1 Introduction↩︎

Electric dipole moment (EDM) experiments are extremely sensitive probes of new sources of charge-parity (CP) violation and indirectly probe beyond-the-Standard-Model (BSM) physics at very high scales of up to \(\sim 100\) TeV [1], [2]. Recent years have seen impressive experimental progress using polar molecules which benefit from large internal electric fields that amplify the CP-violating signal [3]–[7]. EDMs of paramagnetic systems, which have one unpaired electron, are mainly interpreted in terms of the electron EDM. Current measurements lead to a strong bound on the electron EDM, \(|d_e|<4.1\cdot 10^{-30}\, e\) cm, and future experiments aim to improve this by one to two orders of magnitude [6]–[11]. This constraint is 4 orders of magnitude more stringent than the neutron EDM limit [12].

Traditionally, paramagnetic systems have not been used to constrain hadronic sources of CP violation, such as the quantum chromodynamics (QCD) \(\bar \theta\) term within the SM or higher-dimensional quark-gluon operators that arise from heavy BSM physics. This is because of the assumption that far stricter limits can be obtained through the EDMs of the neutron or diamagnetic atoms. That being said, paramagnetic systems are sensitive to hadronic sources of CP violation through the CP-odd electron-nuclear force they induce [13]–[15]. While this force is typically strongly suppressed, the rapid progress in paramagnetic EDM experiments might make it the best way to search for hadronic sources of CP violation in the future. However, the current theoretical description of the CP-odd electron-nuclear force is still at a very rudimentary stage.

In this Letter, we systematically derive this force as induced by the EDMs of neutrons and protons, making it possible to constrain these EDMs with paramagnetic molecular EDM experiments. As the problem involves a multitude of well-separated energy scales, it can be systematically described using effective-field-theory (EFT) techniques. We show that this connection requires the calculation of a set of nuclear matrix elements (NMEs) that are different from the ones involved in the Schiff moments of diamagnetic systems [16]–[18]. As an explicit example, we compute the NMEs for the polar molecule BaF, which is being targeted by the NL-eEDM collaboration [8].

2 Effective Field Theory↩︎

The calculation of molecular EDMs in terms of fundamental sources of CP violation involves widely separated energy scales. These range from the BSM and electroweak scales \((\Lambda\) and \(M_W)\) to low-energy scales such as the electron mass or electron binding energy \(\mathcal{O}(\alpha_{\mathrm{em}}^2m_e)\). The atomic nucleus gives rise to additional scales associated with the chiral-symmetry-breaking scale \(\Lambda_\chi \sim m_N\sim 1\) GeV (comparable to the nucleon mass), the pion mass \(m_\pi \sim \gamma \sim 100\) MeV (comparable to the nuclear binding momentum) and the scale of nuclear excitations \(m_\pi^2/m_N \sim \mathcal{O}(\mathrm{MeV})\).

Within the SM, the most relevant source of CP violation is the QCD \(\bar \theta\) term, as CKM-induced (paramagnetic) EDMs are orders of magnitude too small to be detected by current and envisioned experiments [19]. BSM sources of hadronic CP violation can, at energies well below \(\Lambda\), be described by effective operators of space-time dimension six. They have been classified and evolved to lower energies in a series of previous works [20]–[23]. At energies slightly above \(\Lambda_\chi\), the most relevant hadronic operators are the (chromo-)electric dipole moments of quarks, the Weinberg three-gluon operator [24], and several CP-odd four-quark interactions [20]. At energies \(<\Lambda_\chi\), these effective operators can be matched to a \(\chi\)EFT Lagrangian describing CP-violating interactions among the relevant low-energy degrees of freedom (light mesons, nucleons, photons, electrons) [1], [20]. For our purposes, the most relevant interactions are given by \[\begin{align} \mathcal{L}_{\chi} &=& \bar{g}_0 \bar{N} \tau^a N \pi^a + \bar{g}_1 \bar{N}\!N \pi^0 + \bar{g}_{0\eta} \bar{N}\!N \eta\nonumber\\ &&+ 2 \bar N ( d_0 + d_1 \tau^3) v^\mu S^\nu N F_{\mu\nu}\,, \label{eq:gbarterms} \end{align}\tag{1}\] where the first line describes three CP-odd meson-nucleon interactions, and the second line, respectively, the isoscalar and isovector nucleon EDM. We use the non-relativistic nucleon doublet \(N=(p,\,n)^T\) with spin \(S^\mu=(0,\boldsymbol{\sigma}/2)\) and velocity \(v^\mu=(1,\, \mathbf{0})\), as well as the pion triplet \(\pi^a\) and the eta meson \(\eta\).

