in the Standard Model at the Dawn of the 2020s

We reanalyse the ratio $$ in the Standard Model (SM) using most recent hadronic matrix elements from the RBC-UKQCD collaboration in combination with most important NNLO QCD corrections to electroweak penguin contributions and the isospin-breaking corrections. We illustrate the importance of the latter by using their latest estimate from chiral perturbation theory (ChPT) based on the octet approximation for lowest-lying mesons and a very recent estimate in the nonet scheme that takes into account the contribution of $$ . We find $$ and $$ , respectively. Despite a very good agreement with the measured value $$ , the large error in $$ still leaves room for significant new physics (BSM) contributions to this ratio. We update the 2018 master formula for $$ valid in any extension beyond the SM without additional light degrees of freedom. We provide new values of the penguin parameters $$ and $$ at the $$ -scales used by the RBC-UKQCD collaboration and at lower scales $$ used by ChPT and Dual QCD (DQCD). We present semi-analytic formulae for $$ in terms of these parameters and $$ that summarizes isospin-breaking corrections to this ratio. We stress the importance of lattice calculations of the $$ contributions to the hadronic matrix elements necessary for the removal of renormalization scheme dependence at $$ in the present analyses of $$ .