  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, one-third at 7 Tev center-of-mass 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 non-resonant 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.
