Limits on $CP$-violating hadronic interactions and proton EDM from paramagnetic molecules

Experiments with paramagnetic molecules (ThO, HfF$^+$, YbF, YbOH, BaF, etc.) provide strong constraints on electron electric dipole moment (EDM) and coupling constant $C_{SP}$ of contact semileptonic interaction. We compute contributions to $C_{SP}$ arising from the nucleon EDMs and from the $P,T$-violating nucleon-nucleon interactions. This allows us to derive limits on the $CP$-violating parameters, such as the proton EDM, $|d_p|<4.8\times 10^{-25} e\cdot $cm, the QCD vacuum angle, $|\bar \theta|<7.6\times 10^{-10}$, as well as the quark chromo-EDMs and $\pi$-meson-nucleon couplings.

In paramagnetic atoms, an atomic EDM may be induced by the following contact CP -odd semileptonic operators where G F is the Fermi coupling constant, e, p and n are respectively the electron, proton and neutron fields; C p SP and C n SP are the electron couplings to the proton and neutrons, respectively. The subscript SP denotes the nucleon-scalar and electron pseudoscalar two-fermion bilinears.
In polarised polar molecules, the interaction (1) induces shifts of energy levels. The measurement of these shifts places constraints on the value of C SP ≡ C p SP Z/A+ C n SP N/A, where A and Z are the nuclear mass and charge numbers, and N = A−Z is the number of neutrons. The most stringent constraint on C SP is placed by the ACME collaboration [31], which used the molecule 232 ThO, (90% C.L.) The coupling constant C SP receives contributions from various CP -violating interactions, including the CP -odd nuclear forces with coupling constants ξ p,n and the nucleon EDM d p,n effects. The parameters ξ p,n and d p,n , in turn, may be expressed in terms of more fundamental ones, namely, the π-meson-nucleon coupling constants g 0,1,2 , the quark chromo-EDMsd d,u and the QCD vacuum angleθ. Our aim is to determine the leading dependence of C SP on the parameters d p,n , ξ p,n ,ḡ 0,1,2 andd d,u andθ for 232 Th and several other atoms of experimental interest, including Ba, Yb, Hf, Pb and Ra.
In the recent paper [32], the contributions to C SP from the two-photon and π, η-meson exchanges between electrons and nucleons were calculated. These contributions led to a limit on the QCD vacuum angle |θ| 3 × 10 −8 . In this letter, we will take into account nuclear structure effects in the electron-nucleon interaction. As we will demonstrate further, this allows us to find stronger limits on the CP -violating hadronic parameters.
It is important to note that, according to a theorem by Schiff [33], atomic electrons completely shield the atomic nucleus from any constant external electric field, thus diminishing the effects of nuclear EDM. The CP -odd nuclear forces can be measured through the nuclear Schiff moments and magnetic quadrupole moments [34][35][36], or by means of applying an oscillating electric field and observing nuclear spin rotations as argued in the Refs. [37][38][39][40][41]. However, since the Schiff theorem applies only to a system which interacts electrically, the interaction of the atomic electrons with the magnetic dipole moment of the nucleus allows for a non-zero atomic EDM induced by a nuclear EDM [33]. This letter is devoted to the study of the mechanism for the production of the atomic EDM from the combined electric and magnetic interaction between the atomic electrons and the nucleus.
In Ref. [42] it was argued that for atoms with vanishing nuclear spins there are no non-vanishing contributions to the atomic EDM from CP -violating nuclear scalar polarizability. However, the analysis of Ref. [42] did not take into account specific near-nucleus electronic contributions which are significantly enhanced by relativistic effects in heavy atoms. Indeed, since the electronic s and p Dirac wave functions for a point-like nucleus are singular at the origin, the electronic matrix elements be-tween these states are formally divergent and thus make significant contributions to the induced atomic EDM. In this letter, we will systematically analyze the contributions to the atomic EDM from such matrix elements in atoms with vanishing nuclear spins and compare them with those arising from the contact electron-nucleon interaction (1). This will allow us to obtain the leadingorder dependence C SP (d p,n , ξ p,n ,ḡ 0,1,2 ,d d,u ,θ) for several atoms of experimental interest and deduce improved limits on the CP -violating hadronic parameters.
