Lepton flavor violating decays of mesons to lepton-pairs in a gauge group $S{U}_L(2)\times U_{Y_1}(1)\times S{U}_X(2)$

The lepton flavor conserving and lepton flavor violating decays of $K$ and $B$ mesons to lepton pairs in the gauge group $G=SU_L(2)\times U_{Y_1}(1) \times SU_X(2)$ are discussed. The quark-lepton transitions mediated by the lepto-quark bosons $X_{\mu }^{\pm 2\mathrm{/}3}$ of the group $S{U}_X(2)$ provide a framework to construct an effective Hamiltonian for these decays. The effective coupling constant $\frac{G_X}{\sqrt{2}}=\frac{g_X^2}{8m_X^2}$; $m_X$ is the mass at which the group $G$ is broken to the SM group. The upper bound on ($G_X\mathrm{/}{G}_F$)is obtained from the most stringent experimental limit on the $B\mathrm{.}R(K_L^0\to {\mu }^{\mp }{e}^{\pm })$. Several cases of pairing three generations of leptons and quarks in the representation $(2,\bar{2})$ of the group are analyzed. For some pairing, the upper bound on $(G_X\mathrm{/}{G}_F)$ is of the order $(6-9)\times {10}^{-6}$ and is compatible with the upper limits on various LF violating $K$-decays . In particular for these cases, we find the upper limit on the branching ratio $B.R(K^{0}_{L}\to\mu^{-}\mu^{+})\sim (1.9-8.3)\times10^{-9}$. It is shown that for LF violating B-decays to lepton pairs, the time integrated decay rate $B_{d,s}^0\to {\mathrm{\ell }}^{-}{\tau }^{+}({\tau }^{-}{\mathrm{\ell }}^{+})$ is a promising area to test the model.