Anatomy of \(\varepsilon '/\varepsilon \) beyond the standard model
 165 Downloads
 1 Citations
Abstract
We present for the first time a modelindependent 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 \(K\rightarrow \pi \nu {\bar{\nu }}\) decays are briefly discussed. Building on our EFT analysis and the modelindependent 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.
1 Introduction
As far as the shortdistance contributions are concerned, they have been known already for 25 years at nexttoleading order (NLO) [4, 5, 6, 7, 8, 9]. First steps towards nexttonexttoleading order (NNLO) predictions for \(\varepsilon '/\varepsilon \) have been made in [10, 11, 12, 13] and further progress towards a complete NNLO result is under way [14].

The analysis of \(\varepsilon '/\varepsilon \) by the RBCUKQCD collaboration based on their lattice QCD calculation of \(K\rightarrow \pi \pi \) matrix elements [15, 16], as well as the analyses performed in [17, 18] that are based on the same matrix elements but also include isospin breaking effects, find \(\varepsilon '/\varepsilon \) in the ballpark of \((12) \times 10^{4}\). This is by one order of magnitude below the data, but with an error in the ballpark of \(5\times 10^{4}\). Consequently, based on these analyses, one can talk about an \(\varepsilon '/\varepsilon \) anomaly of at most \(3\sigma \).

An independent analysis based on hadronic matrix elements from the Dual QCD (DQCD) approach [19, 20] gives a strong support to these values and moreover provides an upper bound on \(\varepsilon '/\varepsilon \) in the ballpark of \(6\times 10^{4}\).

A different view has been expressed in [21] where, using ideas from chiral perturbation theory, the authors find \(\varepsilon '/\varepsilon = (15 \pm 7) \times 10^{4}\). While in agreement with the measurement, the large uncertainty, that expresses the difficulties in matching long distance and short distance contributions in this framework, does not allow for clearcut conclusions. Consequently, values above \(2\times 10^{3}\), that are rather unrealistic from the point of view of lattice QCD and DQCD, are not excluded in this approach.
Based on the results from RBCUKQCD and the DQCD approach of 2015 and without the inclusion of NNLO corrections mentioned above, a number of analyses have been performed in specific models beyond the SM (BSM) with the goal to obtain a sufficient upward shift in \(\varepsilon '/\varepsilon \) and thereby its experimental value. These include in particular treelevel \(Z^\prime \) exchanges with explicit realization in 331 models [22, 23] or models with treelevel \(Z^0\) exchanges [24, 25] with explicit realization in models with mixing of heavy vectorlike fermions with ordinary fermions [26] and the Littlest Higgs model with Tparity [27]. Also simplified \(Z^\prime \) scenarios [28, 29], the MSSM [30, 31, 32, 33, 34], the typeIII TwoHiggs Doublet model (2HDM) [35, 36], a \(SU(2)_L \otimes SU(2)_R \otimes U(1)_{BL}\) model [37, 38] and the one based on SU(8) symmetry [39] are of help here. On the other hand, as demonstrated in [40], it is very unlikely that leptoquarks are responsible for the \(\varepsilon '/\varepsilon \) anomaly when the constraints from rare semileptonic and leptonic K decays are taken into account.
An important limitation of the recent literature is that it addressed the \(\varepsilon '/\varepsilon \) anomaly only in models in which new physics (NP) entered exclusively through modifications of the Wilson coefficients of SM operators. However, generally, BSM operators with different Dirac structures – like the ones resulting from treelevel scalar exchanges and leading to scalar and tensor operators – or chromomagnetic dipole operators could play a significant role in \(\varepsilon '/\varepsilon \). Until recently, no quantitative judgment of the importance of such operators was possible because of the absence of even approximate calculations of the relevant hadronic matrix elements in QCD. This situation has been changed through the calculation of the matrix elements in question for the chromomagnetic dipole operators by lattice QCD [41] and DQCD [42] and in particular through the calculation of matrix elements of all fourquark BSM operators, including scalar and tensor operators, by DQCD [43]. The first application of these new results for chromomagnetic dipole operators can be found in [36] and in the present paper we will have a closer look at all BSM operators.
Another important question is which of the operators in the lowenergy effective theory can be generated in a shortdistance BSM scenario. A powerful tool for this purpose is the standard model effective field theory (SMEFT) [44, 45], where the SM Lagrangian above the electroweak scale \({\mu _\mathrm {ew}}\sim 100\) GeV and below the scale of new physics \({\mu _{\Lambda }}\gg {\mu _\mathrm {ew}}\) is supplemented by all dimension five and six operators that are invariant under the SM gauge group \(G_\text {SM} = SU(3)_c \otimes SU(2)_L \otimes U(1)_Y\). As we will show, matching the SMEFT at tree level on the \(\Delta S = 1\) effective field theory (EFT) at \({\mu _\mathrm {ew}}\), not all operators that are allowed by the QCD and QED gauge symmetry \(SU(3)_c \otimes U(1)_Q\) are generated.
The goal of the present paper is to perform a general BSM analysis of \(\varepsilon '/\varepsilon \), taking into account all possible operators and exploiting the SMEFT to single out the operators that can be generated in highscale BSM scenarios. In this manner, one can obtain a general view on possible BSM physics behind the emerging \(\varepsilon '/\varepsilon \) anomaly and point out promising directions to be explored in concrete models and exclude those in which the explanation of the data in (1) is unlikely. In the context of SMEFT, constraints from other processes, in particular from \(\varepsilon _K\), \(D^0\)\({\bar{D}}^0\) mixing, and electric dipole moments, play an important role and we will discuss them in the present paper.
One of the highlights of our paper is the derivation of a master formula for \(\varepsilon '/\varepsilon \), recently presented in [46], which can be applied to any theory beyond the SM in which the Wilson coefficients of the operators have been calculated at the electroweak scale. The relevant hadronic matrix elements of BSM operators entering this formula are taken from the DQCD approach and for the SM ones from lattice QCD.
The outline of our paper is as follows. In Sect. 2 we present a complete modelindependent anatomy of \(\varepsilon '/\varepsilon \) from the point of view of the \(\Delta S = 1\) EFT and provide the master formula of \(\varepsilon '/\varepsilon \) beyond the SM. We give also the treelevel matching of SMEFT on the \(\Delta S = 1\) EFT relevant for \(\varepsilon '/\varepsilon \). In Sect. 3 we discuss correlations that arise in SMEFT between \(\varepsilon '/\varepsilon \) and other processes, in particular \(\varepsilon _K\), \(D^0\)\({\bar{D}}^0\) mixing, the electric dipole moment of the neutron, and the decays \(K_{L}\rightarrow \pi ^0\nu {\bar{\nu }}\) and \(K^+\rightarrow \pi ^+\nu {\bar{\nu }}\). Based on the previous section, we derive lessons for model building in Sect. 4 to facilitate the identification of classes of models that are constrained by \(\varepsilon '/\varepsilon \) as well as singling out prime candidates for new physics scenarios behind the \(\varepsilon '/\varepsilon \) anomaly. We summarize the main virtues of our analysis in Sect. 5. In several appendices we collect our conventions, recall useful definitions, and provide the necessary material for the numerical analysis of \(\varepsilon '/\varepsilon \) beyond the SM.
2 Modelindependent anatomy of \(\varepsilon '/\varepsilon \)

Section 2.1 discusses the relevant operators in \({\mathcal {H}}_{\Delta S = 1}^{(3)}\) and their \(K\rightarrow \pi \pi \) matrix elements.

Section 2.2 discusses the RG evolution between the lowest scale \({\mu }\) and \({\mu _\mathrm {ew}}\) and the additional operators in \({\mathcal {H}}_{\Delta S = 1}^{(5)}\) that can play a role.

Section 2.3 summarizes the results of Sects. 2.1 and 2.2 in the form of a convenient master formula of \(\varepsilon '/\varepsilon \).

Section 2.4 discusses the matching of SMEFT onto \({\mathcal {H}}_{\Delta S = 1}^{(5)}\) at \({\mu _\mathrm {ew}}\), singling out the operators that arise at the dimensionsix level, and briefly discusses RG effects in SMEFT above \({\mu _\mathrm {ew}}\).
2.1 \(K\rightarrow \pi \pi \) matrix elements
Since the number of active quark flavours is \(N_f=3\), in principle the fourquark operators with \(q=u,d,s\) are present in (7). However, we expect the contribution to \(K\rightarrow \pi \pi \) matrix elements from operators with flavour structure \(({\bar{s}}d) ({\bar{s}}s)\) to be strongly suppressed^{4} and we will neglect them.
Using Fierz relations to eliminate redundant operators (see “Appendix A” for details), it then follows that there are only \(10 + 10'\) \(({\bar{s}}d)({\bar{u}}u)\) and \(5 + 5'\) \(({\bar{s}}d)({\bar{d}}d)\) linearly independent fourquark operators that contribute to \(\varepsilon '/\varepsilon \) via a nonvanishing \(K\rightarrow \pi \pi \) matrix element and in addition the chromomagnetic dipole operators (\(1 +1'\)). In the amplitude \(A_0\), there are then in total 16 independent matrix elements, seven of which are the ones of the SM fourquark operators and one the chromomagnetic dipole matrix element. In the amplitude \(A_2\), further simplifications arise as the chromomagnetic dipole operator cannot generate a \(\Delta I=3/2\) transition, neither can an operator of the form \({O}_{XAB}^u + {O}_{XAB}^d\), leaving only five linearly independent matrix elements, three of which are present in the SM. We write the number of total matrix elements in the \(I=0,2\) amplitudes as \(16_0 + 5_2\). In “Appendix B”, we specify a nonredundant basis for them.

First lattice calculations for the \(7_0 + 3_2\) matrix elements^{5} generated in the SM have recently been performed by the RBCUKQCD collaboration [15, 16]. These results are in good agreement with the pattern of matrix elements of the relevant QCD and QED penguin operators obtained in the DQCD approach [19, 50, 51, 52].

