators of Landau gauge Yang-Mills theory

A bstractWe solve the Dyson-Schwinger equations of the ghost and gluon propagators of Landau gauge Yang-Mills theory together with that of the ghost-gluon vertex. The latter plays a central role in many truncation schemes for functional equations. By including it dynamically we can determine its influence on the propagators. We also suggest a new model for the three-gluon vertex motivated by lattice data which plays a crucial role to obtain stable solutions when the ghost-gluon vertex is included. We find that both vertices have a sizable quantitative impact on the mid-momentum regime and contribute to the reduction of the gap between lattice and Dyson-Schwinger equation results. Furthermore, we establish that the three-gluon vertex dressing turns negative at low momenta as suggested by lattice results in three dimensions.