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. 
Fig1.pdf [9 KiB] HiDef png [198 KiB] Thumbnail [39 KiB] *.C file 

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. 
Fig2.pdf [27 KiB] HiDef png [132 KiB] Thumbnail [30 KiB] *.C file 

Fourbody invariant mass distribution on a (left) linear and (right) logarithmic scale superimposed with the mass fit model. 
Fig3a.pdf [30 KiB] HiDef png [315 KiB] Thumbnail [104 KiB] *.C file 

Fig3b.pdf [28 KiB] HiDef png [308 KiB] Thumbnail [86 KiB] *.C file 

Distribution of the two $(K\pi)$ pair invariant masses, with the signal weights applied, after all of the selection requirements. 
Fig4.pdf [34 KiB] HiDef png [522 KiB] Thumbnail [195 KiB] *.C file 

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. 
Fig5a.pdf [16 KiB] HiDef png [154 KiB] Thumbnail [72 KiB] *.C file 

Fig5b.pdf [15 KiB] HiDef png [111 KiB] Thumbnail [51 KiB] *.C file 

Fig5c.pdf [15 KiB] HiDef png [118 KiB] Thumbnail [59 KiB] *.C file 

Fig5d.pdf [16 KiB] HiDef png [154 KiB] Thumbnail [59 KiB] *.C file 

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. 
Fig6.pdf [15 KiB] HiDef png [112 KiB] Thumbnail [43 KiB] *.C file 

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. 
Fig7a.pdf [16 KiB] HiDef png [211 KiB] Thumbnail [78 KiB] *.C file 

Fig7b.pdf [16 KiB] HiDef png [186 KiB] Thumbnail [69 KiB] *.C file 

Fig7c.pdf [15 KiB] HiDef png [160 KiB] Thumbnail [65 KiB] *.C file 

Fig7d.pdf [15 KiB] HiDef png [160 KiB] Thumbnail [66 KiB] *.C file 

Fig7e.pdf [15 KiB] HiDef png [146 KiB] Thumbnail [58 KiB] *.C file 

Fig7f.pdf [15 KiB] HiDef png [138 KiB] Thumbnail [55 KiB] *.C file 

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. 
Fig8a.pdf [26 KiB] HiDef png [155 KiB] Thumbnail [57 KiB] *.C file 

Fig8b.pdf [26 KiB] HiDef png [136 KiB] Thumbnail [49 KiB] *.C file 

Animated gif made out of all figures. 
PAPER2017048.gif Thumbnail 
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 ???. 
Table_1.pdf [54 KiB] HiDef png [86 KiB] Thumbnail [13 KiB] tex code 

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}. 
Table_2.pdf [54 KiB] HiDef png [124 KiB] Thumbnail [21 KiB] tex code 

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. 
Table_3.pdf [37 KiB] HiDef png [54 KiB] Thumbnail [10 KiB] tex code 

Results of the decaytimedependent amplitude fit to data. The first uncertainty is statistical and the second uncertainty is systematic. 
Table_4.pdf [63 KiB] HiDef png [281 KiB] Thumbnail [51 KiB] tex code 

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}. 
Table_5.pdf [63 KiB] HiDef png [240 KiB] Thumbnail [42 KiB] tex code 

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 $50\%$. 
Table_6.pdf [34 KiB] HiDef png [97 KiB] Thumbnail [15 KiB] tex code 

Coefficients used in the modelindependent parameterisation of the scalar $K\pi$ massdependent amplitude. 
Table_7.pdf [36 KiB] HiDef png [62 KiB] Thumbnail [11 KiB] tex code 
Supplementary material full pdf 
supple[..].pdf [141 KiB] 

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. 
Fig1S.pdf [52 KiB] HiDef png [2 MiB] Thumbnail [268 KiB] *C file 

Fig2S.pdf [52 KiB] HiDef png [2 MiB] Thumbnail [270 KiB] *C file 
Created on 20 April 2019.Citation count from INSPIRE on 25 April 2019.