$varepsilon/varepsilon$ in the Standard Model at the Dawn of the 2020s


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We reanalyse the ratio $varepsilon/varepsilon$ in the Standard Model (SM) using most recent hadronic matrix elements from the RBC-UKQCD collaboration in combination with most important NNLO QCD corrections to electroweak penguin contributions and the isospin-breaking corrections. We illustrate the importance of the latter by using their latest estimate from chiral perturbation theory (ChPT) based on the $octet$ approximation for lowest-lying mesons and a very recent estimate in the $nonet$ scheme that takes into account the contribution of $eta_0$. We find $(varepsilon/varepsilon)^{(8)}_text{SM} = (17.4 pm 6.1) times 10^{-4}$ and $(varepsilon/varepsilon)^{(9)}_text{SM} = (13.9 pm 5.2) times 10^{-4}$, respectively. Despite a very good agreement with the measured value $(varepsilon/varepsilon)_text{exp} = (16.6 pm 2.3) times 10^{-4}$, the large error in $(varepsilon/varepsilon)_text{SM}$ still leaves room for significant new physics (BSM) contributions to this ratio. We update the 2018 master formula for $(varepsilon/varepsilon)_text{BSM}$ valid in any extension beyond the SM without additional light degrees of freedom. We provide new values of the penguin parameters $B_6^{(1/2)}(mu)$ and $B_8^{(3/2)}(mu)$ at the $mu$-scales used by the RBC-UKQCD collaboration and at lower scales $mathcal{O}(1,text{GeV})$ used by ChPT and DQCD. We present semi-analytic formulae for $(varepsilon/varepsilon)_text{SM}$ in terms of these parameters and $hat{Omega}_text{eff}$ that summarizes isospin-breaking corrections to this ratio. We stress the importance of lattice calculations of the $mathcal{O}(alpha_text{em})$ contributions to the hadronic matrix elements necessary for the removal of renormalization scheme dependence at $mathcal{O}(alpha_text{em})$ in the present analyses of $varepsilon/varepsilon$.

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