The paramagnetic EDMs induced by the meson-nucleon interactions in Eq. 1 arise mainly through intermediate CP-odd electron-nucleon interactions, which take on the form \[\mathcal{L} = \frac{G_F}{\sqrt{2}}\bar{e}i\gamma_5 e\, \bar N\left( C_{\mathrm{SP}}^0 + C_{\mathrm{SP}}^1 \tau^3\right)N\,. \label{eq:CSPLagrangian}\tag{2}\] The nucleon EDMs in Eq. 1 give rise to contributions at longer distance scales through the diagrams in Fig.1. They induce effective interactions between the nucleus and the electrons, i.e.the nuclear equivalent of \(C_{\rm SP}^{0,1}\), which we denote by \(\bar C_\text{SP}\), see Eq.@eq:eq:CSPandbetav . To systematically compute the various contributions, it is useful to consider different photon modes depending on the scaling of their momentum \(q_\gamma^\mu = (q_\gamma^0,\,\mathbf{q}_\gamma)\). We identify three regions that give relevant contributions

1. soft photons: \(q^0_\gamma \sim |\mathbf{q}_\gamma| \sim m_\pi\),

2. ultrasoft photons: \(q^0_\gamma \sim |\mathbf{q}_\gamma| \sim m_\pi^2/m_N\),

3. potential photons: \(q^0_\gamma \sim \mathbf{q}_\gamma^2/m_N|\mathbf{q}_\gamma| \sim m_\pi\) ,

and we define \(Q\sim m_\pi \sim \gamma\) and \(q\sim Q^2/m_N\).

The CP-odd meson-nucleon interactions in Eq. 1 contribute to \(C_\text{SP}^{0,1}\) through diagrams involving a meson exchange or a pion loop in combinations with the exchange of two photons in the ultrasoft or soft region. These diagrams were first considered in Ref. [13] and later computed with heavy-baryon chiral perturbation theory in Ref. [15]. In addition, integrating out the mesons leads to renormalization of nucleon EDMs [25]–[27], effectively shifting \(d_{0,1}\rightarrow \bar d_{0,1}\), where the bar denotes the renormalized LECs. In what follows, we use \(\bar d_{0,1}\) as the physical nucleon EDMs.

In this Letter, we focus on additional contributions to \(\bar C_{\mathrm{SP}}\) from the nucleon EDMs, which arise through the topologies shown in Fig. 1 (a) and 1 (b). These diagrams are captured by an effective action of the form \[\begin{align} \label{eq:Seff} &&\langle h_f (p_f)e(p_e') | iS_{\rm eff} | h_i(p_i) e(p_e)\rangle = \frac{e^3}{2} \int_{x_i} \langle h_f e|\\ &&\times T\Big[\bar e \cancel{A} e(x_1)\, \bar e \cancel{A} e(x_2) \mathcal{L}^{(d_{0,1})}_{\chi}(x_3) \, \left( A_\mu J_{\rm em}^\mu \right)(x_4) \Big]|e h_i\rangle \,, \nonumber \end{align}\tag{3}\] where we integrate over all \(x_{1,2,3,4}\), \(h_{i,f}\) denote the initial and final nuclear states (for EDMs we have the nuclear ground state \(|h_i\rangle = |h_f\rangle = |0^+\rangle\)) and \(J_{\rm em}^\mu\) denotes the nuclear electromagnetic current. Diagrams involving nucleon EDMs and photons with soft momenta are subleading as they require the inclusion of additional pions. Power counting gives the expected size of the potential and ultrasoft contributions \[\begin{align} \bigg\{C_\text{SP}^\text{(pot)},\,C_\text{SP}^\text{(usoft)}\bigg\} = \frac{m_e \alpha^2 \mu_i \bar d_i}{e G_F m_N}\,\bigg\{\frac{4\pi}{Q},\,\frac{1}{q}\bigg\}, \end{align}\] where \(\mu_i\) are the nucleon magnetic dipole moments (MDMs) in units of the nuclear magneton. Numerically \(4\pi q \sim Q\) and these estimates are rather close, but, as we will see, they do not capture possible coherent enhancements.

