An angular analysis of the decay $B_s^0 \rightarrow K^{*0}\bar{K}{}^{*0}$ is performed using $pp$ collisions corresponding to an integrated luminosity of $1.0$ ${fb}^{1}$ collected by the LHCb experiment at a centreofmass energy $\sqrt{s} = 7$ TeV. A combined angular and mass analysis separates six helicity amplitudes and allows the measurement of the longitudinal polarisation fraction $f_L = 0.201 \pm 0.057 {(stat.)} \pm 0.040{(syst.)}$ for the $B_s^0 \rightarrow K^*(892)^0 \bar{K}{}^*(892)^0$ decay. A large scalar contribution from the $K^{*}_{0}(1430)$ and $K^{*}_{0}(800)$ resonances is found, allowing the determination of additional $CP$ asymmetries. Triple product and direct $CP$ asymmetries are determined to be compatible with the Standard Model expectations. The branching fraction $\mathcal{B}(B_s^0 \rightarrow K^*(892)^0 \bar{K}{}^*(892)^0)$ is measured to be $(10.8 \pm 2.1 {\ \rm (stat.)} \pm 1.4 {\ \rm (syst.)} \pm 0.6 \ (f_d/f_s) ) \times 10^{6}$.
Definition of the angles involved in the analysis of $ B ^0_ s \rightarrow K ^{*0} \overline{ K }{} {}^{*0} $ decays. 
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Invariant mass distribution for selected $K^+\pi^ K^ \pi^+$ candidates. The (blue) solid line is the result of the fit explained in the text. The $ B ^0_ s $ and $ B ^0$ signal peaks are shown as as dasheddotted lines (pink and dark green, respectively). The various peaking background components are represented as dotted lines: (red) $ B ^0 \rightarrow \phi K ^{*0} $ , (green) $\Lambda^0_b \rightarrow p\pi^ K^ \pi^+$ and (light blue) partially reconstructed decays. The (grey) dotted line is the combinatorial background component. The normalised residual (pull) is shown below. 
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Results of the simultaneous fit to $ B ^0_ s \rightarrow K^+\pi^K^\pi^+$ candidates (blue solid line) in the three helicity angles. The dots represent the data after background subtraction and acceptance correction. The red dashed line is the {Pwave} component, the green dashed line is the {Swave} component and the lightblue dashed line represents the $\mathcal{A}^+_s \mathcal{A}_0$ interference term. 
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Projections of the model fitted to $ B ^0_ s \rightarrow K^+\pi^K^\pi^+$ candidates (blue solid line) in the (top) invariant mass of $K\pi$ pairs and (bottom) $\cos\theta$ asymmetries as functions of $K\pi$ mass ($m_1$, $m_2$). The dots represent the data after background subtraction and acceptance correction. The red dashed line is the {Pwave} component, the green dashed line is the {Swave} component and the lightblue dashed line represents the ${A}^+_s {A}_0$ interference term. 
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(Left) Profile likelihood for the parameter $f_L$. (Right) Regions corresponding to $\Delta\ln\mathcal{L} =$ 0.5, 2 and 4.5 (39%, 87% and 99% confidence level) in the $A_s^^2$  $f_L$ plane. 
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Invariant mass of the selected $K^+K^K^{\pm}\pi^{\mp}$ combinations and result of the fit to the data. The points represent the data and the (blue) solid line is the fit model. The $ B ^0_ s $ and $ B ^0$ signal peaks are shown as as dasheddotted lines (pink and dark green, respectively). The contribution from partially reconstructed decays is represented as a (light blue) dashed line. The (grey) dotted line is the combinatorial background component. The normalised residual (pull) is shown below. 
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Animated gif made out of all figures. 
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Untagged timeintegrated terms used in the analysis, under the assumption of no $ C P$ violation. 
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Parameters of the mass propagators used in the fit. 
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Triple product and direct $ C P$ asymmetries measured in this analysis. The first uncertainties are statistical and the second systematic. 
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Results of the simultaneous fit to $ B ^0_ s \rightarrow K^+\pi^K^\pi^+$ TOS and nonTOS candidates with $M(K^+, \pi^, K^, \pi^+)  m_{ B ^0_ s } < 30 {\mathrm{ Me V /}c^2} $ (phases are measured in radians). The first uncertainties are statistical and the second systematic. 
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Systematic uncertainties in the measurement of the magnitude and phase of the different amplitudes contributing to the $ B ^0_ s \rightarrow K^+\pi^K^\pi^+$ decay. 
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Summary of relevant quantities in the $ {\cal B}( B ^0_ s \rightarrow K ^{*0} \overline{ K }{} {}^{*0} )$ calculation. The factor $\kappa( B ^0_ s \rightarrow K ^{*0} \overline{ K }{} {}^{*0} )$ is defined as $\lambda_{f_L}( B ^0_ s \rightarrow K ^{*0} \overline{ K }{} {}^{*0} )/f_{ B ^0_ s \rightarrow K ^{*0} \overline{ K }{} {}^{*0} }$, and equivalently for the $ B ^0 \rightarrow \phi K ^{*0} $ decay. The first uncertainty is statistical, the second systematic. 
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Created on 20 February 2021.