On the failure of the $12\%$ rule in $J/ψ$ and $ψ(2S)$ decays

Abstract: The $12\%$ rule relates hadronic decays of the $J/ψ$ and $ψ(2S)$ through the ratio of their wave functions at the origin, but is strongly violated in several vector--pseudoscalar ($VP$) channels, most prominently in the $ρπ$ one. We study these decays within an effective chiral framework including strong and electromagnetic interactions, with the latter implemented through vector meson dominance. The $ψ(2S)\to VP$ data are described well by a relatively minimal model with three free parameters, yielding $χ^2_{\rm red}\simeq 6.71/(11-3)=1.12$. In contrast, the corresponding description of the $J/ψ$ decays is insufficient ($χ^2{\rm <em>{red}}\simeq 132/(11-3)=16.5$) and requires additional $\mathrm{SU}(3)$ flavor breaking and electromagnetic contributions for a fair agreement with data: $$χ²</em>{\rm red}\simeq 9.87/(11-6)= 1.97$$. In the past, a small mixing of $J/ψ$ with an invisible vector glueball was considered as a way to describe the decay patterns. However, we will show that our extended fits are `de facto' equivalent to those where a vector glueball mixes with the $J/ψ$, so that we cannot distinguish whether or not the vector glueball is present. Our results suggest that an important part of the $ρπ$ puzzle may arise from additional contributions to the $J/ψ\to VP$ amplitudes rather than from a suppression mechanism specific to the $ψ(2S)$.