The decay $\overline{B}_s^0 \rightarrow \psi(2S) K^+ \pi^$ is observed using a data set corresponding to an integrated luminosity of $3.0fb^{1}$ collected by the LHCb experiment in $pp$ collisions at centreofmass energies of 7 and 8 TeV. The branching fraction relative to the $B^0\rightarrow \psi(2S) K^+ \pi^$ decay mode is measured to be \begin{equation} \frac{{\cal B}(\overline{B}^0_s \rightarrow \psi(2S) K^+ \pi^)}{{\cal B}(B^0 \rightarrow \psi(2S) K^+ \pi^)} = 5.38 \pm 0.36 (stat) \pm 0.22 (syst) \pm 0.31 \, (f_s/f_d) \, \%,\nonumber \end{equation} where $f_s/f_d$ indicates the uncertainty due to the ratio of probabilities for a $b$ quark to hadronise into a $B_s^0$ or $B^0$ meson. Using an amplitude analysis, the fraction of decays proceeding via an intermediate $K^*(892)^0$ meson is measured to be $0.645 \pm 0.049 (stat) \pm 0.049 (syst)$ and its longitudinal polarisation fraction is $0.524 \pm 0.056 (stat) \pm 0.029 (syst)$. The relative branching fraction for this component is determined to be \begin{equation} \frac{{\cal B}(\overline{B}^0_s \rightarrow \psi(2S) K^*(892)^0)}{{\cal B}(B^0 \rightarrow \psi(2S) K^*(892)^0)} = 5.58 \pm 0.57 (stat) \pm 0.40 (syst) \pm 0.32 \, (f_s/f_d) \, \%. \nonumber \end{equation} In addition, the mass splitting between the $B_s^0$ and $B^0$ mesons is measured as \begin{equation} M(B^0_s)  M(B^0) = 87.45 \pm 0.44 (stat) \pm 0.07 (syst) MeV/c^2. \nonumber \end{equation}
Tree (left) and penguin (right) topologies contributing to the $ B ^0_{( s )} \rightarrow \psi V$ decays where $\psi = J/\psi, \psi(2S)$ and $V = \phi, K^*(892)^0$. 
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Invariant mass distribution for selected $\psi(2S)K^+ \pi^$ candidates in the data. A fit to the model described in the text is superimposed. The full fit model is shown by the solid (red) line, the combinatorial background by the solid (yellow) and the sum of background from the exclusive $b \rightarrow \psi(2S)X$ modes considered in the text by the shaded (blue) area. The maximum of the $y$scale is restricted so as to be able to see more clearly the $\overline{ B }{} {}^0_ s \rightarrow \psi(2S)K^+ \pi^$ signal. The lower plot shows the differences between the fit and measured values divided by the corresponding uncertainty of the measured value, the socalled pull distribution. 
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Dalitz plot for the selected $\overline{ B }{} {}^0_ s \rightarrow \psi(2S) K ^+ \pi^$ candidates in the signal window $m(\psi(2S) K ^+ \pi^) \in [5350, 5380] {\mathrm{\,Me V\!/}c^2} $. 
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Definition of the helicity angles. 
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Distributions of (a) $\cos\theta_\mu$, (b) $\phi$, (c) $\cos\theta_K$ and (d) $m( K ^+ \pi^)$ of simulated $B^0_s\rightarrow \psi(2S) K ^+ \pi^$ decays in a phase space configuration (black points) with the parameterisation of the efficiency overlaid (blue lines). 
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Distributions of (a) $\cos\theta_\mu$, (b) $\phi$, (c) $\cos\theta_K$ and (d) $m( K ^+ \pi^)$ of $B^0_{(s)}\rightarrow \psi(2S) K ^+ \pi^$ candidates with $m(\psi(2S) K ^+ \pi^) > 5390 {\mathrm{\,Me V\!/}c^2} $ (black points), with the parameterisation of the background distribution overlaid (blue lines). 
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Distributions of (a) $\cos\theta_\mu$, (b) $\phi$, (c) $\cos\theta_K$ and (d) $m( K ^+ \pi^)$ for selected $\overline{ B }{} {}^0_ s \rightarrow \psi(2S) K ^+ \pi^$ candidates (black points) with the projections of the fitted amplitude model overlaid. The following components are included in the model: $K^*(892)^0$ (red dashed), LASS Swave (green dotted), and background (grey dasheddotted). The residual pulls are shown below each distribution. 
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Animated gif made out of all figures. 
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Summary of systematic uncertainties on the $K^*(892)^0$ fit fraction and $f_{\rm L}$. Rows marked with (*) refer to uncorrelated sources of uncertainty between the $ B ^0_ s $ and $ B ^0 $ modes for the computation of the ratio of branching fractions. 
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Systematic uncertainties on the ratio of branching fractions. 
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Supplementary material full pdf 
supple[..].pdf [123 KiB] 

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Created on 20 April 2019.Citation count from INSPIRE on 24 April 2019.