The charmless three-body decays $B^0_{(s)} \to K_{\mathrm{\scriptscriptstyle
S}}^0 h^+ h^{\prime -}$ (where $h^{(\prime)} = \pi, K$) are analysed using a
sample of $pp$ collision data recorded by the LHCb experiment, corresponding to
an integrated luminosity of $3 fb^{-1}$. The branching fractions are
measured relative to that of the $B^0 \to K_{\mathrm{\scriptscriptstyle S}}^0
\pi^{+} \pi^{-}$ decay, and are determined to be: \begin{eqnarray*}
\frac{{\cal B}(B^0\rightarrow K^0_{\rm S}K^{\pm}\pi^{\mp})}{{\cal
B}(B^0\rightarrow K^0_{\rm S}\pi^{+}\pi^{-})}
= {} & 0.123 \pm 0.009 \; \mathrm{ (stat)}\; \pm 0.015 \;
\mathrm{ (syst)} ,
\frac{{\cal B}(B^0\rightarrow K^0_{\rm S}K^{+}K^{-})} {{\cal
B}(B^0\rightarrow K^0_{\rm S}\pi^{+}\pi^{-})}
= {} & 0.549 \pm 0.018 \; \mathrm{ (stat)}\; \pm 0.033 \;
\mathrm{ (syst)} ,
\frac{{\cal B}(B_{s}^0\rightarrow K^0_{\rm S}\pi^{+}\pi^{-})}{{\cal
B}(B^0\rightarrow K^0_{\rm S}\pi^{+}\pi^{-})}
= {} & 0.191 \pm 0.027 \; \mathrm{ (stat)}\; \pm 0.031 \;
\mathrm{ (syst)}\; \pm 0.011 \; (f_s/f_d) ,
\frac{{\cal B}(B_{s}^0\rightarrow K^0_{\rm S}K^{\pm}\pi^{\mp})} {{\cal
B}(B^0\rightarrow K^0_{\rm S}\pi^{+}\pi^{-})}
= {} & 1.70\phantom{0} \pm 0.07\phantom{0} \; \mathrm{ (stat)} \; \pm
0.11\phantom{0} \; \mathrm{ (syst)}\; \pm 0.10\phantom{0} \; (f_s/f_d) ,
\frac{{\cal B}(B_{s}^0\rightarrow K^0_{\rm S}K^{+}K^{-})}{{\cal
B}(B^0\rightarrow K^0_{\rm S}\pi^{+}\pi^{-})} \in {} & [0.008 - 0.051] \rm
at 90\% confidence level,
\end{eqnarray*} where $f_s/f_d$ represents the ratio of hadronisation
fractions of the $B^0_s$ and $B^0$ mesons.
Invariant mass distributions of, from top to bottom, $ K ^0_{\mathrm{ \scriptscriptstyle S}} K ^+ K ^- $ , $ K ^0_{\mathrm{ \scriptscriptstyle S}} K ^\pm \pi ^\mp $
and $ K ^0_{\mathrm{ \scriptscriptstyle S}} \pi ^+ \pi ^- $ candidates, summing the three periods of
data taking, with the selection optimisation chosen
for the favoured decay modes for (left) downstream and
(right) long $ K ^0_{\mathrm{ \scriptscriptstyle S}}$ reconstruction categories.
In each plot, data are the black points with error bars and the total fit model is overlaid (solid blue line).
The $ B ^0$ ( $ B ^0_ s $ ) signal components are the pink (orange) short-dashed (dotted) lines,
while fully reconstructed misidentified decays are the green and dark blue dashed lines
close to the $ B ^0$ and $ B ^0_ s $ peaks.
The sum of the partially reconstructed contributions from $ B $ to open charm decays,
charmless hadronic decays, $ B ^0 \rightarrow \eta ^{\prime} K ^0_{\mathrm{ \scriptscriptstyle S}} $ and charmless radiative decays are
the red dash triple-dotted lines.
The combinatorial background contribution is the gray long-dash dotted
line.
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Invariant mass distributions of, from top to bottom, $ K ^0_{\mathrm{ \scriptscriptstyle S}} K ^+ K ^- $ , $ K ^0_{\mathrm{ \scriptscriptstyle S}} K ^\pm \pi ^\mp $
and $ K ^0_{\mathrm{ \scriptscriptstyle S}} \pi ^+ \pi ^- $ candidates, summing the three periods of
data taking, with the selection optimisation chosen
for the favoured decay modes for (left) downstream and
(right) long $ K ^0_{\mathrm{ \scriptscriptstyle S}}$ reconstruction categories.
In each plot, data are the black points with error bars and the total fit model is overlaid (solid blue line).
The $ B ^0$ ( $ B ^0_ s $ ) signal components are the pink (orange) short-dashed (dotted) lines,
while fully reconstructed misidentified decays are the green and dark blue dashed lines
close to the $ B ^0$ and $ B ^0_ s $ peaks.
