The resonant structure of the reaction $\overline{B}^0\rightarrow J/\psi \pi^+\pi^$ is studied using data from 3 fb$^{1}$ of integrated luminosity collected by the LHCb experiment, onethird at 7 Tev centerofmass energy and the remainder at 8 Tev. The invariant mass of the $\pi^+\pi^$ pair and three decay angular distributions are used to determine the fractions of the resonant and nonresonant components. Six interfering $\pi^+\pi^$ states: $\rho(770)$, $f_0(500)$, $f_2(1270)$, $\rho(1450)$, $\omega(782)$ and $\rho(1700)$ are required to give a good description of invariant mass spectra and decay angular distributions. The positive and negative CP fractions of each of the resonant final states are determined. The $f_0(980)$ meson is not seen and the upper limit on its presence, compared with the observed $f_0(500)$ rate, is inconsistent with a model of tetraquark substructure for these scalar mesons at the eight standard deviation level. In the $q\overline{q}$ model, the absolute value of the mixing angle between the $f_0(980)$ and the $f_0(500)$ scalar mesons is limited to be less than $17^{\circ}$ at 90% confidence level.
(a) Tree level and (b) penguin diagram for $\overline{ B }{} ^0 $ decays into $J/\psi \pi^+\pi^$. 
feyn3.pdf [58 KiB] HiDef png [96 KiB] Thumbnail [21 KiB] *.C file 

Illustration of the three angles used in this analysis. 
4variables.pdf [50 KiB] HiDef png [303 KiB] Thumbnail [30 KiB] *.C file 

Invariant mass of $J/\psi \pi^+\pi^$ combinations together with the data fit. The (red) solid curve shows the $\overline{ B }{} ^0 $ signal, the (brown) dotted line shows the combinatorial background, the (green) shortdashed line shows the $B^$ background, the (purple) dotdashed curve is $\overline{ B }{} ^0_ s \rightarrow J/\psi \pi^+\pi^$, 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. The points at the bottom show the difference between the data points and the total fit divided by the statistical uncertainty on the data. 
fitmpull.pdf [48 KiB] HiDef png [590 KiB] Thumbnail [111 KiB] *.C file 

Distribution of $m^2(\pi ^+ \pi ^ )$ versus $m^2( { J \mskip 3mu/\mskip 2mu\psi \mskip 2mu} \pi ^+ )$ for all events within $\pm20$ MeV of the $\overline{ B }{} ^0 $ mass. 
Dalitz[..].pdf [964 KiB] HiDef png [1 MiB] Thumbnail [208 KiB] *.C file 

Distributions of (a) $m( { J \mskip 3mu/\mskip 2mu\psi \mskip 2mu} \pi ^+ )$ and (b) $m(\pi ^+ \pi ^ )$ for $\overline{ B }{} ^0 \rightarrow J/\psi\pi^+\pi^$ candidates within $\pm20$ MeV of the $\overline{ B }{} ^0 $ mass. The red points with error bars show the background contributions obtained by fitting the $m( { J \mskip 3mu/\mskip 2mu\psi \mskip 2mu} \pi ^+ \pi ^ )$ distribution in bins of the plotted variables. 
overla[..].pdf [41 KiB] HiDef png [294 KiB] Thumbnail [55 KiB] *.C file 

Projections of invariant mass squared (a) $s_{12}\equiv m^2(J/\psi \pi^+)$ and (b) $s_{23}\equiv m^2(\pi^+\pi^)$ of the simulated Dalitz plot used to measure the efficiency parameters. The points represent the simulated event distributions and the curves the polynomial fit. 
Fig6_a.pdf [38 KiB] HiDef png [380 KiB] Thumbnail [71 KiB] *.C file 

Fig6_b.pdf [38 KiB] HiDef png [208 KiB] Thumbnail [50 KiB] *.C file 

The variation of $\varepsilon_1$ is shown as a function of $m(\pi^+\pi^)$ and $\cos \theta_{\pi^+\pi^}$. 
eff_vs[..].pdf [186 KiB] HiDef png [1 MiB] Thumbnail [177 KiB] *.C file 

Distributions of $m(\pi^+\pi^)$ of background components. The (blue) histogram shows the likesign combinations added with additional backgrounds using simulations. The (black) points with error bars show the background obtained from the fits to the $m( { J \mskip 3mu/\mskip 2mu\psi \mskip 2mu} \pi ^+ \pi ^ )$ mass spectrum in each bin of $\pi^+\pi^$ mass. 
bkgfi[..].pdf [42 KiB] HiDef png [255 KiB] Thumbnail [67 KiB] *.C file 

