  We examine the ratio of the decay rate of the eta_c into light hadrons to the
decay rate into photons and find that most of the large next-to-leading-order
(NLO) correction is associated with running of the strong coupling alpha_s. We
resum such contributions by analyzing final-state chains of vacuum-polarization
bubbles. We show that the nonperturbative parts of the bubble chains can be
absorbed into a color-octet matrix element, once one has used contour
deformations of the phase-space integrals to cancel certain contributions. We
argue that these contributions are incompatible with the uncertainty principle.
We also argue that perturbation theory is reliable only if one carries out the
phase-space integrations before the perturbation summation. Our results are in
good agreement with experiment and differ considerably from those that one
obtains by applying the scale-setting method of Brodsky, Lepage, and Mackenzie
to the NLO result.
