Determination of the Decay Coupling Constants and Mixing Angle ofJP=1+Mesons

The mixing angle between the two $J^P=1^{+}$ kaons ($K_A$ and $K_C$) and the decay coupling constants of the $A_1$, $B$, $K_A$ and $K_C$ mesons are calculated with the $\mathrm{SU}\left(3\right)\mathrm{}\times \mathrm{SU}\left(3\right)\mathrm{}$ chiral algebra. These results are used to calculate the widths of the $K_A$ and $K_C$ mesons, which are found to be in reasonable agreement with present experiments. The method used is the asymptotic $\mathrm{SU}\left(3\right)\mathrm{}$ symmetry due to Matsuda and Oneda—i.e., the charge operator $V_K$, which is an $\mathrm{SU}\left(3\right)\mathrm{}$ raising or lowering operator in the symmetry limit, still acts as a generator in the broken symmetry in an appropriately chosen infinite-momentum limit. For the axial charge $A_{\pi }$, the hypothesis of partially conserved axial-vector current is used. One uses the commutators $[V_{K^0},A_{{\pi }^{-}}]=0\mathrm{}$ and $[V_{K^0},A_{{\pi }^{-}}]=0\mathrm{}$, and inserts them between the $K^{*0\mathrm{}}$ and $A_1^{}{}_{}{}^{+}$ states on the one hand and the $K^{*0\mathrm{}}$ and $B^{+}$ states on the other. This gives eight sum rules involving the decay coupling constants of the $A_1$, $B$, $K_C$, and $K_A$ mesons, the masses of the $J^P=1^{+},1^{-}$ mesons, and the mixing angle between $K_C$ and $K_A$. From these sum rules and the ratio $\frac{\Gamma (A_1\to \rho \pi )}{\Gamma (B\to \omega \pi )}$, one can derive the value of the mixing angle. Using the values of $\Gamma (A_1\to \rho \pi )$ and $\Gamma (B\to \omega \pi )$, one obtains all the coupling constants and can calculate the rates of $K_C$ and $K_A$ decays.