Constraints on CP-odd ALP couplings from EDM limits of fermions

We discuss constraints on soft CP-violating couplings of axion-like particles (ALPs) with photon and fermions by using data on electric dipole moments (EDMs) of Standard Model (SM) particles. In particular, we derive bounds on CP-odd ALP-photon-photon coupling from data of the {\tt ACME} Collaboration on electron EDM. We also discuss prospects of the Storage Ring experiment to constrain the ALP-photon-photon coupling from data on proton EDM. The regarding constraints from experimental bounds on the muon and neutron EDMs are weak. We set constraint on the CP-odd ALP coupling with electron and derive bounds on combinations of coupling constants, which involve soft CP-violating terms. Finally, we discuss a scenario of inducing of Yukawa-like interaction of SM fermions with ALP through the electro-weak symmetry breaking mechanism.


I. INTRODUCTION
Since resolving strong CP violation problem using Peccei-Quinn (PQ) mechanism [1] the axion-like particles (ALPs) proposed by Weinberg and Wilczek [2,3] play important role in hadron phenomenology and searching for New Physics (NP) beyond Standard Model (SM) [4,5]. In this vein the important step was formulation of the effective Lagrangian approach with explicit manifestation of the invisible axion [6]. In particular, Lagrangian involving couplings of axion with SM gauge fields and fermions has been proposed. It was shown that the couplings of axion with SM gauge fields (G = g, W, B) are generated using anomalous coupling of the ALP to GG gauge field currents, where G andG are generic strength of gauge field and its dual. In particular, the part describing the coupling of ALPs with photons and fermions ψ = e, µ, p reads [6,7] L ⊃ 1 2 (∂ µ a) 2 − m 2 a 2 a 2 + g aγγ 4 a F µνF µν + ψ=e,µ,p where g aγγ = c aγγ /Λ and g aψψ = c ψψ m f /Λ are the couplings of ALP with photons and fermions, Λ is the NP scale, which is much larger than the electroweak scale Λ EW : Λ Λ EW . One should stress that the coupling of axion with SM fermions are suppressed by 1/Λ. Such coupling can be generated from the coupling of axion to the scalar fields (dimension-5 operator) after spontaneous breaking of electroweak symmetry [8]. In the present paper we address that issue in detail.
In addition to this CP-even coupling, let us consider CP-odd coupling of ALP with photons whereḡ aγγ has dimension of GeV −1 . Such coupling was recently discussed in Ref. [9]. On the other hand, ALP is accompanied by a scalar field, dilaton φ, in extra dimension theories. In particular, these degrees of freedom play important role in phenomenology of black holes and hadrons [10][11][12][13]. Coupling of the dilaton with photons has similar structure as the CP-odd one for the axion: L ⊃ g φγγ 4 φ F µν F µν . Note, analogous couplings with two photons in case of light scalar mesons f 0 (600) and a 0 /f 0 (980) have been studied in Refs. [14][15][16]. The coupling of the dilaton with fermions has Yukawa type, which is manifestly CP invariant: L ⊃ḡ φψψ φψ ψ. Note, the dilaton plays the role of the Nambu-Goldstone-like boson responsible for spontaneous breaking of conformal/scale invariance [17]. Its mass is expected below the typical conformal symmetry breaking scale m φ 1/g φγγ .
Constraints on φγγ coupling from collider experiments are widely discussed in the literature [18][19][20][21][22][23][24][25][26] for the mass range 1 GeV m φ 1 TeV. In addition, authors of Ref. [27] provided a detailed analysis of light dilaton scenarios (1 keV m φ 10 GeV) and estimated the bounds on radion-photon-photon coupling g φγγ from Supernova SN1987a, cosmology, Horizontal Branch stars, and beam-dump experiments. The latter analysis reveals an unconstrained window below g φγγ 10 −5 GeV −1 for the regarding mass range. However, we note that emerging a CP violating coupling in the dilaton model L ⊃ g φψψ φψ iγ 5 ψ will require a proper recasting of the relevant bounds. That task however is beyond the scope of present paper. Instead, we study CP-violating sce-nario (2) for light sub-GeV pseudo-scalar particle and analyze in detail its implication for EDM physics of charged leptons and nucleons. In addition, for the certain ALP mass range we also set the limits on soft CP-violating coupling of ALP with electron: In our previous paper [28] we discussed NP phenomenology of hidden scalar, pseudoscalar, vector, and axial-vector particles coupled to nucleons and leptons, which could give contributions to different puzzles in particle physics (like proton charge radius, (g − 2) µ , 8 Be-4 He anomaly, electric dipole moments (EDMs) of SM particles).
In the present paper we derive new limits on the couplings of ALPs with SM fermions using data on fermion EDMs. In particular, we consider the contribution of diagrams to fermion EDMs generated by the CP-even coupling of ALP with fermions and CP-odd coupling of ALP with photons. In this vein we do not require a universality of the coupling of ALP with leptons and quarks, which means that the limits on quark couplings with ALP are not necessary applicable to corresponding couplings in lepton sector.
In addition, in our paper we discuss an exotic model of ALP interaction with Higss field. In that scenario we describe inducing of ALP coupling with SM fermions. In particular, ALP-Higgs coupling after spontaneous breaking of electroweak symmetry gives rise the coupling of axion to the Z 0 gauge boson and then Yukakawa-like coupling to the leptons and quarks.
The paper is structured as follows. In Sec. II we consider constraints on CP-even ALP-lepton couplings for the mass range of interest from 100 keV to 1 GeV. In Sec. III we discuss existing bounds from fixed target experiments on CP-odd couplings of axion with photons. In Sec. IV we obtain more stringent limits on ALP couplings using data on electron EDM. The expected bounds on ALP couplings from proton EDM are derived in Sec. V, we also discuss bounds on ALP coupling from muon and neutron EDM in that section. In Sec. VI we discuss bounds on CP-odd couplings associated with aγZ 0 and aee interaction. Combined bounds on products of ALP couplings are dicsussed in Sec. VII. In Sec. VIII we discuss inducing of Yukawa-like coupling of SM fermions through the interaction of ALP with neutral gauge field.

