Z0decay into two gluons and a photon for massive quarks

We have determined the decay rates for the rare decays $Z^0\to \mathrm{gg}\gamma$ ($\mathrm{ggg}$) including quark masses. We find a coherence effect in the amplitude for these processes which is sensitive to the top-quark mass. Inclusion of a top-quark mass of $m_t=20\mathrm{}$ GeV decreases the rate for $Z^0\to \mathrm{gg}\gamma$ by a factor of 2 and enhances the rate for $Z^0\to \mathrm{ggg}$ by a factor of 3. The branching ratios are $\frac{\Gamma (Z^0\to \mathrm{gg}\gamma )}{{\Gamma }_0}=1.8\mathrm{}\times {10}^{-6\mathrm{}}$ and $\frac{\Gamma (Z^0\to \mathrm{gg}\gamma )}{{\Gamma }_0}>0.8\mathrm{}\times {10}^{-52\mathrm{}}$, where the latter is a lower bound since the axial-vector contribution has yet to be computed, and ${\Gamma }_0$ is the total hadronic decay width.