On the oscillating electric dipole moment induced by axion-fermion couplings

It has been recently claimed that the axion coupling to fermions is responsible for an oscillating electric dipole moment (EDM) in the background of axion dark matter. In this work, we re-examine the derivation of this effect. Contrary to previous studies, we point out the physical relevance of an axion boundary term, which is crucial in restoring the axion shift symmetry and drastically affects the EDM phenomenology. To describe the latter, we introduce the notion of a time-integrated effective axion EDM, which encodes the boundary term and whose magnitude depends on the oscillating regime. For slow oscillations the boundary term washes out the standard oscillating EDM, resulting in an exact cancellation in the static limit. Conversely, during fast oscillations, the boundary term amplifies the effective EDM. This new observable is especially interesting in the case of the electron EDM. Remarkably, for an O (1) axion-electron coupling, the overall size of the effective EDM in the intermediate oscillations regime is comparable to the present static EDM limit.


Introduction
The QCD axion [1][2][3][4] represents a most compelling paradigm for the physics beyond the Standard Model (SM), by simultaneously providing an elegant solution to the strong CP problem and an excellent dark matter candidate [5][6][7].Current axion searches have started to probe the region of parameter space predicted by the QCD axion (see Ref. [8] for a comprehensive overview of axion models) while next-generation experiments will explore regions of parameters space deemed to be unreachable until a decade ago [9,10].
Among novel and ingenious axion detection strategies, it was proposed in Refs.[11][12][13] to employ nuclear magnetic resonance techniques to look for an oscillating neutron electric dipole moment (EDM) induced by axion dark matter.Other experimental approaches have been suggested in order to search for oscillating EDMs of atoms, molecules and nuclei [14][15][16], as well as that of protons/deuterons [17][18][19][20] and electrons [21] employing storage ring facilities.As a theoretical setup, most of these works focussed on the oscillating nucleon EDM induced by the defining property of the QCD axion, namely an axion coupling to the G G operator.
On the other hand, it has been recently pointed out by Smith [22] (see also Refs.[23,24] for earlier similar claims) that the model-dependent axion-fermion coupling, here denoted as g = C m/f a (where C is an adimensional coupling, m the fermion mass and f a the axion decay constant) leads to an axion-dependent EDM of the type defined in terms of the non-relativistic (NR) Hamiltonian, H EDM ⊃ −d(a) ⃗ σ • ⃗ E. Eq. (1.1) would have striking experimental consequences, since in the background of axion dark matter, with the amplitude of the oscillation fixed in terms of the local dark matter relic density, one predicts in the case of electrons an oscillating EDM, d e (a) ≃ 7 • 10 −30 C cos(m a t) e cm [22], that for C ∼ 1 is comparable the current limit on the static EDM, |d e | ≲ 1.1 • 10 −29 e cm [25].
This paper aims to re-examine the derivation of the axion-induced EDM in Eq. (1.1), given its potential impact for EDM searches.While the NR expansion of the axion Dirac equation in Ref. [22] was obtained by means of the Foldy-Wouthuysen approach [26], which consists of a block-diagonalization of the Dirac equation, we here employ the more direct Pauli elimination method [27] in order to provide an independent derivation.These two techniques are known to yield equivalent results (see e.g.[28]), although the Pauli elimination method requires careful treatment of spinor normalization.
Throughout our derivation, we pay a special attention to verifying that the expected properties of the axion theory, such as the equivalence of the axion formulation in the derivative and exponential bases, as well as the axion shift symmetry, are properly satisfied.While we were able to reproduce most of the results of Ref. [22] (as well as [23,24]), we disagree on the physical interpretation of the axion-fermion coupling in terms of a standard axion EDM, as given by Eq. (1.1).The crucial point consists in the identification of a previously overlooked axion boundary term which is needed in order to restore the axion shift symmetry, irrespective of the chosen axion basis.Without this term, the axion shift symmetry would be explicitly broken by Eq. (1.1).Remarkably, the inclusion of this boundary term in time-dependent perturbation theory drastically affects the phenomenology of the standard axion EDM.
Nonetheless, in the presence of a constant electric field, it is still possible to define a timeaveraged effective EDM, which is made of the standard EDM of Eq. (1.1) plus a contribution arising from the axion boundary term.Depending on the oscillation regime, different behaviors emerge.For slow oscillations the axion boundary term washes out the standard oscillating EDM, resulting in an exact cancellation in the static limit.In the regime of fast oscillations, where the standard EDM contribution is suppressed, the axion boundary term instead maintains a constant amplitude of the effective EDM.For intermediate or fast oscillations, the effective EDM is similar in size to the standard one in Eq. (1.1).
The paper is structured as follows.We start in Section 2 by presenting the axion Lagrangian in the exponential and derivative bases.In Section 3 we provide the NR expansion of the axion Dirac equation in the derivative basis, while we refer to Appendix A for the calculation in the exponential basis.In Section 4 we discuss the formal equivalence of the axion Hamiltonian as derived in the exponential and the derivative bases.We discuss this by means of a unitary transformation that is derived from the NR limit of the field redefinition connecting the two bases at the Lagrangian level.Next, in Section 5, we consider the consequences of the axion Hamiltonian in time-dependent perturbation theory, we argue that the axion boundary term is physically relevant, and finally, we introduce the concept of a time-averaged effective axion EDM.We conclude in Section 6, summarizing the main findings of our study.

