Non-resonant four-body decay of $B \to D^- \pi^+\pi^+\pi^- $

Abstract

We calculate the branching ratio of the non-resonant $B \to D^- \pi^+\pi^+\pi^- $ decay using a simple model based on the framework of the factorization approach. In naive factorization approach, there are only tow tree diagrams for this decay mode. In the first diagram, the matrix element of decay mode is factorized into a $B\to D$ form factor multiplied by a $3\pi$ decay constant and in the second diagram, the matrix element is factorized into a $B\to D\pi$ form factor multiplied by a $2\pi$ decay constant, We assume that in the rest frame of B meson, the $D$ meson remains stationary, so we obtain the value $(3.47\pm0.14)\times 10^{-3}$ for the branching ration of the $B \to D^- \pi^+\pi^+\pi^- $ decay mode, while the experimental results are $(3.9\pm1.9)\times10^{-3}$.