The first observation of the $B_s^0 \to \overline{D}^0 K^+ K^$ decay is reported, together with the most precise branching fraction measurement of the mode $B^0 \to \overline{D}^0 K^+ K^$. The results are obtained from an analysis of $pp$ collision data corresponding to an integrated luminosity of $3.0 \textrm{fb}^{1}$. The data were collected with the LHCb detector at centreofmass energies of $7$ and $8$ TeV. The branching fraction of the $B^0 \to \overline{D}^0 K^+ K^$ decay is measured relative to that of the decay $B^0 \to \overline{D}^0 \pi^+ \pi^$ to be $$\frac{\mathcal{B}(B^0 \to \overline{D}^0 K^+ K^)}{\mathcal{B}(B^0 \to \overline{D}^0 \pi^+ \pi^)} = (6.9 \pm 0.4 \pm 0.3)\%,$$ where the first uncertainty is statistical and the second is systematic. The measured branching fraction of the $B_s^0 \to \overline{D}^0 K^+ K^$ decay mode relative to that of the corresponding $B^0$ decay is $$\frac{\mathcal{B}(B_s^0 \to \overline{D}^0 K^+ K^)}{\mathcal{B}(B^0 \to \overline{D}^0 K^+ K^)} = (93.0 \pm 8.9 \pm 6.9)\%.$$ Using the known branching fraction of ${B^0 \to \overline{D}^0 \pi^+ \pi^}$, the values of ${{\mathcal B}(B^0 \to \overline{D}^0 K^+ K^ )=(6.1 \pm 0.4 \pm 0.3 \pm 0.3) \times 10^{5}}$, and ${{\cal B}(B_s^0 \to \overline{D}^0 K^+ K^)=}$ $(5.7 \pm 0.5 \pm 0.4 \pm 0.5) \times 10^{5}$ are obtained, where the third uncertainties arise from the branching fraction of the decay modes ${B^0 \to \overline{D}^0 \pi^+ \pi^}$ and $B^0 \to \overline{D}^0 K^+ K^$, respectively.
Example Feynman diagrams that contribute to the ${B^0_{(s)} \rightarrow \overline{ D }{} {}^0 K ^+ K ^ }$ decays via (a) $W$exchange, (b) nonresonant three body mode, (c) and (d) rescattering from a coloursuppressed decay. 
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Distributions of the Fisher discriminant, for preselected ${ B ^0 \rightarrow \overline{ D }{} {}^0 \pi ^+ \pi ^ }$ data candidates, in the mass range $[5240,5320]$ $ {\mathrm{ Me V /}c^2}$ : (black line) unweighted data distribution, and sWeighted training samples: (blue triangles) signal, (red circles) background, and (green squares) their sum. The training samples are scaled with a factor of two to match the total yield. The cyan (magenta) filled (hatched) histogram displays the simulated ${ B ^0 ( B ^0_ s ) \rightarrow \overline{ D }{} {}^0 K ^+ K ^ }$ decay signal candidates that are normalised to the number of ${ B ^0 \rightarrow \overline{ D }{} {}^0 \pi ^+ \pi ^ }$ normalisation channel candidates (blue triangles). The (magenta) vertical dashed line indicates the position of the nominal selection requirement. 
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Fit to the ${m_{\overline{ D }{} {}^0 \pi ^+ \pi ^ }}$ invariantmass distribution with the associated pull plot. 
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Fit to the ${m_{\overline{ D }{} {}^0 K ^+ K ^ }}$ invariantmass distribution with the associated pull plot. 
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Fit to the (left) ${m_{\overline{ D }{} {}^0 \pi ^+ \pi ^ }}$ invariant mass and (right) ${m_{\overline{ D }{} {}^0 K ^+ K ^ }}$ invariant mass, in logarithmic vertical scale (see the legend on Figs. 3 and 4). 