Atomic EDM due to contact electron-nucleon interaction: In an atom, the CP -odd interaction (1) between a valence electron and the nucleus is described by the Hamiltonian H c = G F √ 2 C SP γ 0 γ 5 ρ(r), where γ 0 and γ 5 are the Dirac matrices, r is the position vector of the electron and ρ(r) is the normalized nuclear charge density. In the leading approximation, ρ(r) is constant inside the nucleus of radius R 0 and vanishes outside.
The matrix element p 1/2 | H c |s 1/2 has a large relativistic amplification factor because of the small-distance singularity of the s 1/2 and p 1/2 wave functions [43]. As a result, to the leading order, the atomic EDM arising from the interaction H c may be written as where E s 1/2 and E p 1/2 are the energies of the s 1/2 and p 1/2 states, respectively; α is the fine structure constant and γ = √ 1 − Z 2 α 2 . Here we have taken into account that for r a B /Z the s 1/2 and p 1/2 Dirac-Coulomb wave functions have the form (see, e.g., Ref. [44]) where Ω κ µ is the spherical spinor and c s 1/2 , c p 1/2 are normalization constants. We do not specify the values of these normalization constants, as our final results will be independent of c s 1/2 and c p 1/2 .
Contribution to the atomic EDM from nucleon permanent EDM: Let d τ = d τ σ and µ τ = µ 0 (g l τ l + g s τ s) be the operators of electric and magnetic dipole moments of the nucleon. Here d τ = (d p , d n ) is the nucleon permanent EDM, µ 0 is the nuclear magneton, g l τ and g s τ are the orbital and spin g-factors of the nucleon. The operators d τ and µ τ couple with the electric and magnetic fields of the electron, yielding the interaction Hamiltonian where α are the electron's Dirac matrices. The unperturbed atomic states will be denoted by |mm = |m |m , where |m and |m are electronic and nuclear states, respectively. The first-order contributions to the atomic EDM due to the interaction Hamiltonian (5) vanish for spinless nuclei which we consider in this letter, 0 |s|0 = 0. non-vanishing contributions to the atomic EDM arise in the second order of the perturbation theory (see Ref. [42] for details), where the sum is taken over m = 0 and nn = 00 . Note also that we employ the notation E nn ≡ E n +E n , where E n and E n are energies of electronic and nuclear excitations, respectively. Similarly to the contact interaction considered in the previous section, the expression (6) receives leading contributions from terms with |0 = |s 1/2 and |m = |p 1/2 . As a result, Eq. (6) may be cast in the form (3a), with the contact interaction operator H c replaced with an effective interaction Hamiltonian H eff defined by Upon substitution of the operators (5) into Eq. (7), the contribution to the matrix element of H eff which is linear in d τ may be written as where β n accounts for the nuclear matrix elements, and the function M (E n ) represents the contributions due to the electronic matrix elements, For each nuclear energy E n , the value of the function M (E n ) is calculated numerically using the method described, e.g., in Ref. [43]. Within this method, one first solves for the corrections to the wave function |s 1/2 due to the perturbations e (r × α) /r 3 and r/r 3 , respectively, and then integrates with the |p 1/2 wave function. Further details of these computations are given in Appendix B.
