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Register a new userBSM Master Formula for $varepsilon/varepsilon$ in the WET Basis at NLO in QCD
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As an important step towards a complete next-to-leading order (NLO) QCD analysis of the ratio $varepsilon/varepsilon$ within the Standard Model Effective Field Theory (SMEFT), we present for the first time the NLO master formula for the BSM part of this ratio expressed in terms of the Wilson coefficients of all contributing operators evaluated at the electroweak scale. To this end we use the common Weak Effective Theory (WET) basis (the so-called JMS basis) for which tree-level and one-loop matching to the SMEFT are already known. The relevant hadronic matrix elements of BSM operators at the electroweak scale are taken from Dual QCD approach and the SM ones from lattice QCD. It includes the renormalization group evolution and quark-flavour threshold effects at NLO in QCD from hadronic scales, at which these matrix elements have been calculated, to the electroweak scale.
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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.
Motivated by the recent findings that the ratio $varepsilon/varepsilon$ in the Standard Model (SM) appears to be significantly below the data we investigate whether the necessary enhancement of this ratio can be obtained in 331 models, in which new physics (NP) contributions to $varepsilon/varepsilon$ and other flavour observables are dominated by tree-level exchanges of a $Z^prime$. NP contributions to $varepsilon/varepsilon$ in these models are governed by the electroweak operator $Q_8$ for which the hadronic matrix element is rather well known so that our analysis of NP contributions is subject to much smaller uncertainties than within the SM. We consider seven 331 models selected in our earlier analysis on the basis of electroweak precision data. Imposing the constraints from $Delta F=2$ transitions we find that only three of these models can provide a significant positive shift in $varepsilon/varepsilon$ up to $6times 10^{-4}$ for $M_{Z^prime}=3$ TeV. Two of them allow simultaneously a supression of ${cal B}(B_{s}to mu^+mu^-)$ by $20%$, bringing the theory closer to the data without any significant impact on the Wilson coefficient $C_9$. The third one provides also the shift $Delta C_9=-0.6$, softening the anomalies in $Bto K^*mu^+mu^-$, without any significant impact on $B_{s}to mu^+mu^-$. NP effects in rare $K$ decays and in $Bto K(K^*) ubar u$ turn out to be small. The flavour structure of 331 models implies that even for $M_{Z^prime}=30$ Tev a shift of $varepsilon/varepsilon$ up to $8times 10^{-4}$ and a significant shift in $varepsilon_K$ can be obtained, while the effects in other flavour observables are small.
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.
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 $varepsilon/varepsilon$.
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.
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