Shape Function Effects in $B \to X_c \ell \barν_\ell$

Owing to the fact that $m_c^2 \sim m_b \Lambda_{\rm QCD}$, the endpoint region of the charged lepton energy spectrum in the inclusive decay $B \to X_c \ell \bar{\nu}_\ell$ is affected by the Fermi motion of the initial-state $b$ quark bound in the $B$ meson. This effect is described in QCD by shape functions. Including the mass of the final-state quark, we find that a different set of operators as employed in Ref. \cite{Bauer:2002yu} is needed for a consistent matching, when incorporating the subleading contributions in $B \to X_q \ell \bar{\nu}_\ell$ for both $q = u$ and $q = c$. In addition, we modify the usual twist expansion in such a way that it yields a description of the lepton energy spectrum which is not just valid in the endpoint region, but over the entire phase space.