Nucleon electric dipole moments in high-scale supersymmetric models

The electric dipole moments (EDMs) of electron and nucleons are promising probes of the new physics. In generic high-scale supersymmetric (SUSY) scenarios such as models based on mixture of the anomaly and gauge mediations, gluino has an additional contribution to the nucleon EDMs. In this paper, we studied the effect of the CP -violating gluon Weinberg operator induced by the gluino chromoelectric dipole moment in the high-scale SUSY scenarios, and we evaluated the nucleon and electron EDMs in the scenarios. We found that in the generic high-scale SUSY models, the nucleon EDMs may receive the sizable contribution from the Weinberg operator. Thus, it is important to compare the nucleon EDMs with the electron one in order to discriminate among the high-scale SUSY models.


Introduction
The standard model (SM) is established by the discovery of the Higgs boson with mass of 125 GeV at the LHC run 1 [1][2][3]. However, there are several reasons to expect that the SM must be an effective theory of a certain full theory; no candidate of the dark matter (DM), no reason of the gauge anomaly cancellation, and so on. Supersymmetry (SUSY) is one of the attractive extensions of the SM. The lightest supersymmetric particle (LSP) is the candidate of the DM. The gauge coupling unification is improved due to the additional matters, that is, the SUSY partners of the SM particles. This unification may imply that the SM gauge groups are embedded in a larger gauge group, such as SU (5), SO (10), and E 6 .
However, the LHC run 1 has also reported that there is no signal of new physics around electroweak (EW) scale. Besides, the observed Higgs boson is too heavy in the minimal supersymmetric standard model (MSSM) if the SUSY particle masses are smaller than O(1) TeV. It is needed to introduce the large quantum correction to the Higgs mass or the additional tree-level contribution. Several extensions of the MSSM are proposed; introduction of additional vector-like matters [4], specific mass spectra (large A-term or Next-to-MSSM) [5], and high-scale SUSY scenarios [6][7][8].
In the high-scale SUSY scenarios, sfermions have masses around 10 2 TeV, while the gaugino masses lie around several TeV. This mass spectrum leads to the fascinating results: the SUSY flavor and CP problems are eased due to heavy sfermions [9], and the neutral wino behaves as the LSP with mass of several TeV, which is favored in the thermal DM scenario [10]. Recently, the sfermion flavor structure is focused attention on in order to survey these models by using indirect searches [11][12][13]. In the grand unified theories JHEP11(2015)085 (GUTs) based on the high-scale SUSY scenarios, the specific mass spectrum yields the several features: the gauge couplings unify at the GUT scale with higher accuracy [14], and the dangerous proton decay via color-triplet Higgs exchange is suppressed due to the heavy sfermions [12,15].
The simplest model for the high-scale SUSY scenarios is based on the anomaly mediation [16,17]. On the other hand, the generic models may include the gauge mediated contribution to the SUSY-breaking terms [18,19]. In the extensions, the pattern of gaugino masses differs from the simplest model. The vector-like multiplets for messengers are naturally introduced since they may obtain masses proportional to the gravitino mass via the Giudice-Masiero mechanism [20]. If we assume that the vector-like multiplets are in SU(5) multiplets, the gauge coupling unification is maintained in these models. Thus, the extensions should be considered equally to the simplest model.
The electric dipole moments (EDMs) are important to investigate the additional CP violation in the SUSY breaking terms. In the SM, the EDMs for fundamental fermions are small [21], and thus the EDMs have high sensitivities on the new physics. In high-scale SUSY breaking scenarios, the generic CP -violating phases are still allowed thanks to the heavy sfermions, even if the EDMs are generated at one-loop level. However, the future experiments for EDM searches may have sensitivities to the high-scale SUSY models.
In the high-scale SUSY scenarios, the dominant contribution to the EDMs in the MSSM comes from the Barr-Zee two-loop diagrams, especially the chargino/neutralino two-loop diagrams [22]. When the higgsino and wino are around a few TeV, the current experimental upper bound on the electron EDM has already given the constraint on the models. The ratios of electron and nucleon EDMs are predictive so that the measurements of the ratios would lead to determination of the mass spectrum.
