Extracting $R_b$ and $R_c$ Without Flavor Tagging

At present, two outstanding discrepancies between experiment and the standard model are the measurements of the hadronic branching fractions $R_b$ and $R_c$. We note that an independent measurement of these branching fractions may be obtained from the width of hadronic $Z$ decays with a prompt photon, $\Gamma_{qq\gamma}$, along with the total hadronic decay rate, $\Gamma_{had}$, and an additional theoretical assumption. Such an analysis requires no flavor tagging. We consider several plausible theoretical assumptions and find that the current value of $\Gamma_{qq\gamma}$ favors larger $R_b$ and smaller $R_c$ relative to standard model predictions, in accord with the direct measurements. If $\Gamma_{qq\gamma}$ and $\Gamma_{had}$ are combined with the direct measurements, generation-blind corrections to all up-type and all down-type quark widths are most favored. An updated measurement of $\Gamma_{qq\gamma}$ with the currently available LEP data is likely to provide an even stronger constraint on both the branching fraction discrepancies and their possible non-standard model sources.