Potential region: To evaluate the potential contributions, we can use the so-called method of regions to expand the amplitude in small ratios of scales, such as \(q_\gamma^0/|\mathbf{q}_\gamma|\). After doing so, there are no contributions from diagrams where the nucleon EDM and \(J_{\rm em}\) attach to the same nucleon (the potential region arises from picking up the poles of nucleon propagators, which can always be avoided in these one-body diagrams). There are, however, two-nucleon effects through the diagram in Fig. 1 (a). Due to spin and parity constraints, the first contributions arise from the nucleon magnetic moments, which appear in \(J_{\rm em}^\mu\) at next-to-leading order \({\cal O}(Q/m_N)\). This results in the following contribution to the amplitude¹ \[\begin{align} {\cal A}_{\rm pot}&=&- \langle h_f | V | h_i \rangle \,\bar u ({p_e'} )\left(1-\frac{v\cdot (p_e'-p_e)}{2m_e}\cancel{v} \right)i\gamma_5 u({p_e})\,, \nonumber \end{align}\] where we take the limit \(p_e'-p_e\ll m_e\) in what follows, while \(V\) denotes the potential between the two interacting nucleons. In momentum space² \[\begin{align} \label{eq:Vq} V &=&\frac{4e^4 m_e}{9m_N } \sum_{i\neq j}\frac{ \mu^{(i)} D^{(j)} }{|\mathbf{q}|^4}\left[ \boldsymbol{\sigma}^{(i)}\cdot \boldsymbol{\sigma}^{(j)}-\frac{1}{4}S^{(ij)}\right] \,, \end{align}\tag{4}\] where \(i,j\) label the nucleons, \(\mathbf{q}_i = \mathbf{p}_i'-\mathbf{p}_i\) is the exchanged momentum and we define the combination \(\mathbf{q}= (\mathbf{q}_i - \mathbf{q}_j)/2\). In addition, \(D = ( \bar{d}_0 + \bar{d}_1\tau_3)/e\), \(\mu \equiv \frac{1+\kappa_0}{2}+\frac{1+\kappa_1}{2}\tau_3\) describe the EDM and MDM operators (with \(\kappa_0=-0.12\) and \(\kappa_1=3.7\)), while \(S^{(ij)} =\boldsymbol{\sigma}^{(i)}\cdot \boldsymbol{\sigma}^{(j)} -3 \mathbf{q}\cdot \boldsymbol{\sigma}^{(i)}\, \mathbf{q}\cdot \boldsymbol{\sigma}^{(j)}/\mathbf{q}^2\). These contributions are thus determined by the NME of \(V\), which scales as \(1/|\mathbf{q}|^4\). The form of this two-body potential is similar to the NMEs appearing in neutrinoless double-\(\beta\) decay [29] or radiative corrections to superallowed \(\beta\) decays [28], [30] although with different isospin and/or \(\mathbf{q}\) dependence. The many-body techniques needed to compute such NMEs, including ab-initio approaches, have developed significantly in the last decade and can be directly applied to these EDM calculations [29], [31].

Ultrasoft region: In this region, we expand in small ratios of scales, such as \(q_\gamma^0/m_\pi\sim |\mathbf{q}_\gamma|/m_\pi\) and \(m_e/m_\pi\). Photons with this (small) momentum scaling can be thought of as coupling to the nucleus as a whole, instead of the individual nucleons. After inserting a complete set of (nuclear) states between the hadronic operators and working out the time-ordered product in Eq.@eq:eq:Seff , we again find that the leading contributions involve the magnetic moments,

where \(|n\rangle\) denote intermediate nuclear \(|1^+\rangle\) states with energies \(E_n\), while \(l_{+} = p_i+p_e-k\), \(l_{-} = p_i-p_e'+k\). Furthermore, \(M^\eta\equiv \bar N \mu \, S^\eta N(0)\) and \(D_{\mu\nu} = \bar N D \left(v_\mu S_\nu-v_\nu S_\mu\right)N(0)\) denote the MDM and EDM operators.

