Oscillating nuclear electric dipole moment induced by axion dark matter produces atomic and molecular EDM

According to the Schiff theorem nuclear electric dipole moment (EDM) is completely shielded in a neutral atom by electrons. This makes a static nuclear electric dipole moment (EDM) unobservable. Interaction with the axion dark matter field generates nuclear EDM $d=d_0 \cos (\omega t)$ oscillating with the frequency $\omega= m_a c^2/\hbar$ . This EDM generates atomic EDM proportional to $\omega^2$. This effect is strongly enhanced in molecules since nuclei move slowly and do not produce as efficient screening of oscillating nuclear EDM as electrons do. An additional strong enhancement comes due to a small energy interval between rotational molecular levels. Finally, if the nuclear EDM oscillation frequency is in resonance with a molecular transition, there may be a significant resonance enhancement.

Introduction: It was suggested in Ref. [1] that interaction with the axionic dark matter produces oscillating neutron and nuclear electric dipole moments. However, according to the Schiff theorem [2], the nuclear EDM is completely screened in neutral atomic systems. Atomic and molecular EDMs are actually produced by the nuclear Schiff moment which is suppressed compared to EDM by an additional second power of the nuclear radius which is very small on the atomic scale [3][4][5][6][7] (see also [8][9][10][11][12][13][14] for other effects producing atomic and molecular EDM). The effects produced by the axion-induced Schiff moment have been considered in Ref. [15]. A corresponding experiment in solids has been proposed in Ref. [16]. The first results of the oscillating neutron EDM and Hg atom's EDM measurements are presented in Ref. [17] where the limits on the low-mass axion interaction constant with matter have been improved up to three orders of magnitude.
In the present paper it is shown that an oscillating nuclear EDM such as that produced by the axion dark matter is not completely screened in atoms and molecules and produces atomic and molecular EDMs. The latter case is especially interesting since the effect in molecules is several orders of magnitude larger than in atoms. Indeed, in the screening of the static nuclear EDM, the nuclei in a molecule play as important a role as the electrons. If the nuclear EDM oscillates, because nuclei are not as fast-moving as the electrons, the screening is incomplete. As a result, the residual, partly screened EDM in molecules is M N /m e times larger than that in atoms.
Here M N is the nuclear mass and m e is the electron mass. Enhancement of the oscillating nuclear EDM may happen if the oscillation frequency is in resonance with a molecular transition frequency.
Screening theorem for time-dependent electric field and EDM: As known, a nucleus in a neutral system (atom or molecule) is completely screened from a constant electric field [2]. We will here present a derivation of this fact following the Appendix in Ref. [18]. For definiteness, we assume that the system in question is a neutral atom in a static homogeneous external electric field of an arbitrary strength (we ignore the possibility of atomic ionization and effects of magnetic fields).
The Hamiltonian of an atom placed in a static homogeneous external electric field E 0 is where K i and r i are the kinetic energy and coordinates of the electrons, d is the static nuclear EDM and φ 0 (r i ) is the electrostatic nuclear potential given by where ρ is the nuclear charge distribution. We consider here the case of an infinitely heavy nucleus. The nuclear recoil correction is not enough to generate an atomic EDM [2].

arXiv:1904.07609v3 [hep-ph] 24 Oct 2019
We add to H and auxiliary term which, in the linear approximation in d, does not produce any energy shift, V = 0. Indeed, we have where we have taken into account the fact that the total electron momentum i p i commutes with the electronelectron interaction term. Using Eq. (3) and the fact that (ψ is the wavefunction of the Hamiltonian H), we obtain To find an EDM one needs to measure a linear energy shift in an external electric field. Since V does not contribute to this shift we can add it to the Hamiltoniañ where Note that the HamiltonianH does not contain the direct interaction d · E 0 between the nuclear EDM and external field (Schiff theorem). The dipole term is also canceled out in the multipole expansion of φ (r i ).
