Nuclear EDM from SMEFT flavor-changing operator

We study nuclear electric dipole moments induced by ∆F = 1 effective operators in the Standard Model Effective Field Theory. Such contributions arise through renormalization group evolutions and matching conditions at the electroweak symmetry breaking scale. We provide one-loop formulae for the matching conditions. We also discuss correlations of these effects with ∆F = 2 observables such as ϵK and ΔMBd\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \Delta {M}_{B_d} $$\end{document}.


Introduction
Since we have not discovered new particles at the LHC experiment [1, 2], physics beyond the standard model (SM) is very likely to exist in high energy scale, particularly above the electroweak symmetry breaking (EWSB) scale. Such a high scale can be probed indirectly through flavor and CP violations. In particular, electric dipole moments (EDMs), which are CP -violating observables, are one of the most sensitive observables. Currently, an experimental bound on the EDM of the 199 Hg atom is provided as [3,4] |d Hg | < 6.3 × 10 −30 e cm, (1.1) at the 90% confidence level. Although theoretical calculations suffer from uncertainties in estimating the Schiff moment, it can constrain NP very severely. For the nucleon, the experimental bound of the neutron is [5] |d n | < 3.0 × 10 −26 e cm, (1.2) at the 90% confidence level. On the other hand, the indirect limit on the proton EDM is derived from 199 Hg as [6] |d p | < 2.1 × 10 −25 e cm.
(1. 3) In future, several experiments aim to improve the sensitivity by two orders of magnitudes for the neutron EDM [7,8]. Also, a storage ring experiment is projected to measure the proton EDM at the level of 10 −29 e cm [9]. Although the EDMs are flavor-conserving observables, flavor-violating interactions can contribute to them. In the SM, the W -boson interactions change quark flavors. Thus, a JHEP04(2020)053 class of NP can induce EDMs through quark flavor-changing interactions by exchanging the W boson. Such contributions are represented by the Standard Model Effective Field Theory (SMEFT) [10][11][12]. Here, all the SM particles including the electroweak bosons (W, Z, H) and the top quark (t) are retained. Above the EWSB, NP contributions to flavor and CP violations are encoded in higher dimensional operators in the SMEFT. At the EWSB scale, they are matched onto the effective operators in the low-scale effective field theory (LEFT) by integrating out W , Z, H and t. Low-scale observables such as the EDMs are evaluated by using the LEFT.
In this letter, we study the nuclear EDMs from SMEFT flavor-changing operators. They are induced by ∆F = 1 operators through radiative corrections of the W boson. In particular, we will focus on top-quark loop contributions, because they tend to be large due to the large top quark mass (cf. ref. [13]). The radiative corrections are taken into account by solving the renormalization group equations (RGEs) in the SMEFT [14][15][16]. In addition, the SMEFT operators are matched onto those in the LEFT at the EWSB scale. The one-loop matching conditions are necessary, because the contributions of the ∆F = 1 operators to EDMs are induced by radiative corrections. The one-loop formulae will be provided in this letter. These operators also contribute to ∆F = 2 observables such as K and ∆M d through the W -boson loops. Since these observables are sensitive to NP contributions, we will discuss correlation between the contributions to the EDMs and the ∆F = 2 observables.

Formula
In this section, we provide formulae for evaluating the EDMs induced by flavor-changing operators. By decoupling NP particles, their contributions are encoded in higher dimensional operators in the SMEFT. Then, these operators are evolved by following the RGEs. Anomalous dimensions in the SMEFT are provided at the one-loop level in refs. [14][15][16], and those relevant for the CP and flavor violations are summarized in ref. [13]. At the EWSB scale, they are matched to the LEFT operators. We provide the one-loop matching formulae between the SMEFT ∆F = 1 operators and the LEFT ∆F = 0 CP -violating operators.
In the SMEFT, the ∆F = 1 effects are encoded into higher dimensional operators, which are defined as [11] with the derivative, where q is the SU(2) L quark doublet, d the right-handed down-type quark, u the righthanded up-type quark, and T A the SU(3) C generator with quark-flavor indices i, j, k, l and an At the EWSB scale, the SMEFT operators are matched to the LEFT operators. The latter operators for EDMs are defined as

15)
2 See ref. [17] for an extensive study of the SMEFT operator, ( , where EDMs and flavor observables are examined. Also, the nucleon/nuclear EDM has been discussed within the context of the SMEFT in refs. [18][19][20], where flavor-conserving operators have been studied.