On the other hand, there may exist additional contribution from gluino in the extended models mentioned above. The physical complex phase of the gluino mass may arise from a relative phase of the anomaly and gauge mediated contribution to it. The additional CPviolating source, so-called the gluino chromoelectric dipole moment (CEDM), is induced by the physical phase of the gluino mass and the CP -violating couplings of gluino and vector-like multiplets. The nucleon EDMs are affected by the additional source since the gluino CEDM turns into the CP -violating gluon Weinberg operator [23] below the gaugino threshold. In this paper, we study the effect of the gluino CEDM contribution to nucleon EDMs, and then we show the future prospects for the observation of EDMs (electron, neutron, and proton) in the high-scale SUSY models.
When both left-and right-handed sfermions have flavor-violating soft mass terms, the one-loop diagrams to the EDMs are enhanced by the heavy fermion masses. The contributions may be sizable when the flavor violation is O(1) [24][25][26]. However, they are quickly suppressed when the flavor violation is small. Then, we do not include the contribution to the EDMs from the flavor violation in this paper.
Current status of the EDM experiments is as follows: the bounds on the electron, neutron, and proton EDMs are given by |d e | < 8.7 × 10 −29 [e cm] [27], |d n | < 2.9 × JHEP11(2015)085 10 −26 [e cm] [28], and |d p | < 7.9 × 10 −25 [e cm], respectively. 1 In future experiments, there are several proposals [30,31]: for instance, some neutron EDM measurements may achieve a sensitivity of |d n | ∼ 10 −28 [e cm]. In the proton EDM measurement at COSY [32] and BNL [33], they may achieve a sensitivity of |d p | ∼ 10 −29 [e cm]. For the electron EDM, the final purpose of the ACME experiment is to reach a sensitivity of 3 × 10 −31 [e cm]. If the electron and nucleon EDMs are discovered, we may discriminate models beyond the SM by taking correlation among them. 2 This paper is organized as follows: in section 2, we introduce the high-scale SUSY scenarios with the gauge mediation and show that the physical phase of the gaugino mass appears. The complex gluino mass gives rise to the gluino CEDM, and the CP -violating Weinberg operator [23] is induced by integrating out the gluino field in the scenarios. The gluino CEDM and the Weinberg operator in the scenarios are shown in section 3. In the next section, we briefly introduce our method to estimate the observable EDMs, in particular neutron and proton EDMs. The contributions of (C)EDMs of quarks to the nucleon EDMs are evaluated with the QCD sum rules, while those of the Weinberg operator are based on the naive dimensional analysis. We give the detail of calculations in appendices A and B. In section 5, we study the effect of gluino CEDM to nucleon EDMs. We evaluate the electron, proton, and neutron EDMs in the high-scale SUSY scenarios in the last of this section. Finally, we summarize this paper in section 6.

Complex gaugino mass
In the SUSY breaking sector, we assume that there is no singlet superfield. Under this assumption, the soft parameters are given as follows: mass parameters of scalar components are induced by the Planck-suppressed higher-dimensional operators, and gaugino masses and scalar trilinear couplings are induced with one-loop suppression by the anomaly mediation [16,17]. In particular, the anomaly mediated gaugino masses are given by where the subscripts a = 1-3 denote the gauge groups of the SM, U(1) Y , SU(2) L , and SU(3) C . β(g a ) and m 3/2 denote the beta function for the gauge coupling g a and the gravitino mass, respectively. We note that there also exists the additional contribution from the higgsino-Higgs loops to the wino and bino masses. Below the higgsino threshold, the additional contribution is given as [16] Mh H The parameter L denotes the loop function defined as  where µ H and m A denote the masses of higgsino and heavy Higgs bosons, respectively. tan β denotes the ratio of the vacuum expectation values (VEVs) of MSSM Higgs bosons.