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Dalitz plot for ${ B ^0 \rightarrow \overline{ D }{} {}^0 K ^+ K ^ }$ candidates in the signal region ${m_{\overline{ D }{} {}^0 K ^+ K ^ }\in [5240,5320] {\mathrm{ Me V /}c^2} }$. 
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Dalitz plot for ${ B ^0_ s \rightarrow \overline{ D }{} {}^0 K ^+ K ^ }$ candidates in the signal region ${m_{\overline{ D }{} {}^0 K ^+ K ^ }\in [5327,5407] {\mathrm{ Me V /}c^2} }$. 
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Animated gif made out of all figures. 
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Relative yields, in percent, of the various exclusive $b$hadron decay backgrounds with respect to that of the ${ B ^0 \rightarrow \overline{ D }{} {}^0 \pi ^+ \pi}$ and ${ B ^0_{(s)}\rightarrow \overline{ D }{} {}^0 K ^+ K ^ }$ signal modes. These relative contributions are estimated with simulation in the range $m_{\overline{ D }{} {}^0 h^+ h^}\in [5115, 6000]$ $ {\mathrm{ Me V /}c^2}$ . 
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Fitted yields that are used as Gaussian constraints in the fit to the ${B^0_{(s)} \rightarrow \overline{ D }{} {}^0 h^+ h^}$ invariantmass distributions presented in Sect. 5.2. 
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Parameters from the default fit to ${ B ^0 \rightarrow \overline{ D }{} {}^0 \pi ^+ \pi ^ }$ and ${B^0_{(s)} \rightarrow \overline{ D }{} {}^0 K ^+ K ^ }$ data samples in the invariantmass range ${m_{\overline{ D }{} {}^0 h^+ h^}\in[5115, 6000]}$ $ {\mathrm{ Me V /}c^2}$ . The quantity $\chi^2/{\rm ndf}$ corresponds to the reduced $\chi^2$ of the fit for the corresponding number of degrees of freedom, ndf, while the $p$value is the probability value associated with the fit and is computed with the method of least squares [40]. 
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Total efficiencies ${\varepsilon_{B^0_{(s)} \rightarrow \overline{ D }{} {}^0 h^+h^}}$ and their contributions (before and after accounting for threebody decay kinematic properties) for the each three modes ${ B ^0 \rightarrow \overline{ D }{} {}^0 \pi ^+ \pi ^ }$, ${ B ^0 \rightarrow \overline{ D }{} {}^0 K ^+ K ^ }$, and ${ B ^0_ s \rightarrow \overline{ D }{} {}^0 K ^+ K ^ }$. Uncertainties are statistical only and those smaller than $0.1$ are displayed as $0.1$, but are accounted with their nominal values in the efficiency calculations. 
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Relative systematic uncertainties, in percent, on ${N_{ B ^0 \rightarrow \overline{ D }{} {}^0 \pi ^+ \pi ^ }}$, ${N_{ B ^0 \rightarrow \overline{ D }{} {}^0 K ^+ K ^ }}$ and the ratio ${N_{ B ^0 \rightarrow \overline{ D }{} {}^0 \pi ^+ \pi ^ }}$/${N_{ B ^0 \rightarrow \overline{ D }{} {}^0 K ^+ K ^ }}$ and ${r_{ B ^0_ s / B ^0 }}$, due to PDFs modelling in the $m_{\overline{ D }{} {}^0 \pi ^+ \pi ^ }$ and $m_{\overline{ D }{} {}^0 K ^+ K ^ }$ fits. The uncertainties are uncorrelated and summed in quadrature. 
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Relative systematic uncertainties, in percent, on the ratio of branching fractions $\mathcal{R}_{\overline{ D }{} {}^0 K ^+ K ^ /\overline{ D }{} {}^0 \pi ^+ \pi ^ }$ and $\mathcal{R}_{ B ^0_ s / B ^0 }$. The uncertainties are uncorrelated and summed in quadrature. 
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Created on 14 January 2021.