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

TL;DR

This paper advances the theoretical understanding of inclusive semileptonic B decays by calculating higher-order $1/m_b^5$ terms in the Heavy Quark Expansion, including intrinsic charm effects, and provides a method to compute decay rates and moments with improved precision.

Contribution

It introduces a comprehensive calculation of $1/m_b^5$ corrections in the HQE for B decays, including intrinsic charm effects, and derives a trace formula for decay rate and spectrum moments.

Findings

Intrinsic charm contributions are sizable but partially cancel with genuine $1/m_b^5$ terms.

The trace formula enables precise computation of decay rates and moments up to order $1/m_b^5$.

Estimated overall $1/m_b^5$ effects are small due to cancellations.

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 $1/(m_b^3m_c^2)$ 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.