Weak annihilation and the end-point spectrum in semileptonic B decays

A general relationship is formulated between the contributions of `Weak Annihilation' (WA) to inclusive semileptonic decays of heavy flavour hadrons and the matrix elements of four fermion operators. We argue that nonperturbative contributions from low energy hadronic final states provide the dominant impact of WA on the semileptonic $b\ra u$ width of $B~-$ mesons. In that case WA affects the lepton spectrum mainly in its endpoint region -- i.e. beyond the kinematical boundary for $b\ra c$ transitions -- and we expect thelater to look quite different in $B~-$ than in $B_d$ decays. At present we are unable to make a truly quantitative prediction; yet a detailled experimental comparison of the $B~-$ and the $B_d$ lepton spectra will enable one to determine the size of the relevant matrix elements and to check directly to what extent the factorization approximation works here. Using universal analyticity properties of the decay amplitude we analyze WA in the perturbative regime and rederive our earlier results about the absence of a power enhancement due to gluon emission in a more general way. We address also the problem of the invariant mass of hadrons in the final state in semileptonic $B$ decays and identify nontrivial cancellations in $\aver{M~2\ind{hadr}}$.