Heavy Quark Distribution Function in QCD and the AC$^2$M$^2$ Model

We show that the phenomenological \ACM ansatz is consistent with QCD through order $1/m_b$ in the description of $B\ra l\bar \nu_l+X_u$ and $B\ra \gamma +X_s$ transitions, including their energy spectra and differential distributions. This suggests a concrete realization for the QCD distribution function, which we call the ``Roman'' function. On the other hand the \ACM model description of the end-point domain in $B\ra l\bar \nu_l + X_c$ is incompatible with QCD: a different distribution function enters the description of $b\ra c$ decays as compared to the transitions to the massless quarks. Both observations -- the validity of the {\ACM}-like description for heavy-to-light transitions and the emergence of the new distribution function in the $b\ra c$ case -- are in contradiction to a recent claim in the literature. The intrinsic limitation of the \ACM model could reveal itself in different values of the effective $b$ quark mass from fits of the $\Lambda _b$ and $B$ decays.