Perturbative QCD Fragmentation Functions as a Model for Heavy-Quark Fragmentation

The perturbative QCD fragmentation functions for a heavy quark to fragment into heavy-light mesons are studied in the heavy-quark limit. The fragmentation functions for S-wave pseudoscalar and vector mesons are calculated to next-to-leading order in the heavy-quark mass expansion using the methods of heavy-quark effective theory. The results agree with the $m_b\to \mathrm{\infty }$ limit of the perturbative QCD fragmentation functions for $\bar{b}$ into $B_c$ and $B_c^{\ast }$. We discuss the application of the perturbative QCD fragmentation functions as a model for the fragmentation of heavy quarks into heavy-light mesons. Using this model, we predict the fraction $P_V$ of heavy-light mesons that are produced in the vector meson state as functions of the longitudinal momentum fraction $z$ and the transverse momentum relative to the jet axis. The fraction $P_V$ is predicted to vary from about 1/2 at small $z$ to almost 3/4 near $z=1$.