Although the appearing integrals can be evaluated using known techniques [32], the expressions are rather unwieldy. They greatly simplify if there is a hierarchy between the nuclear excited states and the electron momenta, \(p_e\sim m_e\ll \Delta_n=|E_n-E_i|\), which is a good approximation for \(^{138}\)Ba as discussed in Sec.3. Likewise, the relevant excited states in magnetic-dipole transitions —also driven by the spin operator— in isotopes of Yb, Hf, and Th with an even number of neutrons also enter at about 2 MeV or higher energies [33]–[35]. In this case, Eq.@eq:eq:usoftMag can be captured by a low-energy nuclear EFT in which the excited nuclear states have been integrated out, but still contains electrons, ultrasoft photons, and the ground state of the nucleus. The relevant interactions in this theory can be written as \[\begin{align} \label{eq:CSPandbetav} {\cal L}_{\Psi} =\Psi_i^\dagger\left[ \frac{G_F}{\sqrt{2}}\bar C_{SP} \, \bar e i\gamma_5 e +\beta_v\,v^\alpha F_{\alpha \beta} v_\lambda \epsilon^{\beta\lambda\mu\nu}F_{\mu\nu}\right] \Psi_i\,, \end{align}\tag{6}\] where \(\Psi_i\) denotes the spin-0 field describing the nucleus,³ \(\bar C_{SP}\) describes the nuclear version of \(C_{SP}^{0,1}\), while \(\beta_v\) has a similar form as the nuclear polarizability but violates CP. At the scale \(\mu=\Delta_n\), \(\beta_v\) obtains a contribution from integrating out the excited states at tree level, while \(\bar C_{SP}\) arises from Eq.@eq:eq:usoftMag . After expanding in \(m_e/ |E_n-E_i|\), this expression simplifies and the remaining integrals are of the form \[I_n \equiv \int \frac{d^dk}{(2\pi)^d} \frac{1}{(k^2)^n}\frac{1}{v\cdot k-\Delta}\,,\] which are evaluated as [36] \[\begin{align} I_n = 2i \frac{(-1)^{n+1}(2\Delta)^{d-2n-1}}{(4\pi)^{d/2}}\frac{\Gamma(2n+1-d)\Gamma(d/2-n)}{\Gamma(n)}\,. \end{align}\] All in all, matching the nucleon-level theory to the EFT without excited states then gives at a scale \(\mu\simeq \Delta_n\), \[\begin{align} &\frac{G_F}{\sqrt{2}} \bar C_{SP}&= {-}\frac{e^4m_e}{{4}\pi^2 m_N}\sum_n\frac{A_n}{\Delta_n}\left(4-3\log\frac{4\Delta _n^2}{\mu^2}\right)\,,\\ & \beta_v =& \frac{e^2}{m_N}\sum_n \frac{A_n}{\Delta_n}\,,\quad A_n =- \frac{\langle h_i | D \boldsymbol{\sigma}| n \rangle \cdot \langle n | \mu \boldsymbol{\sigma}|h_i \rangle}{12}\,.\nonumber \end{align}\]

These interactions can be evolved from \(\mu\sim \Delta_n\) to lower energies, \(\mu_e\sim m_e\), using the renormalization group equation (RGE), \[\begin{align} \frac{d \bar C_{SP}(\mu)}{d\ln\mu} = \frac{3 \sqrt{2} e^2 m_e\beta_v}{4\pi^2 G_F}\,. \end{align}\] This RGE arises through the loop diagram of Fig.1 (c), which allows \(\beta_v\) to contribute to \(\bar C_{SP}\).

The amplitude at low scales, \(\mu_e\sim m_e\), can finally be expressed as the sum of \(\bar C_{SP}(m_e)\) and a loop contribution due to \(\beta_v\). We capture the total combination of the ultrasoft and potential contributions by an effective contact interaction, such that \({\cal A}_{\rm total} = \frac{G_F}{\sqrt{2}} \bar C_{SP}^{\rm eff} \bar u(p_e') i\gamma_5 u(p_e)\) with \[\begin{align} \label{CSPeffTOT} \bar C_{SP}^{\rm eff} ={-}\frac{\sqrt{2}}{G_F}\left[\frac{ {4}\alpha^2 m_e}{m_N}\sum_n \frac{A_n}{\Delta_n}\left(3\ln \frac{m_e^2}{4\Delta_n^2}-1\right){+} \langle h_i| V|h_i\rangle\right] \end{align}\tag{7}\] which is independent of the renormalization scale \(\mu\). We stress that this is the effective interaction between electrons and the nucleus as a whole and differs from Eq. 2 , which is the coupling to individual nucleons. Evaluating the ultrasoft region thus requires the excited state energies, \(\Delta_n\), and the set of nuclear matrix elements of the one-body operator \(\sim \langle h_i |\boldsymbol{\sigma}|n\rangle\) contained in \(A_n\). These have a form similar to the leading two-neutrino double-\(\beta\), and double magnetic-dipole NMEs [37]–[39] and of subleading NMEs of the neutrinoless double-\(\beta\) decay [40], [41]. Therefore, similar many-body methods used in these studies can be applied here. Eq. 7 is the main result of this work and makes it possible to connect nucleon EDMs to measurements of paramagnetic molecules.

