The first observation of the rare decay$B^0_s \to \phi\pi^+\pi^$ and evidence for $B^0 \to \phi\pi^+\pi^$ are reported, using $pp$ collision data recorded by the LHCb detector at centreofmass energies $\sqrt{s} = 7$ and 8 TeV, corresponding to an integrated luminosity of $3{ fb^{1}}$. The branching fractions in the $\pi^+\pi^$ invariant mass range $400<m(\pi^+\pi^)<1600{\mathrm{ Me V /}c^2}$ are $[3.48\pm 0.23\pm 0.17\pm 0.35]\times 10^{6}$ and $[1.82\pm 0.25\pm 0.41\pm 0.14]\times 10^{7}$ for $B^0_s \to \phi\pi^+\pi^$ and $B^0 \to \phi\pi^+\pi^$ respectively, where the uncertainties are statistical, systematic and from the normalisation mode $B^0_s \to \phi\phi $. A combined analysis of the $\pi^+\pi^$ mass spectrum and the decay angles of the finalstate particles identifies the exclusive decays $B^0_s \to \phi f_0(980) $, $B_s^0 \to \phi f_2(1270) $, and $B^0_s \to \phi\rho^0$ with branching fractions of $[1.12\pm 0.16^{+0.09}_{0.08}\pm 0.11]\times 10^{6}$, $[0.61\pm 0.13^{+0.12}_{0.05}\pm 0.06]\times 10^{6}$ and $[2.7\pm 0.7\pm 0.2\pm 0.2]\times 10^{7}$, respectively.
Feynman diagrams for the exclusive decays (a) $B_s^0 \rightarrow \phi f _{0}(980) $ and (b) $B_s^0 \rightarrow \phi \rho^0$. 
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The $ K^+ K^ \pi^+ \pi^$ invariant mass distribution for candidates in the mass range $0.4 < m_{\pi\pi} < 1.6$ $ {\mathrm{ Ge V /}c^2}$ . The fit described in the text is overlaid. The solid (red) line is the total fitted function, the dotted (green) line the combinatorial background, the dashed (blue) line the $ B ^0_ s $ and the dotdashed (black) line the $ B ^0 $ signal component. 
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The $K^+ K^ K^+ K^$ invariant mass distribution after all selection criteria. The solid (red) line is the total fitted function including the $ B ^0_ s \rightarrow \phi \phi$ signal, and the dashed (green) line is the combinatorial background. 
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The $ K^+ K^ $ invariant mass distribution for backgroundsubtracted $ B ^0_ s \rightarrow \phi \pi ^+ \pi ^ $ signal events with a fit to the dominant Pwave $\phi$ meson shown as a solid (red) line, and a small Swave $ K^+ K^ $ contribution shown as a hatched (blue) area. 
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The definition of the decay angles $\theta_1$, $\theta_2$ and $\Phi$ for the decay $ B ^0_ s \rightarrow \phi \pi ^+ \pi ^ $ with $\phi\rightarrow K^+K^$ and taking $f_0(980)\rightarrow \pi^+\pi^$ for illustration. 
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Onedimensional projections of the detection efficiency parameterised using Legendre polynomials (solid red lines) as a function of (a) $\cos\theta_1$, (b) $\cos\theta_2$, (c) $\Phi$ and (d) $m(\pi^+ \pi^ )$, superimposed on the efficiency determined from the ratio of the accepted/generated $ B ^0_ s \rightarrow \phi \pi ^+ \pi ^ $ events. 
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Projections of (a) $\cos\theta_1$, (b) $\cos\theta_2$, (c) $\Phi$, and (d) $m(\pi^+ \pi^ )$ for the preferred fit. The $\rho$ contribution is shown by the dotted (black) line, the $f_0(980)$ by the dotdashed (blue) line, the $f_2(1270)$ by the doubledotdashed (magenta) line and the $f_0(1500)$ by the dashed (cyan) line. Note that the expected distributions from each resonance include the effect of the experimental efficiency. The solid (red) line shows the total fit. The points with error bars are the data, where the background has been subtracted using the $ B ^0_ s $ signal weights from the $ K^+ K^ \pi^+ \pi^ $ invariant mass fit. 
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Animated gif made out of all figures. 
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Possible resonances contributing to the $m(\pi^+ \pi^ )$ mass distribution. The shapes are either relativistic BreitWigner (BW) functions, or empirical threshold functions for the $f_0(500)$ proposed by Bugg [26] based on data from BES, and for the $f_0(980)$ proposed by Flatt\'e [27] to account for the effect of the $ K^+ K^ $ threshold. 
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The individual terms $i=1$ to $i=6$ come from the Swave and Pwave $\pi^+ \pi^ $ amplitudes associated with the $f_0(980)$ and $\rho$, and the terms $i=7$ to $i=12$ come from the Dwave amplitudes associated with the $f_2(1270)$. See the text for definitions of $T_i$, $f_i$ and $\mathcal{M}_i$, and for a discussion of the interference terms omitted from this table. 
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The resonance amplitudes and phases from the preferred fit to the $m(\pi^+ \pi^ )$ and decay angle distributions of the $ B ^0_ s $ candidates, including the $\rho$, $f_0(980)$, $f_2(1270)$ and $f_0(1500)$. See text for definitions of the amplitudes and phases. 
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Fit fractions in % and event yields for the resonances contributing to $ B ^0_ s \rightarrow \phi \pi ^+ \pi ^ $. Results are quoted for the preferred model with a $\rho$, and for an alternative model without a $\rho$ which is used to evaluate systematic uncertainties. 
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Selection efficiencies for the signal and normalisation modes in %, as determined from simulated event samples. Here "Initial selection" refers to a loose set of requirements on the four tracks forming the $ B $ candidate. The "Offline selection" includes the charm and $\phi K^*{^0}$ vetoes, as well as the BDT. Angular acceptance and decay time refer to corrections made for the incorrect modelling of these distributions in the inclusive and $ B ^0_ s \rightarrow \phi f _{0}(980) $ simulated event samples. 
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Systematic uncertainties in % on the branching fractions of $ B ^0_ s $ and $ B ^0$ decays. All the uncertainties are taken on the ratio of the signal to the normalisation mode. Uncertainties marked by a dash are either negligible or exactly zero. The asymmetric uncertainties on $\phi f_0(980)$ and $\phi f_2(1270)$ come from the differences in yields between the fits with and without the $\rho^0$ contribution. 
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Created on 06 March 2021.