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We evaluate the contribution of pion-pion rescattering to the $p p \to p p\pi^+ \pi^-$ and $p \bar p \to p \bar p \pi^+ \pi^-$ reactions. We compare ourresults with the close-to-threshold experimental data. The pion-pionrescattering contribution is found there to be negligible. The predictions forfuture experiments with PANDA detector at High Energy Storage Ring (HESR) inGSI Darmstadt are presented. It is discussed how to cut off the dominant Roperresonance and double-$\Delta$ excitation mechanisms leading to the $p p \pi^+\pi^-$ channel in final state. Differential distributions in momentumtransfers, transverse momentum, effective two-pions mass, relative azimuthalangle between pions, and pion rapidities are presented.

Low-energy pion-pion scattering in the $pp \to pp\pi^{+}\pi^{-}$ and $p\bar p \to p\bar p\pi^{+}\pi^{-}$ reactions