Axion Lagrangian: exponential vs. derivative basis
Let us consider a toy model with a massless axion interacting with a Dirac fermion field charged under an abelian gauge group (e.g. an electron).In specific ultraviolet completions, the axion can be represented as the orbital Goldstone mode of a complex scalar field, ϕ ⊃ f a e ia/f a , whose radial mode has been integrated out.This defines the axion effective Lagrangian in the exponential basis where we have introduced the coupling g, defined via g/m = 1/f a . 1 The subscript E is a reminder that these fields are defined in the exponential basis and L and R indicate left and right chiral components, respectively.The U(1) PQ symmetry is implemented as with α the global transformation parameter.The last term in the bracket of Eq. (2.1) can be also expressed as where in the last step we have expanded at the first non-trivial order in the axion field, which defines the so-called linear basis.Note that the linear approximation might lead to incorrect results if more than one axion is involved in a given process (for an explicit example, cf. the discussion around Eq. (A.3)).Eq. (2.1) can be brought into the derivative basis via an axion-dependent field redefinition: so that L E is mapped into the equivalent Lagrangian In realistic QCD axion models with axion couplings to SM fermions, such as the DFSZ model [29,30], the axion spans over different Higgs multiplets and g/m is related to the axion decay constant via g/m = C/f a , where C is an O(1) parameter proportional to the PQ charge of the associated Higgs doublet (see e.g.Sect.2.7.2 in [8]).
with the last term arising from the non-invariance of the path-integral measure [31] under the anomalous field redefinition in Eq. (2.4).In the derivative basis the original U(1) PQ symmetry defined in Eq. (2.2) is implemented as with the ψ D field that is left invariant.
In the following, we will perform the NR limit of the axion Dirac equation in the derivative basis, and compare this to the calculation in the exponential basis.After showing the equivalence of the two formulations, we will finally discuss the physical consequences of the derived axion Hamiltonian in time-dependent perturbation theory.