The nuclear matrix elements (9) correspond to the spin-flip M1 nuclear transitions. The computations of these matrix elements slightly differ for spherical and deformed nuclei. For the former, it is appropriate to describe the nuclear states in the spherical basis, in which the matrix elements of the total momentum j vanish for fine structure doublets. As a result, substituting the operators d τ and µ τ into Eq. (9) and using the identity l = j − s, the nuclear matrix elements may be written as 0 |d τ |n n |µ τ |0 = 2µ 0 d τ (g s τ − g l τ )| n |s|0 | 2 . For deformed nuclei, the states are usually represented within the Nilsson basis [45], in which 0 | l |n n | s |0 = 0. Thus, for deformed nuclei in Eq. (9) we have 0 |d τ |n n |µ τ |0 = 2µ 0 d τ g s τ | n |s|0 | 2 . These two cases may be combined in one expression where = 1 for spherical nuclei and = 0 for deformed ones. In Eq. (11), the values of the g-factors for the proton and neutron are g l p = 1, g s p = 5.586, g l n = 0 and g s n = −3.826, respectively. The energies E n and matrix elements 0 | s |n of the spin-flip M1 transitions are listed in Appendix A for different nuclei of interest.
It is convenient to denote the energies and matrix elements for proton M1 spin-flip transitions as E π and 0 | s |n π , and for neutron transitions as E ν and 0 | s |n ν . With β n given by Eq. (11), this allows us to represent the matrix element (8) in a compact form where the quantities M p and M n are defined by Note that for deformed nuclei this expression simplifies, because in the Nilsson basis | 0 | s |n | 2 = 1. The details of numerical computations of the coefficients M p and M n are given in Appendix B. In particular, for 232 Th they are M p = 0.053c Contributions to the atomic EDM from P, Todd nuclear forces: The P, T -odd nucleon-nucleon interaction may be effectively taken into account by introducing the perturbed nuclear wave functions [34] |n → |ñ = (1 + ξ τ σ · ∇) |n , where ξ τ = (ξ p , ξ n ) is the effective coupling constant. This P, T -odd nucleon-nucleon interaction amounts to the atomic EDM which arises due to the interaction Hamiltonian (5), with d τ = e τ r being the nuclear electric dipole operator. Here e τ = (e p , e n ) is the effective charge of the nucleon, e p = (N/A)e, e n = −(Z/A)e, which appears due to the recoil effect. The leading contribution to the atomic EDM may be represented by an equation similar to (3a), in which the Hamiltonian H c is replaced with the effective Hamiltonian (7) possessing the matrix element of the form (8). The only new feature is that the nuclear matrix elements (9) are now given by β n = 0 |d τ |ñ ñ |µ τ |0 . Substituting here the P, T -perturbed nuclear states (14) and taking into account the comments above Eq. (11), we find where p is the nucleon momentum operator. The last term in Eq. (15) involves the E1 matrix element 0 |r|n which may be accounted for within the giant resonance model. In this model, we assume that all the E1 matrix elements have approximately the same energyĒ, which is the energy of the E1 giant resonance. In this case, after the summation over n , the last term in Eq. (15) is proportional to n 0 |r|n n |s × p|0 = − 0 |l · s|0 ≡ − l · s , where we have taken into account the completeness of the states |n . The expectation value of the l · s operator may be estimated using Nilsson nuclear model [45,46]. The values of these matrix elements as well as the energiesĒ are listed for different nuclei of interest in Appendix A.
Taking into account the above comments about the nuclear matrix elements, one may substitute the expression (15) into Eq. (8) and find the matrix element of the effective interaction Hamiltonian where l · s p and l · s n are expectation values of the l · s operator for the proton and neutron states, respectively. The values of the coefficients M p and M n for 232 Th and 180 Hf were given at the end of the previous section.  Table III in Appendix B.
Constraints on CP -odd hadronic parameters: In Eqs. (12) and (16), we computed the contributions to the atomic EDM (3a) due to the nucleon EDMs and P, Todd nuclear interactions. It is natural to compare these contributions with that due to the contact interaction H c . This amounts to comparing the sum of the matrix elements (12) and (16) with the matrix element (3b). Equating these matrix elements gives the following relation between the constant C SP and the parameters d p , d n and ξ p , ξ n : We point out that the right-hand side of Eq. (17) is independent of the wave functions normalization constants c s 1/2 and c p 1/2 . Indeed, as is seen from the definition (10), the function M (E) contains the same normalization constants, so that they cancel out in Eqs. (17).

