Analytic Structure of the Gluon and Quark Propagators in Landau Gauge

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. The standard model of particle physics consists of gauge field theories. These have been postulated on the basis of symmetries and their elementary excitations do not reflect the observed particle spectrum. In the quantum formulation of these theories, especially in Poincaré-covariant gauges, an intricate problem is posed by the separation of physical and unphysical degrees of freedom. In Quantum Electrodynamics (QED) in linear covariant gauges, the electromagnetic field can be decomposed into transverse, longitudinal and timelike photons, however, only transverse polar-izations are observed. From a purely mathematical point of view, this can be understood from the representations of the Poincaré group for massless states: massless particles have only two possible polarizations [1]. The apparent contradiction is resolved by the fact that timelike and longitudinal photons cancel exactly in the S-matrix [2]. In this context we emphasize that the states of quantum gauge field theories in covariant gauges necessarily constitute an indefinite metric space. In covariant gauge QED one thus has to sacrifice the principle of positivity of the representation space. * Summary of a talk given at several occasions; to be published in the proceedings of the international conference In Quantum Chromodynamics (QCD) in linear covariant gauges, the cancellation of unphys-ical degrees of freedom in the S-matrix is substantially complicated by the self-interaction of the gauge fields and by the ghost fields that are necessarily present in the quantum formulation of these theories [3]. To order α 2 S in perturbation theory, one obtains amplitudes for the scattering of two transverse gluons into one transverse and one longitudinal gluon. However, at the same order , a ghost loop appears and cancels the various gluon loops, and scattering of transverse to longitudinal gluons does not occur. It is possible to prove this cancellation to all orders in perturbation theory on the basis of the BRS [4] symmetry of the covariantly gauge fixed theory. This symmetry …