II. CONSTRAINTS ON ALP COUPLINGS WITH CHARGED LEPTONS
In this section we discuss constraints on CP-even ALPfermion coupling L ⊃ ig aψψ aψγ 5 ψ for the mass range of interest 100 keV m a 1 GeV. In particular, we refer to analysis of Ref. [7] on leptophilic coupling of ALP Let us consider existing constraints on ALP coupling with electrons g aee . It is appropriate to rewrite corresponding coupling g aee in Yukawa-like term as follows g aee = c ee m e /Λ. Indeed, Eq. (4) on lepton mass shell implies that Furthermore, authors of Ref. [7] provided current limits on c ll /Λ from beam-dump experiments [29] and BaBar facility [30] as well as from astro-particle physics and cosmological observations [31], assuming lepton universality of couplings, c ee c µµ c τ τ . In particular, in our estimate we use a benchmark conservative value c ee /Λ 10 −1 GeV −1 from coupling loop hole in the ALP mass range 100 keV m a 200 MeV. Moreover, we show corresponding constraints on c ll /Λ in Fig. 2. That plot is adapted from right panel in Fig. 4 of Ref. [7]. Finally, this implies g aee 5 × 10 −5 for electron-ALP coupling.
However, one remark should be added. For concreteness in our study we consider non-universal ALP coupling with leptons and quarks, c ll = c qq , which means that limits on c qq /Λ coming from meson decays [29,32] are not directly applicable to c ll /Λ bounds.

III. BOUNDS FROM FIXED TARGET EXPERIMENTS
In this section we discuss existing bounds from fixed target experiments on CP-even couplings of axion with photons. For completeness, we consider CP-even interaction of ALP with photons L ⊃ gaγγ 4 aF µνF µν . That coupling has been already constrained for mass range 1 MeV m a 1 GeV by electron and proton fixed target experiments associated with ALP production in Primakoff process, γN → N a, followed by visible decay a → γγ within fiducial volume of regarding facilities [33]. Furthermore, one can also relate bounds FIG. 2: Exclusion limits for leptophilic ALP coupling c ll /Λ ≡ g all /m l (see, e.g., Eq. (4) for details) adapted from Ref. [7] from g aγγ toḡ aγγ as followsḡ aγγ = g aγγ . In particular, latter relation implies equivalent ALP decay width into photons, a → γγ, and its production crosssection via Primakoff reaction, γN → N a (see, e.g., Ref. [33] for details). This means that operators O 1 = gaγγ 4 aF µν F µν and O 2 = gaγγ 4 aF µνF µν yield the equivalent observables at three level after averaging (summation) of amplitudes squared |M aγγ | 2 over photon polarizations: pol |M aγγ | 2 = 1 2 g 2 aγγ m 4 a . In Fig. 3 we show limits onḡ aγγ − m a adapted from Ref. [34] for SLAC137, CHARM, NuCal and SLAC141 experimental facilities, assumingḡ aγγ = g aγγ and Br(a → γγ) = 1.

