$$\varepsilon '/\varepsilon $$ ε′/ε in the Standard Model at the Dawn of the 2020s

Abstract

Abstract We reanalyse the ratio $$\varepsilon '/\varepsilon $$ ε′/ε 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 $$\eta _0$$ η0 . We find $$(\varepsilon '/\varepsilon )^{(8)}_\text {SM} = (17.4 \pm 6.1) \times 10^{-4}$$ (ε′/ε)SM(8)=(17.4±6.1)×10-4 and $$(\varepsilon '/\varepsilon )^{(9)}_\text {SM} = (13.9 \pm 5.2) \times 10^{-4}$$ (ε′/ε)SM(9)=(13.9±5.2)×10-4 , respectively. Despite a very good agreement with the measured value $$(\varepsilon '/\varepsilon )_\text {exp} = (16.6 \pm 2.3) \times 10^{-4}$$ (ε′/ε)exp=(16.6±2.3)×10-4 , the large error in $$(\varepsilon '/\varepsilon )_\text {SM}$$ (ε′/ε)SM still leaves room for significant new physics (BSM) contributions to this ratio. We update the 2018 master formula for $$(\varepsilon '/\varepsilon )_\text {BSM}$$ (ε′/ε)BSM valid in any extension beyond the SM without additional light degrees of freedom. We provide new values of the penguin parameters $$B_6^{(1/2)}(\mu )$$ B6(1/2)(μ) and $$B_8^{(3/2)}(\mu )$$ B8(3/2)(μ) at the $$\mu $$ μ -scales used by the RBC-UKQCD collaboration and at lower scales $$\mathcal {O}(1\, \text {GeV})$$ O(1GeV) used by ChPT and Dual QCD (DQCD). We present semi-analytic formulae for $$(\varepsilon '/\varepsilon )_\text {SM}$$ (ε′/ε)SM in terms of these parameters and $$\hat{\Omega }_\text {eff}$$ Ω^eff that summarizes isospin-breaking corrections to this ratio. We stress the importance of lattice calculations of the $$\mathcal {O}(\alpha _{\text {em}})$$ O(αem) contributions to the hadronic matrix elements necessary for the removal of renormalization scheme dependence at $$\mathcal {O}(\alpha _{\text {em}})$$ O(αem) in the present analyses of $$\varepsilon '/\varepsilon $$ ε′/ε .