The \(K\rightarrow \pi \pi \) matrix element of the chromomagnetic dipole operator is presently not accessible directly on the lattice, but can only be estimated by relating it to the analogous \(K\rightarrow \pi \) matrix element via SU(3) chiral symmetry [41]. Recently, the \(K\rightarrow \pi \pi \) matrix element of this operator has been calculated directly for the first time in the DQCD approach in the SU(3) chiral limit [42]. Both results are in good agreement with each other and show that the relevant matrix element is by a factor of three to four smaller than previously expected in the chiral quark model [53], thereby decreasing the impact of these operators on \(\varepsilon '/\varepsilon \). Nevertheless there are NP scenarios where they play an important role (see e.g. [36, 54]).

The matrix elements of the remaining \(8_0+2_2\) linearly independent BSM fourquark matrix elements in Table 5 have only been calculated very recently in DQCD in the SU(3) chiral limit [43] and it will still take some time before corresponding results in lattice QCD will be available. Yet already these approximate results from DQCD can teach us a lot about the relevance of various operators. The scalar and tensor operators \(X=S,T\) belong to this group and their matrix elements cannot be expressed in terms of the SM ones.

the matrix elements present in the SM,

the chromomagnetic dipole matrix element,

the matrix elements of BSM scalar and tensor operators.
2.2 Renormalization group evolution below the electroweak scale
In the previous subsection, we have seen that \(15+15'\) fourquark operators in \({\mathcal {H}}_{\Delta S=1}^{(3)}\) can contribute to \(K\rightarrow \pi \pi \) at the scale \({\mu }\). However, additional fourquark operators are present in the fiveflavour Hamiltonian \({\mathcal {H}}_{\Delta S=1}^{(5)}\) at \({\mu _\mathrm {ew}}\), namely the fourquark operators with flavour structures \(({\bar{s}} d)({\bar{q}} q)\) where \(q=c\) and b. They can contribute to \(\varepsilon '/\varepsilon \) indirectly if they undergo QCD and/or QED RG mixing with \(q = u, d\) operators. The same is true for the operators with \(q=s\) that were already present for \(N_f=3\), but did not contribute directly (at least in our approximation). In principle, also semileptonic operators can contribute, since they mix under QED into fourquark operators, but we will neglect them in the following, since they are typically strongly constrained from semileptonic kaon decays (as demonstrated for leptoquark models in [40]).
To evolve the Wilson coefficients from \({\mu _\mathrm {ew}}\) down to the scale \({\mu }\) where the matrix elements are evaluated, the anomalous dimension matrices (ADMs) are required. The QCD and QED oneloop ADMs for the linearly independent set of fourquark and dipole operators can be extracted from the literature [55, 56, 57] and we have implemented them in the open source tool wilson [58] that allows to solve the RG equations numerically.

The vector operators \(O^{c,b}_{VAB}\) and their colourflipped counterparts, as well as the operators \(O^{s,d}_{SAB}\) with \(A\ne B\), mix into \(O^{u,d}_{VAB}\) at one loop in QCD and QED, specifically into the QCD and QED penguin operators present in the SM.

For scalar and tensor operators \(O^{q}_{XAA}\) (\(X=S\) or T) there is instead no mixing among operators with different q. This implies in particular that the operators \(O^{s,c,b}_{XAA}\) cannot mix into fourquark operators that have nonvanishing \(K\rightarrow \pi \pi \) matrix elements. However, they do mix at one loop in QCD into the chromomagnetic dipole operators \(O_{8g}^{(\prime )}\) and in QED into the electromagnetic ones.
2.3 Master formula for \(\varepsilon '/\varepsilon \) beyond the SM
Having both the RG evolution and all matrix elements at the lowenergy scale \({\mu }\) for the first time at hand allowed us recently [46] to present in a letter a master formula for \((\varepsilon '/\varepsilon )_\text {BSM}\) that exhibits its dependence on each Wilson coefficient at the scale \({\mu _\mathrm {ew}}\) and consequently is valid in any theory beyond the SM that is free from nonstandard light degrees of freedom below the electroweak scale. We will now discuss various ingredients and technical details which led to this formula.
The dimensionless coefficients \(p_{ij}^{(I)}({\mu _\mathrm {ew}},{\mu })\) include the QCD and QED RG evolution from \({\mu _\mathrm {ew}}\) to \({\mu }\) for each Wilson coefficient as well as the relative suppression of the contributions to the \(I=0\) amplitude due to \({\text {Re}A_2} / {\text {Re}A_0}\ll 1\) for the matrix elements \(\langle O_j ({\mu }) \rangle _I\) of all the operators \(O_j\) present at the lowenergy scale, see “Appendix B”. The index j includes also i so that the effect of selfmixing is included. The \(P_i({\mu _\mathrm {ew}})\) do not depend on \({\mu }\) to the considered order, because the \({\mu }\)dependence cancels between matrix elements and the RG evolution operator. Moreover, it should be emphasized that their values are modelindependent and depend only on the SM dynamics below the electroweak scale, which includes short distance contributions down to \({\mu }\) and the long distance contributions represented by the hadronic matrix elements. The BSM dependence enters our master formula in (18) only through the Wilson coefficients \(C_i({\mu _\mathrm {ew}})\) and \(C^\prime _i({\mu _\mathrm {ew}})\). That is, even if a given \(P_i\) is nonzero, the fate of its contribution depends on the difference of these two coefficients. In particular, in models with exact leftright symmetry this contribution vanishes as first pointed out in [62].
The numerical values of the \(P_i({\mu _\mathrm {ew}})\) are collected in the tables in “Appendix C”. As seen in (19), the \(P_i\) depend on the hadronic matrix elements \(\langle O_j ({\mu }) \rangle _I\) and the RG evolution factors \(p_{ij}^{(I)}({\mu _\mathrm {ew}}, {\mu })\). The numerical values of the hadronic matrix elements rely on lattice QCD in the case of SM operators and DQCD in the case of BSM operators as summarized above. Consequently, the uncertainties of the \(P_i\) are of the order of 5–7% resulting from SM matrix elements and at the level of \(20\%\) coming from BSM matrix elements.

The large \(P_i\) values for operators with flavour content \(({\bar{s}}d)({\bar{u}}u)\) and \(({\bar{s}}d)({\bar{d}}d)\) in Class A can be traced back to the large values of the matrix elements \(\langle Q_{7,8}\rangle _2\), the dominant electroweak penguin operators in the SM, and the enhancement of the \(I=2\) contributions relative to \(I=0\) ones by \(\omega \approx 22\).

The small \(P_i\) values in Class B are due to the fact that they are all proportional to \(\langle O_{8g} \rangle _0\), which has recently been found to be much smaller than previously expected [41, 42]. Moreover, as \(\langle O_{8g}\rangle _2=0\), all contributions in this class are suppressed by the factor \(1/\omega \) relative to contributions from other classes.

The large \(P_i\) values in Classes C–D can be traced back to the large hadronic matrix elements of scalar and tensor operators calculated recently in [43]. Due to the smallness of \(\langle O_{8g} \rangle _0\), the contribution of the chromomagnetic dipole operator in Classes C–D is negligible.

While the \(I=0\) matrix elements of the operators in Class E cannot be expressed in terms of SM ones, the \(I=2\) matrix elements can, and the large \(P_i\) values can be traced back to the large SM matrix elements \(\langle Q_{7,8}\rangle _2\).
2.4 Matching from SMEFT onto \(\Delta S = 1\) EFT
The SMEFT operators and accordingly their Wilson coefficients are defined in terms of the gauge and fermion fields in the unbroken phase of the SM, see also “Appendix D” for notation and definitions. In contrast to the \(\Delta S=1\) EFT discussed above, there is no preferred weak basis for the (massless) fermion fields in SMEFT and the wouldbe mass basis is not \(SU(2)_L\) invariant. Instead, in the following we use the freedom of SU(3)flavour rotations to work in a weak basis where the running downtype quark mass matrix is diagonal at the electroweak scale (cf. [63]).
Number of linearly independent fourquark operators with flavour content \(({\bar{s}}d)({\bar{u}}_i u_i)\) and \(({\bar{s}}d)({\bar{d}}_i d_i)\) in the \(\Delta S = 1\) EFT with \(N_f=5\) that contribute to \(\varepsilon '/\varepsilon \) (first row). Number of nonvanishing matching contributions from SMEFT due to fourquark operators and modified righthanded \(W^\pm \) couplings for dimensionsix operators at tree level (second row)
\(u_i = u\)  \(u_i = c\)  \(d_i = d\)  \(d_i=s\)  \(d_i = b\)  \(\Sigma \)  