Non-relativistic limit of the axion Dirac equation
The axion Dirac equation stemming from Eq. (2.5) in the derivative basis reads where we dropped for simplicity the subscript D on the spinor field.To perform the NR limit of the axion Dirac equation, it is convenient to employ the Dirac representation for the gamma matrices with σ i=1,2,3 denoting the Pauli matrices, and adopt the subsequent parametrization for the Dirac spinor The mass term in Eq. ( 3.3) provides the dominant time evolution in the NR limit, while the bi-spinors χ = χ (⃗ x, t) and Φ = Φ (⃗ x, t) are slowly varying functions which exhibit a small energy dependence suppressed by 1/m.In terms of two-component spinors the axion Dirac equation (3.1) can be written as with where operators are denoted with a hat, ⃗ To perform the NR expansion we employ the Pauli elimination method [32], which consists in substituting the lower equation Φ = − D−1 [ Ĉ[χ]] from Eq. (3.4) into the upper one, 2 thus obtaining: where to keep track of the operatorial nature of ⃗ p and ε we have introduced the notation X[...], meaning that "X" acts on "...".
We next expand the inverse of . Therefore, when applied to "x", we obtain: This expression provides a solution for the inverse of D at second order, with further terms in the expansion accounting for O(1/m 3 ) higher-order corrections.Including all the previous ingredients in Eq. (3.6), we match the latter onto the form of a time-independent Schrödinger equation, given by Ĥ where the superscript (i) denotes terms at O(1/m i ).
In the following, it will be useful to split , respectively into a piece with and without the axion.At O(1/m) we obtain the axion Hamiltonian, which corresponds to the so-called axion-wind term (see e.g.[11]).Note that in the last step of Eq. (3.9) we exploited the relation with f an auxiliary function and The previous calculation can be extended at O(1/m 2 ).From Eq. (3.6) we obtain Ĥ where in the last step we used a relation analogous to Eq. (3.10) and {a, b} c ≡ a Note that the ȧ ⃗ σ • ⃗ p term stemming from Eq. (3.11) leads to the so-called axioelectric effect, discussed e.g. in Ref. [33].
On the other hand, the calculation in the exponential basis leads to a different Hamiltonian starting at O(1/m 2 ), see Appendix A for details. 3 Denoting the axion Hamiltonian in the 2 Since Φ ≃ (⃗ σ • ⃗ p + . . .)/(2m) χ ≪ χ in the NR limit, χ and Φ go under the name of large and small component spinors, respectively.
3 A non-trivial issue that one has to deal with in the exponential basis is the fact that, by simply employing the Pauli elimination method, the Hamiltonian at O(1/m 2 ) is not Hermitian.An analogous problem arises for the correct identification of the Darwin term in the NR Hamiltonian describing the hydrogen fine structure (see e.g.[34]) and its solution requires an extra normalization of the Schrödinger spinor.In Appendix A, we provide a comprehensive discussion of this normalization issue, both for the exponential and derivative bases.For the latter basis, the problem arises only at O(1/m 3 ) and hence it does not show up in the present calculation.
exponential basis as ĤEa , we obtain which differs from Eq. (3.11) by the last two terms and, in particular, it features an axion EDM term proportional to the electric field,

Equivalence of bases from unitary transformations
Before addressing the physical consequences of the axion Hamiltonian at O(1/m 2 ), we wish to discuss the equivalence of the two formulations in the derivative and exponential bases.At the formal level, it is possible to show that the Hamiltonians in Eqs.(3.11)-(3.12)are connected by a unitary transformation, that is nothing but the NR limit of the field redefinition in Eq. (2.4), allowing us to go from the exponential to the derivative basis.
To this end, let us first rewrite Eq. (2.4) in terms of bi-spinors as where we have explicitly labeled the states in the exponential (E) and derivative (D) bases.
Expanding the above expression in powers of 1/m, we obtain: The equation in the first row of (4.2) yields where Φ D can be expressed in terms of χ D by using the Dirac equation: with This results in Eq. (4.6) can be understood as the linear term of an exponentiation, which reconstructs the unitary transformation A unitary transformation does not alter the underlying physics, as long as both the states and the Hamiltonian are transformed.Applying the transformation in Eq. (4.7) to the Schrödinger equation reveals a shift in the Hamiltonian Ĥ(2) a of the following form: which, after expanding to O(1/m 2 ) and using results in which precisely reproduces the exponential basis Hamiltonian in Eq. (3.12).This argument can also be seen as a non-trivial check of the more involved calculation of the axion Hamiltonian in the exponential basis, that is provided in Appendix A. More generally, it is possible to perform the following unitary transformation on the state in the derivative basis [22] with β being a free parameter.Following similar steps as for the derivation below Eq. (4.7), one arrives at the following Hamiltonian In particular, choosing β = 1, the Hamiltonian in Eq. (4.12) corresponds to that of an axiondependent EDM [22], to which we will refer to in the following as the "EDM picture".However, in the next section it will be show that, when doing time-dependent perturbation theory, the transformation of the state in Eq. (4.11) implies the presence of an axion boundary term which does not allow to interpret the net effect of Eq. (4.12) with β = 1 solely in terms of an axiondependent EDM.