Conclusions:
In this letter, we demonstrated that the experiments measuring the electron electric dipole moment (eEDM) with paramagnetic atoms and molecules are also sensitive to the nucleon EDMs and P, T -violating nucleon-nucleon interactions. Dominant contributions to   the atomic EDM in such atoms arise from the combined electric and magnetic electron-nucleus interaction. Taking into account nuclear structure effects, we derived relations between the electron-nucleus contact interaction constant C SP and CP -violating parameters. Using the limit on C SP from ThO experiment [31], we placed independent limits on the proton EDM, |d p | < 4.8 × 10 −25 e·cm, and the QCD vacuum angle, |θ| < 7.6×10 −10 , and other CP -odd parameters. We expect that the obtained relations may place the most stringent limits on these parameters once improved constraints on C SP are available from next generations of eEDM experiments [10,11,[29][30][31]53]. In this appendix, we estimate the nuclear matrix elements and corresponding energies of M1 spin-flip nuclear single-particle transitions. The details of these computations slightly differ for (nearly) spherical and deformed nuclei. Therefore, we consider these two cases separately.

Spherical nuclei
In this section, we focus on the 208 Pb, 206 Pb and 138 Ba nuclei, which are nearly spherical, i.e., they have deformation δ < 0.1. For such a nucleus, proton and neutron single-particle states may be labeled as |n, l, j, m , where n is the oscillator quantum number, l and j are the orbital and total momentum numbers, m is magnetic quantum number. In this basis, the nuclear spin operator s provides transitions between fine structure doublets.
In the 208 Pb nucleus, the non-vanishing matrix elements of the spin operator are 5h 9 2 |s|5h 11 2 for protons and 6i 11 2 |s|6i 13 2 for neutrons. The isotope 206 Pb has additional contributions from the 5p 1 2 |s|5p 3 2 neutron matrix elements. For 138 Ba, non-vanishing proton contributions arise from the matrix elements 4d 3 2 |s|4d 5 2 and 4g 9 2 |s|4g 7 2 whereas neutron contributions come from 5h 9 2 |s|5h 11 2 . All these matrix elements are calculated using the properties of spherical spinors (see, e.g., Ref. [54]). The energies of all these transitions are estimated with the use of Fig. 5 in Ref. [46]. When the energies are (nearly) degenerate, we give the sum of matrix elements corresponding to the same energy. In the tables below, we collect the values of such matrix elements with the corresponding energies for 208 Pb, 206 Pb and 138 Ba. These tables also include the expectation values of the l · s operator and the energies of the giant electric dipole resonance estimated by the empirical formulā E = 95(1 − A −1/3 )A −1/3 MeV, with A being the mass number of a heavy nucleus [55,56].

Proton transitions
Neutron transitions | n |s|0 p | 2 E n (MeV) | n |s|0 n | 2 E n (MeV) For deformed heavy nuclei with δ > 0.1, it is convenient to use the Nilsson basis [45,46]. In this basis, proton and neutron single-particle states are labeled as |n 3 , n ⊥ , Λ, Ω , where n 3 and n ⊥ are the oscillator quantum numbers, Λ and Ω are the projections of angular and total momenta on the deformation axis. Note that Ω = Λ+Σ where Σ is the projection of the nucleon's spin on the deformation axis. The dependence of the energy levels on the deformation parameter δ in this model is represented in Fig. 5 in Ref. [46]. From such dependence, one may estimate the energies of the spin-flip transitions. Note that in the basis |n 3 , n ⊥ , Λ, Ω each spin-flip M1 matrix element is m | s + |0 = 1, and the corresponding energy level is doubly degenerate since each quantum number Σ corresponds to ±Λ.
