Meson Spectrum, Mass Formula, and the Quark-Antiquark Interaction

Higher mesons fit into an $L=1\mathrm{}$, $U\left(6\right)\mathrm{}\otimes U\left(6\right)\mathrm{}$ "35"-plet, $J^P=0^{+}, 1^{+}, \mathrm{and} 2^{+}$ splitting by tensor and spinorbit interactions. Quantum numbers are assigned with $G$ parities that check and missing ones predicted. Mesons are placed on quark-antiquark ($q\overline{q}$) Regge trajectories. A general mass formula is derived which includes modifications of $\mathrm{SU}\left(6\right)\mathrm{}$-type splittings in higher supermultiplets, as well as additional quantum numbers related to dynamical groups. General features of an effective ($q\overline{q}$) interaction indicated by the mass spectrum are deduced. Predicted mesons are a ${\pi }^{\prime }(640\mathrm{}\mp 40)$, an ${\omega }^{\prime }(567\mathrm{}\mp 40)$, and a ${\phi }^{\prime }(835\mathrm{}\mp 40)$ all with $J^P=0^{+}$; an $\hat{f}(1000\mathrm{}\mp 20)$ and a $\hat{\phi }(1268\mathrm{}\mp 60)$ with $J^P=1^{+}$; and a ${\phi }^{\prime \prime }(1520\mathrm{}\mp 60)$ with $J^P=2^{+}$. There is also an $L=1\mathrm{}$, $\mathrm{SU}\left(6\right)\mathrm{}$ singlet with $J^P=1^{+}$ and mass about 1600 MeV. Resonances that the above system would lead to at still higher energies are discussed.