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where i, j are quark-flavor indices. The effective operators are defined as 3 where α, β are color indices, and F µν (G A µν ) is the electromagnetic (gluon) field strength.
The matching conditions are derived by integrating out SM heavy degrees of freedom, such as W, Z, H and t. At the tree level, we obtain the conditions, where the Wilson coefficients are evaluated at the EWSB scale, µ = µ W . The other LEFT operators are not induced at the tree level. In addition, the SMEFT ∆F = 1 operators can generate ∆F = 0 amplitudes through the one-loop matching conditions at the EWSB scale. We focus on the contributions from the loop diagrams with the top quark and the W boson (cf. ref. [13]). The conditions in the Feynman-'t Hooft gauge are obtained as Besides, there is a strong CP phase,θ. In this letter, we assumeθ = 0, for simplicity. JHEP04(2020)053 where the parameters are defined as Here, V ij is the CKM matrix, and s W = sin θ W with the Weinberg angle θ W . The loop functions are defined as All the Wilson coefficients are evaluated at the EWSB scale, µ = µ W . The other LEFT operators for the EDMs do not receive one-loop corrections at this scale. The SMEFT ∆F = 1 operators also generate LEFT ∆F = 2 operators. The latter operators are defined as We follow the analysis in ref. [13], where the SMEFT RGEs and the matching formulae at the one-loop level are provided. Below the EWSB scale, the LEFT ∆F = 0, 2 operators are evolved by the RGEs. Then, the low-scale observables are evaluated around the hadron scale.

Observables
In this section, low-scale observables are summarized. We consider the EDMs, K and ∆M B d . All of them are very sensitive to NP contributions to CP violations.

Nuclear EDMs
The CP -violating operators of the down-type quarks induce the nuclear EDMs. 4 Then, hadronic matrix elements are necessary to evaluate their contributions. There are many types of the SMEFT four-quark operators. Contributions ofÕ ds 1 andÕ sd 1 are evaluated by the effective chiral Lagrangian technique [29]. Those operators generate CP -violating baryon-meson interactions through vacuum-expectation values (VEVs) of pseudoscalar mesons. Then, the 199 Hg EDM is induced at the tree level as [30] d Hg In addition, from the baryon-meson loop diagrams, we obtain 5 where the Wilson coefficients are estimated at the hadron scale, µ = 1 GeV. Here and hereafter, we setθ = 0 for simplicity. 6 The derivations of eqs. (3.2) and (3.3) are given in appendix A. Four-quark operators,Õ db andÕ bd , involve the bottom quark. In order to derive their contributions to the neutron and proton EDMs, we follow the strategy explored in refs. [31][32][33]. The result becomes where the Wilson coefficients are estimated at the hadron scale, µ = 1 GeV. Here, the contribution to the proton EDM, (3.5), is derived by multiplying a ratio of the magnetic moments, µ p /µ n , to that of the neutron EDM, (3.4) (cf., ref. [33]). On the other hand, O sb andÕ bs are much less constrained by the EDMs, because they do not depend on the down quark.
Let us summarize the current experimental limits and future prospects. The current bounds are obtained as [3][4][5][6] |d In the analysis, CP -violating baryon-meson interactions are considered to discuss the nuclear EDMs (see appendix A). They can also induce the electron EDM, e.g., via the Barr-Zee diagram, which will be explored in future. 5 The nucleon EDMs are also induced by baryon-meson diagrams at the tree level [30]. However, we confirmed that they are sub-dominant. 6 The Peccei-Quinn (PQ) mechanism is not assumed for realizingθ = 0. It is straightforward to extend the case forθ = 0. Then, the PQ mechanism is introduced to avoid the strong CP problem. The following conclusions do not change qualitatively.