One may also compute the ground state expectation value 0 | l · s |0 = 0 | ΛΣ |0 with the corresponding energies by taking the sum over Λ and Σ quantum numbers for all protons and neutron, respectively, 0 |l · s|0 p,n = p,n Λ · Σ.
The values of the single-nucleon spin-flip transition energies, the ground state expectation values 0 | l · s |0 , the electric dipole giant resonance energies and the deformation parameter δ for several nuclei of interest are presented in the corresponding tables below.
convenience, we use the spherical basis (e + , e − , e 0 ). The components of the vectors in this basis will be labeled by the (+, −, 0) subscripts. Due to spherical symmetry, Eq. (10) may be conveniently rewritten as where the wave functions |0 B and |0 E are defined by By construction, these wave functions obey the equations where H 0 is the unperturbed electronic Hamiltonian, H 0 |n = E n |n . In this appendix, we solve the equations (B3) for |0 B and |0 E and determine the values of the electronic matrix element (B1) for particular nuclear energies E n tabulated in Appendix A. From this point on, without loss of generality, we set E s 1/2 0 = 0. Equations (B3) may be solved in different ranges of the radial variable r. As will be demonstrated in Sect. B1, analytical solutions to Eqs. (B3) exist for r Zα/E n , i.e., in the region very close to the nucleus. However, the leading contribution to Eq. (B1) comes from distances beyond the vicinity of the nucleus. We will compute this contribution numerically in Sect. B2.

B1: Short-range contribution
At small distances from the nucleus, r a B /Z, the inter-electron interaction is negligible as compared to the Coulomb interaction of the electrons with the nucleus. Therefore, in this region, the electron unperturbed Hamiltonian is simply H 0 = α · p + βm e − Zα/r. Also, in this regime, the unperturbed s 1/2 and p 1/2 wave functions have the simple form (4). In this section, we will employ these approximate wave functions to determine the short-range contributions to the electronic matrix element (B1).
With the wave functions (4), the right-hand sides in Eqs. (B3) may be written as where the coefficients λ κ µ and χ κ µ are defined via the algebraic equations (n × σ) 0 Ω 1 with some yet unknown radial functions f κ B , g κ B , f κ E and g κ E . Substituting the solutions (B5) into Eqs. (B3), we obtain the differential equations for these radial functions where we have neglected the electron mass m e in comparison with the nuclear energy E n . In general, Eqs. (B6) and (B7) are hard to solve analytically. However, for small distances r Zα/E n , the nuclear energies in these equations may be discarded in comparison with the other terms. In this case, the solutions to Eqs. (B6) and (B7) may be written explicitly as .
Substituting the solutions (B8) into Eq. (B1) and integrating over radial and angular coordinates, we obtain the short-range contribution to the electronic matrix element M M (s) = 4e 2 c s 1/2 c p 1/2 3 where I is the integral over the radial variable,  With the above boundary conditions, Eqs. (B12) and (B13) may be solved numerically in the region R 0 ≤ r ≤ qR 2 . In Fig. 1, we show that the values of electronic matrix elements are not sensitive to a particular choice of the parameter q subject to the constraint q 25. The values of the nuclear energies needed for computation are taken from the tables presented in the Sect. B1. The obtained solutions f κ B,E and g κ B,E are then integrated with the p 1/2 wave function with radial parts as given in Eqs.  Table. III. In this table, we give also the short-range contributions to these coefficients labeled by the superscript (s). As is seen from this table, the mid-range contribution dominates over the short-range one.
As an illustration, we present also a plot in Figure  1, which shows the dependence of the electronic matrix element M (5 MeV), corresponding to the 5-MeV neutron spin-flip transition in 232 Th on the upper limit qR 2 of the radial integrals. Clearly, near-nucleus contributions are negligible in comparison with mid-range ones. It is also clear that long-range contributions vanish and the result saturates as we increase the cut-off.