Leptoquark on $P\to \ell^{+} \nu$, FCNC and LFV
  Motivated by the disagreement between the experimental data and lattice calculations on the decay constant of the $D_{s}$ meson, we investigate leptoquark (LQ) contributions to the purely leptonic decays of a pseudoscalar (P). We concentrate on the LQs which only couple to the second-generation quarks before the electroweak symmetry breaking and we discuss in detail how flavor symmetry breaking effects are brought into the extension of the standard model after the spontaneous symmetry breaking. We find that the hermiticity of fermion mass matrices can not only reproduce the correct Cabibbo-Kobayashi-Maskawa and Maki-Nakagawa-Sakata matrices, but also reduce the number of independent flavor mixing matrices and lead to $V^{R}_{f}=V^{L}_{f}$ with $L(R)$ denoting the chirality of the f-type fermion. Accordingly, it is found that the decays $D_{s,d}\to \ell^{+} \nu$, $B^{+}\to \tau^{+} \nu$ and $B_c\to \ell^{+} \nu$ have a strong correlation in parameters. We predict that the decay constant of the $B_c$ meson calculated by the lattice could be less than the experimental data by 23%. Intriguingly, the resultant upper limits of branching ratios for $D\to \mu^{+} \mu^{-}$ and $\tau\to \mu (\pi^0, \eta, \eta', \rho, \omega)$ are found to be around $ 5.1\times 10^{-7}$ and $(2.6, 1.5, 0.6, 7.4, 4.8)\times 10^{-8}$, which are below and close to the current experimental upper bounds, respectively. 
Rachid Benbrik, Chuan-Hung Chen