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In this work, we provide a simultaneous and accurate description of the$\pi^+\pi^-$ and $\pi^{\pm} J/\psi$ invariant mass distributions of the recentBESIII data on $e^+ e^- \to J/\psi \; \pi^+ \pi^-$ together with the $e^+ e^-\to J/\psi \; K^+K^-$ cross sections at $e^+e^-$ center-of-mass energies$q=4.23$ GeV and $q=4.26$ GeV. The rescattering effects between pions in the Sand D-waves are taken into account through the Muskhelishvili-Omn\`esformalism. Since the physical region of the $\pi\pi$ invariant mass extendsabove 1 GeV, the important $K\bar{K}$ intermediate state in the S-wave isimplemented through coupled-channel unitarity. For the left-hand cuts, weaccount for the well established charged exotic state $Z_c(3900)$ in $t$- and$u$-channels, while the other contributions are absorbed in the subtractionconstants. For the $e^+ e^- \to J/\psi\, K \bar{K}$ we provide the predictionof the two-kaon invariant mass distribution. The constructed amplitudes serveas an essential framework to interpret the present and forthcoming measurementsby the BESIII and Belle II Collaborations.

Simultaneous description of the $e^+e^- \to J/ \psi \, \pi \pi\, (K \bar{K})$ processes