Axion EDM in time-dependent perturbation theory
To describe the time evolution of a physical system in NR quantum mechanics, we can employ the interaction picture in which the Hamiltonian is split into Ĥ(t) = Ĥ0 + V(t), where Ĥ0 = ⃗ p 2 /(2m) is the free Hamiltonian in the NR limit and V(t) is a perturbation which depends explicitly on time, encoding in particular the axion dependence.The state in the interaction picture is defined as |χ I (t)⟩ = e i Ĥ0 t |χ(t)⟩ and its time evolution is governed by |χ I (t)⟩ = Û (t, t 0 ) |χ I (t 0 )⟩, in terms of the time evolution operator where T is the time-ordered product.
In the following, we will adopt the temporal gauge 5 A 0 = 0 and further consider the approximation ⃗ ∇a = 0, which is phenomenologically motivated in the case of NR dark matter axions, for which ⃗ ∇a ≃ m a ⃗ v and v/c ≃ 10 −3 .In such a case, focusing on the axion-dependent terms, V(t) = Ĥ(2) a , and hence VI (t) = e i Ĥ0 t V(t)e −i Ĥ0 t = V(t) + O(1/m 3 ) [22].Hereafter, we will drop for simplicity the subscript I when referring to the interaction picture.

Check of equivalence theorem and axion shift symmetry
In the spirit of the equivalence theorem in quantum field theory [35][36][37], which implies the exact matching of S-matrix elements in different bases connected by field redefinitions, it is instructive to verify that ⟨ψ holds in perturbation theory, where we have explicitly labeled the states and the time evolution operator in the derivative (D) and exponential (E) bases.
In order to check Eq. ( 5.2) at the first order in time-dependent perturbation theory, let us consider first the left-hand side (LHS) where in the second step we have employed the axion Hamiltonian in the derivative basis from Eq. (3.11).On the other hand, taking into account the shift of the state from Eq. (4.6) as well as the axion Hamiltonian in the exponential basis from Eq. (3.12), the right-hand side (RHS) of Eq. ( 5.2) can be rewritten as where in the second to last step we have taken into account the definition of the electric field in the temporal gauge ( ⃗ E = −∂ t ⃗ A), integrated by parts, and used that ⃗ p is time-independent to add and subtract a total derivative.Note that to prove the equivalence of the two bases it was crucial to take into account the transformation of the state in Eq. (5.4), which effectively leads to a boundary term in the time-integrated Hamiltonian, which exactly cancels out the boundary term arising from the integration by parts.
Similarly, we can verify that the axion shift symmetry is properly preserved in time-dependent perturbation theory.While this is manifest in the derivative basis (cf.Eq. (2.6) and Eq.(3.11)), being the Hamiltonian itself shift-invariant, that is not the case for the exponential basis.Since in the exponential basis also the fermion field transforms under the U(1) PQ symmetry (cf.Eq. (2.2)), we expect that the invariance under the shift symmetry can be recovered in the matrix element ⟨ψ E (t)| ÛE (t, t 0 ) |χ E (t 0 )⟩ after including the shift of the state.
To verify this last statement, let us first identify how the U(1) PQ symmetry acts on the state |χ E (t 0 )⟩.Combining Eq. (2.2), (4.1) and (4.6) we have Hence, applying the shift in Eq. (5.6) to the matrix element in the exponential basis at the first order in time-dependent perturbation theory, we find (see also Eq. (5.5)) up to O(1/m 2 ) corrections.Also in this case the transformation of the state, resulting in an effective boundary term, was crucial for ensuring invariance under the axion shift symmetry.

