CPviolation and kaon-pion interactions inB→Kπ+π−decays

We study $CP$ violation and the contribution of the strong kaon-pion interactions in the three-body $B\ensuremath{\rightarrow}K{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}$ 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 $K\ensuremath{\pi}$ scalar ($S$-wave) and vector ($P$-wave) form factors. These are determined from Muskhelishvili-Omn\`es 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 $(32\ifmmode\pm\else\textpm\fi{}5)\text{ }\text{ }\mathrm{MeV}$. The additional phenomenological amplitudes are fitted to reproduce the $K\ensuremath{\pi}$ effective mass and helicity angle distributions, the $B\ensuremath{\rightarrow}{K}^{*}(892)\ensuremath{\pi}$ 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 ${K}^{*}(892)\ensuremath{\pi}$. Our predicted ${B}^{\ifmmode\pm\else\textpm\fi{}}\ensuremath{\rightarrow}{K}_{0}^{*}(1430){\ensuremath{\pi}}^{\ifmmode\pm\else\textpm\fi{}}$, ${K}_{0}^{*}(1430)\ensuremath{\rightarrow}{K}^{\ifmmode\pm\else\textpm\fi{}}{\ensuremath{\pi}}^{\ensuremath{\mp}}$ branching fraction, equal to $(11.6\ifmmode\pm\else\textpm\fi{}0.6)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}6}$, is smaller than the result of the analyzes of both collaborations. For the neutral ${B}^{0}$ decays, the predicted value is $(11.1\ifmmode\pm\else\textpm\fi{}0.5)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}6}$. In order to reduce the large systematic uncertainties in the experimental determination of the $B\ensuremath{\rightarrow}{K}_{0}^{*}(1430)\ensuremath{\pi}$ branching fractions, a new parametrization is proposed. It is based on the $K\ensuremath{\pi}$ scalar form factor, well constrained by theory and experiments other than those of $B$ decays.