Searching for New Physics with $overline{mathcal{B}}(B_{s,d}tomubarmu)/Delta M_{s,d}$

We reemphasize that the ratio $R_{smu} equiv overline{mathcal{B}}(B_stomubarmu)/Delta M_s$ is a measure of the tension of the Standard Model (SM) with latest measurements of $overline{mathcal{B}}(B_stomubarmu)$ that does not suffer from the persistent puzzle on the $|V_{cb}|$ determinations from inclusive versus exclusive $bto cellbar u$ decays and which affects the value of the CKM element $|V_{ts}|$ that is crucial for the SM predictions of both $overline{mathcal{B}}(B_stomubarmu)$ and $Delta M_s$, but cancels out in the ratio $R_{smu}$. In our analysis we include higher order electroweak and QED corrections und adapt the latest hadronic input to find a tension of about $2sigma$ for $R_{smu}$ measurements with the SM independently of $|V_{ts}|$. We also discuss the ratio $R_{dmu}$ which could turn out, in particular in correlation with $R_{smu}$, to be useful for the search for New Physics, when the data on both ratios improves. Also $R_{dmu}$ is independent of $|V_{cb}|$ or more precisely $|V_{td}|$.