  Based on the nonrelativistic QCD factorization theorem, we resum QCD
corrections to the inclusive decay rate of $\eta_c$ and $\eta_b$ in the
large-$n_f$ limit using bubble chain resummation. By employing dimensional
regularization, we show explicitly the cancellation of the infrared renormalon
ambiguity in the factorization formula at leading order in $v$ in the
large-$n_f$ limit, where $v$ is the typical heavy quark velocity inside the
meson. We also make predictions of the ratio of the inclusive decay rate to the
decay rate into two photons. By comparing our results with a fixed-order
calculation we conclude that resummation of QCD corrections is crucial in
making an unambiguous prediction. We also find significant corrections beyond
the large-$n_f$ limit for the decay of $\eta_c$, which may imply that QCD
corrections need to be resummed beyond the large-$n_f$ limit to make an
accurate prediction of the decay rate.
