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

## TL;DR

This paper investigates the virtual contributions of off-shell $D^*$ mesons in $B$ decays to $D o ext{pions/kaons}$ using perturbative QCD, revealing small virtual effects and confirming flavor symmetry with implications for $D^*$ decay widths.

## Contribution

It provides a detailed analysis of virtual $D^*$ contributions in $B$ decays and introduces ratios to test flavor-$SU(3)$ symmetry and constrain $D^*$ decay widths.

## Key findings

- Virtual $D^*$ contributions are about 5% of quasi-two-body results.
- Flavor-$SU(3)$ symmetry holds with minimal breaking.
- Constraints on $D^{*0}$ decay width are consistent with previous predictions.

## Abstract

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