Computation of the one-loop spectral QCD running coupling using covariant spectral regularization

Abstract

Methods described in the literature for the computation of the QCD running coupling are essentially all defined with respect to the renormalization group equations and these equations are associated with the method of renormalization for dealing with infinities in Feynman integrals. The problem with the renormalization group equations is their prediction of the unphysical Landau pole which, for QCD occurs at an energy of the order of a few hundred MeV. The models described in the literature generally interlace high energy renormalization group predictions with modified low energy formulations. It would be desirable to have a method for the computation of the running strong coupling which is not {\em ad hoc} but is unified over the whole range of energies and is based on a single mathematically rigorous formulation which is guided by physical principles. In this paper we describe a method using the regularization technique called covariant spectral regularization for which renormalization is not required. The densities associated with the quark and gluon bubbles are computed without requiring renormalization and hence the spectral QCD vacuum polarization tensor is determined. It is found that the position space spectral QCD running coupling is an analytic function which does not manifest a Landau pole, but instead it manifests what might be called a ``Landau peak", and has the property known as ``freezing of $α_s$" in the infrared. We determine a spectral bare strong coupling constant of $α_b\approx(411)^{-1}$ which can be used in higher order QCD computations (this is to be compared with QCD using renormalization where the bare strong coupling is infinite). Thus it seems that one can conclude that, when analyzed using covariant spectral regularization, QCD is perturbative at all energies.