Nonresonant Cabibbo suppressed decay ${\mathit{B}}^{\pm }$→${\mathrm{\pi }}^{+}$${\mathrm{\pi }}^{\mathrm{-}}$${\mathrm{\pi }}^{\pm }$ and signal for CP violation

We consider various contributions to the nonresonant decay ${\mathit{B}}^{\pm }$→${\mathrm{\pi }}^{+}$ ${\mathrm{\pi }}^{\mathrm{-}}$ ${\mathrm{\pi }}^{\pm }$, both of the long-distance and short-distance types, with the former providing for most of the branching ratio, predicted to be scrB(${\mathit{B}}^{\pm }$→${\mathrm{\pi }}^{+}$ ${\mathrm{\pi }}^{\mathrm{-}}$ ${\mathrm{\pi }}^{\pm }$)=(1.0–5.0)×${10}^{\mathrm{-}5}$. We also discuss an application to CP violation resulting from the interference of that nonresonant background [with m(${\mathrm{\pi }}^{+}$ ${\mathrm{\pi }}^{\mathrm{-}}$)≊3.4 GeV] and ${\mathit{B}}^{\pm }$→${\mathrm{\chi }}_{\mathit{c}0}$ ${\mathrm{\pi }}^{\pm }$ followed by ${\mathrm{\chi }}_{\mathit{c}0}$→${\mathrm{\pi }}^{+}$ ${\mathrm{\pi }}^{\mathrm{-}}$. The resulting value of the partial rate asymmetry is (0.44–0.49)sinγ, where γ=arg(${\mathit{V}}_{\mathit{u}\mathit{b}}^{\mathrm{*}}$).