The sum of the partially reconstructed contributions from $ B $ to open charm decays,
charmless hadronic decays, $ B ^0 \rightarrow \eta ^{\prime} K ^0_{\mathrm{ \scriptscriptstyle S}} $ and charmless radiative decays are
the red dash triple-dotted lines.
The combinatorial background contribution is the gray long-dash dotted
line.
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Results of the simultaneous fit to data (downstream, 2011) with the BDT optimisation corresponding to the favoured decay modes. The modes $ K ^0_{\mathrm{ \scriptscriptstyle S}} K ^+ K ^- $ , $ K ^0_{\mathrm{ \scriptscriptstyle S}} K ^+ \pi ^- $ , $ K ^0_{\mathrm{ \scriptscriptstyle S}} \pi ^+ K ^- $ and $ K ^0_{\mathrm{ \scriptscriptstyle S}} \pi ^+ \pi ^- $ are shown from top to bottom. The left-hand side plots show the results with a logarithmic scale and the right-hand side with a linear scale. Legend is similar to that of plots shown in Fig. ???.
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Results of the simultaneous fit to data (downstream, 2012a) with the BDT optimisation corresponding to the favoured decay modes. The modes $ K ^0_{\mathrm{ \scriptscriptstyle S}} K ^+ K ^- $ , $ K ^0_{\mathrm{ \scriptscriptstyle S}} K ^+ \pi ^- $ , $ K ^0_{\mathrm{ \scriptscriptstyle S}} \pi ^+ K ^- $ and $ K ^0_{\mathrm{ \scriptscriptstyle S}} \pi ^+ \pi ^- $ are shown from top to bottom. The left-hand side plots show the results with a logarithmic scale and the right-hand side with a linear scale. Legend is similar to that of plots shown in Fig. ???.
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Results of the simultaneous fit to data (downstream, 2012b) with the BDT optimisation corresponding to the favoured decay modes. The modes $ K ^0_{\mathrm{ \scriptscriptstyle S}} K ^+ K ^- $ , $ K ^0_{\mathrm{ \scriptscriptstyle S}} K ^+ \pi ^- $ , $ K ^0_{\mathrm{ \scriptscriptstyle S}} \pi ^+ K ^- $ and $ K ^0_{\mathrm{ \scriptscriptstyle S}} \pi ^+ \pi ^- $ are shown from top to bottom. The left-hand side plots show the results with a logarithmic scale and the right-hand side with a linear scale. Legend is similar to that of plots shown in Fig. ???.
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Results of the simultaneous fit to data (downstream, 2011) with the BDT optimisation corresponding to the favoured decay modes. The modes $ K ^0_{\mathrm{ \scriptscriptstyle S}} K ^+ K ^- $ , $ K ^0_{\mathrm{ \scriptscriptstyle S}} K ^+ \pi ^- $ , $ K ^0_{\mathrm{ \scriptscriptstyle S}} \pi ^+ K ^- $ and $ K ^0_{\mathrm{ \scriptscriptstyle S}} \pi ^+ \pi ^- $ are shown from top to bottom. The left-hand side plots show the results with a logarithmic scale and the right-hand side with a linear scale. Legend is similar to that of plots shown in Fig. ???.
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Results of the simultaneous fit to data (downstream, 2012a) with the BDT optimisation corresponding to the favoured decay modes. The modes $ K ^0_{\mathrm{ \scriptscriptstyle S}} K ^+ K ^- $ , $ K ^0_{\mathrm{ \scriptscriptstyle S}} K ^+ \pi ^- $ , $ K ^0_{\mathrm{ \scriptscriptstyle S}} \pi ^+ K ^- $ and $ K ^0_{\mathrm{ \scriptscriptstyle S}} \pi ^+ \pi ^- $ are shown from top to bottom. The left-hand side plots show the results with a logarithmic scale and the right-hand side with a linear scale. Legend is similar to that of plots shown in Fig. ???.
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Results of the simultaneous fit to data (downstream, 2012b) with the BDT optimisation corresponding to the favoured decay modes. The modes $ K ^0_{\mathrm{ \scriptscriptstyle S}} K ^+ K ^- $ , $ K ^0_{\mathrm{ \scriptscriptstyle S}} K ^+ \pi ^- $ , $ K ^0_{\mathrm{ \scriptscriptstyle S}} \pi ^+ K ^- $ and $ K ^0_{\mathrm{ \scriptscriptstyle S}} \pi ^+ \pi ^- $ are shown from top to bottom. The left-hand side plots show the results with a logarithmic scale and the right-hand side with a linear scale. Legend is similar to that of plots shown in Fig. ???.
