Leading-order SM Feynman diagram of the $ { B ^0_ s \rightarrow K ^{*0} \overline{ K }{} {}^{*0} }$ decay.

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$), with each final-state quasi-two-body meson decaying to pseudoscalars ($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.

Four-body invariant mass distribution on a (left) linear and (right) logarithmic scale superimposed with the mass fit model.

Distribution of the two $(K\pi)$ pair invariant masses, with the signal weights applied, after all of the selection requirements.

Kinematic acceptance and decay-time distributions evaluated with simulated vector-vector $ { B ^0_ s \rightarrow K ^{*0} \overline{ K }{} {}^{*0} }$ and pure phase-space $ { B ^0_ s \rightarrow ( K ^+ \pi ^- )( K ^- \pi ^+ )}$ candidates scaled by the mean acceptance. In the bottom right plot the decay-time 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.

Per-event decay-time resolution, $\sigma_t$, versus the estimated per-event decay-time uncertainty, $\delta_t$, obtained from simulated samples containing both vector-vector resonant $ { B ^0_ s \rightarrow K ^{*0} \overline{ K }{} {}^{*0} }$ and phase-space $ { B ^0_ s \rightarrow K ^+ \pi ^- K ^- \pi ^+ }$ events.

One-dimensional projections of the decay-time-dependent, flavour-tagged 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 decay-time. The solid gray line represents the total fit model along with the $ C P$ -averaged components for each contributing decay.

Line-shapes of the (left) modulus and (right) phase of the scalar $K\pi$ mass-dependent amplitude. The nominal model is shown with a solid blue line and the model-independent parameterisation, used in systematic studies, is shown with a dashed red line.

Animated gif made out of all figures.

Quasi-two-body 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 tensor-tensor component denote different spin states leading to a parallel or a perpendicular configuration, as discussed in Appendix ???.

Yields of the signal decay and the various background components considered in the four-body 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}.

The flavour-tagging 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.

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 ???.

Results of the decay-time-dependent amplitude fit to data. The first uncertainty is statistical and the second uncertainty is systematic.

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}.

Parameters used in the nominal model for the scalar $K\pi$ mass-dependent amplitude. The correlations among them are found to be small, the largest ones been of the order of $50\%$.

Coefficients used in the model-independent parameterisation of the scalar $K\pi$ mass-dependent amplitude.

Supplementary material full pdf

This ZIP file contains supplementary material for the publication LHCb-PAPER-2017-048. The files are: supplementary.pdf : A description of the extra material, consisting on correlation matrices. *.pdf, *.eps, *.C : The correlation matrices in various formats.