The first observation of the decay $\bar{B}^0_s \to D^0 K^{*0}$ using $pp$ data collected by the LHCb detector at a centreofmass energy of 7 TeV, corresponding to an integrated luminosity of 36 pb$^{1}$, is reported. A signal of $34.4 \pm 6.8$ events is obtained and the absence of signal is rejected with a statistical significance of more than nine standard deviations. The $\bar{B}^0_s \to D^0 K^{*0}$ branching fraction is measured relative to that of $\bar{B}^0 \to D^0 \rho^0$: $\frac{{\cal B}(\bar{B}^0_s \to D^0 K^{*0})}{{\cal B}(\bar{B}^0 \to D^0 \rho^0)} = 1.48 \pm 0.34 \pm 0.15 \pm 0.12$, where the first uncertainty is statistical, the second systematic and the third is due to the uncertainty on the ratio of the $B^0$ and $B^0_s$ hadronisation fractions.
The invariant mass distribution for the $\overline{ B }{} ^0 \rightarrow D ^0 \rho ^0 $ decay mode for the $\texttt{TOSOnly}$ (left) and $\texttt{TIS}$ (right) trigger categories with the result of the fit superimposed. The black points correspond to the data and the fit result is represented as a solid line. The signal is fitted with a double Gaussian (dashed line), the partially reconstructed background with an exponential function (light grey area) and the combinatorial background with a flat distribution (dark grey area) as explained in the text. The contributions from crossfeed are too small to be visible. 
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The invariant mass distribution for the $\overline{ B }{} ^0_s \rightarrow D ^0 K ^{*0} $ decay mode with the result of the fit superimposed. The black points correspond to the data and the fit result is represented as a solid line. The signal is fitted with a double Gaussian (dashed line), the partially reconstructed background with an exponential function (light grey area), the combinatorial background with a flat distribution (dark grey area) and the crossfeed from $\overline{ B }{} ^0 \rightarrow D ^0 \rho ^0 $ (intermediate grey area) as explained in the text. 
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The $\rho ^0$ (on the left) and $ K ^{*0}$ (on the right) invariant mass distributions obtained from data using an sPlot technique. The level of non $ K ^{*0}$ combinations in the $\overline{ B }{} ^0_s \rightarrow D ^0 K ^{*0} $ peak is negligible. Despite being mainly due to $ D ^0 \rho ^0 $ combinations, the $\overline{ B }{} ^0 \rightarrow D ^0 \rho ^0 $ contains a significant contribution of "non $\rho ^0$ " events. The black points correspond to the data and the fit result is represented as a solid line. The resonant component is fitted with a BreitWigner convoluted with a Gaussian (dashed line) and the nonresonant part, if present, with a secondorder polynomial (grey area). 
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Animated gif made out of all figures. 
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Summary of the fitted yields for the different categories. The background yields are quoted for the full mass regions. 
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Summary of the contributions to the systematic uncertainties. The uncertainty on the $r$ ratio gives the range used for the systematic uncertainty extraction on the ratios of the branching fractions. 
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Created on 06 March 2021.