Measurement of the amplitude ratio of $B^0 \to D^0K^{*0}$ and $B^0 \to \bar{D^0}K^{*0}$ 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 $B^0 \to D^0K^{*0}$ and $B^0 \to \bar{D^0}K^{*0}$ decays with a model-independent Dalitz plot analysis using $D\to K_S^0\pi^+\pi^-$ decays. Using the full data sample of $772\times10^6$ $B\bar{B}$ pairs collected at the $\Upsilon(4S)$ 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 $\phi_3$ because the statistical uncertainty is proportional to $1/r_S$. 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 $\phi_3 (\delta_S)$ are the weak (strong) phase difference between $B^0 \to D^0K^{*0}$ 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.