  The $J/\psi$ meson has negative $G$-parity so that, in the limit of isospin
conservation, its decay into $\pi^+\pi^-$ should be purely electromagnetic.
However, the measured branching fraction $\mathcal{B}(J/\psi\to\pi^+\pi^-)$
exceeds by more than 3.9 standard deviations the expectation computed according
to BaBar data on the $e^+e^-\to\pi^+\pi^-$ cross section. The possibility that
the two-gluon plus one-photon decay mechanism is not suppressed by $G$-parity
conservation is discussed, even by considering other multi-pion decay channels.
As also obtained by phenomenological computation, such a decay mechanism could
be responsible for the observed discrepancy. Finally, we notice that the BESIII
experiment, having the potential to perform an accurate measurement of the
$e^+e^-\to\pi^+\pi^-$ cross section in the 3 GeV energy region, can definitely
prove or disprove this strong $G$-parity-violating mechanism by confirming or
confuting the BaBar data.
