${\epsilon }^{\prime }/\epsilon$ in the Standard Model at the Dawn of the 2020s

Abstract

We reanalyse the ratio ${\epsilon }^{\prime }/\epsilon$ 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 $({\epsilon }^{\prime }/\epsilon {)}_{\text{SM}}^{(8)}=(17.4\pm 6.1)\times 10^{-4}$ and $({\epsilon }^{\prime }/\epsilon {)}_{\text{SM}}^{(9)}=(13.9\pm 5.2)\times 10^{-4}$, respectively. Despite a very good agreement with the measured value $({\epsilon }^{\prime }/\epsilon {)}_{\text{exp}}=(16.6\pm 2.3)\times 10^{-4}$, the large error in $({\epsilon }^{\prime }/\epsilon {)}_{\text{SM}}$ still leaves room for significant new physics (BSM) contributions to this ratio. We update the 2018 master formula for $({\epsilon }^{\prime }/\epsilon {)}_{\text{BSM}}$ 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 $({\epsilon }^{\prime }/\epsilon {)}_{\text{SM}}$ 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 ${\epsilon }^{\prime }/\epsilon$.

arXiv comment26 pages, 1 figure; v2: Analysis extended to the new value of Omega. More details on the numerical analysis given. Semi-analytic formula slightly modified

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