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

Abstract

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 $1/(m_b^3{m}_c^2)$ 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.

Comments: 36 pages, 4 figures; In this updated version, we update the trace formulae and q2 moments attachments in light of a mistake spotted in our old version in [hep-ph/2501.09090], resulting in small differences in the X8 and X10 terms at 1/mb5, while all other contributions (including the intrinsic charm) are unchanged. Consequently, the conclusions are unaffected