Finite $W$ mass corrections to inclusive $B$ decays

We compute the effects of a finite $W$ boson mass on the inclusive $b\rightarrow c \ell \nu$ decay rate and the semileptonic moments. To this end we keep terms of $\mathcal{O}\left(q^2/m_W^2 \right)$ when integrating out the $W$ boson in the Weak Effective Theory. In the third hadronic invariant mass moment such corrections reach the $0.5\%$ level but they are smaller than the present-day experimental uncertainties in all observables.