Let us now consider the case where the nuclear EDM is time-dependent d = d (t). In this case, Eq. (5) becomes Therefore, the contribution due to V is zero in the first order in d. As a result, just as in the case of a static nuclear EDM, there is no direct interaction between a time-dependent nuclear EDM and a static external electric field, hence, no nuclear spin rotation. Indeed, the external electric field does not penetrate to the nucleus (since an atom and its nucleus are not accelerated by a static homogeneous electric field), so the nuclear EDM has nothing to interact with. Now consider the case of a time-dependent electric field. In this case, we have since the external field now penetrates to the nucleus [19][20][21]. Indeed, the external electric field forces the electron shells to oscillate and since the atom's center of mass stays at rest, the nucleus must move, so the electric field on it is not zero. Therefore, the nuclear EDM interacts with this electric field and nuclear spin rotation happens. Note that the absence of nuclear spin rotation in the case of a static electric field does not mean that the oscillating nuclear EDM does not produce any effect. An oscillating nuclear EDM excites the electrons and produces atomic and molecular EDMs (as demonstrated below). This effect is particularly clear in the case where the nuclear EDM's frequency of oscillation is in resonance with some atomic or molecular frequency, in which case the electronic wavefunciton is a linear combination of two states of opposite parities and thus gives rise to oscillating atomic and molecular EDMs. Oscilalting nuclear EDMs may be detected using the atomic and molecular transitions they induced, as investigated in Ref. [22,23].
The case where both the nuclear EDM and the external electric field are time-dependent, particularly when they are oscillating, is of special interest. As demonstrated in Refs. [19][20][21], an external electric field which oscillates with a frequency ω, E 0 ∼ cos ωt, induces an electric field on the nucleus which oscillates with the same frequency. The interaction of this field with a nuclear EDM which itself oscillates with a frequency Ω, d ∼ cos Ωt, is proportional to cos ωt cos Ωt. If ω = Ω then this interaction contains a time-independent component and the nuclear spin rotation angle grows linearly with time.
Nuclear EDM produced by the axion dark matter field: It has been noted in Ref. [24] that the neutron EDM may be produced by the QCD θ term. Numerous references and recent results for the neutron and proton EDMs are summarised in Ref. [25]: Calculations of the nuclear EDM produced by the P,Todd nuclear forces have been performed in the Refs. [5][6][7]26]. For a general estimate of the nuclear EDM it is convenient to use a single-valence-nucleon formula from Ref. [5] and express the result in terms of θ following Ref. [27]: where ξ = 7 × 10 −16 θcm.
Here q = 1 for the valence proton, q = 0 for the valence neutron, the nuclear spin matrix element σ = 1 if I = l + 1/2 and σ = −I/ (I + 1) if I = l − 1/2. Here, I and l are the total and orbital momenta of the valence nucleon.
It was noted in Ref. [1] that the axion dark matter field may be an oscillating θ term and thus generates the oscillating neutron EDM. To reproduce the density of dark matter, following Ref. [15] we may substitute θ(t) = θ 0 cos(ωt) where θ 0 = 4 × 10 −18 , ω = m a c 2 / and m a is the axion mass. In the following sections, we estimate the electric dipole moment of atoms and molecules induced by the oscillating nuclear EDM.
Atomic EDM induced by an oscillating nuclear EDM: The Hamiltonian of an atom in the field of an oscillating nuclear EDM d = d 0 cos(ωt) may be written as where H 0 is the Schrödinger or the Dirac Hamiltonian for the atomic electrons in the absence of d, N e is the number of electrons, Ze is the nuclear charge, Z i = Z −N , −e is the electron charge, r k is the electron position relative to the nucleus, P = Ne k=1 p k is the total momentum of all atomic electrons (which commutes with the electronelectron interaction but not with the nuclear-elect in- Here we assumed that the nuclear mass is infinite and neglect very small effects of the Breit and magnetic interactions.
Using H 0 |n = E n |n we obtain the matrix element of V between atomic states |n and |m where E nm = E n − E m . Using the time dependent perturbation theory [28] for the oscillating perturbation V = V 0 cos ωt and Eq. (14) we obtain a formula for the induced atomic EDM where = ω and D = −e N k=1 r k . The energy dependent factor may be presented as The energy independent term 1 on the right hand side allows us to sum over states |n in Eq. (15). Using the closure condition and the commutator relation [P, D] = −ie N e , this term gives We observe that, in agreement with the Schiff theorem, the atomic electric dipole moment D atom vanishes in a neutral atom (Z i = Z − N = 0) with static nuclear EDM ( = ω = 0).
