K*resonance effects on directCPviolation inB→ππK

Abstract

${B}^{\ifmmode\pm\else\textpm\fi{}}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}{K}^{\ifmmode\pm\else\textpm\fi{}}$ and ${\overline{B}}^{0}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}{\overline{K}}^{0}$ decay channels are analyzed within the QCD factorization scheme where final state interactions before and after hadronization are included. The ${K}^{*}(892)$ and ${K}_{0}^{*}(1430)$ resonance effects are taken into account using the presently known $\ensuremath{\pi}K$ strange vector and scalar form factors. The weak decay amplitudes, which are calculated at leading power in ${\ensuremath{\Lambda}}_{\mathrm{QCD}}/{m}_{b}$ and at the next-to-leading order in the strong coupling constant, include the hard-scattering and annihilation contributions. The end-point divergences of these weak final state interactions are controlled by two complex parameters determined through a fit to the available effective mass and helicity angle distribution, $CP$ asymmetry, and ${K}^{*}(892)$ branching ratio data. The predicted ${K}_{0}^{*}(1430)$ branching ratios and the calculated direct $CP$ violation asymmetries are compared to the Belle and BABAR Collaboration data.