\(\Delta S = 1\) EFT  \(10 + 10'\)  \(8 + 8'\)  \(5 + 5'\)  \(5 + 5'\)  \(8 + 8'\)  \(36 + 36'\) 
SMEFT  \(9 + 9'\)  \(8 + 8'\)  \(3 + 3'\)  \(3 + 3'\)  \(4 + 4'\)  \(27 + 27'\) 

fourquark operators,

\(\psi ^2 H^2 D\) operators describing modified \(Z^0\) or \(W^\pm \) couplings, and

chromomagnetic dipole operators.
A nontrivial consequence of SMEFT is that none of the operators \(O_{SLR}^{u_i}\), \(O_{SLL}^{d_i}\), \(O_{TLL}^{d_i}\), or their chirality and colourflipped counterparts, are generated in the lowenergy EFT in the treelevel matching of SMEFT fourquark operators. The reason is that these operators conserve only electric charge, but not hypercharge. Only the operator \({\widetilde{O}}_{SLR}^{u}\) eventually contributes to \(\varepsilon '/\varepsilon \), namely through the righthanded \(W^\pm \) coupling discussed in “Appendix E.2”. This contribution is not subject to the hypercharge constraint, as it only arises after electroweak symmetry breaking. Below \({\mu _\mathrm {ew}}\) this leads to vanishing Wilson coefficients of \(9 + 9'\) linearly independent operators in the \(\Delta S = 1\) EFT with \(N_f= 5\), reducing the number of nonredundant \(\Delta S=1\) fourquark operators that contribute to \(\varepsilon '/\varepsilon \) from \(36 + 36'\) to \(27 + 27'.\)^{9} At the oneloop level, QCD and QED running from \({\mu _\mathrm {ew}}\) down to \({\mu }\) does not regenerate these operators. This is summarized in Table 1. Consequently, in SMEFT the number of linearly independent operators that contribute directly to \(\varepsilon '/\varepsilon \) via nonvanishing \(K\rightarrow \pi \pi \) matrix elements is reduced from \(15 + 15'\) to \(12 + 12'\), out of which only \(5 + 5'\) are nonstandard. The chromomagnetic dipole operators are not subject to these considerations and their number equals in SMEFT and \(\Delta S = 1\) EFT.
Consequently, in SMEFT only the operators in Classes A–C in (8)–(12) contribute to \(\varepsilon '/\varepsilon \) through fourquark operators, and a single operator from Class E in (14) (and its chiralityflipped counterpart) through the righthanded \(W^\pm \) coupling. Inspecting the matching relations listed in “Appendix E”, these three classes, expressed in terms of the SMEFT operators of Tables 11 and 12, are as follows
Effective scales of SMEFT operators contributing to \(\varepsilon _K\) and CP violation in \(D^0\)\({\bar{D}}^0\) mixing, defined as in (25) and (31), respectively. These scales give an indication of the sensitivity to the individual operators. Note however that the normalization is different for \(\Delta S=2\) and \(\Delta C=2\)
\({\mathcal {C}}_i\)  \(\sigma _i\)  \(\Lambda _i\)  \({\mathcal {C}}_i\)  \(\sigma _i\)  \(\Lambda _i\)  \({\mathcal {C}}_i\)  \(\sigma _i\)  \(\Lambda _i\) 

\(\Delta S = 2\)  
\(\big [{{\mathcal {C}}}_{qq}^{(1)}\big ]_{2121}\)  −  13.3 PeV  \(\big [{{\mathcal {C}}}_{qd}^{(1)}\big ]_{2121}\)  \(+\)  104.6 PeV  \(\big [{{\mathcal {C}}}_{dd}^{}\big ]_{2121}\)  −  13.3 PeV 
\(\big [{{\mathcal {C}}}_{qq}^{(3)}\big ]_{2121}\)  −  13.3 PeV  \(\big [{{\mathcal {C}}}_{qd}^{(8)}\big ]_{2121}\)  \(+\)  126.5 PeV  
\(\Delta C = 2\)  
\(\big [\widehat{{\mathcal {C}}}_{qq}^{\,(1)}\big ]_{1212}\)  −  14.1 PeV  \(\big [\widehat{{\mathcal {C}}}_{qu}^{\,(1)}\big ]_{1212}\)  \(+\)  29.2 PeV  \(\big [\widehat{{\mathcal {C}}}_{uu}^{\,}\big ]_{1212}\)  −  14.1 PeV 
\(\big [\widehat{{\mathcal {C}}}_{qq}^{\,(3)}\big ]_{1212}\)  −  14.1 PeV  \(\big [\widehat{{\mathcal {C}}}_{qu}^{\,(8)}\big ]_{1212}\)  \(+\)  33.3 PeV 
3 Modelindependent constraints in SMEFT

by \(SU(2)_L\) relations between operators involving lefthanded quark doublets that require a CKM rotation to go to the mass basis for the up or downtype quarks,

by flavourdependent RG effects due to the mixing of operators in SMEFT given in Sect. 2.4.
3.1 \(\Delta S=2\)
As the SM describes the experimental value of \(\varepsilon _K\) rather well, \(\text {Im}\,{\mathcal {C}}_i({\mu _\mathrm {ew}})\) corresponding to the largest \(\Lambda _i\) must be suppressed most strongly, thereby probing the largest NP scales. Given the experimental measurement and theory uncertainty of this ratio (25) can be used to constrain SMEFT Wilson coefficients from \(\varepsilon _K\) in phenomenological analyses.
Given these huge scales probed by \(\varepsilon _K\), any model predicting sizable direct CP violation in \(\Delta S=1\) can only be viable if it does not induce too large contributions to indirect CP violation in \(\Delta S=2\).
As discussed above, an important source of constraints are \(SU(2)_L\) relations between operators with lefthanded quark fields, involving a CKM rotation between the mass bases for up and downtype quarks. The first four operators in Table 2 are a prime example of this effect. \(\big [{{\mathcal {C}}}_{qq}^{(1,3)}\big ]_{2121}\) contribute to the matching of \(C_{VLL}^{u_i}\) and \(\big [{{\mathcal {C}}}_{qd}^{(1,8)}\big ]_{2121}\) to the matching of \(C_{VRL}^{u_i}\), as seen from the matching conditions in Sect. 2.4. For both cases \(i = 1, 2\), the suppression by the Cabibbo angle \(V_{us}^{} V_{ud}^* \sim V_{cs}^{} V_{cd}^* \sim 0.23\) is of first order in \(\varepsilon '/\varepsilon \) and furthermore the operators with \(i = 2\) have no direct \(K\rightarrow \pi \pi \) matrix elements, which introduces for them another suppression of \(\alpha _{s,e}/(4\pi )\) from RG mixing in \(\varepsilon '/\varepsilon \) compared to \(i = 1\). When considering NP effects in only a single operator, clearly the strong constraint from \(\varepsilon _K\) excludes any visible effect in \(\varepsilon '/\varepsilon \) induced by imaginary parts of these Wilson coefficients.
3.2 \(\Delta C=2\)
Although the SM contribution to the \(D^0\)\({\bar{D}}^0\) mixing amplitude is dominated by poorly known longdistance contributions, the structure of the CKM matrix implies that the SM contribution to CP violation in mixing can at most reach the percent level [82]. This fact can be used to constrain the imaginary part of the mixing amplitude.
3.3 Interplay of \(\Delta S=2\) and \(\Delta C=2\)
We finally note that, in principle, since each of the observables only probes a single direction in the space of Wilson coefficients, cancellations could be arranged that remove these constraints. In view of the severeness of the constraints and the fact that delicate cancellations are not invariant under the RG evolution, we consider such cancellations unrealistic.
3.4 Neutron electric dipole moment
Since \(\varepsilon '/\varepsilon \) probes CP violation associated to the first two generations of quarks, it is natural to ask whether there is any constraint from the electric dipole moment (EDM) of the neutron, which is a sensitive probe of flavourdiagonal CP violation involving up and down quarks. In principle, CPviolating fourquark operators can directly induce a neutron EDM. Correlations of the neutron EDM with \(\varepsilon '/\varepsilon \) from these operators have been considered recently in [37, 85]; they require the knowledge of the matrix elements of these operators, which are relatively poorly known.
We finally note that beyond the neutron EDM, also the EDMs of diamagnetic atoms are sensitive to CP violation in dipole operators and fourquark operators, in addition to leptonic and semileptonic CP violation. In principle a global analysis of the various EDM measurements to disentangle the different shortdistance sources of CP violation would be useful, but currently suffers from many unknown longdistance contributions, see [93] for a recent review.
3.5 \(K\rightarrow \pi \nu {\bar{\nu }}\) and \(K\rightarrow \pi \ell ^+\ell ^\)
In specific NP models one often finds correlations between BSM contributions to \(\varepsilon '/\varepsilon \) and rare kaon decays, in particular with \(K^+\rightarrow \pi ^+\nu {\bar{\nu }}\) and \(K_{L}\rightarrow \pi ^0\nu {\bar{\nu }}\). In fact in all papers that addressed the \(\varepsilon '/\varepsilon \) anomaly listed in the introduction [22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40] such correlations have been investigated. Such correlations will play an important role in distinguishing various models when the theoretical status of \(\varepsilon '/\varepsilon \) improves and the branching ratios for rare kaon decays will be well measured.

modified \(Z^0\) or \(W^\pm \) couplings contributing to \(\varepsilon '/\varepsilon \) and neutral or charged current semileptonic decays, respectively,

semileptonic operators that contribute directly to semileptonic decays and mix into \(\Delta S=1\) fourquark operators by QED or electroweak RG effects, thereby contributing indirectly to \(\varepsilon '/\varepsilon \),

fourquark operators mixing into semileptonic operators by QED or electroweak RG effects and contributing directly to \(\varepsilon '/\varepsilon \).
The coefficients in (43), which contribute to \(\varepsilon '/\varepsilon \) only via modified lefthanded \(W^\pm \) couplings, contribute also to FCNC B decays via modified \(Z^0\) couplings. Barring unrealistic cancellations, visible effects in \(\varepsilon '/\varepsilon \) induced by these couplings are excluded since they would lead to excessive effects e.g. in the decays \(B_s\rightarrow \mu ^+\mu ^\) and \(B^0\rightarrow \mu ^+\mu ^\).
3.6 \(\Delta C = 1\)
Fourquark SMEFT operators containing downtype quarks generated by the exchange of scalar or vector mediators at tree level. The second column gives the representation under \(G_\text {SM} = SU(3)_c \otimes SU(2)_L \otimes U(1)_Y\)
Spin  Rep.  \({\mathcal {O}}^{(1)}_{qq}\)  \({\mathcal {O}}^{(3)}_{qq}\)  \({\mathcal {O}}^{(1)}_{qu}\)  \({\mathcal {O}}^{(8)}_{qu}\)  \({\mathcal {O}}^{(1)}_{qd}\)  \({\mathcal {O}}^{(8)}_{qd}\)  \({\mathcal {O}}^{(1)}_{ud}\)  \({\mathcal {O}}^{(8)}_{ud}\)  \({\mathcal {O}}_{dd}\)  \({\mathcal {O}}^{(1)}_{quqd}\)  \({\mathcal {O}}^{(8)}_{quqd}\) 

