The cross section of $e^+e^- \to \Lambda \overline \Sigma{}^0+{\rm c.c.}$ as a litmus test of isospin violation in the decays of vector charmonia into $\Lambda \overline \Sigma{}^0+{\rm c.c.}$

Under the aegis of isospin conservation, the amplitudes in Born approximation, i.e., considering the only one-photon-exchange mechanism, of the decay $\psi\to \Lambda \overline \Sigma{}^0+{\rm c.c.}$, where $\psi$ is a vector charmonium, and of the reaction $e^+e^- \to \Lambda \overline \Sigma{}^0+{\rm c.c.}$ at the $\psi$ mass, are parametrized by the same electromagnetic coupling. It follows that, the modulus of such a coupling can be extracted the data on the two observables: the decay branching fraction and the annihilation cross section. By considering the first two vector charmonia, $J/\psi$ and $\psi(2S)$, it is found that, especially in the case of $\psi(2S)$, there is a substantial discrepancy between the values of the modulus of the same electromagnetic coupling extracted from the branching ratio and the cross section. We propose, as a possible explanation for such a disagreement, the presence in the decay amplitude of an isospin-violating contribution driven by a mechanism based on physical (on-shell) intermediate states, that, due to their own nature, should be more effective in the $\psi(2S)$ decay than in that of the $J/\psi$.