QCD Green functions and their application to hadron physics

Abstract

In a functional approach to QCD the infrared behaviour of Landau gauge Green functions is investigated. It can be proven that the ghost Dyson-Schwinger equation implies the Gribov-Zwanziger horizon condition. Its relation to the Kugo-Ojima confi nement scenario is elucidated. Positivity violation for gluons is demonstrated, and the analytic structure of the gluon propagator is studied. Quark confi nement is related to an infrared divergence of the quark-gluon vertex. It is shown that in the latter various components are non-vanishing due to the dynamical breaking of chiral symmetry. As a result an infrared fi nite running coupling in the Yang-Mills sector is derived whereas the running coupling related to the quark-gluon vertex is infrared divergent. In Coulomb gauge QCD already the one-gluon-exchange (over-)confi nes. This leads to a vanishing quark propagator, and thus quarks are confi ned. Nevertheless colour singlet quantities derived from the quark propagator are welldefi ned. Especially the expression for the quark condensate proves that chiral symmetry is dynamically broken. As expected the properties of mesons can be directly calculated whereas the mass of coloured diquarks diverges, and thus diquarks are confi ned. The latter nevertheless possess a well-defi ned size. In the third part the results obtained so far will be used to formulate a covariant Faddeev approach to nucleons. The resulting amplitudes describe the quark core of the nucleon. Besides the mass of this state also the electromagnetic form factors are calculated. The results for charge radii and magnetic moments as a function of the quark current mass provide some indication what the missing pion cloud may contribute to the nucleons' properties.Keywords: Confi nement; Infrared behaviour of Gluons and Quarks; Chiral Symmetry Breaking; Nucleons I. ON CONFINEMENT IN THE COVARIANT GAUGE: ANALYTIC PROPERTIES OF THE LANDAU GAUGE GLUON AND QUARK PROPAGATORS A. MotivationHadrons are believed to consist of quarks and gluons. However, every attempt to break a hadron into its constituents has failed so far. Thus the situation is very much different as e.g. in atomic physics. There studies of seperated atomic nuclei and electrons directly verify the nature of atoms being bound states. In hadronic physics, however, the evidence of physical states being bound states is indirect. Especially the plethora of hadrons fi nds a natural explanation in the assumption of hadrons being composite but their constituents do not exist as free particles. This phenomenon is called confinement.The focus of these lectures is the infrared behaviour of QCD Green functions (for recent reviews see e.g. refs. [1,2]) and what they tell us about confi nement. It will be seen that from this perspective the confi nement mechanisms for gluons, quarks, and colored composites show some distinctive features. Another virtue of the presented approach is the fact that QCD Green functions can be employed directly in hadron phenomenology. This opens up a road to the ambitious aim of verif...

Lattice Results on Gluon and Ghost Propagators in Landau Gauge

“…Thus, our largest lattice size 32 4 corresponds to a volume (6.7 fm) 4 . In order to fix the Landau gauge for each lattice gauge field {U } generated by means of a MC procedure, the gauge functional…”

“…They have proposed that the Landau gauge ghost propagator is infrared diverging while the gluon propagator is infrared vanishing. The interest in these propagators was stimulated in part by the progress achieved in solving Dyson-Schwinger equations (DSE) for these propagators (for a recent review see [4]). Recently it has been argued that a unique and exact power-like infrared asymptotic behavior of all Green functions can be derived without truncating the hierarchy of DSE [5].…”

Landau gauge ghost propagator and running coupling inSU(2)lattice gauge theory

“…From the phenomenological point of view the propagators can serve as input to bound state equations as there are Bethe-Salpeter or Faddeev equations for hadron phenomenology [8,24,25]. In the ultraviolet limit they allow a determination of phenomenogically relevant parameters such as Λ MS or condensates ψψ , A 2 , .…”

A covariant view on the nucleons’ quark core

“…Besides and together with lattice QCD and chiral perturbation theory functional methods have proven to be a method of choice when investigating the infrared properties of QCD, for some recent brief overview see e.g. [4]. The investigation to be reported here will build on such functional methods in the context of Dyson-Schwinger equations, and we will relate our results to corresponding ones obtained by a synthesis of lattice and effective field theory calculations.…”