Inclusive semi-leptonic $B_{(s)}$ mesons decay at the physical $b$ quark mass

We address the non-perturbative calculation of the decay rate of inclusive semi-leptonic $B_{(s)}$ mesons decays from lattice QCD. Precise theoretical Standard Model predictions are key ingredients in searches for new physics. This type of computation may eventually provide new insight into the long-standing tension between the inclusive and exclusive determinations of the CKM matrix elements $|V_{cb}|$ and $|V_{ub}|$. We perform a pilot lattice computation for $B_s \rightarrow X_c \, l \nu_l$ and improve on existing techniques. The valence-quark masses in our simulations are approximately physical for the domain-wall strange and charm quarks as well as for the bottom quark, for which we use a relativistic heavy quark effective action. We report on our progress and discuss future plans towards a first study with fully controlled systematic effects.