If there exist vector-like superfields, so-called messenger multiplets of the gauge mediation, the soft parameters differ from the simple high-scale SUSY breaking scenario. In order to maintain the gauge coupling unification, we assume that messenger superfields are in 5 + 5 or 10 + 10 representation. The mass terms of the messenger superfields arise from the Giudice-Masiero mechanism [20], if the Kähler potential is given as where Φ and Φ denote the messenger chiral superfields. On the other hand, the superpotential W (Φ, Φ) may have the mass terms of the messenger superfields, In those cases, the mass matrix of the scalar components of Φ and Φ is given as where the term proportional to m 3/2 arises from the Giudice-Masiero mechanism. For simplicity, the mass parameters are re-parametrized by M and F in the last form, and θ F denotes the complex phase of F . In the above models, the gaugino masses are induced by the anomaly and gauge mediation mechanisms at one-loop order. The anomaly mediated gluino mass is given by [16,17]: is induced by a diagram in figure 1 [35]: where x ≡ |F/M 2 | and n 3 (Φ) is the sum of Dynkin indices of the pair of chiral multiplets, Φ and Φ. The loop function g(x) is given by For later use, we define the real gluino mass parameter M g and the phase of the gluino mass θ as: (2.10) By the chiral rotation of the gluino field g a → g ′a = e −iθγ 5 /2 g a , this additional complex phase appears in the interaction terms between gluino and messengers. In next section, we show that the additional complex phase gives rise to the gluino CEDM, and then the CP -violating Weinberg operator also arises from the gluino CEDM operator below the gluino threshold scale.

Gluino CEDM and Weinberg operator
In our models, the gluino CEDM is generated by the messenger loop diagrams (figure 2) since there exists the non-vanishing CP -violating phase in the gluino-messenger interaction after the chiral rotation of the gluino. The gluino CEDM d g is given as We estimate the relevant CP -violating terms at the gluino mass scale (M g ) from those at the messenger mass scale (M mess ) by using the renormalization group equation (RGE) analysis. It is useful to define the dimension-six gluino CEDM operator in order to estimate the RGE evolution. The gluino CEDM operator O g and its Wilson coefficient C g are defined as By evaluating a diagram in figure 2, we obtain the Wilson coefficient of O g as where m 2 ± = |M | 2 ± |F | are the mass eigenvalues of the mass matrix for the scalar components of Φ and Φ, and r ± = |M | 2 /m 2 ± . θ and θ F are respectively the phases of the complex gluino mass and the off-diagonal element F of the mass matrix m 2 φ , as defined in the previous section. The loop functions A(r) and B(r) are given as Now, we estimate the gluino CEDM at the gluino mass scale by using the RGEs between the messenger and the gluino mass scales. The RGE for the Wilson coefficient C g (µ) at the leading order is given as where γ O g = 12N C and N C (= 3) is the number of colors. This anomalous dimension is found by substituting the Casimir invariant C F = N C for C F = 4/3 in the anomalous dimension of the dipole operator for b → sg [36,37]. The gluino CEDM at the gluino mass scale is obtained as follows: Here, b 3 = −7| SM + 2| gluino where the subscripts "SM" and "gluino" indicate the contribution from the SM particles and gluino, respectively. The exponents −3N C /b 3 and 1/2 are due to the one-loop renormalization-group evolution of the gluino mass M g and the strong gauge coupling g 3 , respectively.

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The CP -violating Weinberg operator is induced through the gluino one-loop diagram in figure 3. The effective Lagrangian for the Weinberg operator is defined as [23] Here, ǫ µνρσ is the totally-antisymmetric tensor with ǫ 0123 = +1. At the gluino mass scale, we obtain the Wilson coefficient of the Weinberg operator by matching the amplitude as follows: 3 Notice that quark (C)EDMs are induced as a three-loop contribution after integrating out the messenger multiplets. The Weinberg operator could be also induced directly after squarks and messenger multiplets are integrated out, not via the gluino CEDM. These contributions are suppressed by the masses of squarks and messenger multiplets. Thus, the gluino CEDM contribution is dominant for the Weinberg operator as far as gluino is lighter than them.

Nucleon electric dipole moments
In previous section, we show the expression of the CP -violating Weinberg operator induced by the gluino CEDM at the gluino mass scale. The quark EDMs are also induced through the Barr-Zee diagrams [42], which are dominated by only the chargino and neutralino loops in the MSSM based on the high-scale SUSY scenarios [22]. The quark CEDMs are radiatively induced from the Weinberg operator so that they are subdominant.
