Z0decay into charmonium via charm quark fragmentation

In decays of the $Z^0$, the dominant mechanism fopr tddirect production of charmonium states is the decay of the $Z^0$ into a charm quark or antiquark followed by its fragmentation into the charmonium state. We calculate the fragmentation functions describing the splitting of charm quarks into $S$-wave charmonium states to leading order in the QCD coupling constant. Leading logarithms of $\frac{M_Z}{m_c}$ are summed up using Altarelli-Parisi evolution equations. Our analytic result agrees with the complete leading order calculation of the rate for $Z^0\to \psi c\overline{c}$. We also use our fragmentation functions to calculate the production rate of heavy quarkonium states in $W^{\pm }$, top quark, and Higgs boson decays.