Study of the $B^{\pm} \rightarrow { (K \pi \pi^{0} )}_{ D^{0} / \overline{D^{0}}} K^{\pm}$ through the ADS method

We report on our study of $B^{\pm}$ mesons decaying into the final states $K^{+} \pi^{-}\pi^{0}K^{\pm}$ and $K^{-} \pi^{+}\pi^{0}K^{\pm} $, where $K\pi\pi^{0}$ come from a $D^{0}$ or $\overline{D^{0}}$ meson, using $205~{\rm fb}^{-1}$ of data collected by the BaBar detector at the PEP-II asymmetric-energy $e^+e^-$ $B$ Factory at SLAC. We aim to measure the ratio $R_{ADS}= \frac{\mathcal{BR}(B^{+}\rightarrow K^{-}\pi^{+}\pi^ {0} K^{+}) + \mathcal{BR}(B^{-}\rightarrow K^{+}\pi^{-}\pi^{0} K^{-})} { \mathcal{BR}(B^{+}\rightarrow K^{+}\pi^{-}\pi^{0} K^{+}) + \mathcal{BR}(B^{-}\rightarrow K^{-}\pi^{+}\pi^{0} K^{-}) }$, from which we can extract information on the parameter $r_ {B}=\frac{A(B^{+}\rightarrow D^{0}K^{+})}{A(B^{+}\rightarrow \overline{D^{0}}K^{+})}$. Since $r_{B}$ is the ratio between a $V_{ub}$ and the $V_{cb}$ decay amplitude, it's knowledge corresponds to a constraint on $\rho$--$\eta$ plane in the $\gamma$ sector.