The resonant structure of the reaction ¯ B 0 → J / ψ π + 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 ) , f 0 ( 500 ) , f 2 ( 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 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 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 f 0 ( 980 ) and the f 0 ( 500 ) scalar mesons is limited to be less than 17° at 90% confidence level.