Relation of Charged-KDecay toCPViolation

An $\mathrm{SU}\left(3\right)\mathrm{}$ model is described which relates the rate of the charged-$K$ decay $K^{+}\to {\pi }^{+}+{\pi }^0$ to the magnitude of $\mathrm{CP}$ violation. A weak-interaction Lagrangian for nonleptonic processes is constructed from octet currents in a Cabibbo frame under the hypothesis that the $\mathrm{CP}$-conserving processes are described predominantly by an octet-dominant interaction Lagrangian, and that the $\mathrm{CP}$-violating processes enter solely through the 27-dimensional representation of 8≈8. Three types of octet currents are used: $L_{\mu }$, which is linear in the field operators, and $J_{\mu }$ and $K_{\mu }$, which are bilinear and possess opposite $C$-transformation properties. It is shown that if only $I=2\mathrm{}$ contributions enter into the 27 portion of the Lagrangian, then, approximating $|K_S〉$ and $|K_L〉$ by the $\mathrm{CP}$ eigenstates $|K_1〉$ and $|K_2〉$, the observed rate of the charged-$K$ decay and $\theta =0.26\mathrm{}$ for the Cabibbo angle fix $\left|{\eta }_{+-}\right|=5.38\mathrm{}\pm {10}^{-3\mathrm{}}$ and $\left|{\eta }_{00}\right|=3.12\mathrm{}\times {10}^{-3\mathrm{}}$, of the proper order of magnitude. Introducing the experimentally observed departure of $|K_S〉$ and $|K_L〉$ from $\mathrm{CP}$ eigenstates and a renormalization constant for the $K$ current yields $\left|{\eta }_{+-}\right|=1.97\mathrm{}\times {10}^{-3\mathrm{}}$ and $\left|{\eta }_{00}\right|=2.92\mathrm{}\times {10}^{-3\mathrm{}}$, in good agreement with experiment.