$G$-parity violating amplitudes in the $J/\psi \to \pi^+ \pi^-$ decay

The decays of the negative $G$-parity meson $J/\psi$ into even numbers of pions violate $G$-parity. Such decays, as well as other $G$-parity decays into hadrons, can be parametrized in terms of three main intermediate virtual states: one photon, one photon plus two gluons, and three gluons. Since the electromagnetic interaction does not conserve $G$-parity, $J/\psi$ decays into positive $G$-parity final states should be dominantly electromagnetic. Nevertheless, the one-photon contribution to $J/\psi \to \pi^+ \pi^-$, that can be estimated by exploiting the cross section $\sigma(e^+e^-\to \pi^+ \pi^-)$, differs from the observed decay probability for {more than} 4.5 standard deviations. We present a computation of the $gg\gamma$ amplitude based on a phenomenological description of the decay mechanism in terms of dominant intermediate states $\eta\gamma$, $\eta'\gamma$ and $f_1(1285)\gamma$. The obtained value is of the order of the electromagnetic contribution.