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

In this work, we provide a simultaneous and accurate description of the $π^+π^-$ and $π^{\pm} J/ψ$ invariant mass distributions of the recent BESIII data on $e^+ e^- \to J/ψ\; π^+ π^-$ together with the $e^+ e^- \to J/ψ\; K^+K^-$ 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 S and D-waves are taken into account through the Muskhelishvili-Omnès formalism. Since the physical region of the $ππ$ invariant mass extends above 1 GeV, the important $K\bar{K}$ intermediate state in the S-wave is implemented through coupled-channel unitarity. For the left-hand cuts, we account for the well established charged exotic state $Z_c(3900)$ in $t$- and $u$-channels, while the other contributions are absorbed in the subtraction constants. For the $e^+ e^- \to J/ψ\, K \bar{K}$ we provide the prediction of the two-kaon invariant mass distribution. The constructed amplitudes serve as an essential framework to interpret the present and forthcoming measurements by the BESIII and Belle II Collaborations.