Generalized Nonet Representation ofSU(3)⊗SU(3)and Its Applications

A class of representations of the nonchiral $\mathrm{SU}\left(3\right)\mathrm{}\otimes \mathrm{SU}\left(3\right)\mathrm{}$ is worked out. These consist of a sequence of self-conjugate representations of $\mathrm{SU}\left(3\right)\mathrm{}$, starting always with a singlet and with each $\mathrm{SU}\left(3\right)\mathrm{}$ representation occurring once. An analog of the Gell-Mann-Okubo mass formula, valid for these representations of $\mathrm{SU}\left(3\right)\mathrm{}\otimes \mathrm{SU}\left(3\right)\mathrm{}$, is obtained. When applied to the lowest nontrivial representation, this formula correctly explains $\omega -\phi$ mixing, thus providing a justification of Okubo's ansatz. Possible use of the next higher representation is indicated. From the same construction, the corresponding unitary irreducible representations of $\mathrm{SL}(3,C)\mathrm{}$ and ${\mathrm{T}}_8\times \mathrm{SU}\left(3\right)\mathrm{}$ are simultaneously obtained.