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

In this work we address the issue of large perturbative uncertainties in the theory prediction for $\mathrm{\Delta }{\mathrm{\Gamma }}_s$, 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 $\mathrm{\mid }\mathrm{\Delta }B\mathrm{\mid }=1$ and $\mathrm{\mid }\mathrm{\Delta }B\mathrm{\mid }=2$ 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 $\mathrm{\Delta }{\mathrm{\Gamma }}_s$.