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The resonant structure of the decay $\overline{B}_s^0\to J/\psi\pi^+\pi^-$ isstudied 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 $f_2^{\prime}(1525)$. Analternative model including these states and a non-resonant $J/\psi \pi^+\pi^-$component also provides a good description of the data. Based on the differenttransversity components measured for the spin-2 intermediate states, the finalstate is found to be compatible with being entirely $CP$-odd. The $CP$-evenpart is found to be $<2.3$% at 95% confidence level. The $f_0(500)$ state isnot observed, allowing a limit to be set on the absolute value of the mixingangle with the $f_0(980)$ of $<7.7^{\circ}$ at 90% confidence level, consistentwith a tetraquark interpretation of the $f_0(980)$ substructure.

Measurement of resonant and $CP$ components in $\overline{B}_s^0\rightarrow J/\psi \pi^+\pi^-$ decays