IV. ALP BOUNDS FROM ELECTRON EDM
First, let us consider Lagrangian describing the coupling of ALP with SM photons and electrons L ⊃ḡ These operators induce a finite contribution to the EDM of electron. Furthermore, it must be point out that CPeven couplings L ⊃ gaγγ 4 aF µνF µν don't generate electron EDM operators at one-loop level. In particular, in Ref. [28] authors showed that ALP Lagrangian (6) induces the electron EDM, which has the following form where  Red line shows parameter space of ALP at 90% CL constrained by ACME that corresponds to the electron EDM limits |de/e| < 1.1 × 10 −29 cm and gaee 5.0 × 10 −5 (cee/Λ 10 −1 GeV −1 ). Black solid lines are expected bounds onḡaγγ for planing sensitivity of SRE to the proton EDM at the level of |dp/e| < 10 −29 cm. In addition, it is assumed that EDM limits from ACME data are calculated for Br(a → e + e − ) = 1 and fixed target bounds are obtained for Br(a → γγ) = 1.
and for light ALP, m a /m e 1, it is given by In Fig. 1 we show corresponding contribution of 1-loop diagram to the operator of fermion EDM.
Therefore, for g aee 5 × 10 −5 it follows from Eqs. (7), (9), and (10) that for m a m e . Moreover for m a m e one has the following allowed limit on ALP-photon-photon couplinḡ .
In Fig. 3 we show ALP parameter space constrained by ACME experiment atḡ aγγ − m a plane.

V. BOUNDS FROM PROTON, MUON AND NEUTRON EDM CONSTRAINTS
Now let us consider prospects of the Storage Ring experiment (SRE) (see, e.g., Ref. [37]) to probe ALP We address to Ref. [38] and Ref. [39] for experimental constraints on muon and proton EDM respectively (for recent review see, e.g., Ref. [40]).
scenario with coupling to proton L ⊃ ig app apγ 5 p. In particular, in our analysis we consider the following typical bound on ALP-proton-proton coupling g app 10 −10 − 10 −9 for the mass range of interest 200 MeV m a 1 GeV. In addition, the limit on g app is expected to be reasonable due to ruled out limits on light pseudoscalar universal coupling with quarks [32] at the level of g aqq 10 −8 . The Storage Ring experiment is expected to be sensitive to the proton EDM at the order of |d p /e| 10 −29 cm. This implies the following conservative bound on ALP-photon-photon interaction The relevant expected limits for SRE are shown in Fig. 3. We note that existing limit on muon EDM, |d µ /e| < 1.5 × 10 −19 cm, provides relatively weak bound on g aγγ 10 −2 GeV −1 , which is ruled out by LEP constraints,ḡ aγγ 8×10 −3 GeV −1 . Moreover, current limit on neutron electric charge, e n < (−2 ± 8) × 10 −22 e and its coupling to ALP at the level of g ann 10 −10 don't induce an experimentally favored constraints onḡ aγγ at one-loop level.

VI. CONSTRAINTS ON ALP COUPLING WITH Z 0 -BOSON AND ELECTRON
It is worth mentioning, that one can also consider dimension-5 operator of ALP coupling with photon and Z 0 -boson L ⊃ḡ  (1) and (2) black boxes correspond toḡ aψψ aψψ vertices, black rounds denote ig aψψ aψγ5ψ vertices. In diagrams (3-6) black box (the aγZ vertex) corresponds to the interaction (14).
In our previous paper [28] we derived constraints on product of the couplingsḡ aee and g aee from EDM bounds of electron. Corresponding 1-loop diagrams, which induce electron EDM, are labeled by (1) and (2) in Fig. 5. However, it is instructive to obtain limits on CP-odd couplingḡ aee for certain values of benchmark coupling g aee and ALP mass m a . Indeed, one has the following esti-mate for the electron EDM [41] |d e /e| =ḡ aee g aee 8π 2 m e I(m a /m e ), (15) where I(m a /m e ) can be approximated as Therefore, for g aee = 5 · 10 −5 and the ALP mass range 100 keV m a 200 MeV one has the following conservative limits onḡ aee at 90% CL It is worth mentioning that the limitḡ aee 4.3·10 −13 for m a 100 keV is better than the bound onḡ aee from nonresonant production of new scalars in Horizontal branch star core during helium burning (for detail, see, e. g. Ref. [42] and corresponding left panel in Fig. 5).

