Lepton Electric Dipole Moment and Strong CP Violation

Contribution of the strong CP angle, $\bar\theta$, to the Wilson Coefficients of electron and muon electric dipole moment (EDM) operators are discussed. Previously, $\bar\theta$ contribution to the electron EDM operator was calculated by Choi and Hong. However, the effect of CP-violating three meson coupling was missing there. We include this missing contribution for the first time in the literature, and reevaluate the Wilson coefficients of the lepton EDM operator. We obtain $d_e = - (2.2-8.6) \times 10^{-28} \bar\theta$ e-cm which is 15 - 70 % of the result obtained in Choi and Hong. We also estimated the muon EDM as $d_\mu = - (0.5-1.8) \times 10^{-25} \bar\theta$ e-cm. Using $|\bar\theta| \lesssim 10^{-10}$ suggested by the neutron EDM measurements, we obtain $|d_e| \lesssim 8.6 \times 10^{-38}$ e-cm and $|d_\mu| \lesssim 1.8 \times 10^{-35}$ e-cm. The $\bar\theta$ contribution to the muon EDM is much below the sensitivities of the current and near future experiments. Our result shows that the $\bar\theta$ contribution to $d_{e,\mu}$ can be larger than the CKM contributions by many orders of magnitude.


Introduction
Precise measurement of EDMs is an important probe of CP violation. In particular, the lepton EDMs have recently received much attention because the experimental sensitivity is expected to improve considerably in near future. The electron EDM will be searched by ACME-II and III experiments [2], and the muon EDM will be searched in J-PARC [3] and Fermilab [4]. These experiments will probe interesting regions of parameter space of many well motivated models beyond the Standard Model (SM) [2,5,6].
Before discussing EDMs in models beyond the SM, it is important to first know the SM predictions precisely. The SM has two possible sources of CP violation: the phase of Cabibbo-Kobayashi-Maskawa (CKM) matrix and the QCD θ term (by QCD θ term, we always refer to the field redefinition independent combination,θ ). In this paper, we discuss the Wilson Coefficient of the lepton EDM operator: 1 1 The electron EDM experiments actually measure the EDM of an atom or a molecule. If we consider the effect ofθ to such measurements, CP violating electron-nucleon interaction (ēiγ 5 e)(N N ) gives the dominant contribution, and the true lepton EDM operator contribution (i.e., from de) is negligible. See e.g., refs. [7,8]. On the other hand, the muon EDM experiments measure spin precession of a single muon. Hence, unlike the electron EDM experiments, these experiments actually measure the Wilson Coefficient dµ.
The CKM contribution to d e is 10 −44 e-cm, see for example, [9] and the references therein. Thē θ contribution to electron EDM was first discussed by Choi and Hong in [1]. They obtained d e 1.4 × 10 −27θ e-cm. In their work, they considered CP violating meson-baryon-baryon coupling in chiral Lagrangian [10], and estimated chiral logarithm contribution to d e . However, there also exists CP violating meson-meson-meson couplings, and contributions from these couplings were not discussed in [1]. In this paper, we calculate d e and d µ taking into account both the CP violating meson-baryon-baryon and meson-meson-meson couplings.
In section 2, we briefly review the chiral Lagrangian for the calculation of EDMs. The details of the calculations of d e and d µ are presented in section 3. In section 4, we briefly discuss our results and summarize.

Chiral Lagrangian for the calculation of EDMs
In this section, we briefly review the chiral Lagrangian for the calculation of EDMs induced by the the strong CP phaseθ [10,11] (see also [12]). According to the CCWZ prescription [13,14], one introduces the coset fields ξ L,R and the baryon field B transforming under SU (3) L × SU (3) R chiral symmetry as where h is a compensator field, The quark mass matrix M is introduced as a spurion for explicit SU (3) L × SU (3) R breaking. The transformation rules for U and M are

Meson
The Meson field U is written as where f π = 93 MeV. The constant B 0 is determined by the pion and quark masses. We write the quark mass matrix M as The imaginary part of M has the role of a spurion for explicit P and CP breaking. Since Im(M ) in this basis does not break SU (3) V flavor symmetry, we do not have tadpole term for π 0 and η 0 [10,15].

CP violating couplings of mesons
gives CP violating interaction terms involving mesons. We obtain the triple pion coupling to be where, d abc is defined in terms of the Gell-Mann matrices λ a which satisfy {λ a , λ b } = (4/3)δ ab + 2d abc λ c . See also Refs. [10,11] for an alternative derivation. For interactions involving the η and K mesons, we obtain In section 3.1, we will calculate the CP violating π 0 F µν F µν and ηF µν F µν couplings by using the above interaction terms.

Baryon
The Baryon octet field is decomposed as We defineM andM + asM It can be easily seen thatM andM + transform as SinceM is transformed asM →M † under parity transformation,M + is parity invariant.

Baryon masses
It is always possible to choose a basis such that ξ L = ξ † R , i.e., ξ L = U 1/2 and ξ R = U −1/2 . In this basis,M Baryon mass splitting is generated by the following parity invariant terms, The mass splittings can now be written as Here we have defined The parameters b 1 and b 2 can now be determined in terms of the Baryon masses. The are given by

CP violating couplings of baryons
The Lagrangian in Eq. (14) also gives rise to CP violating interaction terms between mesons and baryons. For example, CP violating interaction terms which are relevant to neutron EDM are whereb 1 andb 2 are defined as Here we took m u = 2.15 MeV, and m d = 4.70 MeV [16]. The above formulae ofb 1 andb 2 hold at the leading order of the chiral perturbation theory. For the discussion on the next leading order corrections, see Ref. [17]. The interaction terms of π 0 /η 0 with the charged baryons induce the CP violating π 0 F µν F µν and ηF µν F µν couplings. The relevant interaction terms are given by π 0 /η 0 K ± Figure 1: An example diagram that contributes to the lepton EDM. The black circle is the CP-violating coupling and the grey circle is the CP-conserving interaction arising from the chiral anomaly matching.