Assume that nuclear EDM d is directed along the z-axis. Using the non-relativistic commutator relation where m e is the electron mass), we can express the atomic EDM in terms of the atomic dynamical polarisability α zz (ω) The axion field oscillation frequency may be very small on the atomic scale, therefore, we may use static polarisabilities in this expression which are known for all atoms. The formula (18) may be rewritten, with the energy and the polarizabilty expressed in atomic units˜ = e 2 /a B and α zz = αzz a 3 B (where a B is the Bohr radius), as: Since the atomic EDM D atom is proportional to 1/Z, it appears that the shielding is stronger in heavy atoms. This, however, is not necessary the case since, for example in hydrogen and heliumα zz ∼ 1 whereasα zz ∼ 400 in caesium (Z=55). Indeed, the numerical value of the polarizabilityα zz in atomic units often exceeds the value of the nuclear charge Z, therefore, the suppression of EDM in a neutral atom mainly comes from the small frequency of the dark matter field oscillations in atomic units,˜ .
Molecular EDM induced by oscillating nuclear EDM: We see from the first line in Eq. (18) that the residual EDM in a neutral system Z i = 0 is proportional to the mass m of the particle which produces the screening of the nuclear EDM d. Masses of nuclei M N in a molecule are up to 5 orders of magnitude larger than the mass of electron m e . In addition, the interval between molecular rotational energy levels (∼ m e /M N atomic units) are many orders of magnitude smaller than typical energy intervals in atoms and this may give an additional enormous advantage, see the denominator in the second line in Eq. (18). Finally, since the molecular spectra are very rich, the energy intervals are small and may be tuned by electric and magnetic fields, it is easier to bring them into resonance with the small oscillation frequency of the axion dark matter field.
Calculations presented in Appendix A give the following results for the induced electric dipole of a neutral diatomic molecule when is smaller or of the order of the first rotational energy E rot where is the reduced nuclear mass,X is the ground state inter-nuclear distance,d is the ground state intrinsic electric dipole of a polar molecule and E rot ≈ 2 µ −1 NX −2 is the energy of the first rotational state and d 1,2 are the nuclear EDMs. In writing Eq. (20), we have assumed that the molecular ground state has total angular momentum 0.
Note that traditionally, the interaction of the nuclear EDMs and a molecule is expressed in terms of the nuclear spin-molecular axis interaction. To do this, we need to rewrite Eq. (20) in terms of the polarization degree of the molecule in an electric field E, P =dE/(3E rot ), and the energy shift ∆E = D EDM mol E. Substituting these quantities into Eq. (20), we have For d 1 ∼ d 2 , we see that the lighter nucleus gives dominating contribution. In other words, if Z 1 Z 2 then the term d 2 /Z 2 drops out. We assume this is the case. In the limits E rot and E rot , Eq. (20) gives We see that in the small axion mass limit ( = m a c 2 E rot ), heavy molecules have an advantage (µ 2 N /Z 1 ). In the large axion mass limit ( = m a c 2 E rot ), the ratio of the EDMs is independent of and has asymptotic value 2d/ 3eZ 1X < 2/ (3Z 1 ) ≤ 2/3 (d ∼ eX for polar molecule) so molecules with at least one light nucleus are more advantageous.
The result (20) applies for the off-resonance case. If = E rot then we have the following relation between the oscillation amplitudes of D EDM mol and d 1 ; which is the large axion mass asymtotic value in Eq. (22) multiplied by the resonace enhancement factor E rot /Γ where Γ is the width. Again, we see that molecules with at least one light nucleus give bigger effect. There may be different contributions to Γ: natural width (which is typically small), Doppler width, collision width and time of flight (if the experiment is done with molecular beam). If, however, the experiment uses a trapped molecule then Γ is mainly due to the velocity distribution of the axion: Γ/E rot ≈ v 2 /c 2 ∼ 10 −6 where v is the mean axion velocity.
If molecules with Cesium or heavier nuclei are used then the contribution to the total D mol due to the Schiff moment may becomes significant. Still assuming that < ∼ E rot , the contribution to D mol from the Schiff moment S = SI/I (I is the nuclear total angular momentum) is where W S is the effective strength of the interaction between S and the molecular axis. We note that since W S scales as Z n with n > 2 [5], the contribution due to the heavier nucleus dominates: S ≈ S 2 .
To compare the effects of the nuclear EDMs and nuclear Schiff moments, it is convenient to form the ratio We see that the effect of the nuclear EDMs dominates for large axion mass. Also, as noted above, for light nuclei, W S is typically small so the effect of the nuclear Schiff moment is negligible compared to that of the nuclear EDM.
