The Nonresonant Cabibbo Suppressed Decay $B^{\pm }\to {\pi }^{+}{\pi }^{-}{\pi }^{\pm }$ and Signal for CP Violation

We consider various contributions to the nonresonant decay $B^\pm\rightarrow \pi^+\pi^-\pi^\pm$, both of the long-distance and short-distance types with the former providing for most of the branching ratio, predicted to be $BR(B^{\pm }\to {\pi }^{+}{\pi }^{-}{\pi }^{\pm })=(1.5-8.4)\times {10}^{-5}$. We also discuss an application to CP violation resulting from the interference of that nonresonant background (with $m({\pi }^{+}{\pi }^{-})\approx 3.4$ GeV) and $B^{\pm }\to {\chi }_{c0}{\pi }^{\pm }$ followed by $\chi_{c0}\rightarrow \pi^+\pi^-$\mathrm{.}Theresultingvalueofthepartialrateasymmetryis$(0.40\sim 0.48)\mbox{sin}\gamma$,where$\gamma = \mbox{arg}(V_{ub}^*)$.