Study of the K + π + π - final state in B + →J/ψK + π + π - and B + →ψ ' K + π + π -

Using $535\\ifmmode\\times\\else\\texttimes\\fi{}{10}^{6}$ $B$-meson pairs collected by the Belle detector at the KEKB ${e}^{+}{e}^{\\ensuremath{-}}$ collider, we measure branching fractions of $(7.16\\ifmmode\\pm\\else\\textpm\\fi{}0.10(\\mathrm{stat})\\ifmmode\\pm\\else\\textpm\\fi{}0.60(\\mathrm{syst})\\ifmmode\\times\\else\\texttimes\\fi{}{10}^{\\ensuremath{-}4}$ for ${B}^{+}\\ensuremath{\\rightarrow}J/\\ensuremath{\\psi}{K}^{+}{\\ensuremath{\\pi}}^{+}{\\ensuremath{\\pi}}^{\\ensuremath{-}}$ and $(4.31\\ifmmode\\pm\\else\\textpm\\fi{}0.20(\\mathrm{stat})\\ifmmode\\pm\\else\\textpm\\fi{}0.50(\\mathrm{syst}))\\ifmmode\\times\\else\\texttimes\\fi{}{10}^{\\ensuremath{-}4}$ for ${B}^{+}\\ensuremath{\\rightarrow}{\\ensuremath{\\psi}}^{\\ensuremath{'}}{K}^{+}{\\ensuremath{\\pi}}^{+}{\\ensuremath{\\pi}}^{\\ensuremath{-}}$. We perform amplitude analyses to determine the resonant structure of the ${K}^{+}{\\ensuremath{\\pi}}^{+}{\\ensuremath{\\pi}}^{\\ensuremath{-}}$ final state in ${B}^{+}\\ensuremath{\\rightarrow}J/\\ensuremath{\\psi}{K}^{+}{\\ensuremath{\\pi}}^{+}{\\ensuremath{\\pi}}^{\\ensuremath{-}}$ and ${B}^{+}\\ensuremath{\\rightarrow}{\\ensuremath{\\psi}}^{\\ensuremath{'}}{K}^{+}{\\ensuremath{\\pi}}^{+}{\\ensuremath{\\pi}}^{\\ensuremath{-}}$ and find that the ${K}_{1}(1270)$ is a prominent component of both decay modes. There is significant interference among the different intermediate states, which leads, in particular, to a striking distortion of the $\\ensuremath{\\rho}$ line shape due to the $\\ensuremath{\\omega}$. Based on the results of the fit to the ${B}^{+}\\ensuremath{\\rightarrow}J/\\ensuremath{\\psi}{K}^{+}{\\ensuremath{\\pi}}^{+}{\\ensuremath{\\pi}}^{\\ensuremath{-}}$ data, the relative decay fractions of the ${K}_{1}(1270)$ to $K\\ensuremath{\\rho}$, $K\\ensuremath{\\omega}$, and ${K}^{*}(892)\\ensuremath{\\pi}$ are consistent with previous measurements, but the decay fraction to ${K}_{0}^{*}(1430)$ is significantly smaller. Finally, by floating the mass and width of the ${K}_{1}(1270)$ in an additional fit of the ${B}^{+}\\ensuremath{\\rightarrow}J/\\ensuremath{\\psi}{K}^{+}{\\ensuremath{\\pi}}^{+}{\\ensuremath{\\pi}}^{\\ensuremath{-}}$ data, we measure a mass of $(1248.1\\ifmmode\\pm\\else\\textpm\\fi{}3.3(\\mathrm{stat})\\ifmmode\\pm\\else\\textpm\\fi{}1.4(\\mathrm{syst}))\\text{ }\\text{ }\\mathrm{MeV}/{c}^{2}$ and a width of $(119.5\\ifmmode\\pm\\else\\textpm\\fi{}5.2(\\mathrm{stat})\\ifmmode\\pm\\else\\textpm\\fi{}6.7(\\mathrm{syst}))\\text{ }\\text{ }\\mathrm{MeV}/{c}^{2}$ for the ${K}_{1}(1270)$.