0  \(\left( 1,2\right) _{\frac{1}{2}}\)  \(\times \)  \(\times \)  \(\times \)  \(\times \)  \(\times \)  
\(\left( 8,2\right) _{\frac{1}{2}}\)  \(\times \)  \(\times \)  \(\times \)  \(\times \)  \(\times \)  
1  \(\left( 1,1\right) _0\)  \(\times \)  \(\times \)  \(\times \)  \(\times \)  \(\times \)  
\(\left( 1,1\right) _1\)  \(\times \)  \(\times \)  
\(\left( 8,1\right) _0\)  \(\times \)  \(\times \)  \(\times \)  \(\times \)  \(\times \)  \(\times \)  
\(\left( 8,1\right) _1\)  \(\times \)  \(\times \)  
\(\left( 1,3\right) _0\)  \(\times \)  
\(\left( 8,3\right) _{0}\)  \(\times \)  \(\times \) 
4 Implications for model building
Having discussed the general modelindependent anatomy of \(\varepsilon '/\varepsilon \) below the electroweak scale and the consequences of \(SU(2)_L\otimes U(1)_Y\) gauge invariance within SMEFT, we are now in a position to discuss the implications for the possible effects in BSM scenarios with new sources of CP violation where BSM effects in \(\varepsilon '/\varepsilon \) are encoded in the imaginary part of Wilson coefficients of dimensionsix SMEFT operators.
The size of the coefficients \(P_i\) in our master formula presented in Sect. 2.3, together with the matching conditions in Sect. 2.4, already indicate which scenarios are more promising than others to explain a deviation from the SM in \(\varepsilon '/\varepsilon \). However, in a concrete BSM scenario, the Wilson coefficients with the highest values of \(P_i\) could vanish or be suppressed by small couplings. Consequently without additional dynamical assumptions or specific models no clearcut conclusions can be made. While a comprehensive discussion of models is beyond the scope of this paper, in the following subsections we will discuss a number of general implications on the basis of simplified models with a single treelevel mediator.

through contributions from chromomagnetic dipole operators to \(\varepsilon '/\varepsilon \) that do not affect \(\varepsilon _K\) [36],

through contributions from modified \(Z^0\) couplings to \(\varepsilon '/\varepsilon \) [25, 26, 27] that only enter \(\varepsilon _K\) through topquark Yukawa RG effects [24],

through contributions from modified righthanded \(W^\pm \) couplings to \(\varepsilon '/\varepsilon \) that do not affect \(\varepsilon _K\) [85],

through loopinduced contributions to \(\varepsilon '/\varepsilon \) in conjunction with an accidental suppression of the contributions to \(\varepsilon _K\) arising in models with Majorana fermions like the MSSM [31].
We start by listing all the possible treelevel models in Sect. 4.1. After discussing the scalar scenario in Sect. 4.2, we will comment on the challenges of models with vector mediators in Sect. 4.3 and discuss the generation of modified \(Z^0\) and \(W^\pm \) couplings in Sect. 4.4.
4.1 Treelevel mediators
The simplest models giving rise to a NP contribution to \(\varepsilon '/\varepsilon \) are models with a single treelevel mediator generating a fourquark operator. Given the large scales probed by \(\varepsilon '/\varepsilon \), clearly also models without treelevel FCNCs can give a sizable contribution to \(\varepsilon '/\varepsilon \). Nevertheless, the treelevel models can serve as benchmark cases exhibiting generic features of larger classes of models.
In Table 3, we list all the possible treelevel mediators that can generate any of the fourquark operators that give a matching contribution to \(\Delta S=1\) at \({\mu _\mathrm {ew}}\) [99]. We have omitted states that permit baryon number violating couplings.^{14} These states are either \(SU(3)_c\) triplets (leptoquarks) or sextets (diquarks), and the former are popular scenarios to explain current anomalies in semileptonic B decays. Further we omitted the possibility of a heavy vector doublet \((1, 2)_{\frac{1}{2}}\). For a scalar mediator, the SM gauge quantum numbers then only allow two possible representations: a heavy Higgslike doublet under \(SU(2)_L\) that is either a singlet or an octet under \(SU(3)_c\). For a vector mediator, there are six possibilities, \(SU(3)_c\) singlets or octets that are \(SU(2)_L\) singlets or triplets.
Further treelevel contributions to \(\varepsilon '/\varepsilon \) can arise from models inducing modified \(W^\pm \) or \(Z^0\) couplings and will be discussed in Sect. 4.4.
4.2 Scalar operators from scalar mediators
The novel feature after the calculation of hadronic matrix elements of BSM operators in [43] is the importance of scalar and tensor fourquark operators. As indicated in Fig. 1 and shown in Sect. 2.4, these matrix elements are relevant in scenarios that generate the SMEFT operators \({{\mathcal {O}}}_{quqd}^{(1,8)}\) at the electroweak scale. Table 3 shows that these operators can be mediated at tree level by heavy Higgs doublets, either a coloursinglet or a colouroctet Higgs.
Some of the operators in (53) are also constrained by the \(\Delta F=2\) or \(\Delta F=0\) processes discussed in Sect. 3. In the basis where the downtype quark mass matrix is diagonal, \(\varepsilon _K\) is sensitive to \(\big [{{\mathcal {C}}}_{qd}^{(1)}\big ]_{2121}\). As seen from (53), this Wilson coefficient is proportional to \(X_d^{12*} X_d^{21}\). Interestingly, this means that an imaginary part in one of the couplings \(X_d^{12}\) or \(X_d^{21}\) is not constrained by \(\varepsilon _K\) at all, but could well generate a visible effect in \(\varepsilon '/\varepsilon \). Similar comments apply to the \(\Delta C=2\) constraint on flavour offdiagonal couplings in \(X_u\).
Since the operators of type \({{\mathcal {O}}}_{quqd}^{(1,8)}\) can be generated, in models with scalar mediators also the neutron EDM, induced at the oneloop level as discussed in Sect. 3.4, can be a relevant constraint.
SMEFT operators of type \(\psi ^2H^2D\) inducing corrections to \(W^\pm \) and \(Z^0\) couplings, generated by the treelevel mixing of SM fields with heavy vectorlike quarks or vector fields. The second column gives the representation under \(G_\text {SM} = SU(3)_c \otimes SU(2)_L \otimes U(1)_Y\)
Spin  Rep.  \({\mathcal {O}}^{(1)}_{Hq}\)  \({\mathcal {O}}^{(3)}_{Hq}\)  \({\mathcal {O}}^{}_{Hd}\)  \({\mathcal {O}}^{}_{Hud}\) 

\(\frac{1}{2}\)  \(\left( 3,1\right) _{\frac{2}{3}}\)  \(\times \)  \(\times \)  
\(\left( 3,1\right) _{\frac{1}{3}}\)  \(\times \)  \(\times \)  
\(\left( 3,3\right) _{\frac{1}{3}}\)  \(\times \)  \(\times \)  
\(\left( 3,3\right) _{\frac{2}{3}}\)  \(\times \)  \(\times \)  
\(\left( 3,2\right) _{\frac{1}{6}}\)  \(\times \)  \(\times \)  
\(\left( 3,2\right) _{\frac{5}{6}}\)  \(\times \)  
1  \(\left( 1,1\right) _{0}\)  \(\times \)  \(\times \)  
\(\left( 1,1\right) _{1}\)  \(\times \)  
\(\left( 1,3\right) _{0}\)  \(\times \) 
4.3 Models with vector mediators
As shown in Sect. 3.3, the operators \({{\mathcal {O}}}_{qq}^{(1,3)}\) are strongly constrained by CP violation in \(K^0\)\({\bar{K}}^0\) and \(D^0\)\({\bar{D}}^0\) mixing, precluding any visible effect in \(\varepsilon '/\varepsilon \), barring unrealistic cancellations that are not stable under RG evolution. Consequently, models with a heavy mediator that only couples to lefthanded quark doublets are not among the prime candidates to explain a possible deviation from the SM in \(\varepsilon '/\varepsilon \).
In the case of the operators \({{\mathcal {O}}}_{qd}^{(1,8)}\), more flavour index combinations contribute to the \(\Delta S=1\) matching at \({\mu _\mathrm {ew}}\) as seen in Sect. 2.4, since they can also contribute via righthanded downtype quarks and lefthanded uptype quarks. Nevertheless, a contribution to \(\varepsilon _K\) is generated either by the 12coupling^{16} to righthanded or to lefthanded downtype quarks. Consequently, comparably stringent bounds as in the case of \({{\mathcal {O}}}_{qu}^{(1,8)}\) apply.
We finally note that models where a vector mediator dominantly contributes to \(\varepsilon '/\varepsilon \) via the purely righthanded fourquark operators \({{\mathcal {O}}}_{ud}^{(1,8)}\) or \({{\mathcal {O}}}_{dd}^{}\) are subject to similar constraints from \(\Delta F=2\) and dijets, but their contributions to \(\varepsilon '/\varepsilon \) are not chirally enhanced, as shown in Sect. 2.3, such that a sizable contribution to \(\varepsilon '/\varepsilon \) is even more difficult to attain.
4.4 Models with modified electroweak couplings
Apart from a treelevel exchange of heavy scalar or vector bosons, \(\varepsilon '/\varepsilon \) can also arise at tree level in the SMEFT from the operators of type \(\psi ^2H^2D\) that induce modified couplings to the \(Z^0\) and \(W^\pm \) bosons. In the broken phase of the SM, these contributions can be seen as arising from the mixing between SM fermion or boson fields with heavy vectorlike fermions or vector bosons after electroweak symmetry breaking. In Table 4, we list all the possible vectorlike fermion or vector boson representations that generate any of the \(\psi ^2H^2D\) operators that give a matching contribution to \(\Delta S=1\) at \({\mu _\mathrm {ew}}\) [99].
The vectorlike fermion representations have already been discussed in detail in the context of \(\varepsilon '/\varepsilon \) in [26], with the exception of the state \((3,1)_{2/3}\) that transforms like a righthanded uptype quark singlet. In this case, one gets \(C_{Hq}^{(1)}=C_{Hq}^{(3)}\), such that there is no flavourchanging \(Z^0\) coupling and thus no contribution to semileptonic FCNCs (cf. Sect. 3.5), but a contribution to \(\varepsilon '/\varepsilon \) can nevertheless arise from a modified lefthanded \(W^\pm \) coupling.
For the vector triplet \((1,3)_0\), the analogous contribution to the T parameter is absent, so the \(Z^0\)mediated contribution to \(\varepsilon '/\varepsilon \) could be sizable.
The \(SU(2)_L\) singlet charged gauge boson \((1, 1)_1\) could arise as the lowenergy limit of a broken leftright symmetry (see e.g. [103]). In this case, the contribution to \(\varepsilon '/\varepsilon \) is mediated by a righthanded \(W^\pm \) coupling, such that \(\varepsilon _K\) gives no constraint.
4.5 Models with dipole operators
The chromomagnetic dipole operators \(O_{8g}^{(\prime )}\) can arise in various BSM scenarios. While the corresponding matrix element and thus the value of \(P_i\) in our master formula is small, the absence of modelindependent constraints on this contribution makes it nevertheless interesting.

