A flavourtagged decaytimedependent amplitude analysis of $B_s^0\to(K^+\pi^)(K^\pi^+)$ decays is presented in the $K^{\pm}\pi^{\mp}$ mass range from 750 to 1600 MeV$/c^2$. The analysis uses $pp$ collision data collected with the LHCb detector at centreofmass energies of $7$ and $8$ TeV, corresponding to an integrated luminosity of $3.0$ fb$^{1}$. Several quasitwobody decay modes are considered, corresponding to $K^{\pm}\pi^{\mp}$ combinations with spin 0, 1 and 2, which are dominated by the $K_0^*(800)^0$ and $K_0^*(1430)^0$, the $K^*(892)^0$ and the $K_2^*(1430)^0$ resonances, respectively. The longitudinal polarisation fraction for the $B_s^0\to K^*(892)^0\overline{K}^*(892)^0$ decay is measured as $f_L=0.208 \pm 0.032 \pm 0.046$, where the first uncertainty is statistical and the second is systematic. The first measurement of the mixinginduced $CP$violating phase, $\phi_s^{d\overline{d}}$, in $b\to d\overline{d}s$ transitions is performed, yielding a value of $\phi_s^{d\overline{d}}=0.10$ $\pm$ $0.13$ (stat) $\pm$ $0.14$ (syst) rad.
Leadingorder SM Feynman diagram of the $ { B ^0_ s \rightarrow K ^{*0} \overline{ K }{} {}^{*0} }$ decay. 
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Graphical definition of the angles in the helicity basis. Taking the example of a $ B ^0_ s \rightarrow Q_1 Q_2$ decay (this analysis uses $ B ^0_ s \rightarrow SS$, $ B ^0_ s \rightarrow SV$, $ B ^0_ s \rightarrow VS$, $ B ^0_ s \rightarrow VV$, $ B ^0_ s \rightarrow ST$, $ B ^0_ s \rightarrow TS$, $ B ^0_ s \rightarrow VT$, $ B ^0_ s \rightarrow TV$ and $ B ^0_ s \rightarrow TT$), with each finalstate quasitwobody meson decaying to pseudoscalars ($Q_1 \rightarrow K^+\pi^$ and $Q_2 \rightarrow K^\pi^+$), $\theta_1$ ($\theta_2$) is defined as the angle between the directions of motion of $K^+$ ($K^$) in the $Q_1$ ($Q_2$) rest frame and $Q_1$ ($Q_2$) in the $ B ^0_ s $ rest frame, and $\varphi$ as the angle between the plane defined by $K^+\pi^$ and the plane defined by $K^\pi^+$ in the $ B ^0_ s $ rest frame. 
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Fourbody invariant mass distribution on a (left) linear and (right) logarithmic scale superimposed with the mass fit model. 
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Distribution of the two $(K\pi)$ pair invariant masses, with the signal weights applied, after all of the selection requirements. 
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Kinematic acceptance and decaytime distributions evaluated with simulated vectorvector $ { B ^0_ s \rightarrow K ^{*0} \overline{ K }{} {}^{*0} }$ and pure phasespace $ { B ^0_ s \rightarrow ( K ^+ \pi ^ )( K ^ \pi ^+ )}$ candidates scaled by the mean acceptance. In the bottom right plot the decaytime acceptance obtained from the simulated sample is shown as the black points and the parametric form of the acceptance obtained with cubic splines is shown as the red curve. In the other three plots the black points show the acceptance distribution for the masses and angles. The two $\cos\theta$ variables and the two $m(K\pi)$ masses have been averaged for the purpose of illustration. In the fit, the kinematic acceptance enters via the normalisation weights. 
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Perevent decaytime resolution, $\sigma_t$, versus the estimated perevent decaytime uncertainty, $\delta_t$, obtained from simulated samples containing both vectorvector resonant $ { B ^0_ s \rightarrow K ^{*0} \overline{ K }{} {}^{*0} }$ and phasespace $ { B ^0_ s \rightarrow K ^+ \pi ^ K ^ \pi ^+ }$ events. 
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Onedimensional projections of the decaytimedependent, flavourtagged fit to (black points) the sPlot weighted data for (top row) the two $(K\pi)$ invariant masses, (middle row) the two $(K\pi)$ decay plane angles, (bottom left) the angle between the two $(K,\pi)$ decay planes and (bottom right) the decaytime. The solid gray line represents the total fit model along with the $ C P$ averaged components for each contributing decay. 
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Lineshapes of the (left) modulus and (right) phase of the scalar $K\pi$ massdependent amplitude. The nominal model is shown with a solid blue line and the modelindependent parameterisation, used in systematic studies, is shown with a dashed red line. 
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Animated gif made out of all figures. 
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Quasitwobody decay channels and corresponding polarisation amplitudes contributing to the $ { B ^0_ s \rightarrow ( K ^+ \pi ^ )( K ^ \pi ^+ )}$ final state in the $K\pi$ mass window from 750 to 1600 $ {\mathrm{ Me V /}c^2}$ . The different contributions are identified by the spin $j_1$ ($j_2$) of the $K^+\pi^$ ($K^\pi^+$) pair and the helicity $h$. In cases where more than one amplitude contributes, the polarisations are defined as being longitudinal, parallel, or perpendicular, which are then denoted by $0$, $\parallel$ and $\perp$ respectively, following the definitions given in Ref. \cite{PDG2016}. The subscripts 1 and 2 in the parallel and perpendicular helicities of the tensortensor component denote different spin states leading to a parallel or a perpendicular configuration, as discussed in Appendix ???. 
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Yields of the signal decay and the various background components considered in the fourbody invariant mass fit. The uncertainties are statistical only. The signal region is defined as $\pm 60$ $ {\mathrm{ Me V /}c^2}$ from the known $ B ^0_ s $ meson mass \cite{PDG2016}. 
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The flavourtagging performance of the SS and OS tagging algorithms, as well as the combination of both, for the signal data sample used in the analysis. The quoted uncertainty includes both statistical and systematic contributions. 
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Summary of the systematic uncertainties on the two $ C P$ parameters, the $ C P$ averaged fractions and the strong phase differences (in radians) for each of the components listed in Table ???. 
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Results of the decaytimedependent amplitude fit to data. The first uncertainty is statistical and the second uncertainty is systematic. 
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Functions containing the angular dependence of the amplitudes, as introduced in Eq. (???). For a discussion on some of the angular terms see Ref. \cite{Bhattacharya:2013sga}. 
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Parameters used in the nominal model for the scalar $K\pi$ massdependent amplitude. The correlations among them are found to be small, the largest ones been of the order of $50\%$. 
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Coefficients used in the modelindependent parameterisation of the scalar $K\pi$ massdependent amplitude. 
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Supplementary material full pdf 
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This ZIP file contains supplementary material for the publication LHCbPAPER2017048. The files are: supplementary.pdf : A description of the extra material, consisting on correlation matrices. *.pdf, *.eps, *.C : The correlation matrices in various formats. 
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Created on 20 February 2021.