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.
Categories: hep-ph