The Nonresonant Cabibbo Suppressed Decay $B^\pm\to π^+π^-π^\pm$ 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 $BR(B^\pm\rightarrow \pi^+\pi^-\pi^\pm) = (1.5 - 8.4 )\times 10^{-5}$. 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 $B^\pm\rightarrow \chi_{c0} \pi^\pm$ followed by $\chi_{c0}\rightarrow \pi^+\pi^-$. The resulting value of the partial rate asymmetry is $(0.40\sim 0.48)\mbox{sin}\gamma$, where $\gamma = \mbox{arg}(V_{ub}^*)$.