$B\to K^*$ Transition Form Factors and the Semi-leptonic Decay $B \to K^* \mu^+ \mu^-$

We present a detailed calculation on the $B\to K^*$ transition form factors (TFFs), $A_{0,1,2}$, $V$ and $T_{1,2,3}$, within the QCD light-cone sum rules (LCSR). To suppress the contributions from high-twist light-cone distribution amplitudes, we adopt a right-handed chiral correlator to do the LCSR calculation. In the resultant LCSRs for the TFFs, the transverse leading-twist distribution amplitude $\phi_{2;K^*}^\bot$ provides over $90\%$ contribution, thus those TFFs provide good platforms for testing the property of $\phi_{2;K^*}^\bot$. We suggest a model for $\phi_{2;K^*}^\bot$, in which two parameters $B_{2;K^*}^\bot$ and $C_{2;K^*}^\bot$ dominantly control its longitudinal distribution. With a proper choice of $B_{2;K^*}^\bot$ and $C_{2;K^*}^\bot$, our predictions on $B\to K^*$ TFFs are consistent with those of lattice QCD predictions. As an application, we also calculate the branching fraction of the $B$-meson rare decay $B \to K^* \mu^+ \mu^-$. The predicted differential branching fraction $d{\cal B}/dq^2(B\to K^{*}\mu^+\mu^-)$ is consistent with the LHCb and Belle measurements within errors. After integrating over the allowable $q^2$-region, we get the branching fraction, ${\cal B}(B\to K^*\mu^+\mu^-) = \left(1.088^{+0.261}_{-0.205} \right)\times 10^{-6}$, where the errors are squared average of the mentioned error sources. When the precision of experimental measurements or the other source of theoretical uncertainties for this process have been further improved in the future, we may get a definite conclusion on the behavior of $\phi_{2;K^*}^\bot$.