Projections of (a) $\cos \theta_{\pi\pi}$ and (b) $m(\pi ^+ \pi ^ )$ of the background. The points with error bars show the likesign data combinations added with the $\Lambda ^0_ b $ background and additional simulated backgrounds. The (magenta) dotdashed line shows the $\eta^{(')}\rightarrow \rho\gamma$ background, the (darkblue) dashed line the $\overline{ K }{} ^* $ reflection background, and the (blue) solid line the total. The points at the bottom show the difference between the data points and the total fit divided by the statistical uncertainty on the data. 
cospip[..].pdf [41 KiB] HiDef png [386 KiB] Thumbnail [89 KiB] *.C file 

mpp_bk[..].pdf [42 KiB] HiDef png [399 KiB] Thumbnail [84 KiB] *.C file 

The $\cos\theta_{J/\psi}$ distribution of the datasimulated hybrid background sample. The points with error bars show the likesign data combinations added with the $\Lambda ^0_ b $ background and additional simulated backgrounds. The (magenta) dotdashed line shows the $\rho$ background, the (darkblue) dashed the $\overline{ K }{} ^* $ reflection background and the (blue) solid line the total. 
cosjps[..].pdf [35 KiB] HiDef png [303 KiB] Thumbnail [63 KiB] *.C file 

The $\chi$ distribution of the datasimulated hybrid background sample including the $\Lambda ^0_ b $ background and the fitted function $1+p_{b1}\cos \chi + p_{b2} \cos2 \chi$. The $p$value of this fit is $40\%$. 
chi_bkg.pdf [32 KiB] HiDef png [237 KiB] Thumbnail [50 KiB] *.C file 

Dalitz fit projections of (a) $m(\pi^+\pi^)$, (b) $\cos(\theta_{\pi ^+ \pi ^ })$, (c) $\cos \theta_{J/\psi}$ and (d) $\chi$ for the 5R Model + $\rho(1700)$ (Best Model). The points with error bars are data compared with the overall fit, shown by the (blue) solid line. The individual fit components are signal, shown with a (red) dashed line, background, shown with a (black) dotted line, and $ K ^0_{\rm\scriptscriptstyle S} $, shown with a (green) dashed line. 
Fig12a.pdf [45 KiB] HiDef png [518 KiB] Thumbnail [90 KiB] *.C file 

Fig12b.pdf [42 KiB] HiDef png [421 KiB] Thumbnail [93 KiB] *.C file 

Fig12c.pdf [41 KiB] HiDef png [413 KiB] Thumbnail [91 KiB] *.C file 

Fig12d.pdf [38 KiB] HiDef png [339 KiB] Thumbnail [70 KiB] *.C file 

Fit projection of $m(\pi^+\pi^$) showing the different resonant contributions in the Best Model. 
B0mpp.pdf [49 KiB] HiDef png [636 KiB] Thumbnail [102 KiB] *.C file 

The $\pi^+\pi^$ mass dependence of the spherical harmonic moments of $\cos \theta_{\pi\pi}$ after efficiency corrections and background subtraction: (a) $\langle Y^0_0\rangle$, (b) $\langle Y^0_1\rangle$, (c) $\langle Y^0_2\rangle$, (d) $\langle Y^0_3\rangle$, (e) $\langle Y^0_4\rangle$, (f) $\langle Y^0_5\rangle$. The errors on the black data points are statistical. The (blue) curves show the fit projections. 
shy.pdf [81 KiB] HiDef png [763 KiB] Thumbnail [151 KiB] *.C file 

Animated gif made out of all figures. 
PAPER2014012.gif Thumbnail 
$ C\!P$ parity of the full final state for different spin resonances. Note that spin0 only has the 0 transversity component. 
Table_0.pdf [39 KiB] HiDef png [33 KiB] Thumbnail [6 KiB] tex code 

Efficiency parameters. There are substantial correlations. 
Table_1.pdf [34 KiB] HiDef png [214 KiB] Thumbnail [30 KiB] tex code 

Parameters for the background model. 
Table_2.pdf [42 KiB] HiDef png [263 KiB] Thumbnail [37 KiB] tex code 

Possible resonance candidates in the $\overline{ B }{} ^0 \rightarrow J/\psi \pi^+\pi^$ decay mode. 
Table_3.pdf [65 KiB] HiDef png [81 KiB] Thumbnail [12 KiB] tex code 

The $\chi^2/\text{ndf}$ and the $\rm ln\mathcal{L}$ of different resonance models. The decrease of $\rm ln\mathcal{L}$ is with respect to the 5R model. 
Table_4.pdf [38 KiB] HiDef png [60 KiB] Thumbnail [8 KiB] tex code 

Nonzero interference fractions($\%$) obtained from the fit using the Best Model. The uncertainties are statistical only. 
Table_5.pdf [52 KiB] HiDef png [243 KiB] Thumbnail [38 KiB] tex code 

The fitted resonant phases from the Best Model. The uncertainties are statistical only. 
Table_6.pdf [45 KiB] HiDef png [328 KiB] Thumbnail [47 KiB] tex code 

Fit fractions and transversity fractions of contributing resonances in the Best Model. The first uncertainty is statistical and the second the total systematic. 
Table_7.pdf [54 KiB] HiDef png [64 KiB] Thumbnail [11 KiB] tex code 

Branching fractions for each channel. The first uncertainty is statistical and the second the total systematic. 
Table_8.pdf [50 KiB] HiDef png [112 KiB] Thumbnail [17 KiB] tex code 

Absolute systematic uncertainties on the results of the amplitude analysis estimated using the Best Model except for the $f_0(980)$ where we use the 7R model. 
Table_9.pdf [57 KiB] HiDef png [194 KiB] Thumbnail [31 KiB] tex code 
Created on 09 December 2018.Citation count from INSPIRE on 18 December 2018.