Nucleon EDM from SMEFT flavor-changing operator

We study nucleon electric dipole moments induced by $\Delta F=1$ 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 $\Delta F=2$ observables such as $\epsilon_K$ and $\Delta M_{B_d}$.


Introduction
Since we have not discovered new particles at the LHC experiment [1], 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, the experimental bound of the neutron is [2] |d n | < 3.0 × 10 −26 e cm, (1.1) at the 90% confidence level. On the other hand, the indirect limit on the proton EDM is derived from 199 Hg [3,4] as [5] |d p | < 2.1 × 10 −25 e cm. (1.2) In future, several experiments aim to improve the sensitivity by two orders of magnitudes for the neutron EDM [6,7]. Also, a storage ring experiment is projected to measure the proton EDM at the level of 10 −29 e cm [8].
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 class of new physics (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) [9][10][11]. 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 neucleon 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. [12]). The radiative corrections are taken into account by solving the renormalization group equations (RGEs) in the SMEFT [13][14][15]. In addition, the SMEFT operators are matched onto those in the LEFT at the EWSB scale. The oneloop 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. Theses 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. [13][14][15], and those relevant for the CP and flavor violations are summarized in Ref. [12]. 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 [10] where the first term in the right-hand side is the SM Lagrangian, and the second term represents the higher dimensional operators. Here, the Lagrangian is invariant under the SM gauge symmetry, and all the SM particles including W, Z, H and t are dynamical. In this letter, we consider the operators of the down-type quarks #1 . The dimension-six operators which potentially contribute to the EDMs are shown as #2 with the derivative,

14)
#1 It is straightforward to extend the analysis to ∆F = 1 operators of the up-type quarks. In this case, radiative corrections are likely to be dominated by bottom-quark loops. #2 The neutron EDM has also been discussed within the context of the SMEFT in Ref. [16].
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 At the EWSB scale, the SMEFT operators are matched to the LEFT operators. The latter operators for EDMs are defined as 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. We define F · σ = F µν σ µν , G · σ = G A µν σ µν T A andG A µν = 1 2 µνρσ G A ρσ with σ µν = i 2 [γ µ , γ ν ] and 0123 = +1. Also, f ABC is the structure constant, and m q is a mass for quark q. These operators are mixed through the RGEs, which are found in Refs. [17][18][19][20][21] (see Refs. [22][23][24] for higher order corrections).
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 #3 Besides, there is a strong CP phase,θ. In this letter, we assumeθ = 0, for simplicity. the loop diagrams with the top quark and the W boson (cf. Ref. [12]). The conditions in the Feynman-'t Hooft gauge are obtained as 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. [12], 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.

Nucleon EDMs
The CP -violating operators of the down-type quarks induce the nucleon EDMs. 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 [25]. Those operators generate CP -violating baryon-meson interactions through vacuum-expectation values (VEVs) of pseudoscalar mesons. Then, from baryon-meson loop diagrams, we obtain #4 where the Wilson coefficients are estimated at the hadron scale, µ = 1 GeV. Here and hereafter, we setθ = 0 for simplicity #5 . The derivations of Eqs. (3.1) and (3.2) 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 EDM, we follow the strategy explored in Refs. [27,28]. The result becomes where the Wilson coefficients are estimated at the hadron scale, µ = 1 GeV. Since the proton EDM from these operators has not been evaluated, we analyze only the neutron EDM for them. On the other hand,Õ sb andÕ bs are much less constrained by the EDMs, because they do not depend on the down quark. #4 The nucleon EDMs are also induced by baryon-meson diagrams at the tree level [26]. However, we confirmed that they are sub-dominant. #5 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.
Let us summarize the current experimental limits and future prospects. The current bounds are obtained as [2,5] (3.5) In future, experiments are projected to achieve the sensitivities of |d n | ∼ 10 −28 e cm [6] and |d p | ∼ 10 −29 e cm [8].
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 for O i 1,2 with a = 1, 2 and coefficients α a , β a . By substituting the SMEFT contributions intõ C 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 nucleon 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 with φ = (43.51 ± 0.05) • . The SM prediction is estimated as [29] SM where V cb is determined by the inclusive semileptonic B decays. On the other hand, the experimental result is [30] | exp K | = (2.228 ± 0.011) × 10 −3 .
The first term in the right-hand side denotes the SM contribution, which is estimated as [31] ∆M SM