Physical relevance of the axion boundary term
We next discuss the physical effects stemming from the axion Hamiltonian at O(1/m 2 ).Let us consider the derivative basis, and employ as before the temporal gauge A 0 = 0 and work in the ∇a = 0 approximation.Then at the first order in time-dependent perturbation theory one has Upon integration by parts, the interacting term of Û (t, t 0 ) in Eq. (5.8) can be rewritten as i g which leads to the emergence of a standard axion EDM [22] as well as an axion boundary term.However, differently from Ref. [22], we here argue that the axion boundary term is crucial for correctly describing the physical effect.Note that the LHS of Eq. (5.9) manifestly satisfies two important properties: i) it preserves the axion shift symmetry a → a + α m/g and ii) it yields no static EDM. 6For these two properties to keep holding in the RHS of Eq. (5.9) it is crucial to keep the axion boundary term.Let us show in turn these two properties from the standpoint of the RHS of Eq. (5.9): • Shift invariance (a → a + α m/g) RHS (5.9) → RHS (5.9) + i α 2m • No static EDM (a = a 0 ) RHS (5.9) = i ga 0 Hence, this argument suggests that the axion boundary term should also play a crucial role in the phenomenology of the oscillating axion EDM.Finally, it is worth noting that the results presented in this section can be equivalently obtained in the "EDM picture" discussed at the end of Section 4, upon taking into account the shift of the external states arising from Eq. (4.11) with β = 1.This effectively corresponds to the inclusion of an axion boundary term, which exactly matches the RHS of Eq. (5.9).

Effective axion EDM
To assess the physical consequences of the axion boundary term of Eq. (5.9) for the EDM phenomenology, let us assume the following configuration featuring a constant electric field as well as an oscillating axion dark matter field where we set t 0 = 0, so that t is the measurement time.Here, a 0 and θ 0 are the amplitude and initial phase of the axion field, respectively.Below, without loss of generality, we set θ 0 = 0.The amplitude of axion oscillations is fixed in terms of the local dark matter relic density as where ρ DM ≃ 0.3 GeV/cm 3 = 2 • 10 −42 GeV 4 [38].For a QCD axion, this implies a 0 /f a ≃ 4 × 10 −17 irrespective of the axion mass and decay constant.We are working in a perturbative expansion about a free state, which together with the non-relativistic expansion implies the range of validity |e ⃗ Et| ≪ |⃗ p 0 | ≪ m, where ⃗ p 0 is the initial momentum of the fermion.This implies that our perturbative calculation breaks down once ⃗ E has had sufficient time to significantly alter the fermion momentum.With this setup in mind, Eq. (5.8) becomes .14)In this expression, we are particularly interested in the term proportional to ⃗ σ • ⃗ E, which can be matched onto an EDM.To this end, it is useful to rewrite such a term in the form of an effective time-integrated EDM, defined via the following expression To deal with a quantity with the dimension of an EDM, we introduce the time average where the last step follows directly from Eq. (5.15).Note that the first term in the RHS of Eq. (5.16) corresponds to the standard oscillating EDM (discussed e.g. in [22]), while the second term can be understood to arise from the axion boundary term.Let us consider now the following three oscillation regimes: 1. Slow oscillations (m a t ≪ 1).
The time-averaged effective EDM can be approximated as and in the case of no oscillations, m a t → 0, the static EDM contribution goes to zero.Note that in the m a t ≪ 1 regime, the axion boundary term cancels the leading order contribution to the standard EDM and the overall contribution is suppressed by (m a t) 2 ≪ 1.So we conclude that it is crucial to include the axion boundary term to correctly interpret the physical effect of the time-averaged effective EDM.For non-zero initial axion phase θ 0 , ⟨d eff ⟩ is linear in m a t and still goes to zero.
In this regime, in Eq. (5.16) both the standard EDM contribution and the one arising from the axion boundary term are comparable, corresponding to an effective EDM amplitude a 0 f a . (5.18) In the electron case, taking a unit axion-electron coupling, we obtain ⟨d eff ⟩ ∼ 7•10 −30 e cm.This is notably of the same order as the static electron EDM bound, |d e | ≲ 1.1 • 10 −29 e cm [25].
In this case the boundary term contribution in Eq. (5.17) always dominates, thus leading to a time-averaged effective EDM that avoids the 1/(m a t) ≪ 1 suppression of the standard axion EDM contribution and maintains the amplitude of Eq. (5.18): The behavior of the time-averaged effective EDM of Eq. (5.16) is displayed in Fig. 1 in comparison to the standard EDM contribution discussed e.g. in Ref. [22], i.e. without axion boundary term.We have verified that, as expected from Eq. (5.11), the effective EDM decouples irrespective of the initial axion phase θ 0 because of a cancellation with the boundary term.In the absence of this boundary term, the EDM does not vanish in the static limit but rather takes a generally non-zero value depending on θ 0 .When interpreting these results, it is important to keep in mind the range of validity of the perturbative expansion.Ultimately, the range of validity of our calculation is limited to, taking again the case of an electron, , for a constant electric field as given by Eq. (5.16).In the static limit, m a t → 0, the effective EDM vanishes when taking the axion boundary term into account.In the rapidly oscillating regime, m a t ≫ 1, the axion boundary term dominates and makes the amplitude of the effective EDM independent of m a t, while in contrast the contribution of the standard EDM decouples as 1/(m a t).
We conclude that the axion boundary term contribution crucially affects the phenomenology of the oscillating EDM by suppressing the effect in the small oscillation regime and by enhancing it in the regime of fast oscillations.Remarkably, for an order one axion-electron coupling, as predicted in benchmark axion models [29,30], and oscillations either intermediate or fast, the amplitude of the effective oscillating EDM is of the order of the present static EDM limit.However, a proper discussion of the experimental prospects for measuring such effect is beyond the scope of the present paper and it is left for future work.

