Partial Current Conservation and a Phenomenological Model of Hadrons

Abstract

A method for constructing a phenomenological Lagrangian of the hadrons within the relativistic $\mathrm{SU}(6)$ framework is presented. The method consists of imposing on the Lagrangian the condition (PCAC) that the divergence of the axial-vector current should be proportional to a nonet of pseudoscalar mesons. A particular $\mathrm{SL}(6,C)$-invariant series of meson-meson and meson-baryon interactions are introduced and it is shown that in a first-order calculation a relation between the various couplings is possible if PCAC is maintained. We assume that the model is a phenomenological one, so the interactions consist of fields associated with the physical particles and therefore relations among the couplings can be computed. The effective couplings for all meson-meson and meson-baryon interactions admitted by this model are then obtained to lowest order using one experimental value. If we use the coupling constant for $\ensuremath{\rho}\ensuremath{\pi}\ensuremath{\pi}$ as the sole input, we note particularly the following results: (a) For the quadrilinear meson process ${X}^{0}\ensuremath{\rightarrow}\ensuremath{\eta}\ensuremath{\pi}\ensuremath{\pi}$ we find for the width $\ensuremath{\Gamma}({X}^{0}\ensuremath{\rightarrow}\ensuremath{\eta}\ensuremath{\pi}\ensuremath{\pi})=2$ MeV. (b) For the decay $\ensuremath{\omega}\ensuremath{\rightarrow}3\ensuremath{\pi}$ we find for the width $\ensuremath{\Gamma}(\ensuremath{\omega}\ensuremath{\rightarrow}3\ensuremath{\pi})=8.9$ MeV. (c) The $\overline{N}N\ensuremath{\rho}$ coupling is determined consistent with universality to be $\frac{g_{\overline{N}N\ensuremath{\rho}}^{}{}_{}{}^{2}}{4\ensuremath{\pi}}=0.54$.