A note on spectator effects and quark-hadron duality in inclusive beauty decays

In this paper, we evaluate the expectation values of four-quark operators for the inclusive beauty decays from the differences in the inclusive total decay rates, assuming that the heavy quark expansion converges at $O(1/m^3)$. The obtained expectation values yeilds the ratio $\tau(\Lambda_b)/\tau(B)$ close to the experimental onee. We further point out that the quark-hadron duality violation would be rather small allowing predictions of inclusive quantities.