Approximation scheme for quantum chromodynamics

Abstract

An approximation scheme is proposed for calculating masses of $Q\overline{Q}$ mesons in quantum chromodynamics. The vacuum instability is removed, not by giving the magnetic field an expectation value, but by enhancing the amplitude of the low-frequency components of the virtual magnetic field. A simple truncation of the Schwinger-Dyson equations incorporates such an effect. Rough numerical calculations indicate that the equations for the gluon propagator have a solution which increases logarithmically at large distances. Clustering does not hold for multigluon Green's functions. Instead, if generally believed properties are found to be true for the two-gluon function, they are assumed to be true for the higher functions. Equations for the quark propagator and the $Q\overline{Q}$ bound states are set up. The ladder approximation is totally inadequate in the confinement region, but we can make a reasonable approximation which leads to static forces at large distances. Corrections probably involve vibrational modes of the dual string. At high energy, linearly rising $Q\overline{Q}$ Regge trajectories with the expected quantum numbers are found. The scheme is free from dimensionless parameters if the quarks have zero bare mass. Our equations possess the possibility of chiralsymmetry breaking. Massless "pions" would then appear as $Q\overline{Q}$ bound states; massless $\ensuremath{\eta}$'s would not appear if closed quark loops were included.