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$B \to K^{\ast} l^+ l^-$, $K l^+ l^-$ 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, $B\to πK$, and $B\to X_s μ^{+}μ^{-}$ decays, we look for further constraint on the couplings $Z^{\prime}-μ^{+}μ^{-}$ from $A_{FB}(B\to K^{\ast} μ^{+}μ^{-})_{0 {\rm GeV}^2\leqslant q^2\leqslant 2 {\rm GeV}^2}$ and get numerically $B_{μμ}^{L,R}\sim{\cal O}(10^{-2})$. Moreover, we find that the relations, $B_{μμ}^{L}<B_{μμ}^{R}$ and $B_{μμ}^{L}+B_{μμ}^{R}<0$, with a small negative $ϕ_{s}^{L}$, are crucial to moderate the discrepancy for $A_{FB}(B\to K^{\ast} μ^{+}μ^{-})$ between the SM prediction and the experimental data. Numerically, comparing with the SM prediction, we find that $A_{FB}(B\to K^{\ast} μ^{+}μ^{-})_{0 {\rm GeV}^2\leqslant q^2\leqslant 2 {\rm GeV}^2}$ 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 $ϕ_{s}$ by UTfit collaboration, respectively. However, the results are still about $1.5σ$ lower than the experimental measurement.