Numerical analysis
In this section, we study contributions of the SMEFT ∆F = 1 operators to the nucleon EDMs and ∆F = 2 observables, K and ∆M B d . In Fig. 1, the neutron and proton EDMs are estimated. On each line, one of the Wilson coefficients is set to be C i = i/M 2 NP at the NP scale, M NP . The other coefficients are zero. The effective operators missing in the list do not contribute to the EDMs #6 . Once the operator is set, the RGEs are solved, and the matching conditions are taken into account. In the top panels, the four-quark operators mix the first two generations of the down-type quark. On the other hand, the operators in the bottom panel include the bottom quark. Since there are no estimations of the proton EDM for them, only the neutron EDM is estimated.
In the plots, the horizontal red and blue dotted lines correspond to the current experimental bound and the future sensitivity, respectively. For the latter, we quote |d n | = 10 −28 e cm and |d p | = 10 −29 e cm. Currently, the EDM constraints are weak. The NP contributions are excluded only for M NP 1 − 2 GeV of (C (1,8) qd ) 3312 . The sensitivities are expected to be improved greatly. The neutron and proton EDMs can probe the NP scale up to 2 − 10 TeV. On the other hand, the contributions to the EDMs are suppressed for the operators including the bottom quark. This is because the hadron matrix elements of such operators are small (see Eq. (3.3)). Currently, the constraint is weaker than M NP 100 GeV according to the bottom panel of the figure, and the sensitivity may reach 300 GeV in future. 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 Currently, all the parameter regions are allowed by the EDMs (see also Fig. 1). In future, the EDMs can be sizable for the ∆S = 1 operators. It is noticed that, since the #6 There are operators which can contribute to the EDMs through self-energy corrections. The matching conditions are provided in Section 2, and it is straightforward to analyze them. parameter dependence of the EDMs is different from that of K , they provide an independent information on the effective operators. On the other hand, it is found that the EDMs cannot compete with ∆M B d for all the ∆B = 1 operators in these parameter regions. 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 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.  Figure 3. Same as Fig. 2, but the green region in the right panel corresponds to |d n | < 10 −29 e cm, which is one order of magnitude weaker than the future sensitivity.

Conclusions
We studied the nucleon 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 − 2 GeV by the neutron EDM, and future experiments may be able to probe those in M NP 2 − 10 TeV. Compared with K , it was shown that the nucleon EDMs can provide an complementary information on the ∆S = 1 effective operators in future, though the EDMs from the ∆B = 1 operators are tiny and cannot compete with the constraints from ∆M B d .

A Estimation of neutron and proton EDMs
In this section, let us explain how to estimate the contributions of the four-quark operators to the neutron and proton EDMs with the chiral Lagrangian technique [25]. In particular, we follow the analysis explored in Ref. [32].
We consider the nucleon EDMs through meson condensations induced by the CP -violating four-quark operatorsÕ q q 1 . At the parton level, these operators are represented as Figure 4. Same as Fig. 2, but the pink region in the left panel corresponds to |d n | < 10 −29 e cm, which is one order of magnitude weaker than the future sensitivity. Besides, the deep green region in the right panel is |d n | < 10 −31 e cm, which is much smaller than the future sensitivity.
where the coefficients are defined as Under the chiral rotations of U (3) L × U (3) R , we impose the following transformations,  Figure 5. Same as Fig. 2, but the purple and green regions in the left panel correspond to < 10 −30 e cm, which are weaker than the future sensitivities. Also, the deep green region in the right panel is |d n | < 10 −31 e cm.
where U, χ are defined as Here, F π is the decay constant of the pion, and F 0 is that for η 0 . The mesons matrix U transforms as 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 Figure 6. Same as Fig. 2, but the deep green region in the right panel is |d n | < 10 −31 e cm.
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 andC sd 1 as , (A.13) . (A.14) On the other hand, the baryon chiral Lagrangian is obtained at O(p 2 ) as where the baryon matrix B is defined as with ξ L,R defined as U = ξ R ξ † L , and ξ R = ξ † L . Also, M B is the baryon mass. The definitions of Γ µ , ξ µ , and χ + are By inserting the meson VEVs into the baryon chiral Lagrangian, the CP -violating baryon-meson interactions become L baryons ⊃ḡ npπ −npπ − +ḡ nΣK + Σ + pK + +ḡ π + npp nπ + +ḡ K + Λpp ΛK + +ḡ K + Σ 0 pp Σ 0 K + , (A.20) where the coupling constants are obtained as These couplings contribute to the neutron and proton EDMs through the baryon-meson loop diagrams. By following the analysis in Ref. [33], the EDMs are estimated as where the finite terms of the leading contributions are shown #7 , and the renormalization scale is set to be the nucleon mass, m N . Applying the pion decay constant F π = 86.8 MeV [34], the meson-baryon couplings D = 0.804 and F = 0.463 from the hyperon β decays [35], the low-energy constants b D = 0.161 GeV −1 and b F = −0.502 GeV −1 from the baryon octet mass splittings [36], and the quark masses