If they dominantly match onto the scalar \(\Delta S=1\) operators with flavour \(({\bar{s}}d)({\bar{u}}u)\) in Class C, these have themselves also nonvanishing \(K\rightarrow \pi \pi \) matrix elements and contribute directly to \(\varepsilon '/\varepsilon \), such that the indirect contribution via the dipole operator is negligible.

If they dominantly match onto the scalar \(\Delta S=1\) operators with flavour \(({\bar{s}}d)({\bar{c}}c)\) in Class B, they indeed contribute to \(\varepsilon '/\varepsilon \) exclusively via the dipole Wilson coefficient at the lowenergy scale.

If the scalars couple dominantly to top quarks (see e.g. [36]), these operators do not match at treelevel onto the \(\Delta S=1\) EFT (where top quarks have already been integrated out), but RG evolution above \({\mu _\mathrm {ew}}\) (cf. Sect. 2.4) will generate the SMEFT dipole operator \({{\mathcal {O}}}_{dG}^{}\).
In models with heavy vectors but no heavy fermions, we expect that typically fourquark operator contributions are more important than loopinduced dipole operator contributions, again with the possible exception of top quarks, where RGinduced effects above \({\mu _\mathrm {ew}}\) are relevant.
In models with new heavy vectorlike fermions that couple to the Higgs doublet, sizable contributions to the dipole operator can be generated from a diagram with a SM Higgs in the loop. This gives an important constraint in models with partial quark compositeness [54, 104, 105, 106, 107].
Finally, there can of course also be loop contributions at \({\mu _{\Lambda }}\) with only new heavy particles in the loop. This has been for example studied in MSSM [30, 31, 32, 33, 34], where scalar operators are usually omitted because they are suppressed by lightquark Yukawa couplings, although some might be \(\tan \beta \) enhanced, whereas the oneloop contribution to the dipole operator is not suppressed.
5 Summary
We have presented for the first time a modelindependent anatomy of the ratio \(\varepsilon '/\varepsilon \) in the context of the \(\Delta S = 1\) EFT with operators invariant under QCD and QED and in the context of the SMEFT with the operators invariant under the full SM gauge group. This was only possible thanks to the very recent calculations of the \(K\rightarrow \pi \pi \) matrix elements of BSM operators, namely of the chromomagnetic dipole operators by lattice QCD [41] and DQCD [42] and in particular through the calculation of matrix elements of all fourquark BSM operators, including scalar and tensor operators, by DQCD [43]. Even if the latter calculations have been performed in the chiral limit, they offer for the first time a look into the world of BSM operators contributing to \(\varepsilon '/\varepsilon \).
Our main goal was to identify those new physics scenarios which are probed by \(\varepsilon '/\varepsilon \) and which could help to explain the emerging anomaly in \(\varepsilon '/\varepsilon \), which is signalled both by lattice QCD results and results from the DQCD approach. To this end we have derived a master formula for \(\varepsilon '/\varepsilon \), presented already in [46], which can be applied to any theory beyond the SM 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 DQCD approach and the SM ones from lattice QCD.
In the last three years a number of analyses, addressing the \(\varepsilon '/\varepsilon \) anomaly in concrete models, appeared in the literature (see list at the beginning of our paper) but they concentrated on models in which NP entered exclusively through modifications of the Wilson coefficients of SM operators. In particular the Wilson coefficient of the dominant electroweak penguin operator \(Q_8\) plays an important role in this context as its hadronic matrix element is chirally enhanced and in contrast to the QCD penguin operator \(Q_6\) this contribution is not suppressed by the factor \(1/\omega \approx 22\) related to the \(\Delta I=1/2\) rule. While we confirm these findings through the analysis of models that generate operators of Class A, this is a significant limitation if one wants to have a general view of possible BSM scenarios responsible for the \(\varepsilon '/\varepsilon \) anomaly. In particular, in the absence of even approximate values of hadronic matrix elements of BSM operators, no complete modelindependent analysis was possible until recently.
The recent calculations of BSM \(K\rightarrow \pi \pi \) matrix elements, in particular of those of scalar and tensor operators in [43], combined with the EFT and in particular SMEFT analyses presented in our paper, widened significantly our view on BSM contributions to \(\varepsilon '/\varepsilon \).

It opens the road to the analyses of \(\varepsilon '/\varepsilon \) in any theory beyond the SM and allows with the help of the master formula in (18) [46], with details presented here, to search very efficiently for BSM scenarios behind the \(\varepsilon '/\varepsilon \) anomaly. In particular the values of \(P_i\) collected in “Appendix C” indicate which routes are more promising than others, both in the context of the lowenergy EFT and SMEFT. By implementing our results in the open source code flavio [59], testing specific BSM theories becomes particularly simple.

Through our SMEFT analysis we were able to identify correlations between \(\varepsilon '/\varepsilon \) and various observables that depend sensitively on the operators involved. Here \(\Delta S=2\), \(\Delta C=2\) and electromagnetic dipole moments (EDM) play a prominent role but also correlations with \(\Delta S=1\) and \(\Delta C=1\) provide valuable informations.

Treelevel vector exchanges, like \(Z^\prime \) and \(G^\prime \) contributions, discussed already by various authors, can be responsible for the observed anomaly. In these scenarios one has to face in general important constraints from \(\Delta S=2\) and \(\Delta C=2\) transitions as well as direct searches and often some fine tuning is required. Here the main role is played by the electroweak operator \(Q_8\) with its Wilson coefficient significantly modified by NP.

Models with treelevel exchange of heavy colourless or coloured scalars are a new avenue, opened by the results for BSM operators from DQCD in [43]. In particular scalar and tensor operators, having chirally enhanced matrix elements and consequently large coefficients \(P_i\), are candidates for the explanation of the anomaly in question. Moreover, some of these models, in contrast to models with treelevel \(Z^\prime \) and \(G^\prime \) exchanges, are free from both \(\Delta S=2\) and \(\Delta C=2\) constraints. The EDM of the neutron is an important constraint for these models, depending on the couplings, but does not preclude a sizable NP effect in \(\varepsilon '/\varepsilon \).

