Time-dependent and time-integrated angular analysis of $B\to\phi K^0_S\pi^0$ and $\phi K^\pm\pi^\mp$

We perform a time-dependent and time-integrated angular analysis of the decays $B^0\to\phi K^{*}(892)^0$, $\phi K^{*}_2(1430)^0$, and $\phi(K\pi)^{*0}_{0}$ with the final sample of about 465 million $\BB$ pairs recorded with the $\babar$ detector. Twenty-four parameters are investigated, including the branching fractions, CP-violation parameters, and parameters sensitive to final-state interactions. We use the dependence on the $K\pi$ invariant mass of the interference between the scalar and vector or tensor components to resolve discrete ambiguities of the strong and weak phases. We use the time-evolution of the $B\to\phi K^0_S\pi^0$ channel to extract the CP-violation phase difference $\Delta\phi_{00}=0.28\pm0.42\pm 0.04$ between the $B$ and $\Bbar$ decay amplitudes. When the $B\to\phi K^\pm\pi^\mp$ channel is included, the fractions of longitudinal polarization ${f_L}$ of the vector-vector and vector-tensor decay modes are measured to be $0.494\pm0.034\pm0.013$ and $0.901^{+0.046}_{-0.058}\pm 0.037$, respectively. This polarization pattern requires the presence of a positive-helicity amplitude in the vector-vector decay from a currently unknown source.