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The top quark decay process $\tcups$ is studied in the Standard Model and theMinimal Supersymmetric Standard Model with conserved and violated $R-$parity.In dealing with the heavy quarkonium, $\ups$, production, we take bothcolor-singlet and -octet mechanisms into consideration. With numerical analysisit is found that within the Standard Model the decay rate of the concernedprocess is beyond the border of reachable level of the next round ofexperiment, though it is much larger than that of the rare top decays, whileextra new contributions from beyond the Standard Model could enhance the decayrate by several orders. It is also noticed that a precise measurement of thecharged Higgs mass without $\tan\beta$ dependence is possible via this process.Comment: 16 page

t→cΥ within and beyond the standard model