In this section, we summarize the RGE evolutions of the CP -violating operators and results of the QCD sum rules and the naive dimensional analysis in order to obtain the nucleon EDMs at the hadron scale (µ = 1 GeV) in a compressed way.
The CP -violating operators in the QCD sector below the gluino mass scale are given as Here, F µν and G a µν are the electromagnetic and gluon field strength tensors, respectively, and we define as F · σ = F µν σ µν and G · σ = G a µν σ µν T a . The dual field strength tensor is defined as G a,µν = 1 2 ǫ µνρσ G a ρσ . The first term of eq. (4.1) is the dimension-four CPviolating term, so-called QCD θ-term. Since this operator, however, does not mix with the other operators, we neglect the QCD θ-term in the RGE analysis. The second and third terms of eq. (4.1) correspond to the quark EDMs and CEDMs, respectively. The last

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term is the Weinberg operator [23]. The coefficient of the Weinberg operator w is given as w = −g 3 C W , which is evaluated in the previous section.
In our numerical evaluation of the nucleon EDMs, we include the RGE evolutions of these operators between the gluino mass scale and the scale of µ = 1 GeV. The RGEs for the Wilson coefficients at the leading order are given by ref. [37]. The detail of RGEs with mixing of the quark (C)EDMs and the CP -violating Weinberg operator is given in appendix A.
Next, we show the nucleon EDMs induced by the quark (C)EDMs via the QCD sum rules. In ref. [43], the neutron EDM d n is related to the quark (C)EDMs by using the QCD sum rules at the renormalization scale µ = 1 GeV. Similarly, the proton EDM d p is also associated with the quark (C)EDMs at the scale of µ = 1 GeV. We obtain the relation between the EDMs for light nucleons and the quark (C)EDMs as follows: (4. 2) The explicit formulae and the numerical values are presented in appendix B. The quark EDM contributions to the neutron EDM are consistent with the recent result with the lattice QCD simulation [44]. In the following numerical analyses, we use the results from the QCD sum rules with θ = 0. 4 The nucleon EDMs induced by the Weinberg operator are given by [45]: This is based on the naive dimensional analysis. The sign of the contribution of the Weinberg operator is also ambiguous. We adopt the value d N (w)/e = 20 MeV w(1 GeV) as the nucleon EDMs induced by Weinberg operator in the following numerical analyses.

Numerical results
Now, we estimate the electron and nucleon EDMs in the high-scale SUSY scenarios.
To begin with, let us consider the parameter dependence of nucleon EDMs induced by the gluino CEDM. In this evaluation, we assume that sfermions, heavy Higgs bosons and the gravitino are degenerate in mass M S , and we take M S = 100 TeV. Once we fix M S , we have three parameters; M and |F | which are the mass parameters of the scalar fields of messengers, and θ F which is the complex phase of F . In the following numerical analyses, we choose M , θ F , and x (≡ |F/M 2 |) as independent parameters.  π, there is no contribution from the gluino-induced nucleon EDM, since M GMSB and the couplings of the gluino-messenger interaction are real. The maximal contribution is given by θ F ∼ 0.125π. When the real and imaginary parts of the gluino mass are comparable to each other, the physical phase of the gluino mass θ is maximized. Since the coefficient of the Weinberg operator is proportional to sin(θ + θ F )/M g , the maximum contribution arises when θ + θ F ∼ π/2 and M g is small.
In the right of figure 4, the x dependence of nucleon EDM is shown. This dependence is evaluated with M = 100 TeV and θ F = 0.125π. The gluino CEDM approaches to the maximum as x → 1. On the other hand, one of the scalars of messengers becomes massless when x = 1. In order to avoid this situation, in the following calculations, we set x = 0.99 for simplicity.
