The resonant structure of the decay $\overline{B}_s^0\to J/\psi\pi^+\pi^$ is studied using data corresponding to 3 fb$^{1}$ of integrated luminosity from $pp$ collisions by the LHC and collected by the LHCb detector. Five interfering $\pi^+\pi^$ states are required to describe the decay: $f_0(980),f_0(1500),f_0(1790),f_2(1270)$, and $f_2^{\prime}(1525)$. An alternative model including these states and a nonresonant $J/\psi \pi^+\pi^$ component also provides a good description of the data. Based on the different transversity components measured for the spin2 intermediate states, the final state is found to be compatible with being entirely $CP$odd. The $CP$even part is found to be $<2.3$% at 95% confidence level. The $f_0(500)$ state is not observed, allowing a limit to be set on the absolute value of the mixing angle with the $f_0(980)$ of $<7.7^{\circ}$ at 90% confidence level, consistent with a tetraquark interpretation of the $f_0(980)$ substructure.
Leading order diagram for $\overline{ B }{} ^0_ s $ decays into $J/\psi \pi^+\pi^$. 
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Definition of helicity angles. For details see text. 
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Distributions of the BDT classifier for both training and test samples of $ { J \mskip 3mu/\mskip 2mu\psi \mskip 2mu} \pi^+\pi^$ signal and background events. The signal samples are from simulation and the background samples are from data. 
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Invariant mass of $J/\psi \pi^+\pi^$ combinations. The data have been fitted with double Crystal Ball signal and several background functions. The (red) solid curve shows the $\overline{ B }{} ^0_ s $ signal, the (brown) dotted line shows the combinatorial background, the (green) shortdashed line shows the $B^$ background, the (purple) dotdashed curve is $\overline{ B }{} ^0 \rightarrow J/\psi \pi^+\pi^$, the (light blue) longdashed line is the sum of $\overline{ B }{} ^0_ s \rightarrow J/\psi\eta'$, $\overline{ B }{} ^0_ s \rightarrow J/\psi\phi$ with $\phi\rightarrow \pi^+\pi^\pi^0$ backgrounds and the $\Lambda _b^0 \rightarrow { J \mskip 3mu/\mskip 2mu\psi \mskip 2mu} K^ p$ reflection, the (black) dotlong dashed curve is the $\overline{ B }{} ^0 \rightarrow J/\psi K^ \pi^+$ reflection and the (blue) solid curve is the total. 
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Distribution of $m^2(\pi^+\pi^)$ versus $m^2(J/\psi\pi^+)$ for all events within $\pm20$ MeV of the $\overline{ B }{} ^0_ s $ mass peak. 
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Distributions of (a) $m(J/\psi \pi^+)$ and (b) $m(\pi^+\pi^)$ for $\overline{ B }{} ^0_ s \rightarrow J/\psi \pi^+\pi^$ candidate decays within $\pm20$ MeV of the $\overline{ B }{} ^0_ s $ mass. The (red) points with error bars show the background contribution determined from $m(J/\psi \pi^+\pi^)$ fits performed in each bin of the plotted variables. 
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Distribution of the angle $\chi$ for the $J/\psi \pi^+\pi^$ simulation sample fitted with Eq. (10), used to determine the efficiency parameters. 
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Second order polynomial fit to the acceptance parameter $a(m^2_{hh})$ used in Eq. 11. 
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Projections of invariant mass squared of (a) $m^2(J/\psi \pi^+)$ and (b) $m^2(J/\psi \pi^)$ of the simulated Dalitz plot used to measure the efficiency parameters. The points represent the simulated event distributions and the curves the polynomial fit. 
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Parameterization of the detection efficiency as a function of $\cos\theta_{\pi^+\pi^}$ and $m(\pi^+\pi^)$. The scale is arbitrary. 
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Distribution of $\cos\theta_{J/\psi}$ of the other background and the fitted function $1+\alpha\cos^2\theta_{J/\psi}$. The points with error bars show the background obtained from candidate mass fits in bins of $\cos\theta_{J/\psi}$. 
