Analysis of Bto K^*_J (to K π) μ^+μ^- in the higher kaon resonance region

We study the resonant contributions in the process $\bar{B}^0\to K^- \pi^+\mu^+\mu^-$ with the $K^-\pi^+$ invariant mass square $m_{K\pi}^2\in [1, 5] {\rm GeV}^2$. Width effects of the involved strange mesons, $K^*(1410), K_0^*(1430), K_2^*(1430), K^*(1680), K_3^*(1780)$ and $K_4^*(2045)$, are incorporated. In terms of the helicity amplitudes, we derive a compact form for the full angular distributions, through which the branching ratios, forward-backward asymmetries and polarizations are attained. We propose that the uncertainties in the $B\to K^*_J$ form factors can be pinned down by the measurements of a set of SU(3)-related processes. Using results from the large energy limit, we derive the dependence of branching fractions on the $m_{K\pi}$, and find that the $K^*_2(1430)$ resonance has a clear signature, in particular, in the transverse polarizations.