Shape function effects inB→Xcℓν¯ℓ

Owing to the fact that $m_c^2\sim m_b{\Lambda }_{\mathrm{Q}\mathrm{C}\mathrm{D}}$, the end point region of the charged lepton energy spectrum in the inclusive decay $B\to X_c\ell {\overline{\nu }}_{\ell }$ 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 previous work is needed for a consistent matching, when incorporating the subleading contributions in $B\to X_q\ell {\overline{\nu }}_{\ell }$ 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 end point region, but over the entire phase space.