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Results of the simultaneous fit to data (downstream, 2011) with the BDT optimisation corresponding to the favoured decay modes. The modes $ K ^0_{\mathrm{ \scriptscriptstyle S}} K ^+ K ^- $ , $ K ^0_{\mathrm{ \scriptscriptstyle S}} K ^+ \pi ^- $ , $ K ^0_{\mathrm{ \scriptscriptstyle S}} \pi ^+ K ^- $ and $ K ^0_{\mathrm{ \scriptscriptstyle S}} \pi ^+ \pi ^- $ are shown from top to bottom. The left-hand side plots show the results with a logarithmic scale and the right-hand side with a linear scale. Legend is similar to that of plots shown in Fig. ???.
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Results of the simultaneous fit to data (downstream, 2012a) with the BDT optimisation corresponding to the favoured decay modes. The modes $ K ^0_{\mathrm{ \scriptscriptstyle S}} K ^+ K ^- $ , $ K ^0_{\mathrm{ \scriptscriptstyle S}} K ^+ \pi ^- $ , $ K ^0_{\mathrm{ \scriptscriptstyle S}} \pi ^+ K ^- $ and $ K ^0_{\mathrm{ \scriptscriptstyle S}} \pi ^+ \pi ^- $ are shown from top to bottom. The left-hand side plots show the results with a logarithmic scale and the right-hand side with a linear scale. Legend is similar to that of plots shown in Fig. ???.
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Results of the simultaneous fit to data (downstream, 2012b) with the BDT optimisation corresponding to the favoured decay modes. The modes $ K ^0_{\mathrm{ \scriptscriptstyle S}} K ^+ K ^- $ , $ K ^0_{\mathrm{ \scriptscriptstyle S}} K ^+ \pi ^- $ , $ K ^0_{\mathrm{ \scriptscriptstyle S}} \pi ^+ K ^- $ and $ K ^0_{\mathrm{ \scriptscriptstyle S}} \pi ^+ \pi ^- $ are shown from top to bottom. The left-hand side plots show the results with a logarithmic scale and the right-hand side with a linear scale. Legend is similar to that of plots shown in Fig. ???.
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Results of the simultaneous fit to data (downstream, 2011) with the BDT optimisation corresponding to the favoured decay modes. The modes $ K ^0_{\mathrm{ \scriptscriptstyle S}} K ^+ K ^- $ , $ K ^0_{\mathrm{ \scriptscriptstyle S}} K ^+ \pi ^- $ , $ K ^0_{\mathrm{ \scriptscriptstyle S}} \pi ^+ K ^- $ and $ K ^0_{\mathrm{ \scriptscriptstyle S}} \pi ^+ \pi ^- $ are shown from top to bottom. The left-hand side plots show the results with a logarithmic scale and the right-hand side with a linear scale. Legend is similar to that of plots shown in Fig. ???.
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Results of the simultaneous fit to data (downstream, 2012a) with the BDT optimisation corresponding to the favoured decay modes. The modes $ K ^0_{\mathrm{ \scriptscriptstyle S}} K ^+ K ^- $ , $ K ^0_{\mathrm{ \scriptscriptstyle S}} K ^+ \pi ^- $ , $ K ^0_{\mathrm{ \scriptscriptstyle S}} \pi ^+ K ^- $ and $ K ^0_{\mathrm{ \scriptscriptstyle S}} \pi ^+ \pi ^- $ are shown from top to bottom. The left-hand side plots show the results with a logarithmic scale and the right-hand side with a linear scale. Legend is similar to that of plots shown in Fig. ???.
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Results of the simultaneous fit to data (downstream, 2012b) with the BDT optimisation corresponding to the favoured decay modes. The modes $ K ^0_{\mathrm{ \scriptscriptstyle S}} K ^+ K ^- $ , $ K ^0_{\mathrm{ \scriptscriptstyle S}} K ^+ \pi ^- $ , $ K ^0_{\mathrm{ \scriptscriptstyle S}} \pi ^+ K ^- $ and $ K ^0_{\mathrm{ \scriptscriptstyle S}} \pi ^+ \pi ^- $ are shown from top to bottom. The left-hand side plots show the results with a logarithmic scale and the right-hand side with a linear scale. Legend is similar to that of plots shown in Fig. ???.
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Distribution of sPlot weights in data.
The $ K ^0_{\mathrm{ \scriptscriptstyle S}} K ^+ K ^- $ , $ K ^0_{\mathrm{ \scriptscriptstyle S}} K ^+ \pi ^- $ , $ K ^0_{\mathrm{ \scriptscriptstyle S}} \pi ^+ K ^- $ , and $ K ^0_{\mathrm{ \scriptscriptstyle S}} \pi ^+ \pi ^- $ final states are shown from top to bottom, with a $ B ^0$ parent on the left, and a $ B ^0_ s $ on the right.
All data-taking periods and $ K ^0_{\mathrm{ \scriptscriptstyle S}}$ reconstruction categories are added.
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