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In this work we address the issue of large perturbative uncertainties in thetheory 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 towardsthe full analytic result for the previously unknown Wilson coefficients fromthe matching between $|\Delta B|=1$ and $|\Delta B|=2$ effective Hamiltoniansat next-to-next-leading order (NNLO) in the strong coupling constant. Weprovide a thorough discussion of technical and conceptual difficulties behindthis computation and give an outlook regarding the availability of the new NNLOtheory prediction for $\Delta \Gamma_s$.

Towards two- and three-loop QCD corrections to the width difference in $B_s-\bar{B}_s$ mixing