3 Nuclear Matrix Elements↩︎

We now focus on the polar molecule BaF, which is targeted by the NL-eEDM collaboration [8]. The heaviest atom in the molecule, \({}^{138}\)Ba, has a magic neutron number, and it is just two neutrons away from \({}^{136}\)Ba, the well-studied [42] final state of the double-\(\beta\) decay of \({}^{136}\)Xe. We calculate the nuclear excitation energies and all necessary NMEs with the nuclear shell model [43]. We use a configuration space that comprises the single-particle orbitals \(1d_{5/2}\), \(0g_{7/2}\), \(2s_{1/2}\), \(1d_{3/2}\), and \(0h_{11/2}\) for both neutrons and protons with a \(^{100}\)Sn core. We consider three effective interactions previously tested in this mass region: GCN5082 [44], QX [45] and Sn100pn [46]. We present details of the calculation in the Supplemental Material, and here highlight the main results. The value of the ultrasoft NME is \[\label{usoftBa} \bar C_{SP}^{\rm usoft} = \left( {67} \pm 28\right) \, d_p\, (e\,\mathrm{fm})^{-1}\,,\tag{8}\] where \(d_p = \bar d_0 + \bar d_1\). We only find sensitivity to the proton EDM because, in our calculation, the 82 neutrons form a closed shell, as \(^{138}\)Ba is magic in neutrons. This is also why the first intermediate \(1^+\) excited state appears around 2.5 MeV. The largest contribution to the ultrasoft NME arises from states around \(E_n = 4.5\,\mathrm{MeV} \gg m_e\), justifying our approximation, and higher-energy states only contribute mildly. We show the cumulative contribution from the excited-state spectrum in the Supplemental Material.

The potential contribution evaluates to \[\label{potBa} \bar C_{SP}^{\rm pot} = \left[(-433 \pm 5) \, d_p + (387\pm 0.4)\, d_n \right]\,(e\, \mathrm{fm})^{-1}\,,\tag{9}\] where \(d_{n} = \bar d_0 - \bar d_1\). The small uncertainties are solely from the shell-model calculations but do not capture possible higher-order corrections. Compared to the ultrasoft regime, the potential contribution is dominant. This is because of the coherent nature of the potential NME, which scales linearly with the total number of protons, \(Z\) (\(d_p\) term), or neutrons, \(N\) (\(d_n\) term), in the nucleus. The coherence appears as most NME contributions stem from proton-proton and neutron-neutron pairs, prevalent in nuclei due to the attractive pairing interaction. This scaling is in rough agreement with the estimate of Ref.[13], as well as an evaluation of the potential of Eq.@eq:eq:Vq in a Fermi gas state⁴. Our many-body calculations, which also cover nuclei lighter than \(^{138}\)Ba, suggest that nucleus-dependent effects can correct this estimate by up to \(20\%\). The coherent character makes potential NMEs less dependent on the details of the nuclear structure, reducing their relative uncertainty with respect to ultrasoft NMEs (the very small error in the \(d_n\) potential NME is because in our calculation the 82 neutrons form a closed shell). We provide more details in the Supplemental Material. While we do not expect any breakdown of the scaling behavior discussed above, explicit calculations of ultrasoft and potential NMEs in heavier systems such as Th or Hf are required to confirm the dominance of the potential contributions.

The expected sensitivity of the BaF experiment is an electron EDM equivalent of \(d_e \leq 10^{-30}\) e cm [8]. Using the NME calculations of this Letter, this would correspond to a sensitivity to the nucleon EDMs \(|d_p|_{\text{BaF}}<{8.4} \cdot 10^{-24}\;e\,\mathrm{cm}\) and \(|d_n|_{\text{BaF}}<8\cdot 10^{-24}\;e\,\mathrm{cm}\). While we do not have shell-model calculations for the most precise experiment based on HfF\(^+\), we can use the linear \(Z\) and \(N\) dependence of the potential NME to estimate \[|d_p|_{\text{\tiny HfF}^+}\lesssim 1.6 \cdot 10^{-23}\;e\,\mathrm{cm}\,,\,|d_n|_{\text{\tiny HfF}^+}\lesssim 1.6 \cdot 10^{-23}\;e\,\mathrm{cm}\,,\] roughly two orders of magnitude weaker than the proton EDM limit set by \(^{199}\)Hg [47] and three orders than the direct neutron EDM limit [12]. Considering the past and anticipated progress in molecular EDM experiments, with projected improvements of two to three orders of magnitude within a decade [48], these gaps are not insurmountable.

