First model-independent Dalitz analysis of $B^0 to D^0K^{ast 0}$, $Dto K_S^0pi ^+pi ^-$ 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 $Dto K_S^0pi ^+pi ^-$ 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 $772times 10^6$ $Bbar {B}$ 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 $B^0 to bar {D^0}K^{ast 0}$.