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Projections of (a) $\cos\theta_{\pi\pi}$ and (b) $m(\pi^+\pi^)$ of the total background. The (blue) histogram or curve is projection of the fit, and the points with error bars show the likesign combinations added with additional $\rho^0$ background. 
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Distribution of $\chi$ of the total background and the fitted function. The points with error bars show the likesign combinations added with additional $\rho^0$ background. 
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Projections of (a) $m(\pi^+\pi^)$, (b) $\cos\theta_{\pi\pi}$, (c) $\cos \theta_{J/\psi}$ and (d) $\chi$ for 5R Solution I. The points with error bars are data, the signal fit is shown with a (red) dashed line, the background with a (black) dotted line, and the (blue) solid line represents the total. 
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Projections of (a) $m(\pi^+\pi^)$, (b) $\cos\theta_{\pi\pi}$, (c) $\cos \theta_{J/\psi}$ and (d) $\chi$ for 5R+NR Solution II. The points with error bars are data, the signal fit is shown with a (red) dashed line, the background with a (black) dotted line, and the (blue) solid line represents the total. 
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Distribution of $m(\pi^+\pi^)$ with contributing components labeled from 5R Solution I. 
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Distribution of $m(\pi^+\pi^)$ with contributing components labeled from 5R+NR Solution II. 
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Swave (a) amplitude strength and (b) phase as a function of $m(\pi^+\pi^)$ from the 5R Solution I (open) and 5R+NR Solution II (solid), where the widths of the curves reflect $\pm1\sigma$ statistical uncertainties. The reference point is chosen at 980 MeV with amplitude strength equal to 1 and phase equal to 0. 
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The $\pi^+\pi^$ mass dependence of the spherical harmonic moments of $\cos \theta_{\pi\pi}$ after efficiency corrections and background subtraction: (a) $\langle Y^0_0\rangle$ ($\chi^2$/ndf =78/70), (b) $\langle Y^0_1\rangle$ ($\chi^2$/ndf =37/70), (c) $\langle Y^0_2\rangle$ ($\chi^2$/ndf =79/70), (d) $\langle Y^0_3\rangle$ ($\chi^2$/ndf =42/70), (e) $\langle Y^0_4\rangle$ ($\chi^2$/ndf =43/70), (f) $\langle Y^0_5\rangle$ ($\chi^2$/ndf =35/70). The points with error bars are the data points and the solid curves are derived from the model 5R Solution I. 
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The $\pi^+\pi^$ mass dependence of the spherical harmonic moments of $\cos \theta_{\pi\pi}$ after efficiency corrections and background subtraction: (a) $\langle Y^0_0\rangle$ ($\chi^2$/ndf =73/70), (b) $\langle Y^0_1\rangle$ ($\chi^2$/ndf =36/70), (c) $\langle Y^0_2\rangle$ ($\chi^2$/ndf =72/70), (d) $\langle Y^0_3\rangle$ ($\chi^2$/ndf =43/70), (e) $\langle Y^0_4\rangle$ ($\chi^2$/ndf =41/70), (f) $\langle Y^0_5\rangle$ ($\chi^2$/ndf =34/70). The points with error bars are the data points and the solid curves are derived from the model 5R+NR Solution II. 
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Animated gif made out of all figures. 
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$ C P$ parity for different spin resonances. Note that spin0 only has the transversity component $0$. 
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Possible resonance candidates in the $\overline{ B }{} ^0_ s \rightarrow J/\psi \pi^+\pi^$ decay mode and their parameters used in the fit. 
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Fit $\rm ln\mathcal{L}$ and $\chi^2/\text{ndf}$ of different resonance models. 
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Fit fractions (%) of contributing components for both solutions. 
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Nonzero interference fraction (%) for both solutions. 
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Fitted resonance phase differences ($^\circ$). 
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Other fit parameters. The uncertainties are only statistical. 
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Absolute systematic uncertainties for Solution I. 
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Absolute systematic uncertainties for Solution II. 
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Fit fraction ranges taking $1\sigma$ regions for both solutions including systematic uncertainties. 
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Created on 19 October 2019.