Finally, we show the M dependence of nucleon EDM in the left panel of figure 4. In this figure, we set x = 0.99 and θ F = 0.125π. The Wilson coefficient of Weinberg operator (eq. (3.8)) behaves as follows: When M ≫ M S , the gluino mass mainly comes from the gauge mediation so that the Weinberg operator is highly suppressed by M 2 and also the suppressed CP phase in sin(θ + θ F ). If M ≪ M S , the gluino mass is dominated by the anomaly mediated contribution, and thus, the Weinberg operator is suppressed by M S M . On the other hand, in the region M ∼ M S , the nucleon EDMs are slightly enhanced since the gauge mediated gluino mass is comparable to M AMSB and thus the CP phase of the gluino mass is maximal. Now, let us compare the nucleon and electron EDMs in the high-scale SUSY scenarios, in which the Barr-Zee diagrams and gluino CEDM contribute to the EDMs. In the highscale SUSY scenarios, all MSSM scalar particles except the SM Higgs multiplet are assumed to be heavy. Within the MSSM, the main contributions to the electron and nucleon EDMs arise from the Barr-Zee type two-loop diagram contributions [22]. The diagrams include JHEP11(2015)085  . The CP phase of µ H is also model-dependent and therefore we set the CP phase in the mass matrices of charginos and neutralinos to make the Barr-Zee contributions maximized. 6 We note that we set tan β = 3 since tan β is also model-dependent parameter. 7 In the following numerical analyses, we assume that all MSSM scalar particles except the SM Higgs boson have the same mass M S and three parameters which are associated with the gluino induced nucleon EDMs are set to be M = M S , θ F = 0.125π, and x = 0.99 in order to study the maximal gluino CEDM effects. The higgsino mass µ H is estimated as follows: input value for the higgsino mass is given at the renormalization scale µ = M S , and then, we estimate the higgsino mass at µ = µ H (M S ) by using the one-loop RGEs for higgsino and gauginos.   First, we show electron EDM d e in the high-scale SUSY scenarios (figure 5). We assume that there is no messenger superfield (N 5 = 0; dotted lines in figure 5) and that there is messenger superfields (N 5 = 1; solid lines in figure 5). The dark (light) shaded regions are excluded by the gluino search at the LHC [48,49], that is, M pole g < 1.3 TeV. The red lines correspond to the current bound on the electron EDM measured by the ACME experiment, |d e | < 8.7 × 10 −28 e cm. It is found that the future experiment for the electron EDM may have sensitivities to the SUSY breaking scale of M S ∼ 10 3 TeV in each scenario.

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In the high-scale SUSY models with messengers, there are several differences from the models with no messengers. One is that the constraint on the SUSY breaking scale becomes severe in the models with messengers. This is because that the cancellation between the anomaly and gauge mediated contributions reduce the gluino mass since we choose parameters to maximize the gluino CEDM. Another is that the Barr-Zee contributions are slightly suppressed since the masses of wino and bino become large in the extended models.
Next we investigate the nucleon EDMs in the high-scale SUSY models. First, we compare the nucleon EDMs induced by only the gluino CEDM with the Barr-Zee contribution. In figure 6, we show the nucleon EDMs induced by only the gluino CEDM (solid lines) or the Barr    other hand, if there is no physical phase of the gluino mass, the nucleon EDMs come from the Barr-Zee contributions. The Barr-Zee contributions are suppressed in the region of heavy higgsino and gauginos. Since there remains the ambiguity in the relative sign between the Barr-Zee contribution and the Weinberg operator, the nucleon EDMs are not determined if there exist both contributions. However, it is found that these contributions may be comparable in some region, especially heavy higgsino. Thus, we need to include the gluino CEDM contribution to determine precisely the nucleon EDMs. Now, let us estimate the ratio of neutron EDMs induced by the Barr-Zee diagrams and the Weinberg operator. In this evaluation, we fix the phases of Barr-Zee contributions as mentioned above, and we also estimate the quark (C)EDMs and the Weinberg operator at 1 GeV with the RGEs. In figure 7, we show the ratios of the neutron EDMs induced by the Weinberg operator (d nW ) and induced by the Barr-Zee diagrams (d nBZ ). Since the quark (C)EDMs and the Weinberg operator mix with each other, it is not obvious to discriminate between the neutron EDM derived from quark (C)EDMs and that derived from the Weinberg operator at low energy. We, however, identify the neutron EDM induced by the Barr-Zee contribution (d nBZ ) with d n defined in eq. (4.2), which is dominated by the Barr-Zee contribution since the RGE effect is negligible due to the one-loop suppression. Similarly, one induced by the Weinberg operator (d nW ) is identified with d n (w) defined in eq. (4.3).