4 Discussion↩︎

In this Letter, we have developed a systematic method to compute the contributions of nucleon EDMs to paramagnetic molecular EDMs. Generally, however, there are multiple other sources of CP violation, meaning that the measurement of a nonzero EDM in any system would raise the question of what the underlying mechanism is. It has been shown that the ratio of different paramagnetic systems can be used to unravel the electron EDM contribution from CP-odd electron-nucleon interactions [49], [50], while the ratio of nuclear to nucleon EDMs can separate different CP-odd sources at the quark-gluon level [51]. Based on our results, we devise a new strategy to identify the underlying source of CP violation.

Besides the diagrams calculated in this Letter, there appear contributions from meson-exchange diagrams [13], [15]. Depending on the underlying CP-violating source, the ratio of meson to nucleon EDM contributions varies. We can be most concrete for the QCD \(\bar \theta\) term, where the sizes of the low-energy constants in Eq. 1 are known relatively well [52]–[54]. For BaF, the meson diagrams give a contribution \(\bar C^{\mathrm{meson}}_{SP}\text{(BaF)} = (220 \pm 62)\cdot 10^{-2}\,\bar \theta\) [15]. We can compare this to the nucleon EDM contributions if we insert the lattice-QCD prediction for the nucleon EDMs [55], [56], which results in \[\bar C^{\mathrm{pot+usoft}}_{SP} = -(196\pm 54)\cdot 10^{-2}\,\bar \theta\,.\] This contribution is comparable in size to the meson-exchange diagrams but comes with an opposite sign, so the total contribution is suppressed. This accidental cancellation is specific to the \(\bar \theta\) term and not expected for other mechanisms of CP violation. We combine all contributions to compute the equivalent electron EDM \(d_e^\text{equiv} \equiv r_\text{mol} \bar C_\text{SP}/A\) [57], which is convenient as paramagnetic EDM searches are usually interpreted as limits on \(d_e\). For BaF, \(r_\text{BaF} = 4.46[18]\cdot 10^{-21} e\text{ cm}\) [58], and we obtain \(d_e^\text{equiv}(\bar{\theta}) = (7.5 \pm 27) \cdot 10^{-24} \, \bar{\theta} \,e \text{ cm}\).

For other hadronic sources of CP violation, power counting arguments give insight into the ratio of mesonic to nucleon EDM contributions. For example, the quark chromo-EDM breaks CP and isospin symmetry. The meson-nucleon interactions in Eq. 1 are the leading CP-violating hadronic interactions [20], [60] and their contributions dominate the paramagnetic EDMs. On the other hand, for quark EDMs and the Weinberg operators the mesonic interactions are suppressed by, respectively, \(\alpha/\pi\) (electromagnetic suppression) and \(m_\pi^2/\Lambda_\chi^2\) (chiral suppression)[20]. As such, the ratio of \(\bar C^{\mathrm{eff}}_{SP}\) (and thus \(d_e^\text{equiv}\)) to the neutron EDM is different. We illustrate this in Fig. 2, where we plot the \(d_e^\text{equiv}\), including both mesonic and nucleon EDM contributions, against \(d_n\). The bands correspond to scenarios where \(d_e^\text{equiv}\) and \(d_n\) are sourced, respectively, by the \(\bar{\theta}\) term, the up quark EDM, and the up quark chromo-EDM. Remarkably, the ratio of neutron to paramagnetic EDMs can identify the underlying hadronic source of CP violation.

In conclusion, the EFT approach presented in this Letter allows one to derive the contribution from nucleon EDMs to paramagnetic EDMs in a systematic way. We have identified novel nuclear matrix elements that must be computed in order to interpret paramagnetic EDMs in terms of nucleon EDMs and, ultimately, in terms of hadronic sources of CP violation. While power counting arguments indicate that similar contributions would arise from potential and ultrasoft virtual photons, explicit shell-model calculations show that potential NMEs dominate because of the coherent contribution of most protons and neutrons in the nucleus. Experimental improvements of two-to-three orders of magnitude in paramagnetic molecular systems are needed to set competitive limits on nucleon EDMs. Finally, we have shown that ratios of paramagnetic-to-neutron EDMs can point towards the underlying mechanism of CP violation.

