Measurement of the amplitude ratio of $B^0\to D^0{K}^{\ast 0}$ and $B^0\to \overline{D^0}{K}^{\ast 0}$ decays with a model-independent Dalitz plot analysis using $D\to K_S^0{\pi }^{+}{\pi }^{-}$ decays

We report a measurement of the amplitude ratio $r_S$ of $B^0\to D^0{K}^{\ast 0}$ and $B^0\to \overline{D^0}{K}^{\ast 0}$ decays with a model-independent Dalitz plot analysis using $D\to K_S^0{\pi }^{+}{\pi }^{-}$ decays. Using the full data sample of $772\times {10}^6$ $B\bar{B}$ pairs collected at the $\mathrm{{\rm Y}}(4S)$ resonance with the Belle detector at KEKB accelerator the upper limit is $r_S<0.87$ at the 68 % confidence level. This result is the first measurement of $r_S$ with a model-independent Dalitz analysis, and is consistent with results from other analyses. The value of $r_S$ indicates the sensitivity of the decay to ${\varphi }_3$ because the statistical uncertainty is proportional to $1\mathrm{/}{r}_S$. The $r_S$ result is obtained from observables ($x_{\pm }$, $y_{\pm }$) \begin{eqnarray} x_- &=& +0.4 ^{+1.0 +0.0}{-0.6 -0.1} \pm0.0 \ y- &=& -0.6 ^{+0.8 +0.1}{-1.0 -0.0} \pm0.1 \ x+ &=& +0.1 ^{+0.7 +0.0}{-0.4 -0.1} \pm0.1 \ y+ &=& +0.3 ^{+0.5 +0.0}{-0.8 -0.1} \pm0.1 \ , \end{eqnarray} where $x\pm = r_S \cos(\delta_S \pm \phi_3)$,$y_\pm = r_S \sin(\delta_S \pm \phi_3)$and$\phi_3 (\delta_S)$aretheweak(strong)phasedifferencebetween$B^0 \to D^0K^{*0}$and$B^0 \to \bar{D^0}K^{*0}$. The first uncertainty is statistical, the second is the experimental systematic and the third is the systematic due to the uncertainties on $c_i$ and $s_i$ parameters measured by CLEO.