Lepton electric dipole moment
In this section, we estimate the lepton EDM by using the chiral Lagrangian discussed in the previous section. The dominant source of lepton EDM is hadronic light-by-light diagrams, such as the one shown in Fig. 1. In particular, we estimate the chiral logarithm contributions arising from this class of diagrams.

CP violating meson-photon-photon couplings
As shown in Fig. 1, the relevant diagrams have CP-violating (π 0 /η 0 )-photon-photon subdiagrams. If the charged particles inside loop are heavy enough, these subdiagrams can be written as the following effective interactions, The coefficients c π and c η are obtained by one loop calculation in the chiral Lagrangian. For c π , there are contributions from K ± as well as the baryons (p, Ξ − ). The K ± contribution is denoted by c (K) π , and the baryon contribution is denoted by c π . We obtain The baryon-loop contribution c (b) π was calculated in Ref. [1], and our result is consistent with the reference. The sign of c π . So these contributions are destructive, and we will see in the next subsection that the resulting EDM is smaller than that of Ref. [1]. For c η , the one-loop baryon contribution is Loop diagrams involving π ± or K ± also contribute to η 0 -photon-photon amplitude however, it is not appropriate to consider their contribution as an effective η 0 F µν F µν interaction because π ± and K ± are lighter than η 0 . We will comment on the π ± and K ± loop contributions in section 3.2.3.

Chiral logarithm for lepton EDM
Here we discuss chiral logarithm contribution to lepton EDM. As shown in Fig. 1, the relevant diagrams have either π 0 or η 0 propagator. Below, we discuss π 0 and η 0 exchange diagrams separately.

π 0 contribution
The effective interaction terms relevant for the calculation of d are [1] where σ µν = (i/2)[γ µ , γ ν ]. The renormalization group (RG) equations for d , A 2 , and C 2 are given by [1] (see Fig. 2 for the relevant diagrams) The coefficient A 1 is determined by the chiral anomaly matching, and is given by The coefficient C 1 was calculated in the previous section. They are given below for convenience, As the kaons are integrated out at the scale of m K , K ± loop contributions are included in C 1 (µ) only for µ < m K . Solving the RG equations with a boundary conditions, A 2 (Λ) = C 2 (Λ) = 0, we obtain A 2 (µ) and C 2 (µ) as Using the above formulae, it is now straightforward to calculate the π 0 exchange contribution to the lepton EDM, d π , which is given by The above contributions are the leading chiral logarithm terms. In order to calculate the nextto-leading terms, one has to determine the incalculable counter terms from some observables.
To avoid this procedure, here we estimate d e and d µ by varying µ from m π /2 to m π in Eq. (34). We take the cutoff scale Λ as 4πf π . We obtain The contribution of c (b) π in Eq. (34) corresponds to the contribution which is calculated by Ref. [1]. The size of d π e is smaller than that because K ± loop and the baryon loop contributions are destructive.
Before closing this section, let us comment on the Z boson exchange diagrams. Although Z boson does not contribute to the light-by-light diagram, Z boson exchange gives a contribution to π 0 e + e − coupling at tree level, generating the coupling . The resulting contribution to the EDM, d Z , is For the numerical evaluation, we took Λ = 4πf π and varied µ from m π /2 to m π . Thus, the electroweak contribution is smaller than the hadronic light-by-light diagrams by two orders of magnitude, and we can safely neglect this contribution.

η 0 contribution: baryon loop
Performing a similar calculation as for the π 0 contribution, the baryon loop contribution in η 0 exchange diagrams is estimated to be To evaluate the numerical value, we take Λ = 4πf π and vary µ from m η /2 to m η . This gives, This contribution is smaller but comparable to d π .

η 0 contributions: π and K loop
Here we discuss η 0 exchange diagram with π ± or K ± loop. Since η 0 is the heaviest particle in the diagram, we integrate out η 0 first. After integrating η 0 out, we obtain the following effective interaction, This operator can generate lepton EDM at three-loop level. Its contribution has the same power of the coupling constant and loop factor as in Eq. (34). However, the operator mixing between C 3 and d gives a contribution which is proportional to C 3 log m η /m π , and this is not the leading contribution in the sense of chiral logarithms. So we do not consider this contribution here. We do not discuss K ± loop effect for the same reason.
As mentioned before, d µ in Eq. (42) above is actually equal to the muon EDM which is measured by the spin precession of muon. The experiments in J-PARC [3] and Fermilab [4] will have sensitivities to measure muon EDM up to about d µ = 10 −20 and 10 −21 e-cm respectively. Therefore, θ contribution to the muon EDM is much below sensitivities of the near future experiments. On the other hand, the bound on d e from the ACME experiment [21] assumes vanishing Wilson Coefficient for the operator (ēiγ 5 e)(N N ), and hence, their bound should not be compared directly with the Wilson Coefficient d e of the electron EDM operator given in Eq. (1).
As the CKM contribution to d e (d µ ) is estimated to be of the order of 10 −44 (10 −42 ) e-cm [9], our calculation shows that theθ contribution to lepton EDMs, even after taking into account the strong bound onθ from neutron EDM, can be larger than the CKM contribution by many orders of magnitude.