The resonant structure of the reaction ¯B0→J/ψπ+π− 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 π+π− pair and three decay angular distributions are used to determine the fractions of the resonant and nonresonant components. Six interfering π+π− states, ρ(770), f0(500), f2(1270), ρ(1450), ω(782) and ρ(1700), are required to give a good description of invariant mass spectra and decay angular distributions. The positive and negative charge parity fractions of each of the resonant final states are determined. The f0(980) meson is not seen and the upper limit on its presence, compared with the observed f0(500) rate, is inconsistent with a model where these scalar mesons are formed from two quarks and two antiquarks (tetraquarks) at the eight standard deviation level. In the q¯q model, the absolute value of the mixing angle between the f0(980) and the f0(500) scalar mesons is limited to be less than 17° at 90% confidence level.