The timeintegrated Dalitz plot of the threebody hadronic charmless decay ${{\overline{B}}^0 \to K_{\mathrm{\scriptscriptstyle S}}^0 \pi^+ \pi^}$ is studied using a $pp$ collision data sample recorded with the LHCb detector, corresponding to an integrated luminosity of $3.0\;\mathrm{fb}^{1}$. The decay amplitude is described with an isobar model. Relative contributions of the isobar amplitudes to the ${\overline{B}^0 \to K_{\mathrm{\scriptscriptstyle S}}^0 \pi^+ \pi^}$ decay branching fraction and CP asymmetries of the flavourspecific amplitudes are measured. The CP asymmetry between the conjugate ${\overline{B}^0 \to K^{*}(892)^{}\pi^+}$ and ${\overline{B}^0 \to K^{*}(892)^{+}\pi^}$ decay rates is determined to be $0.308 \pm 0.062$.
Invariant mass distributions of $ K ^0_{\mathrm{ \scriptscriptstyle S}} \pi ^+ \pi ^ $ candidates, summing the two years of data taking and the two $ K ^0_{\mathrm{ \scriptscriptstyle S}}$ reconstruction categories. The sum of the partially reconstructed contributions from $ B $ to open charm decays, charmless hadronic decays, $\overline{ B }{} {}^0 \rightarrow \eta ^{\prime} K ^0_{\mathrm{ \scriptscriptstyle S}} $ and charmless radiative decays are denoted $\overline{ B }{} ^0_{(s)} \rightarrow K ^0_{\mathrm{ \scriptscriptstyle S}} \pi ^+ \pi ^ (X)$. 
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Projections of the sum of all data categories (black points) and the nominal fit function onto the DP variables (left) $m_{K_{\rm S}^0\pi^+}^2$, (right) $m_{K_{\rm S}^0\pi^}^2$ and (bottom) $m_{\pi^+\pi^}^2$, restricted to the twobody low invariantmass regions. The full fit is shown by the solid blue line and the signal model by the dashed red line. The observed difference is due to the (green) combinatorial and (light red) crossfeed background contributions, barely visible in these projections. 
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Animated gif made out of all figures. 
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Components of the DP model used in the fit. The individual amplitudes are referred to by the resonance they contain. The parameter values are given in $ {\mathrm{ Me V /}c^2}$ for the masses and $\mathrm{ Me V}$ for the widths, except for $f_0(980)$ resonance. The parameter $m_0$ is the pole mass of the resonance and $\Gamma_0$ its natural width. The massdependent lineshapes employed to model the resonances are indicated in the third column. Relativistic BreitWigner and GounarisSakurai lineshapes are denoted RBW and GS, respectively. EFKLLM is a parameterisation of the $ K ^0_{\mathrm{ \scriptscriptstyle S}} \pi^{}$ Swave lineshape, $(K\pi)^{}_0$. 
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Created on 06 March 2021.