PQCD study of the $B_{SL}$ and $n_c$ controversy in inclusive $B$ decays
  We calculate the semileptonic branching ratio and the charm counting of inclusive $B$ meson decays using the perturbative QCD formalism. For the nonleptonic decays, we employ the modified factorization theorem, in which Wilson coefficients evolve with the characteristic scales of the decay modes. It is found that the decay rate of the single-charm mode $b\to c{\bar u}d$ is enhanced, and a lower $B_{SL}$ is obtained without increasing $n_c$. We predict a larger $b\to c \tau \bar{\nu}$ branching ratio compared to that from the conventional heavy-quark-effective-theory based operator product expansion. Our result of the $B$ meson lifetime is also consistent with the data. 
Chia-Hung V. Chang, Darwin Chang, We-Fu Chang, Hsiang-nan Li, and   Hoi-Lai Yu