Quarkonium Decays and Light Quark Masses

The $SU(3)$-violating decays $\Phi^{2S} \goto \Phi^{1S} X$, where $X = \pi^0$ or $\eta$ and $\Phi = J/\psi$ or $\Upsilon$ have been recently proposed as a means of probing the light quark masses beyond leading order in chiral perturbation theory. We argue that this analysis is incorrect, even in the heavy quark limit. We show that these decays are governed by an infinite number of matrix elements which are not suppressed by any small parameter, and which cannot be computed with our present understanding of QCD. Furthermore, for sufficiently heavy quarks, we show that the decay amplitudes can be organized into a twist expansion, and that the contributions considered in the above proposal are subleading in this expansion. We also explain how these decays nonetheless give a constraint on the light quark masses valid at {\it leading order} in the chiral expansion. The decays $\Phi^{1S} \goto \eta\gamma$ and $\Phi^{2S} \goto \Phi^{1S} \pi\pi$ also have contributions from infinitely many operators, contrary to claims in the literature.