In the left figure of figure 7, we show the ratio |d nW /d nBZ |, taking the messenger mass to be M S . (We will discuss the right figure of figure 7 below.) In the large µ H limit, the Barr-Zee contributions are suppressed since the EDMs induced by the Barr-Zee diagrams proportional to d f ∼ m f /M 2 µ H as mentioned above. On the other hand, in the small M S region, the neutron EDM induced by the gluino CEDM is enhanced due to the smallness of the gluino mass. Thus, in the region of small M S and large higgsino mass, the dominant contribution arises from gluino CEDM, though such a region is constrained by the gluino search at the LHC.   In the last of this section, let us consider the case of light messengers. The light messengers would realized if some symmetry is imposed to the messengers. As mentioned in the beginning of this section, the gluino CEDM is induced by the CP -violating coupling of gluino and messengers since the gluino mass is dominated by the anomaly mediated one. If the messengers have the mass lighter than M S , the suppression of the gluino CEDM becomes mild. Therefore, the nucleon EDMs induced by the gluino CEDM become large in comparison with the heavy messenger case.
In the light messenger case, since the gaugino masses are dominated by the anomaly mediated contribution, the phase of the gluino mass θ in the case is approximately zero, and then the gluino CEDM is proportional to sin θ F . Thus, in the following numerical evaluation, we set θ F = π/2 in order to maximize the gluino CEDM. If the supersymmetric masses of messengers are sufficiently light, the lightest scalars of messengers may have masses much lighter than the EW scale. In the following analysis, the scale of messengers is set to be M = 0.1M S in order to avoid too light scalars of messengers while x is 0.99.
We show the neutron EDM in the light messenger case in figure 8. The dotted and solid lines respectively describe the cases of no gluino CEDM and no Barr-Zee contribution. In this case, the neutron EDM via the gluino CEDM become larger than in the case of heavy messengers, as expected. On the other hand, the Barr-Zee contributions are the same in size as the case of heavy messengers. Therefore, the gluino CEDM gives the sizable contribution to the neutron EDM. If there exist both of the Barr-Zee and gluino CEDM contributions, the nucleon EDMs may be enhanced or suppressed. However, we do not evaluate the nucleon EDMs in this case since we have the sign ambiguity for the Weinberg operator in the nucleon EDMs.
As is the case for the heavy messenger, we also show the ratio of each contribution with light messengers (in the right figure of figure 7). In this case (M < M S ), the contribution JHEP11(2015)085 from the Weinberg operator is enhanced due to the light messenger mass as mentioned in eq. (5.1). Thus, the d nW dominates the neutron EDMs in a broad region in the light messenger scenario.
In figure 6, we have shown that the proton EDMs induced by the Barr-Zee diagrams and the gluino CEDM behave similar as the neutron EDMs. The behavior of the proton EDM is also similar to the neutron EDM in the light messenger scenario. Since the future experiments of the proton EDM may have sensitivities to |d p | ∼ 10 −29 e cm, the gluino CEDM effect via the proton EDM may be found.
Before we conclude this paper, we mention the uncertainty of numerical calculation. The error of α s (M Z ) = 0.1185 ± 0.0006 gives uncertainty on the results about O(1)% and then it does not appear in the numerical analyses. Thus, the uncertainties mainly come from the QCD sum rules and the naive dimensional analysis which we use in order to obtain the nucleon EDMs from the quark (C)EDMs and the Weinberg operator. Especially, the uncertainty in the naive dimensional analysis should be large so that we might expect larger contributions from the gluino CEDM to the nucleon EDMs. Furthermore, in the above, we assume N 5 = 1 and N 10 = 0 for simplicity. If more messenger fields are introduced, larger contribution from the gluino CEDM to the nucleon EDMs is expected if the CP phases are aligned to be constructive.

