Renormalisable Non-Local Quark–Gluon Interaction: Mass Gap, Chiral Symmetry Breaking and Scale Invariance

Abstract

We derive a Nambu–Jona-Lasinio (NJL) model from a non-local gauge theory and show that it has confining properties at low energies. In particular, we present an extended approach to non-local QCD and a complete revision of the technique of Bender, Milton and Savage applied to non-local theories, providing a set of Dyson–Schwinger equations in differential form. In the local case, we obtain closed-form solutions in the simplest case of the scalar field and extend it to the Yang–Mills field. In general, for non-local theories, we use a perturbative technique and a Fourier series and show how higher-order harmonics are heavily damped due to the presence of the non-local factor. The spectrum of the theory is analysed for the non-local Yang–Mills sector and found to be in agreement with the local results on the lattice in the limit of the non-locality mass parameter running to infinity. In the non-local case, we confine ourselves to a non-locality mass that is sufficiently large compared to the mass scale arising from the integration of the Dyson–Schwinger equations. Such a choice results in good agreement, in the proper limit, with the spectrum of the local theory. We derive a gap equation for the fermions in the theory that gives some indication of quark confinement in the non-local NJL case as well. Confinement seems to be a rather ubiquitous effect that removes some degrees of freedom in the original action, favouring the appearance of new observable states, as seen, e.g., for quantum chromodynamics at lower energies.