In order to estimate the ratio d 1 /S 2 , we need in addition to Eq. (12) for the nuclear EDM, a formula for the nuclear Schiff moment S, which, in the case of a spherical nucleus with one unpaired nucleon, reads [5] S = − eq 10 ξ t I + 1 where q, ξ and t I are defined as in Eq. (12), r 2 is the mean squared radius of the unpaired nucleon and r 2 q is the mean squared charge radius. Approximately,  13 . In LiF, the effect of the Schiff moment comes from the fluorine nucleus which is the heavier of the two whereas in TlF it comes from the thallium nucleus. We demonstrate below that D LiF mol dominates over D SCHIFF LiF for the axion mass ∼ 10 −5 − 10 −3 eV whereas, due to the large Schiff moment of Tl, D SCHIFF TlF dominates over D EDM TlF for < ∼ 10 −4 eV. In the other molecules, the heavier nuclei have zero spin so the Schiff moment contribution comes from the lighter nuclei (F, O and C). As a result, just as in the case of LiF, the Schiff moment contribution in these molecules is negligible in comparison with the nuclear EDM contribution.
We also remark that the last four of the molecules above have 3 ∆ 1 as their ground or metastable state and thus have doublets of opposite parities and very small energy gaps (which may be manipulated by external electric and magnetic fields to scan for resonance with the axionic dark matter field). Accordingly, if the axion mass is of the order of these doublet splittings, the coefficient 2/3 in the results (20)-(23) should be replaced by 1/2 and the first rotational energy E rot by the energy E dbt of the 3 ∆ 1 doublet splitting. The value of E dbt for HfF+ is given in Ref. [29], that for ThF+ in Refs. [30,31], for ThO in Refs. [32,33] and for WC in Ref [34].
The values for the Schiff moment S Tl and interaction strength W S for TlF are taken from Refs. [5,41]. The value of W S for LiF may be estimated by scaling with the nuclear charge Z using the formula in Ref. [5]. The EDM of Li may be estimated using formula (12) Table. I. Note that we have assumed that E rot,dbt /Γ ≈ 10 6 (trapped molecule, Γ is due to axion velocity distribution). We also note that the estimates presented in this paper may be readily extended to the case of polyatomic molecules (see, for example, Ref. [21]). The advantage of polyatomic molecules is that since their spectra are very dense, the probability of a resonance with the axionic dark matter field is higher. Solids also have rich spectra of low-energy excitations and effects of nuclear motion (similar to effect in molecules). by where the nuclear positions R 1,2 , nuclear momenta P 1,2 , electrons position r i and electron momenta p i are defined in the laboratory frame.
A change of coordinates to the center-of-mass frame as described in Ref. [21], gives, after discarding the free motion of the molecule The EDM induced by d 1 and d 2 is given by where is the molecule's total EDM operator. Here, ζ e = e (M N + Z N m e ) /M T and ζ N = e (M 2 Z 1 − M 1 Z 2 ) /M N .
Using the relations (the terms proportional to the molecule's total momentum have been discarded) we may write Substituting formula (32) into Eq. (29), we obtain where we have defined The -independent term in Eq. (33) may be written as 2 e n Im ( 0| Π |n n| d |0 ) where Z T = Z 1 +Z 2 −N e . For neutral molecule (Z T = 0), this term exactly cancels the contribution of d 1 and d 2 to the total molecular EDM. Using the relations and the definition (30) (which may be used to express Ne i=1 x i in terms of d and X), we obtain where and As a result, the -dependent term in Eq. (33) may be written as 2 e n 2 Im ( 0| Π |n n| d |0 ) where is the molecular polarizability tensor and If 1eV then βδ dominates over α∆ because of the factor √ M 1 M 2 /m e . Approximating the sum over states β by the term involved the first rotational state and using the Born-Oppenheimer wavefunction, we obtain the result (20).
If the oscillation of the nuclear EDMs is in resonance with the first rotational energy, = E rot , then, following Refs. [20,21], the formula (29) is replaced (for a neutral molecule) by the following relation between the oscillation amplitude of D EDM mol and d where the ket |1 denotes the first rotational state and Γ is its width. Note that if Γ is the natural width and d 1,2 have time dependence cos ωt then D EDM mol is proportional to sin ωt. Carrying out the same analysis as above, we obtain the result (23).
Finally, we may estimate the contribution to the molecular EDM of the oscillating Schiff moment as D SCHIFF