Conclusions
In this paper, we re-examined the recent claim [22] (see also [23,24]) that the axion-fermion coupling is responsible for an oscillating EDM in the background of axion dark matter, cf.Eq. (1.1).By employing the Pauli elimination method for the NR expansion of the axion Dirac equation, we provided an alternative derivation of the axion Hamiltonian, emphasizing the equivalence between the derivative and exponential bases.
Unlike previous studies, we pointed out the physical relevance of an axion boundary term, which turns out to be crucial in order to restore the axion shift symmetry and plays a critical role for the oscillating EDM phenomenology, e.g. by leading to an exact cancellation of the standard EDM contribution in the static limit.
In the case of a constant electric field, we were able to introduce the notion of a time-averaged effective EDM, which comprises both the standard contribution of Eq. (1.1) and the axion boundary term.As exemplified by Fig. 1, different patterns emerge depending on the oscillation regime.For slow oscillations, the effective EDM is suppressed compared to the standard one, while for fast oscillations it is relatively enhanced, avoiding the suppression experienced by the standard EDM contribution.Only in the intermediate oscillation regime are the effective and standard EDM contributions comparable.
This new observable is particularly relevant for the case of the electron EDM, since for an O(1) axion-electron coupling the amplitude of the effective EDM is comparable to the present static EDM limit if in a regime of intermediate or fast oscillations.The experimental verification of such a scenario remains an interesting open question, which will be addressed elsewhere.

Note added
Simultaneously with our initial submission to the arXiv, a revised version of Ref. [22] appeared, reaching the same conclusions regarding the decoupling of the EDM in the static limit.

. 5 ) 5 A
crucial simplification arising in the temporal gauge consists in the fact that we can describe the axion dynamics at O(1/m 2 ) by doing time-dependent perturbation at the first order, thus neglecting the O(1/m 2 )interference with the leading order term Ĥ(t) = −eA 0 + . . .arising at the second order in time-dependent perturbation theory.

Figure 1 :
Figure1: Time-averaged effective EDM including the axion boundary term (blue) vs. standard EDM (yellow), for a constant electric field as given by Eq. (5.16).In the static limit, m a t → 0, the effective EDM vanishes when taking the axion boundary term into account.In the rapidly oscillating regime, m a t ≫ 1, the axion boundary term dominates and makes the amplitude of the effective EDM independent of m a t, while in contrast the contribution of the standard EDM decouples as 1/(m a t).