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

Following the 1999 analysis of Gambino, Haisch and one of us, we stress that all the recent NLO analyses of epsilon '/epsilon 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 mu W and in particular to mu(t) in mt(mu t). We also reemphasize the important role of isospin-breaking and QED effects in the evaluation of epsilon '/epsilon. Omitting all these effects, as done in the 2015 analysis by RBC-UKQCD collaboration, and choosing as an example the QCD penguin (Q6) and electroweak penguin (Q(8)) parameters B-6((1/2)) and B-8((3/2)) to be B-6((1/2)) = 0.80 +/- 0.08 and B-8((3/2)) = 0.76 +/- 0.04 at mu = m(c) = 1.3 GeV, we find (epsilon '/epsilon)SM = (9.4 +/- 3.5) x 10(-4), whereas including them results in (epsilon '/epsilon) SM = (5.6 +/- 2.4) x 10(-4). This is an example of an anomaly at the 3.3 sigma level, which would be missed without these corrections. NNLOQCDcontributions toQCD penguins are expected to further enhance this anomaly. We provide a table for epsilon '/epsilon for different values of B-6((1/2)) and the isospin-breaking parameter (Omega) over cap (eff), that should facilitate monitoring the values of epsilon '/epsilon in the SM when the RBC-UKQCD calculations of hadronic matrix elements including isospin-breaking corrections andQEDeffectswill improve with time.