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Register a new user$varepsilon/varepsilon$ 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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For direct CP-violation in $Ktopipi$ decays, the usual isospin-breaking effects at the percent level are amplified by the dynamics behind the $Delta I=1/2$ rule and conventionally encoded in $Omega_{rm IB}$ parameters. The updated prediction $Omega_{rm IB}^{(8)}=(15.9pm 8.2)times 10^{-2}$ of the Chiral Perturbation Theory for the strong isospin-breaking due to $pi_3-eta_8$ mixing confirms such a tendency but is quite sensitive to the theoretical input value of the low-energy constant corresponding to the flavour-singlet $eta_0$ exchange contribution in this truncated octet scheme. We rather exploit the phenomenological $eta_8-eta_0$ mixing as a probe for the non-negligible flavour-singlet component of the physical $eta$ pole to find $Omega_{rm IB}^{(9)}=(35pm7)times 10^{-2}$ in a complete nonet scheme. A large central value in the nonet scheme is thus substituted for a large uncertainty in the octet one. Including the experimental $pi^+-pi^0$ mass difference as the dominant electromagnetic isospin-breaking, we obtain for the effective parameter entering the ratio $epsilon/epsilon$ an improved result $hatOmega_{rm eff}^{(9)}=(29pm7)times 10^{-2}$ to be compared with $hatOmega_{rm eff}^{(8)}=(17pm9)times 10^{-2}$ used in recent analyses of $epsilon/epsilon$. Accordingly, we get a reduction from $(epsilon/epsilon)_{rm SM}^{(8)}=(17.4pm 6.1)times 10^{-4}$ to $(epsilon/epsilon)_{rm SM}^{(9)}=(13.9pm 5.2)times 10^{-4}$ and thereby an effective suppression of $(epsilon/epsilon)_{rm SM}$ by isospin-breaking corrections as large as $40%$ relative to the recent RBC-UKQCD value.
Estimates of the CP violating observable $varepsilon/varepsilon$ have gained some attention in the past few years. Depending on the long-distance treatment used, they exhibit up to $2.9sigma$ deviation from the experimentally measured value. Such a deviation motivates the investigation of New Physics (NP) effects in the process $Ktopipi$. In my talk I will review the Standard Model (SM) prediction for $varepsilon/varepsilon$, with a special focus on the Dual QCD approach. On the NP side, I will discuss a recent computation of the hadronic matrix elements of NP operators. Furthermore a master formula for BSM effects in $varepsilon/varepsilon$ is presented. Finally, a treatment of $varepsilon/varepsilon$ using the SM effective theory (SMEFT) will be discussed together with possible correlations to other observables.
We present for the first time a model-independent anatomy of the ratio $varepsilon/varepsilon$ in the context of the $Delta S = 1$ effective theory with operators invariant under QCD and QED and in the context of the Standard Model Effective Field Theory (SMEFT) with the operators invariant under the full SM gauge group. Our goal is to identify the new physics scenarios that are probed by this ratio and which could help to explain a possible deviation from the SM that is hinted by the data. To this end we derive a master formula for $varepsilon/varepsilon$, which can be applied to any theory beyond the Standard Model (BSM) in which the Wilson coefficients of all contributing operators have been calculated at the electroweak scale. The relevant hadronic matrix elements of BSM operators are from the Dual QCD approach and the SM ones from lattice QCD. Within SMEFT, the constraints from $K^0$ and $D^0$ mixing as well as electric dipole moments limit significantly potential new physics contributions to $varepsilon/varepsilon$. Correlations of $varepsilon/varepsilon$ with $Ktopi ubar u$ decays are briefly discussed. Building on our EFT analysis and the model-independent constraints, we discuss implications of a possible deviation from the SM in $varepsilon/varepsilon$ for model building, highlighting the role of the new scalar and tensor matrix elements in models with scalar mediators.
We present for the first time a master formula for $varepsilon/varepsilon$, the ratio probing direct CP violation in $K to pipi$ decays, valid in any theory beyond the Standard Model (BSM). The formula makes use of hadronic matrix elements of BSM operators calculated recently in the Dual QCD approach and the ones of the SM operators from lattice QCD. We emphasize the large impact of several scalar and tensor BSM operators in the context of the emerging $varepsilon/varepsilon$ anomaly. We have implemented the results in the open source code flavio.
Following the recent analysis done in collaboration with Jason Aebischer and Christoph Bobeth, I summarize the optimal, in our view, strategy for the present evaluation of the ratio $varepsilon/varepsilon$ in the Standard Model (SM). In particular, I emphasize the importance of the correct matching of the long-distance and short-distance contributions to $varepsilon/varepsilon$, which presently is only achieved by RBC-UKQCD lattice QCD collaboration and by the analytical Dual QCD approach. An mportant role play also the isospin-breaking and QED effects, which presently are best known from chiral perturbation theory, albeit still with a significant error. Finally, it is essential to include NNLO QCD corrections in order to reduce unphysical renormalization scheme and scale dependences present at the NLO level. Here $mu_c$ in $m_c(mu_c)$ in the case of QCD penguin (QCDP) contributions and $mu_t$ in $m_t(mu_t)$ in the case of electroweak penguin (EWP) contributions play the most important roles. Presently the error on $varepsilon/varepsilon$ is dominated by the uncertainties in the QCDP parameter $B_6^{(1/2)}$ and the isospin-breaking parameter $hatOmega_text{eff}$ We present a table illustrating this. To be published online by the Institute of Physics Proceedings.
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