Completing $1/m_b^3$ corrections to non-leptonic bottom-to-up-quark decays

We compute the contributions of dimension six two-quark operators to the non-leptonic decay width of heavy hadrons due to the flavor changing bottom-to-up-quark transition in the heavy quark expansion. Analytical expressions for the Darwin term $\rho_D$ and the spin-orbit term $\rho_{\rm LS}$ are obtained with leading order accuracy.