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Register a new user$epsilon^prime/epsilon$-2018: A Christmas Story
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Andrzej J. Buras
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I was supposed to review the status of $epsilon^prime/epsilon$ both at the CKM Workshop in September in Heidelberg and recently at the Discrete 2018 Conference in Vienna. Unfortunately I had to cancel both talks for family reasons. My main goal in these talks was to congratulate NA48 and KTeV collaborations for the discovery of new sources of CP violation through their heroic efforts to measure the ratio $epsilon^prime/epsilon$ in the 1980s and 1990s with final results presented roughly 16 years ago. As I will not attend any other conferences this year I will reach this goal in this writing. In this context I will give arguments, why I am convinced about the presence of new physics in $epsilon^prime/epsilon$ on the basis of my work with Jean-Marc Gerard within the context of the Dual QCD (DQCD) approach and why RBC-UKQCD collaboration and in particular Chiral Perturbation Theory practitioners are still unable to reach this conclusion. I will demonstrate that even in the presence of pion loops, as large as advocated recently by Gilbert and Pich, the value of $epsilon^prime/epsilon$ is significantly below the data, when the main non-factorizable QCD dynamics at long distance scales, represented in DQCD by {the meson evolution}, is taken into account. As appriopriate for a Christmas story, I will prophesy the final value of $epsilon^prime/epsilon$ within the SM, which should include in addition to the correct matching between long and short distance contributions, isospin breaking effects, NNLO QCD corrections to both QCD penguin and electroweak penguin contributions and final state interactions. Such final SM result will probably be known from lattice QCD only in the middle of 2020s, but already in 2019 we should be able to see some signs of NP in the next result on $epsilon^prime/epsilon$ from RBC-UKQCD.
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The ratio $epsilon/epsilon$ measures the size of the direct CP violation in $K_Ltopipi$ decays $(epsilon^prime)$ relative to the indirect one described by $epsilon$ and is very sensitive to new sources of CP violation. As such it played a prominent role in particle physics already for 45 years. Due to the smallness of $epsilon/epsilon$ its measurement required heroic efforts in the 1980s and the 1990s on both sides of the Atlantic with final results presented by NA48 and KTeV collaborations 20 years ago. Unfortunately, even 45 years after the first calculation of $epsilon/epsilon$ we do not know to which degree the Standard Model agrees with this data and how large is the room left for new physics contributions to this ratio. This is due to significant non-perturbative (hadronic) uncertainties accompanied by partial cancellation between the QCD penguin contributions and electroweak penguin contributions. While the significant control over the short distance perturbative effects has been achieved already in the early 1990s, with several improvements since then, different views on the non-perturbative contributions to $epsilon/epsilon$ have been expressed by different authors over last thirty years. In fact even today the uncertainty in the room left for NP contributions to $epsilon/epsilon$ is very significant. My own work on $epsilon/epsilon$ started in 1983 and involved both perturbative and non-perturbative calculations. This writing is a non-technical recollection of the steps which led to the present status of $epsilon/epsilon$ including several historical remarks not known to everybody. The present status of the $Delta I=1/2$ rule is also summarized. This story is dedicated to Jean-Marc Gerard on the occasion of the 35th anniversary of our collaboration and his 64th birthday.
Recently, the standard model prediction of $epsilon/epsilon$ was improved, and a discrepancy from the experimental results was reported at the $2.9sigma$ level. We study the chargino contributions to $Z$ penguin especially with the vacuum stability constraint. The vacuum decay rate is investigated, and it is shown that the discrepancy can be explained if superparticles are lighter than 4-6 TeV. Correlations with $mathcal{B}(K_Ltopi^0 ubar u)$ and other experimental constraints are also discussed.
Three years after the completion of the next-to-leading order calculation, the status of the theoretical estimates of $epsilon/epsilon$ is reviewed. In spite of the theoretical progress, the prediction of $epsilon/epsilon$ is still affected by a 100% theoretical error. In this paper the different sources of uncertainty are critically analysed and an updated estimate of $epsilon/epsilon$ is presented. Some theoretical implications of a value of $epsilon/epsilon$ definitely larger than $10^{-3}$ are also discussed.
Recently the RBC-UKQCD lattice collaboration presented new results for the hadronic matrix elements relevant for the ratio $varepsilon/varepsilon$ in the Standard Model (SM). With the present knowledge of the Wilson coefficients and isospin breaking effects there is still much room for new physics (NP) contributions to $varepsilon/varepsilon$ which could both enhance or suppress this ratio to agree with the data. The new SM value for the $K^0-bar K^0$ mass difference $Delta M_K$ from RBC-UKQCD is on the other hand by $2sigma$ above the data hinting for NP required to suppress $Delta M_K$. Simultaneously the most recent results for $K^+rightarrowpi^+ ubar u$ from NA62 and for $K_{L}rightarrowpi^0 ubar u$ from KOTO still allow for significant NP contributions. We point out that the suppression of $Delta M_K$ by NP requires the presence of new CP-violating phases with interesting implications for $Ktopi ubar u$, $K_Stomu^+mu^-$ and $K_Ltopi^0ell^+ell^-$ decays. Considering a $Z^prime$-scenario within the SMEFT we analyze the dependence of all these observables on the size of NP still allowed by the data on $varepsilon/varepsilon$. The NP QCD penguin scenario for $varepsilon/varepsilon$ is excluded by SMEFT renormalization group effects in $varepsilon_K$ so that NP effects in $varepsilon/varepsilon$ are governed by electroweak penguins. We also investigate for the first time whether the presence of a heavy $Z^prime$ with flavour violating couplings could generate through top Yukawa renormalization group effects FCNCs mediated by the SM $Z$-boson. The outcome turns out to be very interesting.
We present new results for the matrix elements of the Q_6 and Q_4 penguin operators, evaluated in a large-Nc approach which incorporates important O(N_c^2frac{n_f}{N_c}) unfactorized contributions. Our approach shows analytic matching between short- and long-distance scale dependences within dimensional renormalization schemes, such as MS-bar. Numerically, we find that there is a large positive contribution to the Delta I =1/2 matrix element of Q_6 and hence to the direct CP-violation parameter epsilon/epsilon. We also present results for the Delta I = 1/2 rule in K -> pi pi amplitudes, which incorporate the related and important ``eye-diagram contributions of O(N_c^2frac{1}{N_c}) from the Q_2 operator (i.e. the penguin-like contraction). The results lead to an enhancement of the Delta I = 1/2 effective coupling. The origin of the large unfactorized contributions which we find is discussed in terms of the relevant scales of the problem.
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