B to K^(ast) l^+ l^-, K l^+ l^- decays in a family non-universal Z^(prime) model

Motivated by the observed forward-backward asymmetry in $B\to K^{\ast} l^+ l^-$ decay, we perform a detailed analysis of this decay mode within a family non-universal $Z^{\prime}$ model. With the related coupling $Z^{\prime}-\bar{s}b$ constrained by $\bar{B}_s-B_s$ mixing, $B\to \pi K$, and $B\to X_s \mu^{+}\mu^{-}$ decays, we look for further constraint on the couplings $Z^{\prime}-\mu^{+}\mu^{-}$ from $A_{FB}(B\to K^{\ast} \mu^{+}\mu^{-})_{0 {\rm GeV}^2\leqslant q^2\leqslant 2 {\rm GeV}^2}$ and get numerically $B_{\mu\mu}^{L,R}\sim{\cal O}(10^{-2})$. Moreover, we find that the relations, $B_{\mu\mu}^{L}<B_{\mu\mu}^{R}$ and $B_{\mu\mu}^{L}+B_{\mu\mu}^{R}<0$, with a small negative $\phi_{s}^{L}$, are crucial to moderate the discrepancy for $A_{FB}(B\to K^{\ast} \mu^{+}\mu^{-})$ between the SM prediction and the experimental data. Numerically, comparing with the SM prediction, we find that $A_{FB}(B\to K^{\ast} \mu^{+}\mu^{-})_{0 {\rm GeV}^2\leqslant q^2\leqslant 2 {\rm GeV}^2}$ could be enhanced about 80% and 50% by $Z^{\prime}$ contribution at most in scenarios S1 and S2, corresponding to the two fitted results of $\phi_{s}$ by UTfit collaboration, respectively. However, the results are still about $1.5\sigma$ lower than the experimental measurement.