Inclusive Semileptonic $b \to c \ell \bar{\nu}$ Decays to Order $1/m_b^5$

Inclusive semileptonic $B\to X_c \ell\bar{\nu}$ decays can be described in the Heavy Quark Expansion (HQE) and allow for a precision determination of the CKM element $|V_{cb}|$. We calculate the terms of $1/m_b^5$ and derive a ``trace formula'' which allows for the computation of the decay rate and kinematic moments of the spectrum up to this order in the HQE. We focus specifically on the reparametrization invariant (RPI) dilepton invariant mass $q^2$ moments of the spectrum, which depend on a reduced set of HQE parameters. At this order, ``intrinsic charm'' (IC) contributions proportional to $1/(m_b^3m_c^2)$ enter, which are numerically expected to be sizeable. Using the ``lowest-lying state saturation ansatz'' (LLSA), we estimate the size of these contributions. Within this approximation, we observe a partial cancellation between the IC and the ``genuine'' $1/m_b^5$ contributions, resulting in a small overall contribution.