Conclusion and discussion
In this paper, we estimated the nucleon and electron EDMs in the high-scale SUSY models. Even if the gaugino masses induced by the anomaly mediation are real, the additional contributions to gaugino masses may give rise to the physical phases of gaugino masses. We show the case in the high-scale SUSY models which is based on the mixture of anomaly and gauge mediations. In particular, the gluino CEDM is induced by the physical phase of the gluino mass mass and the CP -violating couplings of gluino, and then it generates the CP -violating Weinberg operator.
We estimated the effect of the gluino CEDM in the extension of high-scale SUSY scenarios, and we showed the nucleon EDMs induced by the gluino CEDM and the Barr-Zee contributions. The dominant contribution to the nucleon EDMs within the MSSM comes from the Barr-Zee contribution of chargino/neutralino loops at two-loop level, while one-loop diagrams of the messenger particles generate the gluino CEDM. We revealed that the gluino CEDM may affect on the prediction of EDMs of nucleons in the high-scale SUSY models, especially in the cases of the light messengers or heavy higgsino in comparison with M S . We do not determine the total EDMs for nucleons since we still have large ambiguities in the naive dimensional analysis for the Weinberg operator, including the sign. If it is determined precisely, the proton and neutron EDMs may be found to behave differently in the extended models, and thus, it would be more important to detect nucleon EDMs and electron EDM or the ratios of them.

A Renormalization group equations
The (flavor-diagonal) CP -violating operators below the electroweak scale are given up to dimension-six operators as defined in eq. (4.1). For considering the RGE evolutions of these operators, it is convenient to define the quark (C)EDM operators O q i (i = 1, 2) as e q denotes the electric charge of the quark q and g 3 is the QCD coupling constant. The Weinberg operator is also given by O W as in eq. (3.7). Then, by using these operators, the effective Lagrangian is rewritten as follows: The relation between the coefficients in the Lagrangian in eq. (4.1) and the Wilson coefficients is given by: The RGEs for the Wilson coefficients at the leading order are given as [37] ∂ ∂ ln µ Here, N C (= 3) and N f are the number of colors and quark flavors, respectively. C F = (N 2 C − 1)/(2N C ) is the Casimir invariant and b 0 = −11N C /3 + 2N f /3 is the coefficient of the one-loop beta function for g 3 .

B QCD sum rules for nucleon EDMs
In order to get predictions for the nucleon and electron EDMs, we have to estimate the contribution to them from the parton-level interactions. In ref. [43], the neutron EDM d n is related to the quark EDMs and CEDMs by using the QCD sum rules at the renormalization scale µ = 1 GeV. Similarly, the proton EDM d p is also associated with the quark (C)EDMs at the renormalization scale µ = 1 GeV. The nucleon EDMs are given by using the QCD sum rules as Here, c 0 = 0.234, qq = −m 2 π f 2 π /(m u + m d ) = −(0.262 GeV) 3 is the quark condensate, and λ N relates the interpolation fields with the proton and neutron fields. m N denotes the mass of nucleon N . Θ N is calculated through the operator product expansions (OPE) for the correlator of interpolation fields and are just the coefficients proportional to qq . Without the Peccei-Quinn (PQ) mechanism [50] for the strong CP problem, we obtain Here, e q , m q , and d q ( d q ) denote the electric charge for quark q, the mass of quark q, and the (C)EDM for quark q, respectively. χ, κ, and ξ are parameters which relate the quark condensates on the electromagnetic background with qq ; qσ µν q F = e q χF µν qq , g s qG A µν T A q F = e q κF µν qq , 2g s q G A µν T A q F = ie q ξF µν qq , where · · · F denotes the vacuum expectation value on the electromagnetic background. These parameters are estimated by refs. [51,52], and then the values are given by χ = −5.7 GeV −2 , ξ = −0.74, and κ = −0.34. ρ u and ρ d are defined as The parameter m 2 0 which is associated with the VEV of q(G · σ)q is estimated by Belyaev and Ioffe [51]: Even if the PQ mechanism works, the theta parameter is induced as and then, we find the OPE coefficients Θ N as (B.8) Thus, we find the relation between the nucleon EDMs and the quark (C)EDMs under the PQ symmetry as follows: Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.