Models with modified \(W^\pm \) or \(Z^0\) couplings can induce sizable effects in \(\varepsilon '/\varepsilon \) without appreciable constraints from semileptonic decays such as \(K^+\rightarrow \pi ^+\nu {\bar{\nu }}\) or \(K_L\rightarrow \pi ^0\ell {\bar{\ell }}\). In the case of a SM singlet \(Z'\) mixing with the \(Z^0\), sizable \(Z^0\)mediated contributions are disfavoured by electroweak precision tests.
On the shortdistance side the NNLO results for QCD penguins should be available soon [14]. The dominant NNLO corrections to electroweak penguins have been calculated almost 20 years ago [11] and, as we have pointed out, play a significant role in removing the scale uncertainty in \(m_t(\mu )\) and the uncertainty due to renormalization scheme dependence. Moreover, as we have seen, its inclusion increases the size of the \(\varepsilon '/\varepsilon \) anomaly. With present technology a complete NNLO calculation, using the results in [12], should be feasible in a not too distant future. As far as BSM operators are concerned, a NLO analysis of their Wilson coefficients is in progress, but its importance is not as high as of hadronic matrix elements due to significant additional parametric uncertainties residing in any NP model. In any case, in the coming years the ratio \(\varepsilon '/\varepsilon \) is expected to play a significant role in the search for NP. In this respect, the results presented here will be helpful in disentangling potential models of new CP violating sources beyond the SM as well as constraining the magnitude of their effects.
Footnotes
 1.
Isospin breaking corrections have been considered in [47, 48] and have been taken into account in the SM analyses in [17, 18]. There they play a significant role in suppressing the \(\text {Im}A_0\) contribution relatively to the \(\text {Im}A_2\) one, making the cancellation between these two contributions stronger. However, in BSM scenarios, such a strong cancellation is not expected and typically contributions to \(\text {Im}A_2\) dominate as they are not suppressed by the factor \(1/\omega \approx 22\) in contrast to \(\text {Im}A_0\). Therefore, the inclusion of isospin breaking effects in the BSM contributions calculated by us is insignificant and it is justified to neglect them in view of the remaining uncertainties in hadronic matrix elements that affect the dominant contributions to \(\text {Im}A_2\).
 2.
It is common to omit the subscript K on \(\varepsilon \equiv \varepsilon _K\) when writing the ratio \(\varepsilon '/\varepsilon \).
 3.
For \(\Gamma _T\) there is only \(P_A = P_B\) in four dimensions but not \(P_A \ne P_B\).
 4.
The \(N_f= 3\) lattice results [15, 16] of the \(K\rightarrow \pi \pi \) matrix elements in principle include these contributions but from these results they cannot be disentangled from the \(({\bar{s}}d) ({\bar{u}}u)\) and \(({\bar{s}}d) ({\bar{d}}d)\) ones. In this regard it is desirable that lattice collaborations provide in the future separately the matrix element for each \(({\bar{s}}d)({\bar{q}}q)\) operator for \(q = u, d, s\).
 5.
Note that the 10 operators in the traditional SM basis are not linearly independent and correspond only to 7 linearly independent operators for \(N_f= 3\) [8].
 6.
As stated above, we neglect electromagnetic dipole operators.
 7.
 8.
We denote Wilson coefficients of SMEFT by caligraphic \({{\mathcal {C}}}_{k}^{}\) and of lowenergy EFTs by \(C_k\).
 9.
 10.
We do not count redundant operators such as \([{{\mathcal {O}}}_{qd}^{(1,8)}]_{1212} \equiv [{{\mathcal {O}}}_{qd}^{(1,8)}]_{2121}^\dagger \), but adopt the basis of nonredundant operators defined in [70]. Contributions from \(\psi ^2 H^2 D\) operators corresponding to modified \(Z^0\) and \(W^\pm \) couplings arise at oneloop from topquark Yukawa mixing [24] and those from \(h^0\) couplings count as beyond dimension six [65].
 11.
This basis is denoted Warsaw up in the WCxf standard [63].
 12.
We neglect a numerically subleading part from the strange quark, since \(g_T^s \ll g_T^{u,d}\), and assume that the contribution from the strange quark CEDM can be neglected as well.
 13.
Note that the signs on the righthand sides of (36) depend on the sign convention for the covariant derivative. We use \(D_\mu =\partial _\mu + ieQ_fA_\mu +ig_s G_\mu ^a T^a\). Our convention for \(\sigma ^{\mu \nu }\) is \(\sigma ^{\mu \nu }=\frac{i}{2}[\gamma _\mu ,\gamma _\nu ]\).
 14.
 15.
We again omit redundant operators.
 16.
The only way to generate a \(\Delta S=1\) operator at \({\mu _\mathrm {ew}}\) without a 12coupling is via the operators \([{{\mathcal {O}}}_{qd}^{(1,8)}]_{1332}\); however, they match onto \(C_{SLR}^b\) and \({\widetilde{C}}_{SLR}^b\), which contribute to \(\varepsilon '/\varepsilon \) neither directly nor indirectly, as shown in Sect. 2.2.
 17.
In the second line, the Fierzredundant operators \({\widetilde{O}}_{XLB}^d\) are used for the sake of notational brevity.
 18.
Omitting coefficients that are redundant due to \({{\mathcal {C}}}_{Hq}^{(3)}\) being hermitian.
Notes
Acknowledgements
We would like to thank JeanMarc Gérard for discussions and Aneesh Manohar for clarifying communications. This work was supported by the DFG cluster of excellence “Origin and Structure of the Universe”.
References
 1.NA48 Collaboration, J.R. Batley et al., A Precision measurement of direct CP violation in the decay of neutral kaons into two pions. Phys. Lett. B 544, 97–112 (2002). arXiv:hepex/0208009
 2.KTeV Collaboration, A. AlaviHarati et al., Measurements of direct CP violation, CPT symmetry, and other parameters in the neutral kaon system. Phys. Rev. D 67, 012005 (2003). arXiv:hepex/0208007. [Erratum: Phys. Rev.D70,079904(2004)]
 3.KTeV Collaboration, E. Abouzaid et al., Precise measurements of Direct CP Violation, CPT symmetry, and other parameters in the neutral Kaon System. Phys. Rev. D 83, 092001 (2011). arXiv:1011.0127
 4.A.J. Buras, M. Jamin, M.E. Lautenbacher, P. H. Weisz, Effective Hamiltonians for \(\Delta S = 1\) and \(\Delta B = 1\) nonleptonic decays beyond the leading logarithmic approximation. Nucl. Phys. B 370, 69–104 (1992). [Addendum: Nucl. Phys.B375,501(1992)]Google Scholar
 5.A.J. Buras, M. Jamin, M.E. Lautenbacher, P.H. Weisz, Two loop anomalous dimension matrix for \(\Delta S = 1\) weak nonleptonic decays I: \({\cal{O}}(\alpha _s^2)\). Nucl. Phys. B 400, 37–74 (1993). arXiv:hepph/9211304 Google Scholar
 6.A.J. Buras, M. Jamin, M.E. Lautenbacher, Two loop anomalous dimension matrix for Delta S = 1 weak nonleptonic decays. 2. O(\(\alpha \alpha _s\)). Nucl. Phys. B 400, 75–102 (1993). arXiv:hepph/9211321
 7.M. Ciuchini, E. Franco, G. Martinelli, L. Reina, \(\varepsilon ^{\prime } / \varepsilon \) at the Nexttoleading order in QCD and QED. Phys. Lett. B 301, 263–271 (1993). arXiv:hepph/9212203 Google Scholar
 8.A.J. Buras, M. Jamin, M.E. Lautenbacher, The anatomy of \(\varepsilon ^{\prime } / \varepsilon \) beyond leading logarithms with improved hadronic matrix elements. Nucl. Phys. B 408, 209–285 (1993). arXiv:hepph/9303284 Google Scholar
 9.M. Ciuchini, E. Franco, G. Martinelli, L. Reina, The \(\Delta S = 1\) effective Hamiltonian including nexttoleading order QCD and QED corrections. Nucl. Phys. B 415, 403–462 (1994). arXiv:hepph/9304257 Google Scholar
 10.C. Bobeth, M. Misiak, J. Urban, Photonic penguins at two loops and \(m_t\)dependence of \(BR(B\rightarrow X_s \ell ^+ \ell ^)\). Nucl. Phys. B 574, 291–330 (2000). arXiv:hepph/9910220 Google Scholar
 11.A.J. Buras, P. Gambino, U.A. Haisch, Electroweak penguin contributions to nonleptonic \(\Delta F=1\) decays at NNLO. Nucl. Phys. B 570, 117–154 (2000). arXiv:hepph/9911250 Google Scholar
 12.M. Gorbahn, U. Haisch, Effective Hamiltonian for nonleptonic \(\Delta F = 1\) decays at NNLO in QCD. Nucl. Phys. B 713, 291–332 (2005). arXiv:hepph/0411071 Google Scholar
 13.J. Brod, M. Gorbahn, \(\varepsilon _K\) at nexttonexttoleading order: the charmtopquark contribution. Phys. Rev. D 82, 094026 (2010). arXiv:1007.0684 Google Scholar
 14.M. CerdàSevilla, M. Gorbahn, S. Jäger, A. Kokulu, Towards NNLO accuracy for \(\varepsilon ^{\prime } / \varepsilon \). J. Phys. Conf. Ser. 800(1), 012008 (2017). arXiv:1611.08276
 15.RBC, UKQCD Collaboration, Z. Bai et al., Standard model prediction for direct CP violation in \(K\rightarrow \pi \pi \) decay. Phys. Rev. Lett. 115(21), 212001 (2015). arXiv:1505.07863
 16.T. Blum et al., \(K \rightarrow \pi \pi \) \(\Delta I=3/2\) decay amplitude in the continuum limit. Phys. Rev. D 91(7), 074502 (2015). arXiv:1502.00263
