  In this work we address the issue of large perturbative uncertainties in the
theory prediction for $\Delta \Gamma_s$, the width difference in the $B^0_s -
\bar{B}^0_s$ mixing process. To this aim we complete important steps towards
the full analytic result for the previously unknown Wilson coefficients from
the matching between $|\Delta B|=1$ and $|\Delta B|=2$ effective Hamiltonians
at next-to-next-leading order (NNLO) in the strong coupling constant. We
provide a thorough discussion of technical and conceptual difficulties behind
this computation and give an outlook regarding the availability of the new NNLO
theory prediction for $\Delta \Gamma_s$.