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In future, experiments are projected to achieve the sensitivities of |d n | ∼ 10 −28 e cm [7] and |d p | ∼ 10 −29 e cm [9]. Although the 199 Hg EDM constraint is the strongest at this moment, the neutron/proton EDMs can provide severer bound by the future experiments. Before closing this section, let us comment on contributions to the electric and chromoelectric dipole moments, O i 1 and O i 2 . As mentioned in the previous section, the SMEFT ∆F = 1 operators contribute only to the LEFT four-quark operators,Õ ij 1,2 . Below the EWSB scale, they induce O i 1,2 through radiative corrections. However, according to the RGEs in the LEFT, their contributions appear as linear combinations ofC ij 1 +C ji 1 and C ij 2 +C ji 2 as for O i 1,2 with a = 1, 2 and coefficients α a , β a . By substituting the SMEFT contributions intoC ij 1,2 in the right-hand side, all the contributions are found to vanish (see eqs. (2.25)-(2.28)). Consequently, the SMEFT ∆F = 1 operators do not generate the electric or chromoelectric dipole moment. Hence, we will study the nuclear EDMs directly from the four-quark operators.

∆F = 2 observables
The ∆F = 2 operators contribute to the oscillations of the neutral mesons. In particular, the indirect CP violation of the neutral K mesons, K , and the mass difference of the neutral B q mesons are sensitive to NP contributions. The former is sensitive to flavor violations between the first two generations of the down-type quark. The SM and NP contributions are represented as where V cb is determined by the inclusive semileptonic B decays. The NP contribution is represented as  [37]. The Wilson coefficients are evaluated with the NLO-QCD RGEs [38], and hadron matrix elements in ref. [39] are used. On the other hand, the experimental result is [37] | exp K | = (2.228 ± 0.011) × 10 −3 . at the 2σ level.

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Next, flavor violations including the bottom quark are constrained by the oscillations of the neutral B q mesons. In particular, those between the first and third generations contribute to ∆M B d . The SM and NP contributions are represented as where GeV [37]. The first term in the right-hand side denotes the SM contribution, which is estimated as [40] ∆M SM d = (4.21 ± 0.34) × 10 −13 GeV. (3.16) The Wilson coefficients are evaluated with the NLO-QCD RGEs [38], and hadron matrix elements in ref. [40] are used. On the other hand, the experimental result is obtained as [37] ∆M exp

Numerical analysis
In this section, we study contributions of the SMEFT ∆F = 1 operators to the nuclear EDMs and ∆F = 2 observables, K and ∆M B d . In figure 1 JHEP04(2020)053 (C Hd ) 13 (C ud (1) ) 3313 (C qd (8) ) 3313 neutron/proton EDMs are expected to be improved greatly. They can probe the NP scale up to 2-10 TeV, which are beyond the limit of the 199 Hg EDM.
The contributions to the nuclear EDMs are suppressed for the operators including the bottom quark. This is because the hadron matrix elements of such operators are small (see eq. (3.4)). Currently, the constraint is weaker than M NP 100 GeV according to the middle panels of the figure, and the sensitivity may reach at most 3 TeV in future. On the other hand, the deep green region in the right panel corresponds to |d n | < 10 −29 e cm, which is below the future sensitivity. In the left panel, the blue region is allowed by K at the 2σ level, and the region in the right panel is allowed by ∆M B d at the 2σ level.  Figure 5. Same as figure 2, but the deep green region in the right panel is |d n | < 10 −29 e cm, which is one order of magnitude weaker than the future sensitivity.
Let us study correlations between the EDMs and the ∆F = 2 observables. The results depend on the SMEFT operators. The ∆S = 1 operators of (C  Figure 6. Same as figure 2, but the deep green region in the right panel is |d n | < 10 −29 e cm, which is one order of magnitude weaker than the future sensitivity.
at the 2σ level. On the other hand, contours of the neutron, proton and 199 Hg EDMs are shown by the bands with different colors.
From the figures, it is noticed that the 199 Hg EDM gives a bound on the ∆S = 1 operators, and the proton EDM can provide a better sensitivity for them. For some of the ∆B = 1 operators especially (C (8) qd ) 3313 and (C (8) ud ) 3313 , future measurements of the proton EDMs will also be able to compete with the constraint from ∆M B d . We want to emphasize that the parameter dependence of the EDMs is different from that of K . Thus, the NP contributions to the effective operators can be specified by combining the EDMs with the flavor observables.
Next, let us consider C Hq ) 12,13 . We found that they do not contribute to the EDMs because of the Lorentz structures of these operators. In fact, they generate only the vector-type operators of the four quarks below the EWSB scale, which do not violate the CP symmetry.
Similarly, the operators of (C Hq ) 12,13 do not contribute to the EDMs through the fourquark operators. Let us consider another contribution. It is noticed that these operators include W boson interactions by taking the Higgs VEV as in the Feynman-'t Hooft gauge, where G ± is the NG bosons. Here, all the quark fields are left-handed in these interactions. Then, they seem to generate the electric and chromoelectric dipole moments through penguin diagrams of the W boson loops. However, it can JHEP04(2020)053 be checked that such contributions vanish by paying attention to the chirality structure of the quark. Hence, the operators of (C Hq ) 12,13 do not contribute to the nucleon EDMs. Finally, let us comment on C dd . This operator can also contribute to the EDMs through the RGEs and matching conditions. However, these contributions are found to be very small, and we do not discuss them anymore.

