Relation Between Masses of Pseudoscalar Octet and Vector Octet

The effect of the splitting of a pseudoscalar octet on the splitting of a vector octet is investigated using a simple effective-range formula for the coupling of a vector to two pseudoscalars. Taking the observed masses of the pseudoscalar octet to be $\ensuremath{\eta}(548)$, $K(496)$, and $\ensuremath{\pi}(140)$, it is found that the mass differences among the members of the vector octet $\ensuremath{\phi}$, ${K}^{*}$, and $\ensuremath{\rho}$ give the order $\ensuremath{\phi}>{K}^{*}>\ensuremath{\rho}$. The magnitude of the calculated splitting is approximately twice the observed values. It is also shown that to first order in the mass splitting, if the pseudoscalar octet satisfies the Gell-Mann-Okubo mass formula, then the vector octet also satisfies the GMO formula. Furthermore, a deviation of the pseudoscalar masses from the GMO formula implies a somewhat larger deviation for the vector masses with an opposite sign. This result is in qualitative agreement with experimental values.