Virtual contributions from $D^{\ast }(2007{)}^0$ and $D^{\ast }(2010{)}^{\pm }$ in the $B\to D\pi h$ decays

Abstract

We study the quasi-two-body decays $B\to D^{\ast }h\to D\pi h$ with $h=(\pi ,K)$ in the perturbative QCD approach and focus on the virtual contributions from the off-shell $D^{\ast }(2007{)}^0$ and $D^{\ast }(2010{)}^{\pm }$ in the four measured decays ${\bar{B}}^0\to D^0{\pi }^{+}{\pi }^{-}$, ${\bar{B}}^0\to D^0{\pi }^{+}{K}^{-}$, $B^{-}\to D^{+}{\pi }^{-}{\pi }^{-}$ and $B^{-}\to D^{+}{\pi }^{-}{K}^{-}$. For the ${\bar{B}}^0\to D^{\ast +}{\pi }^{-}\to D^0{\pi }^{+}{\pi }^{-}$ and ${\bar{B}}^0\to D^{\ast +}{K}^{-}\to D^0{\pi }^{+}{K}^{-}$ decays, their branching fractions concentrate in a very small region of $m_{D^0{\pi }^{+}}$ near $D^{\ast +}$ pole mass, and the virtual contributions from $D^{\ast +}$, in the region $m_{D^0{\pi }^{+}}>2.1$ GeV, are about $5\%$ of the corresponding quasi-two-body results. We define two ratios $R_{D^{\ast +}}$ and $R_{D^{\ast 0}}$, from which we conclude that the flavor-$SU(3)$ symmetry will be maintained for the $B\to D^{\ast }h\to D\pi h$ decays with very small breaking at any physical value of the $m_{D\pi }$. The $B^{-}\to D^{\ast 0}{\pi }^{-}\to D^0{\pi }^0{\pi }^{-}$ and $B^{-}\to D^{\ast 0}{K}^{-}\to D^0{\pi }^0{K}^{-}$ decays can be employed as a constraint for the $D^{\ast 0}$ decay width, with preferred values consistent with previous theoretical predictions for this quantity.

12 pages, 5 figures. Published version