Covariant off-shell model for dynamical quark confinement

Abstract

A model for quark propagation is developed in which quarks cannot go on-shell and thus are confined dynamically. The approach is based on a recently proposed time-ordered formulation of the relativistic many-body problem made manifestly covariant by an off-shell modification of Lorentz transformations. It is argued that the present off-shell formalism is better suited for studies of dynamical confinement since it provides a straightforward off-shell extension of the Dirac equation. In the usual Lorentz-covariant formulation, by contrast, spinor solutions do not exist for quarks which cannot go on-shell. Defining the quark propagator by a nonlinear time-ordered one-gluon-loop expansion, similar to a Dyson-Schwinger equation, an energy-dependent quark mass is determined by a nonlinear self-consistency condition. The corresponding time-ordered propagator is found to possess no poles for real energies, thus ensuring confinement. Evidence is given that it is analytic in the entire upper half of the complex energy plane as required by causality. In the model, the self-energy integral exists without regularization, with the convergence being provided by the chiral limit of the quark mass itself. The self-consistency condition exhibits the scaling behavior of the renormalization group and provides an eigenvalue condition for the low-energy value of the quark-gluon coupling constant. Its numerically determined value of g 2/4π= 4.712 can actually be derived in the soft-gluon limit as g 2/4π=3/2π. Implications of these findings and possible further applications are discussed.