5 Ultrasoft contribution to \(\bar{C}^{\rm eff}_{SP}\)↩︎

In the ultrasoft region, the contribution to the CP-odd electron-nucleus interaction is encoded in \[\bar{C}^{\rm usoft}_{SP}=-\frac{\sqrt{2}\alpha^2 m_e}{3 m_N G_F}M^{\rm usoft}_{SP}\;, \label{eq:Csp}\tag{10}\] where \(M_{SP}^{\rm usoft}\) is the NME between initial and final nuclear states, \(\ket{h_{i,f}}\), defined by \[M^{\rm usoft}_{SP}=\sum_n \frac{ \langle h_f | D^{(i)}\vec{\sigma} | n \rangle \cdot \langle n |\mu^{(i)}\vec{\sigma}|h_i \rangle}{\Delta_n}\left(1+3\ln \frac{4\Delta_n^2}{m_e^2}\right)\; \label{eq:Msp}.\tag{11}\] Here \(\Delta_n=E_n-E_i\) is the excitation energy of the intermediate nuclear states \(\ket{n}\), and the nucleon EDMs and MDMs \(D^{(i)}= (\bar d_0 + \bar d_1 \tau_3^{(i)})/e\) and \(\mu^{(i)}=(\mu_0 +\mu_1 \tau_3^{(i)})\) are defined in terms of the isoscalar and isovector nucleon EDMs, \(\bar{d}_{0,1}\), and the isoscalar and isovector anomalous magnetic moments, \(\kappa_{0,1}\), through \(\mu_{i}=(1+\kappa_{i})/2\).

We focus on \(^{138}\)Ba, the heaviest nucleus in the diatomic polar molecule BaF used by the NL-eEDM experiment [8]. We compute the matrix elements involving the one-body spin operator in Eq. 11 to the set of \(1_n^+\) nuclear excited states, as well as the relevant excited-state energies. Nonetheless, we have adjusted these energies to exactly reproduce the one of the first \(1^+\) excited state of \(^{138}\)Ba. This changes \(M^{\rm usoft}_{SP}\) just by 2%-7% depending on the effective interaction used.

Since in our calculation for \(^{138}\)Ba the 82 neutrons completely fill the configuration space, we cannot create any particle-hole excitation involving a neutron orbital, meaning there is no sensitivity to \(d_n\).

Therefore, for \(^{138}\)Ba Eq. 11 reduces to \[\begin{align} M^{\rm usoft}_{SP}=&(\bar{d}_0+\bar{d}_1)(\mu_0+\mu_1) m_{\sigma}=m_{\sigma}\mu_{p}d_{p}\;, \end{align}\] with \(2\mu_{p}=\kappa_0+\kappa_1+2\) (likewise, \(2\mu_{n}=\kappa_0-\kappa_1\)) and \(m_{\sigma}\) defined by \[m_{\sigma}=-\frac{1}{e}\sum_n \frac{\bra{0^+_1}\vec{\sigma}\ket{1_n^+}^2}{\Delta_n}\left(1+3\ln \frac{4\Delta_n^2}{m_e^2}\right)\;, \label{eq:Ms}\tag{12}\] where we drop the isospin operator as only protons contribute to the NME. As indicated by Eq. 12 , different contributions cannot cancel.

Table 1 presents the calculated \(\bar{C}^{\rm usoft}_{SP}\) values for \(^ {138}\)Ba. The results obtained with the three different shell-model Hamiltonians used differ, at most, by about a factor two. This shows a significant sensitivity to nuclear structure for the ultrasoft NME.

Figure 3 shows the cumulative sum of \(\bar{C}^{\rm usoft}_{SP}\) as a function of the excitation energy of the intermediate states. For the three nuclear Hamiltonians used, the behaviour is quite similar: a few states between \(4-5\) MeV dominate, with lower- and higher-energy states contributing little.

Additionally, Table 1 also presents the results for the ultrasoft NME in other nuclei, \(^{106}\)Sn, \(^{104}\)Te and \(^{132}\)Te, using the same configuration space as for \(^{138}\)Ba. Our results indicate NME values comparable to the \(^{138}\)Ba ones. The theoretical uncertainty due to the nuclear Hamiltonian used is also comparable to the one found for \(^{138}\)Ba, highlighting again the sensitivity of the NME to nuclear structure effects.

