Justification of the Okubo-Zweig-Iizuka rule in quantum chromodynamics

Abstract

We present an analysis of the theoretical issues involved in justifying the Okubo-Zweig-Iizuka rule in quantum chromodynamics. We consider, as a model, an $S$-wave bound state of a quark-antiquark pair so heavy that the binding can be taken to be approximately Coulombic. We know from general arguments based on the Kinoshita-Lee-Nauenberg theorems that there is no sensitivity to an infrared cutoff in physical decay rates. Therefore, it is sufficient to analyze the sources of soft gluons with energies ranging down to the binding energy or inverse size of the system. We show that the amplitudes for such soft gluons are suppressed, appearing only as relativistic or binding-energy corrections. Thus, these processes are smaller than their naive level obtained by counting orders of perturbation theory. The leading infrared-sensitive corrections, which are of order ${{\ensuremath{\alpha}}_{s}}^{2}\mathrm{ln}{\ensuremath{\alpha}}_{s}$, arise from relativistic corrections to gluon exchange between the quark-antiquark pair, similar to those in positronium. In our analysis we find a breakdown of perturbation theory from multiple Coulomb-gluon exchange following the emission of a soft gluon from a binding gluon. This can be resolved by summing a class of graphs as in the corresponding analysis of the static potential and the non-Abelian multipole expansion.