First model-independent Dalitz analysis of $B^0 \to D^0K^{\ast 0}$, $D\to K_S^0\pi ^+\pi ^-$ decay

We report a measurement of the amplitude ratio $r_S$ of $B^0 \to D^0K^{\ast 0}$ and $B^0 \to \bar {D^0}K^{\ast 0}$ decays with a Dalitz analysis of $D\to K_S^0\pi ^+\pi ^-$ decays, for the first time using a model-independent method. We set an upper limit $r_S < 0.87$ at the 68% confidence level, using the full data sample of $711\,{\rm fb}^{-1}$ corresponding to $772\times 10^6$ $B\bar {B}$ pairs collected at the $\Upsilon (4S)$ resonance with the Belle detector at the KEKB $e^+e^-$ collider. This result is obtained from observables $x_- = +0.4 ^{+1.0 +0.0}_{-0.6 -0.1} \pm 0.0$, $y_- = -0.6 ^{+0.8 +0.1}_{-1.0 -0.0} \pm 0.1$, $x_+ = +0.1 ^{+0.7 +0.0}_{-0.4 -0.1} \pm 0.1$, and $y_+ = +0.3 ^{+0.5 +0.0}_{-0.8 -0.1} \pm 0.1$, where $x_\pm = r_S \cos (\delta _S \pm \phi _3)$, $y_\pm = r_S \sin (\delta _S \pm \phi _3)$, and $\phi _3\ (\delta _S)$ is the weak (strong) phase difference between $B^0 \to D^0K^{\ast 0}$ and $B^0 \to \bar {D^0}K^{\ast 0}$.