K0−K¯0in theSU(2)L×SU(2)R×U(1)model ofCPviolation

We reexamine the $K^0-{\overline{K}}^0$ system in the ${\mathrm{SU}\mathrm{}\left(2\right)\mathrm{}}_L\times {\mathrm{SU}\mathrm{}\left(2\right)\mathrm{}}_R\times \mathrm{U}\mathrm{}\left(1\right)\mathrm{}$ model of $\mathrm{CP}$ violation with three generations with special emphasis on the minimal model of spontaneous $\mathrm{CP}$ violation. We give a new estimation of the hadronic matrix elements entering in $\frac{{\epsilon }^{\prime }}{\epsilon }$, taking into account constraints from current algebra, final-state interactions, and short-distance QCD corrections. We compute, for an arbitrary number of generations, the phenomenological $\mathrm{CP}$ phases in terms of the quark masses, the parameter $r$ (ratio of Higgs vacuum expectation values), and the single $\mathrm{CP}$ phase $\alpha$ of the model. We make a three-generation phenomenological analysis taking into account the high multiplicity of the sheets for the $\mathrm{CP}$ phases and the uncertainties in the Kobayashi-Maskawa matrix elements and $m_t$. For low $M_{W_R}$ the two-generation $\mathrm{LR}$ part dominates for most of the solutions, but the pure $L$ three-generation contribution can be dominant in some cases.