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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.
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Motivated by the recently improved lattice QCD results on the hadronic matrix elements entering $Delta M_{s,d}$ in $B_{s,d}^0-bar B_{s,d}^0$ mixings and the resulting increased tensions between $Delta M_{s,d}$ and $varepsilon_K$ in the Standard Model and CMFV models, we demonstrate that these tensions can be removed in 331 models based on the gauge group $SU(3)_Ctimes SU(3)_Ltimes U(1)_X$ both for $M_{Z^prime}$ in the LHC reach and well beyond it. But the implied new physics (NP) patterns in $Delta F=1$ observables depend sensitively on the value of $|V_{cb}|$. Concentrating the analysis on three 331 models that have been selected by us previously on the basis of their performance in electroweak precision tests and $varepsilon^prime/varepsilon$ we illustrate this for $|V_{cb}|=0.042$ and $|V_{cb}|=0.040$. We find that these new lattice data still allow for positive shifts in $varepsilon^prime/varepsilon$ up to $6times 10^{-4}$ for $M_{Z^prime}=3~TeV$ for both values of $$|V_{cb}|$ but for $M_{Z^prime}=10~TeV$ only for $|V_{cb}|=0.040$ such shifts can be obtained. NP effects in $B_stomu^+mu^-$ and in the Wilson coefficient $C_9$ are significantly larger in all three models for the case of $|V_{cb}|=0.040$. In particular in two models the rate for $B_stomu^+mu^-$ can be reduced by NP by $20%$ for $M_{Z^prime}=3~TeV$ resulting in values in the ballpark of central values from CMS and LHCb. In the third model a shift in $C_9$ up to $C_9^text{NP}=-0.5$ is possible. We also consider the simplest 331 model, analyzed recently in the literature, in which $X=Y$, the usual hypercharge. We find that in this model NP effects in flavour observables are much smaller than in the three models with $X ot=Y$, in particular NP contributions to the ratio $varepsilon^prime/varepsilon$ are very strongly suppressed.
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.
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 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.
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