On the importance of NNLO QCD and isospin-breaking corrections in $$\varepsilon '/\varepsilon $$ ε′/ε

Abstract

Abstract Following the 1999 analysis of Gambino, Haisch and one of us, we stress that all the recent NLO analyses of $$\varepsilon '/\varepsilon $$ 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 $$\mu _W$$ μW and in particular to $$\mu _t$$ μt in $$m_t(\mu _t)$$ mt(μt) . We also reemphasize the important role of isospin-breaking and QED effects in the evaluation of $$\varepsilon '/\varepsilon $$ ε′/ε . Omitting all these effects, as done in the 2015 analysis by RBC-UKQCD collaboration, and choosing as an example the QCD penguin ($$Q_6$$ Q6 ) and electroweak penguin ($$Q_8$$ Q8 ) parameters $$B_6^{(1/2)}$$ B6(1/2) and $$B_8^{(3/2)}$$ B8(3/2) to be $$B_6^{(1/2)}= 0.80 \pm 0.08$$ B6(1/2)=0.80±0.08 and $$B_8^{(3/2)}= 0.76 \pm 0.04$$ B8(3/2)=0.76±0.04 at $$\mu = m_c=1.3\,\, \text {GeV}$$ μ=mc=1.3GeV , we find $$(\varepsilon '/\varepsilon )_\mathrm{SM} = (9.4 \pm 3.5) \times 10^{-4}$$ (ε′/ε)SM=(9.4±3.5)×10-4 , whereas including them results in $$(\varepsilon '/\varepsilon )_\mathrm{SM} = (5.6\pm 2.4)\times 10^{-4}$$ (ε′/ε)SM=(5.6±2.4)×10-4 . This is an example of an anomaly at the $$3.3\,\sigma $$ 3.3σ 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 $$\varepsilon '/\varepsilon $$ ε′/ε for different values of $$B_6^{(1/2)}$$ B6(1/2) and the isospin-breaking parameter $${\widehat{\Omega }}_\text {eff}$$ Ω^eff , that should facilitate monitoring the values of $$\varepsilon '/\varepsilon $$ ε′/ε in the SM when the RBC-UKQCD calculations of hadronic matrix elements including isospin-breaking corrections and QED effects will improve with time.