QCD corrections for subleading powers in $1/m_b$ for the nonleptonic $b\rightarrow c\bar c s$ transition

We compute next-to-leading order QCD corrections to the $1/m_b^2$ terms of the Heavy Quark Expansion for the contribution to inclusive nonleptonic bottom hadron decays induced by the charged-current operators mediating the transitions $b \to c \bar c q$ ($q = s,d$). Their contributions to lifetimes are Cabibbo favoured and have large Wilson coefficents, but have not been computed before, since the presence of two heavy quarks in the final state makes this calcuation technically demanding. It finally adds the last missing piece of the full NLO terms at $1/m_b^2$ in the heavy quark expansion of $B$-hadron lifetimes. We obtain analytical results with full dependence on the final-state charm quark mass, which we provide in the supplemental file "coefbccs.m".