Abstract
Following the 1999 analysis of Gambino, Haisch and one of us, we stress that all the recent NLO analyses of varepsilon '/varepsilon 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 m_t(mu _t). We also reemphasize the important role of isospin-breaking and QED effects in the evaluation of varepsilon '/varepsilon . Omitting all these effects, as done in the 2015 analysis by RBC-UKQCD collaboration, and choosing as an example the QCD penguin (Q_6) and electroweak penguin (Q_8) parameters B_6^{(1/2)} and B_8^{(3/2)} to be B_6^{(1/2)}= 0.80 pm 0.08 and B_8^{(3/2)}= 0.76 pm 0.04 at mu = m_c=1.3,, text {GeV}, we find (varepsilon '/varepsilon )_mathrm{SM} = (9.4 pm 3.5) times 10^{-4}, whereas including them results in (varepsilon '/varepsilon )_mathrm{SM} = (5.6pm 2.4)times 10^{-4}. This is an example of an anomaly at the 3.3,sigma 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 varepsilon '/varepsilon for different values of B_6^{(1/2)} and the isospin-breaking parameter {widehat{Omega }}_text {eff}, that should facilitate monitoring the values of varepsilon '/varepsilon in the SM when the RBC-UKQCD calculations of hadronic matrix elements including isospin-breaking corrections and QED effects will improve with time.
Highlights
The situation of ε /ε in the Standard Model (SM) by the end of 2017 could be briefly summarized as follows: The analysis of ε /ε by the RBC-UKQCD lattice QCD (LQCD) collaboration based on their 2015 results for K → π π matrix elements [4,5], as well as the analyses performed in [6,7] that are based on the same matrix elements and include isospin breaking effects [8,9], found ε /ε in the ballpark of (1−2)×10−4
Following the 1999 analysis of Gambino, Haisch and one of us, we stress that all the recent NLO analyses of ε /ε 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 μW and in particular to μt in mt
As the values of B6(1/2) and the size of isospin-breaking corrections are expected to dominate the theoretical uncertainties in ε /ε for some time, we present in Sect. 4 a table of the SM values of ε /ε for different B6(1/2) and the isospin-breaking parameter eff that should facilitate monitoring the SM estimates of ε /ε when the LQCD calculations of hadronic matrix elements including isospinbreaking corrections and QED effects will improve with time
Summary
The situation of ε /ε in the SM by the end of 2017 could be briefly summarized as follows:. The analysis of ε /ε by the RBC-UKQCD lattice QCD (LQCD) collaboration based on their 2015 results for K → π π matrix elements [4,5], as well as the analyses performed in [6,7] that are based on the same matrix elements and include isospin breaking effects [8,9], found ε /ε in the ballpark of (1−2)×10−4. This is by one order of magnitude below the experimental world average from NA48 [10] and KTeV [11,12] collaborations,.
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