VII. BOUNDS ON COMBINATION OF COUPLINGS
In this section, for completeness, we summarize current reasonable constraints on combination of couplings, which are ruled out by EDM of SM fermions. In Fig. 4 we show regarding limits associated with muon, proton, and electron EDM. In particular, for relatively light ALP, m a m ψ , one has the following scaling of the limit In addition for heavy ALP, m a m ψ one has One can see from Fig. 4 that most stringent constraint follows from electron EDM bound,ḡ aγγ g aψψ 10 −13 GeV −1 , as expected.
We can estimate sensitivity to ALP physics relatively to interaction with different types of fermion if EDMs are generated by the mechanism discussed here. We can consider ratio of electron and proton EDMs. Then we have the following ratio In case of lightweight ALP we obtain that couplings for electron and proton are proportional to existed data of EDMs. From other hand, ratio of electron and proton couplings has a factor ∼ 4 × 10 6 for case of m a m p , therefore one has enhanced value of g aee /g app . Here we take into account that |d p /e| = 7.9·10 −25 cm and |d e /e| = 1.1 · 10 −29 cm from [35,36].
To conclude this section, we discuss known results on limits for the couplings from EDM bounds. In particular, we note that in Ref. [43,44] authors derived similar constraints on scalar and pseudoscalar coupling constant combinationsḡ aee g aee andḡ aee g app from atomic and molecular EDM experiments for the relatively wide range masses of ALP 10 −6 eV m a 10 8 eV. In particular, first combination of coupling constant corresponds to diagrams (1) and (2) in Fig.5 which were described in [28], second combination connected with exchange of an axion between the atomic electrons and the nucleus. In addition, in Refs. [45][46][47][48] authors provided constraints on CP-violating contact interactionsēeN N from data on EDM of atoms and molecules.

VIII. GENERATION OF COUPLING OF ALPS WITH SM FERMIONS
In electroweak sector we proceed via a coupling of axion with neutral gauge fields: To deal with dimensionless couplings g a and g a we rescale them using the Higgs condensate and the use of dimension factor 1/Λ corresponds to a suppression of the above Lagrangian at electroweak scale. Occurrence of such dimensional factor v 2 /Λ will be clear from discussion below where we will generate the axion-Z 0 mixing via spontaneous breaking of the electroweak symmetry.
Here v = 246 GeV is the Higgs vacuum condensate. Using relations of the W 3 and B with Z 0 and photon one gets To vanish the coupling of axion with single photon we impose the condition: g a sin θ W = g a cos θ W . The similar condition holds for the g and g couplings in SM. Therefore, Now we estimate the coupling of the axion with SM fermions suppose that it occurs via Z 0 exchange between axion and neutral current Note that only the axial coupling of the J µ contributes to the coupling of axion to the SM fermions. Finally we derive the following effective Lagrangian describing the coupling of the axion with SM fermions where c ψ A = −1/2 for ψ = e, µ, τ, d, s, b and c ψ A = 1/2 for ψ = u, c, t. Note that the effective coupling of the axion with SM fermions is suppressed by the NP scale parameter Λ. On fermion mass-shell the "pseudovector" Lagrangian L aψ is conventionally deduced to the "pseudoscalar" one: There could exist different scenarios of emergence of the axion-Z 0 coupling. E.g., it can be induced by the coupling of scalar field with axion (similar coupling of higher dimensions have been discussed in Ref. [8]). In particular, following ideas of Ref. [8]) one can expect that additional coupling. In particular, following ideas of Ref. [8] one can consider the coupling of Higgs φ-scalar fields with axion in the form induced by dimension-5 operator where D µ is the covariant derivative including mixing of electroweak gauge fields. After realization of spontaneous breaking of electroweak symmetry and expressing combination of B and W 3 fields through Z 0 and A one gets [using G aZ 0 = G Zh log(1/2)]: Therefore, we shown that the coupling of axion with Z 0 scales as v 2 /Λ and the matching of our coupling g a and coupling G aZ 0 in Refs. [6,7] reads: by analogy with dark photon we can introduce the coupling of axion with SM fermions starting form mixing of axion and Z 0 boson parameterizing mixing parameter = v 2 /Λ, where = g a / cos θ W , v is Higgs condensate and Λ: Then we shift the Z 0 field as The kinetic term of the axion will get a very small correction with can be eaten by the axion field redifinition: which results a negligible shift of the axion mass. Like in case of dark photon, the shift (32) generates the couplings of the axion with SM fermions. These couplings of the same dimension as in original effective model of the axion [6] and suppressed by the same power of the NP scale parameter Λ:

IX. CONCLUSION
In the present paper we derive constraints on soft CPviolating couplings of ALP from experimental data on EDM bounds of SM fermions. In particular, we derive 90% CL limit on CP-odd ALP coupling with photons by taking into account EDM limits for electron. That analysis is based on simplified phenomenological scenario of leptophilic ALPs in the mass range 100 keV m a 200 MeV. We also obtain expected limits on CP-odd aγγ coupling for SRE experiment on proton EDM. We calculate bounds on soft CP-violating ALP coupling with electron. Finally, we discuss the exotic scenario of inducing Yukawa-like interaction of SM fermions with ALP through the electro-weak symmetry breaking mechanism. That scenario is associated with dimension-5 operator of Higgs and ALP interaction.