  We discuss (i) the evaluation of the expectation values of four-quark
operators assuming that the heavy quark expansion for $b$ sector converges at
the third order in $1/m_Q$, and (ii) the estimation of the duality breaking
short distance nonperturbative corrections to the parton decay rate. We finally
point out the implications of the result obtained for the assumption of
quark-hadron duality in heavy quark expansion.