Conclusions
We studied the nuclear EDMs induced by the SMEFT ∆F = 1 operators and their correlations with the ∆F = 2 observables. These SMEFT operators contribute to them through the W boson loops. The radiative corrections via the RGEs and the matching conditions at the EWSB scale are taken into account. In particular, we provide the one-loop formulae of the matching conditions for the EDMs.
It was found that some of the operators are already excluded for M NP 1-9 GeV by the 199 Hg EDM, and future experiments for the proton EDM may be able to probe those in M NP 2-10 TeV. Compared with K and ∆M B d , it was shown that the nuclear EDMs can provide a complementary information on the ∆F = 1 effective operators in future.
Other nuclear EDMs such as 129 Xe and 225 Ra can also be sensitive to the CP -violating baryon-meson interactions. Although the current bounds are weaker than that of 199 Hg, they would be examined better in future experiments (see e.g., ref. [7]). Although their theoretical calculations suffer from potentially large uncertainties in estimating the the Schiff moment, it is interesting to study future sensitivities to the SMEFT ∆F = 1 operators, which will be explored elsewhere.
Note added: while we are submitting this letter, a new article [41] was published on arXiv; the authors argued that an enhancement factor coming from the strange quark mass which was mentioned in ref. [42] and is quoted in eq. (A.21) disappears. Since this factor can induce a large contribution to the neutron and proton EDMs, the numerical analysis in this article may be affected.

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represented as where the coefficients are defined as Under the chiral rotations of U(3) L × U(3) R , we impose the following transformations, with L, R are transformation matrices of U(3) L and U(3) R , respectively. Then, the righthand side of eq. (A.1) is invariant under this transformation. By adopting this symmetry in the meson chiral Lagrangian, the CP -violating terms are written at O(p 2 ) as where U, χ are defined as

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Here, F π is the decay constant of the pion, and F 0 is that for η 0 . The mesons matrix U transforms as U → RU L † under U(3) L × U(3) R . We approximate as F 0 F π , B 0 m 2 π /(m u + m d ) and 48a 0 /F 2 0 m 2 η + m 2 η − 2m 2 K . By a naive dimensional analysis, we estimate the unknown low-energy constants, c 1 , c 2 and c 3 , as From eq. (A.7), the scalar potential for the neutral mesons, π 0 , η 8 and η 0 , is extracted as Since we are interested only inC ds 1 andC sd 1 , the other Wilson coefficients are set to be zero. Then, the VEVs of the meson fields are obtained at the leading order ofC ds 1 and C sd 1 as , (A.13) . (A.14)

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These couplings contribute to the neutron and proton EDMs through the baryon-meson loop diagrams. By following the analysis in ref. [43], the EDMs are estimated as where the finite terms of the leading contributions are shown, 8 and the renormalization scale is set to be the nucleon mass, m N . Applying the pion decay constant F π = 86.8 MeV [44], the meson-baryon couplings D = 0.804 and F = 0.463 from the hyperon β decays [45], the low-energy constants b D = 0.161 GeV −1 and b F = −0.502 GeV −1 from the baryon octet mass splittings [46], and the quark masses