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) $B_s^0 \rightarrow \phi \rho^0$. 
fig1a.pdf [15 KiB] HiDef png [61 KiB] Thumbnail [28 KiB] *.C file 

fig1b.pdf [12 KiB] HiDef png [41 KiB] Thumbnail [19 KiB] *.C file 

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. 
fig2.pdf [20 KiB] HiDef png [186 KiB] Thumbnail [133 KiB] *.C file 

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. 
fig3.pdf [19 KiB] HiDef png [182 KiB] Thumbnail [142 KiB] *.C file 

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. 
fig4.pdf [12 KiB] HiDef png [191 KiB] Thumbnail [151 KiB] *.C file 

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. 
fig5.pdf [27 KiB] HiDef png [252 KiB] Thumbnail [105 KiB] *.C file 

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. 
fig6a.pdf [15 KiB] HiDef png [186 KiB] Thumbnail [165 KiB] *.C file 

fig6b.pdf [15 KiB] HiDef png [164 KiB] Thumbnail [150 KiB] *.C file 

fig6c.pdf [14 KiB] HiDef png [134 KiB] Thumbnail [134 KiB] *.C file 

fig6d.pdf [17 KiB] HiDef png [202 KiB] Thumbnail [183 KiB] *.C file 

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. 
fig7a.pdf [16 KiB] HiDef png [214 KiB] Thumbnail [172 KiB] *.C file 

fig7b.pdf [15 KiB] HiDef png [233 KiB] Thumbnail [187 KiB] *.C file 

fig7c.pdf [10 KiB] HiDef png [178 KiB] Thumbnail [144 KiB] *.C file 

fig7d.pdf [20 KiB] HiDef png [316 KiB] Thumbnail [232 KiB] *.C file 

Animated gif made out of all figures. 
PAPER2016028.gif Thumbnail 
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. 
Table_1.pdf [33 KiB] HiDef png [106 KiB] Thumbnail [53 KiB] tex code 

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. 
Table_2.pdf [71 KiB] HiDef png [135 KiB] Thumbnail [62 KiB] tex code 

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. 
Table_3.pdf [52 KiB] HiDef png [105 KiB] Thumbnail [55 KiB] tex code 

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. 
Table_4.pdf [38 KiB] HiDef png [71 KiB] Thumbnail [39 KiB] tex code 

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. 
Table_5.pdf [54 KiB] HiDef png [76 KiB] Thumbnail [35 KiB] tex code 

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. 
Table_6.pdf [56 KiB] HiDef png [80 KiB] Thumbnail [37 KiB] tex code 
Created on 22 October 2019.