On the importance of NNLO QCD and isospin-breaking corrections in

Following the 1999 analysis of Gambino, Haisch and one of us, we stress that all the recent NLO analyses of in the Standard Model (SM) suffer from the renormalization scheme dependence present in the electroweak penguin contributions as well as from scale uncertainties in them related to the matching scale and in particular to in . We also reemphasize the important role of isospin-breaking and QED effects in the evaluation of . Omitting all these effects, as done in the 2015 analysis by RBC-UKQCD collaboration, and choosing as an example the QCD penguin () and electroweak penguin () parameters and to be and at , we find , whereas including them results in . This is an example of an anomaly at the level, which would be missed without these corrections. NNLO QCD contributions to QCD penguins are expected to further enhance this anomaly. We provide a table for for different values of and the isospin-breaking parameter , that should facilitate monitoring the values of in the SM when the RBC-UKQCD calculations of hadronic matrix elements including isospin-breaking corrections and QED effects will improve with time.