 17.A.J. Buras, M. Gorbahn, S. Jäger, M. Jamin, Improved anatomy of \(\varepsilon ^{\prime }/\varepsilon \) in the standard model. JHEP 11, 202 (2015). arXiv:1507.06345
 18.T. Kitahara, U. Nierste, P. Tremper, Singularityfree nexttoleading order \(\Delta \)S = 1 renormalization group evolution and \(\varepsilon /\varepsilon \) in the Standard Model and beyond. JHEP 12, 078 (2016). arXiv:1607.06727
 19.A.J. Buras, J.M. Gérard, Upper bounds on \(\varepsilon ^{\prime }/\varepsilon \) parameters \(B_{6}^{(1/2)}\) and \(B_{8}^{(3/2)}\) from large N QCD and other news. JHEP 12, 008 (2015). arXiv:1507.06326
 20.A.J. Buras, J.M. Gérard, Final state interactions in \(K\rightarrow \pi \pi \) decays: \(\Delta I=1/2\) rule vs. \(\varepsilon ^{\prime }/\varepsilon \). Eur. Phys. J. C 77(1), 10 (2017). arXiv:1603.05686
 21.H. Gisbert, A. Pich, Direct CP violation in \(K^0\rightarrow \pi \pi \): standard model status. Rept. Prog. Phys. 81(7), 076201 (2018). arXiv:1712.06147
 22.A.J. Buras, F. De Fazio, \(\varepsilon ^{\prime }/\varepsilon \) in 331 Models. JHEP 03, 010 (2016). arXiv:1512.02869
 23.A.J. Buras, F. De Fazio, 331 Models facing the tensions in \(\Delta F=2\) Processes with the Impact on \(\varepsilon ^\prime /\varepsilon \), \(B_s\rightarrow \mu ^+\mu ^\) and \(B\rightarrow K^*\mu ^+\mu ^\). JHEP 08, 115 (2016). arXiv:1604.02344
 24.C. Bobeth, A.J. Buras, A. Celis, M. Jung, Yukawa enhancement of \(Z\)mediated new physics in \(\Delta S = 2\) and \(\Delta B = 2\) processes. JHEP 07, 124 (2017). arXiv:1703.04753
 25.M. Endo, T. Kitahara, S. Mishima, K. Yamamoto, Revisiting Kaon Physics in General \(Z\) Scenario. Phys. Lett. B 771, 37–44 (2017). arXiv:1612.08839
 26.C. Bobeth, A.J. Buras, A. Celis, M. Jung, Patterns of flavour violation in models with vectorlike quarks. JHEP 04, 079 (2017). arXiv:1609.04783
 27.M. Blanke, A.J. Buras, S. Recksiegel, Quark flavour observables in the Littlest Higgs model with Tparity after LHC Run 1. Eur. Phys. J. C 76(4), 182 (2016). arXiv:1507.06316
 28.A.J. Buras, D. Buttazzo, R. Knegjens, \( K\rightarrow \pi \nu \bar{\nu } \) and \(\varepsilon ^{\prime } / \varepsilon \) in simplified new physics models. JHEP 11, 166 (2015). arXiv:1507.08672
 29.A.J. Buras, New physics patterns in \(\varepsilon ^\prime /\varepsilon \) and \(\varepsilon _K\) with implications for rare kaon decays and \(\Delta M_K\). JHEP 04, 071 (2016). arXiv:1601.00005
 30.M. Tanimoto, K. Yamamoto, Probing SUSY with 10 TeV stop mass in rare decays and CP violation of kaon. PTEP 2016(12), 123B02 (2016). arXiv:1603.07960
 31.T. Kitahara, U. Nierste, P. Tremper, Supersymmetric explanation of CP violation in \(K\rightarrow \pi \pi \) decays. Phys. Rev. Lett. 117(9), 091802 (2016). arXiv:1604.07400
 32.M. Endo, S. Mishima, D. Ueda, K. Yamamoto, Chargino contributions in light of recent \(\varepsilon ^{\prime }/\varepsilon \). Phys. Lett. B 762, 493–497 (2016). arXiv:1608.01444
 33.A. Crivellin, G. D’Ambrosio, T. Kitahara, U. Nierste, \(K\rightarrow \pi \nu \overline{\nu }\) in the MSSM in light of the \(\varepsilon ^{\prime }/\varepsilon \) anomaly. Phys. Rev. D 96(1), 015023 (2017). arXiv:1703.05786
 34.M. Endo, T. Goto, T. Kitahara, S. Mishima, D. Ueda, K. Yamamoto, Gluinomediated electroweak penguin with flavorviolating trilinear couplings. JHEP 04, 019 (2018). arXiv:1712.04959
 35.C.H. Chen, T. Nomura, \(\varepsilon ^{\prime }/\varepsilon \) and \(K \rightarrow \pi \nu {\bar{\nu }}\) in a twoHiggs doublet model. arXiv:1804.06017
 36.C.H. Chen, T. Nomura, \(\varepsilon ^{\prime }/\varepsilon \) from chargedHiggsinduced gluonic dipole operators. arXiv:1805.07522
 37.N. Haba, H. Umeeda, T. Yamada, \(\varepsilon ^{\prime }/\varepsilon \) Anomaly and Neutron EDM in \(SU(2)_L\times SU(2)_R\times U(1)_{BL}\) model with Charge Symmetry. JHEP 05, 052 (2018). arXiv:1802.09903
 38.N. Haba, H. Umeeda, T. Yamada, Direct CP Violation in Cabibbofavored charmed meson decays and \(\varepsilon ^{\prime }/\varepsilon \) in \(SU(2)_L\times SU(2)_R\times U(1)_{BL}\) model. arXiv:1806.03424
 39.S. Matsuzaki, K. Nishiwaki, K. Yamamoto, Simultaneous interpretation of \(K\) and \(B\) anomalies in terms of chiralflavorful vectors. arXiv:1806.02312
 40.C. Bobeth, A.J. Buras, Leptoquarks meet \(\varepsilon ^{\prime }/\varepsilon \) and rare Kaon processes. JHEP 02, 101 (2018). arXiv:1712.01295
 41.ETM Collaboration, M. Constantinou, M. Costa, R. Frezzotti, V. Lubicz, G. Martinelli, D. Meloni, H. Panagopoulos, S. Simula, \(K \rightarrow \pi \) matrix elements of the chromomagnetic operator on the lattice. Phys. Rev. D 97(7), 074501 (2018). arXiv:1712.09824
 42.A.J. Buras, J.M. Gérard, \(K\rightarrow \pi \pi \) and \(K\pi \) Matrix elements of the chromomagnetic operators from dual QCD. JHEP 07, 126 (2018). arXiv:1803.08052
 43.J. Aebischer, A.J. Buras, J.M. Gérard, BSM hadronic matrix elements for \(\epsilon ^{\prime }/\epsilon \) and \(K\rightarrow \pi \pi \) Decays in the Dual QCD approach. JHEP 02, 021 (2019). https://doi.org/10.1007/JHEP02(2019)021 arXiv:1807.01709
 44.W. Buchmüller, D. Wyler, Effective Lagrangian analysis of new interactions and flavor conservation. Nucl. Phys. B 268, 621–653 (1986)ADSCrossRefGoogle Scholar
 45.B. Grzadkowski, M. Iskrzynski, M. Misiak, J. Rosiek, Dimensionsix terms in the standard model lagrangian. JHEP 10, 085 (2010). arXiv:1008.4884 ADSCrossRefGoogle Scholar
 46.J. Aebischer, C. Bobeth, A.J. Buras, J.M. Gérard, D.M. Straub, Master formula for \(\varepsilon ^{\prime }/\varepsilon \) beyond the standard model. arXiv:1807.02520
 47.V. Cirigliano, G. Ecker, H. Neufeld, A. Pich, Isospin breaking in \(K \rightarrow \pi \pi \) decays. Eur. Phys. J. C 33, 369–396 (2004). arXiv:hepph/0310351 Google Scholar
 48.V. Cirigliano, A. Pich, G. Ecker, H. Neufeld, Isospin violation in \(\varepsilon ^\prime \). Phys. Rev. Lett. 91, 162001 (2003). arXiv:hepph/0307030 Google Scholar
 49.G. Buchalla, A.J. Buras, M.E. Lautenbacher, Weak decays beyond leading logarithms. Rev. Mod. Phys. 68, 1125–1144 (1996). arXiv:hepph/9512380 ADSCrossRefGoogle Scholar
 50.W.A. Bardeen, A.J. Buras, J.M. Gérard, A consistent analysis of the \(\Delta I = 1/2\) rule for K decays. Phys. Lett. B 192, 138–144 (1987)Google Scholar
 51.A.J. Buras, J.M. Gérard, \(1/N\) expansion for Kaons. Nucl. Phys. B 264, 371–392 (1986)Google Scholar
 52.A.J. Buras, J.M. Gérard, W.A. Bardeen, Large \(N\) approach to Kaon decays and mixing 28 years later: \(\Delta I = 1/2\) rule, \({\hat{B}}_K\) and \(\Delta M_K\). Eur. Phys. J. C 74, 2871 (2014). arXiv:1401.1385 Google Scholar
 53.S. Bertolini, J.O. Eeg, M. Fabbrichesi, Studying \(\varepsilon ^{\prime }/\varepsilon \) in the chiral quark model: \(\gamma _5\) scheme independence and NLO hadronic matrix elements. Nucl. Phys. B 449, 197–228 (1995). arXiv:hepph/9409437 Google Scholar
 54.O. Gedalia, G. Isidori, G. Perez, Combining direct and indirect Kaon CP violation to constrain the warped KK scale. Phys. Lett. B 682, 200–206 (2009). arXiv:0905.3264 ADSCrossRefGoogle Scholar
 55.A.J. Buras, M. Misiak, J. Urban, Two loop QCD anomalous dimensions of flavor changing four quark operators within and beyond the standard model. Nucl. Phys. B 586, 397–426 (2000). arXiv:hepph/0005183 ADSCrossRefGoogle Scholar
 56.J. Aebischer, M. Fael, C. Greub, J. Virto, B physics beyond the standard model at one loop: complete renormalization group evolution below the electroweak scale. JHEP 09, 158 (2017). arXiv:1704.06639
 57.E.E. Jenkins, A.V. Manohar, P. Stoffer, Lowenergy effective field theory below the electroweak scale: anomalous dimensions. JHEP 01, 084 (2018). arXiv:1711.05270
 58.J. Aebischer, J. Kumar, D.M. Straub, Wilson: a Python package for the running and matching of Wilson coefficients above and below the electroweak scale. arXiv:1804.05033
 59.D.M. Straub et al., Flavio–flavour phenomenology in the standard model and beyondGoogle Scholar
 60.D.M. Straub, Flavio: a Python package for flavour and precision phenomenology in the Standard Model and beyond (2018). arXiv:1810.08132
 61.V. Cirigliano, G. Ecker, H. Neufeld, A. Pich, J. Portoles, Kaon decays in the standard model. Rev. Mod. Phys. 84, 399 (2012). arXiv:1107.6001 ADSCrossRefGoogle Scholar
 62.G.C. Branco, J.M. Frere, J.M. Gerard, The value of \(\epsilon ^\prime / \epsilon \) in Models Based on \(SU(2)_L\times SU(2)_R\times U(1)\). Nucl. Phys. B 221, 317–330 (1983)Google Scholar
 63.J. Aebischer et al., WCxf: an exchange format for Wilson coefficients beyond the standard model. arXiv:1712.05298
 64.J. Aebischer, A. Crivellin, M. Fael, C. Greub, Matching of gauge invariant dimensionsix operators for \(b\rightarrow s\) and \(b\rightarrow c\) transitions. JHEP 05, 037 (2016). arXiv:1512.02830
