Superconvergent Dispersion Relations and Meson Couplings

Superconvergence relations for vector-meson-meson scattering are saturated with nonets of vector and pseudoscalar mesons. It is found that, with this truncation in the saturation, the only meaningful sum rules are the ones selected by the infinite-momentum limit and evaluated near zero momentum transfer. On demanding only isospin invariance at the vertices and considering all possible scattering and reaction processes, one finds a consistent set of equations for the meson coupling constants. The conditions are examined under which collinear $U\left(4\right)\mathrm{}$ or collinear $U\left(6\right)\mathrm{}$ emerge from this complete set of equations. On considering only the equations obtained from vector-pseudoscalar scattering, one is able to relate nonet symmetry breaking at the vertices to nonet breaking in the masses. Use of $\mathrm{SU}\left(3\right)\mathrm{}$ at the vertices unambiguously leads to results consistent with collinear $U\left(6\right)\mathrm{}$ symmetry of the vertices.