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

Abstract

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