 65.E.E. Jenkins, A.V. Manohar, P. Stoffer, Lowenergy effective field theory below the electroweak scale: operators and matching. JHEP 03, 016 (2018). arXiv:1709.04486
 66.C. Bobeth, U. Haisch, Anomalous triple gauge couplings from \(B\)meson and kaon observables. JHEP 09, 018 (2015). arXiv:1503.04829. https://doi.org/10.1007/JHEP09(2015)018
 67.E.E. Jenkins, A.V. Manohar, M. Trott, Renormalization group evolution of the standard model dimension six operators I: formalism and lambda dependence. JHEP 10, 087 (2013). arXiv:1308.2627 ADSCrossRefGoogle Scholar
 68.E.E. Jenkins, A.V. Manohar, M. Trott, Renormalization group evolution of the standard model dimension six operators II: Yukawa dependence. JHEP 01, 035 (2014). arXiv:1310.4838 ADSCrossRefGoogle Scholar
 69.R. Alonso, E.E. Jenkins, A.V. Manohar, M. Trott, Renormalization Group evolution of the standard model dimension six operators III: Gauge coupling dependence and phenomenology. JHEP 04, 159 (2014). arXiv:1312.2014 ADSCrossRefGoogle Scholar
 70.A. Celis, J. FuentesMartin, A. Vicente, J. Virto, Dsix tools: the standard model effective field theory toolkit. Eur. Phys. J. C 77(6), 405 (2017). arXiv:1704.04504
 71.RBC/UKQCD Collaboration, N. Garron, R.J. Hudspith, A.T. Lytle, Neutral Kaon mixing beyond the Standard Model with \(N_f=2+1\) Chiral Fermions Part 1: bare matrix elements and physical results. JHEP 11, 001 (2016). arXiv:1609.03334
 72.RBC, UKQCD Collaboration, P.A. Boyle, N. Garron, R.J. Hudspith, C. Lehner, A.T. Lytle, Neutral kaon mixing beyond the Standard Model with \(N_f = 2 + 1\) chiral fermions. Part 2: non perturbative renormalisation of the \(\Delta F=2\) fourquark operators. JHEP 10, 054 (2017). arXiv:1708.03552
 73.ETM Collaboration, N. Carrasco, P. Dimopoulos, R. Frezzotti, V. Lubicz, G.C. Rossi, S. Simula, C. Tarantino, \(\Delta S=2\) and \(\Delta C=2\) bag parameters in the standard model and beyond from \(N_f=2+1+1\) twistedmass lattice QCD. Phys. Rev. D 92(3), 034516 (2015). arXiv:1505.06639
 74.SWME Collaboration, B.J. Choi et al., Kaon BSM Bparameters using improved staggered fermions from \(N_f=2+1\) unquenched QCD. Phys. Rev. D 93(1), 014511 (2016). arXiv:1509.00592
 75.A.J. Buras, J.M. Gérard, Dual QCD insight into BSM hadronic matrix elements for \(K^0\)\({\bar{K}}^0\) Mixing from Lattice QCD. arXiv:1804.02401
 76.J. Brod, M. Gorbahn, Nexttonexttoleadingorder charmquark contribution to the CP violation parameter \(\varepsilon _K\) and \(\Delta M_K\). Phys. Rev. Lett. 108, 121801 (2012). arXiv:1108.2036 Google Scholar
 77.Z. Bai, N.H. Christ, T. Izubuchi, C.T. Sachrajda, A. Soni, J. Yu, \(K_L\)\(K_S\) mass difference from lattice QCD. Phys. Rev. Lett. 113, 112003 (2014). arXiv:1406.0916 Google Scholar
 78.N.H. Christ, X. Feng, G. Martinelli, C.T. Sachrajda, Effects of finite volume on the \(K_L\)\(K_S\) mass difference. Phys. Rev. D 91(11), 114510 (2015). arXiv:1504.01170
 79.J. Bijnens, J.M. Gérard, G. Klein, The \(K_L\)\(K_S\) mass difference. Phys. Lett. B 257, 191–195 (1991)Google Scholar
 80.A.J. Buras, D. Guadagnoli, Correlations among new CP violating effects in \(\Delta F = 2^{\prime \prime }\) observables. Phys. Rev. D 78, 033005 (2008). arXiv:0805.3887 Google Scholar
 81.A.J. Buras, D. Guadagnoli, G. Isidori, On \(\varepsilon _K\) beyond lowest order in the operator product expansion. Phys. Lett. B 688, 309–313 (2010). arXiv:1002.3612 Google Scholar
 82.M. Bobrowski, A. Lenz, J. Riedl, J. Rohrwild, How large can the SM contribution to CP violation in \(D^0\)\({\bar{D}}^0\) mixing be? JHEP 03, 009 (2010). arXiv:1002.4794 Google Scholar
 83.HFLAV Collaboration, Y. Amhis et al., Averages of \(b\)hadron, \(c\)hadron, and \(\tau \)lepton properties as of summer 2016. Eur. Phys. J. C 77(12), 895 (2017). arXiv:1612.07233
 84.K. Blum, Y. Grossman, Y. Nir, G. Perez, Combining \(K^0\)\({\bar{K}}^0\) mixing and \(D^0\)\({\bar{D}}^0\) mixing to constrain the flavor structure of new physics. Phys. Rev. Lett. 102, 211802 (2009). arXiv:0903.2118 Google Scholar
 85.V. Cirigliano, W. Dekens, J. de Vries, E. Mereghetti, An \(\epsilon ^{\prime }\) improvement from righthanded currents. Phys. Lett. B 767, 1–9 (2017). arXiv:1612.03914
 86.PNDME Collaboration, T. Bhattacharya, V. Cirigliano, S. Cohen, R. Gupta, A. Joseph, H.W. Lin, B. Yoon, Isovector and Isoscalar Tensor Charges of the Nucleon from Lattice QCD. Phys. Rev. D 92(9), 094511 (2015). arXiv:1506.06411
 87.C. Alexandrou et al., Nucleon scalar and tensor charges using lattice QCD simulations at the physical value of the pion mass. Phys. Rev. D 95(11), 114514 (2017). arXiv:1703.08788. [Erratum: Phys. Rev.D96,no.9,099906(2017)]
 88.JLQCD Collaboration, N. Yamanaka, S. Hashimoto, T. Kaneko, H. Ohki, Nucleon charges with dynamical overlap fermions. arXiv:1805.10507
 89.R. Gupta, B. Yoon, T. Bhattacharya, V. Cirigliano, Y.C. Jang, H.W. Lin, Flavor diagonal tensor charges of the nucleon from 2+1+1 flavor lattice QCD. arXiv:1808.07597
 90.M. Pospelov, A. Ritz, Neutron EDM from electric and chromoelectric dipole moments of quarks. Phys. Rev. D 63, 073015 (2001). arXiv:hepph/0010037 ADSCrossRefGoogle Scholar
 91.K. Fuyuto, J. Hisano, N. Nagata, K. Tsumura, QCD Corrections to quark (Chromo)electric dipole moments in highscale supersymmetry. JHEP 12, 010 (2013). arXiv:1308.6493 ADSCrossRefGoogle Scholar
 92.J.M. Pendlebury et al., Revised experimental upper limit on the electric dipole moment of the neutron. Phys. Rev. D D92(9), 092003 (2015). arXiv:1509.04411
 93.N. Yamanaka, B.K. Sahoo, N. Yoshinaga, T. Sato, K. Asahi, B.P. Das, Probing exotic phenomena at the interface of nuclear and particle physics with the electric dipole moments of diamagnetic atoms: a unique window to hadronic and semileptonic CP violation. Eur. Phys. J. A 53(3), 54 (2017). arXiv:1703.01570
 94.V. Cirigliano, S. Gardner, B. Holstein, Beta decays and nonstandard interactions in the LHC Era. Prog. Part. Nucl. Phys. 71, 93–118 (2013). arXiv:1303.6953 ADSCrossRefGoogle Scholar
 95.M. GonzálezAlonso, J Martin Camalich, Global effectivefieldtheory analysis of newphysics effects in (semi)leptonic kaon decays. JHEP 12, 052 (2016). arXiv:1605.07114
 96.M. GonzálezAlonso, O. NaviliatCuncic, N. Severijns, New physics searches in nuclear and neutron \(\beta \) decay. arXiv:1803.08732
 97.K.O.T.O. Collaboration, K. Shiomi, Search for the rare decay \(K_L\rightarrow \pi ^0\nu {\bar{\nu }}\). Talk at ICHEP 2018, 7 (2018)Google Scholar
 98.G. Isidori, J.F. Kamenik, Z. Ligeti, G. Perez, Implications of the LHCb evidence for charm CP violation. Phys. Lett. B 711, 46–51 (2012). arXiv:1111.4987 ADSCrossRefGoogle Scholar
 99.J. de Blas, J.C. Criado, M. PerezVictoria, J. Santiago, Effective description of general extensions of the standard model: the complete treelevel dictionary. JHEP 03, 109 (2018). arXiv:1711.10391
 100.J.M. Arnold, B. Fornal, M.B. Wise, Simplified models with baryon number violation but no proton decay. Phys. Rev. D 87, 075004 (2013). arXiv:1212.4556 ADSCrossRefGoogle Scholar
 101.N. Assad, B. Fornal, B. Grinstein, Baryon number and Lepton Universality violation in leptoquark and diquark models. Phys. Lett. B 777, 324–331 (2018). arXiv:1708.06350
 102.A.J. Buras, F. De Fazio, J. Girrbach, \(\Delta I=1/2\) rule, \(\varepsilon ^{\prime }/\varepsilon \) and \(K\rightarrow \pi \nu \bar{\nu }\) in \(Z^{\prime } (Z)\) and \(G^{\prime } \) models with FCNC quark couplings. Eur. Phys. J. C 74(7), 2950 (2014). arXiv:1404.3824 Google Scholar
 103.S. Alioli, V. Cirigliano, W. Dekens, J. de Vries, E. Mereghetti, Righthanded charged currents in the era of the Large Hadron collider. JHEP 05, 086 (2017). arXiv:1703.04751
 104.K. Agashe, A. Azatov, L. Zhu, Flavor violation tests of warped/composite SM in the twosite approach. Phys. Rev. D 79, 056006 (2009). arXiv:0810.1016 ADSCrossRefGoogle Scholar
 105.C. Delaunay, J.F. Kamenik, G. Perez, L. Randall, Charming CP violation and dipole operators from RS flavor anarchy. JHEP 01, 027 (2013). arXiv:1207.0474 ADSCrossRefGoogle Scholar
 106.N. Vignaroli, \(\Delta \) F=1 constraints on composite Higgs models with LR parity. Phys. Rev. D 86, 115011 (2012). arXiv:1204.0478 Google Scholar
 107.M. König, M. Neubert, D.M. Straub, Dipole operator constraints on composite Higgs models. Eur. Phys. J. C 74(7), 2945 (2014). arXiv:1403.2756]ADSCrossRefGoogle Scholar
 108.WCxf Collaboration, flavio WCxf basis file. https://wcxf.github.io/assets/pdf/WET.flavio.pdf
 109.A.L. Kagan, Righthanded currents, CP violation, and \(B \rightarrow VV\). arXiv:hepph/0407076
Copyright information
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Funded by SCOAP^{3}.