We study $CP$ violation and the contribution of the strong kaon-pion interactions in the three-body $B\to K{\pi }^{+}{\pi }^{-}$ decays. We extend our recent work on the effect of the two-pion $S$- and $P$-wave interactions to that of the corresponding kaon-pion ones. The weak amplitudes have a first term derived in QCD factorization and a second one as a phenomenological contribution added to the QCD penguin amplitudes. The effective QCD coefficients include the leading order contributions plus next-to-leading order vertex and penguins corrections. The matrix elements of the transition to the vacuum of the kaon-pion pairs, appearing naturally in the factorization formulation, are described by the strange $K\pi$ scalar ($S$-wave) and vector ($P$-wave) form factors. These are determined from Muskhelishvili-Omnès coupled channel equations using experimental kaon-pion $T$-matrix elements, together with chiral symmetry and asymptotic QCD constraints. From the scalar form factor study, the modulus of the $K_0^{*}(1430)$ decay constant is found to be $(32\pm 5)\mathrm{MeV}$. The additional phenomenological amplitudes are fitted to reproduce the $K\pi$ effective mass and helicity angle distributions, the $B\to K^{*}(892)\pi$ branching ratios and the $CP$ asymmetries of the recent data from Belle and BABAR collaborations. We use also the new measurement by the BABAR group of the phase difference between the $B^0$ and ${\overline{B}}^0$ decay amplitudes to $K^{*}(892)\pi$. Our predicted $B^{\pm }\to K_0^{*}(1430){\pi }^{\pm }$, $K_0^{*}(1430)\to K^{\pm }{\pi }^{\mp }$ branching fraction, equal to $(11.6\pm 0.6)\times {10}^{-6}$, is smaller than the result of the analyzes of both collaborations. For the neutral $B^0$ decays, the predicted value is $(11.1\pm 0.5)\times {10}^{-6}$. In order to reduce the large systematic uncertainties in the experimental determination of the $B\to K_0^{*}(1430)\pi$ branching fractions, a new parametrization is proposed. It is based on the $K\pi$ scalar form factor, well constrained by theory and experiments other than those of $B$ decays.

• Received 20 February 2009

DOI:https://doi.org/10.1103/PhysRevD.79.094005