As the annihilation contributions play important roles in solving the puzzle of the small longitudinal polarizations in $B\to K^{*}\varphi$ decays, we examine similar effects in the decays of $B\to K_{0,2}^{*}(1430)\varphi$. For the calculations on the annihilated contributions, we adopt that the form factors in $B\to K^{(*)}\varphi$ decays are parameters determined by the observed branching ratios (BRs), polarization fractions, and relative angles in experiments, and we connect the parameters between $B\to K_{0(2)}^{*}\varphi$ and $B\to K^{(*)}\varphi$. We find that the BR of $B_d\to K_0^{*0}(1430)\varphi$ is $(3.69\pm 0.47)\times {10}^{-6}$. We show that the BR of $B_d\to K_2^{*0}(1430)\varphi$ is $(1.70\pm 0.80)\times {10}^{-6}$ in the 2nd version of the Isgur-Scora-Grinstein-Wise (ISGW2) model, whereas it can be a broad range of values in the light-front quark model. In terms of the recent BABAR observations on BRs and polarization fractions in $B_d\to K_2^{*0}(1430)\varphi$, the results in the light-front quark model are found to be more favorable. In addition, due to the annihilation contributions to $B\to K_2^{*}\varphi$ and $B\to K^{*}\varphi$ being opposite in sign, we demonstrate that the longitudinal polarization of $B_d\to K_2^{*0}(1430)\varphi$ is always $O(1)$ with or without including the annihilation contributions.

• Received 9 January 2007

DOI:https://doi.org/10.1103/PhysRevD.75.054010