On the Importance of NNLO QCD and Isospin-breaking Corrections in $\varepsilon'/\varepsilon$

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 $\mu = m_c = 1.3$ GeV, we find $(\varepsilon'/\varepsilon)_\text{SM} = (9.4 \pm 3.5) \times 10^{-4}$, whereas including them results in $(\varepsilon'/\varepsilon)_\text{SM} = (5.6\pm 2.4) \times 10^{-4}$. 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 $\hat{\Omega}_\text{eff}$, 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.