6 Potential contribution to \(\bar{C}^{\rm eff}_{SP}\)↩︎

In the potential region the contribution to the CP-violating electron-nucleus interaction is given by \[\bar{C}^{\rm pot}_{SP}=-\frac{\sqrt{2}}{G_F}\langle h_i | V | h_i \rangle\;, \label{eq:Csp95pot95ori}\tag{13}\] where in coordinate space \[V(\vec{r})=-\frac{e^4m_e}{18\pi m_N}\sum_{i\neq j}\mu^{(i)}D^{(j)}\,|\vec{r}|\left[\sigma^{(i)}\cdot\sigma^{(j)}+\frac{1}{16}S^{(ij)}(r) \right].\] with \(S^{(ij)}=3\hat{r}\cdot\vec{\sigma_i}\hat{r}\cdot\vec{\sigma_j}-\vec{\sigma_i}\cdot\vec{\sigma_j}\). For convenience we define \(N^{(ij)}=\sigma^i\cdot\sigma^j+S^{(ij)}/16\) and rewrite Eq. 13 as \[\bar{C}^{\rm pot}_{SP}=\frac{\sqrt{2}e^4m_e}{18\pi G_Fm_N}M^{\rm pot}_{SP}, \label{eq:Csp95pot}\tag{14}\] with \(M^{\rm pot}_{SP}\) the expectation value of the operator \[\begin{align} O_{SP}&=\;\frac{\bar{d}_0}{e}\sum_{i\neq j}r\left(\mu_0+\frac{\mu_1}{2}(\tau_3^i+\tau_3^j)\right)N^{(ij)}\nonumber\\ &+\frac{\bar{d}_1}{e}\sum_{i\neq j}r\left(\mu_1\tau_3^i\tau_3^j+\frac{\mu_0}{2}(\tau_3^i+\tau_3^j)\right)N^{(ij)}\;,\label{eq:Osp95pot95d095d1} \end{align}\tag{15}\] or in terms of \(d_{p,n}\), \[\begin{align} O_{SP}&=\frac{d_p}{2e}\sum_{i\neq j}r\left(\mu_0+\mu_1\tau_3^i\tau_3^j+\frac{\mu_1+\mu_0}{2}(\tau_3^i+\tau_3^j)\right)N^{(ij)}\nonumber\\ &+\frac{d_n}{2e}\sum_{i\neq j}r\left(\mu_0-\mu_1\tau_3^i\tau_3^j+\frac{\mu_1-\mu_0}{2}(\tau_3^i+\tau_3^j)\right)N^{(ij)}\label{eq:Osp95pot95dp95dn}. \end{align}\tag{16}\]

Again, for \(^{138}\)Ba we calculate \(M_{SP}^{\rm pot}=\bra{0^+}O_{SP}\ket{0^+}\) using the nuclear shell model. We note that both core and valence nucleons contribute to the potential NME. In fact, in the ideal case of a nucleus with fully-closed angular-momentum shells for both neutrons and protons, the \(\sum_{i\ne j}\sigma^{(i)}\cdot\sigma^{(j)}\) operator with the same isospin dependence as in Eq. 15 would just count the number of proton pairs and neutron pairs coupled to spin zero. The corresponding NME is \((-3 Z\,\mu_p d_p-3 N\,\mu_n d_n) (\mathrm{fm}/e)\), with separate coherent contributions of all protons and all neutrons in the nucleus.

Table 2 presents the results of the shell-model calculations for \(^{138}\)Ba for the NMEs corresponding to each component of the operator in Eq. 16 , that is, \(m_{SP}^{\rm pot,p}\) and \(m_{SP}^{\rm pot,n}\) defined as \[M_{SP}^{\rm pot}=m_{SP}^{\rm pot,p}\,d_p+m_{SP}^{\rm pot,n}\,d_n\,.\] The results in Table 2 indicate that, in general, \(m^{\rm pot}_{SP}\) NMEs are significantly larger than ultrasoft NMEs. This difference arises from the coherent contribution of all nucleons, in contrast to the non-coherent ultrasoft NME, which is dominated by a few components where the core does not contribute. Coherence also leads to very similar NMEs across the three shell-model Hamiltonians used, indicating that nuclear structure details are not very relevant for the potential NME. Indeed, for \(m^{\rm pot,n}_{SP}\) the results are almost identical because in our calculations the 82 neutrons in \(^{138}\)Ba form a closed shell.

In addition, Table 2 distinguishes the results for the Gamow-Teller and tensor spin structures. For both neutron and proton parts, the contribution of the tensor is very small. This suggests that the potential NME is mostly driven by pairs of nucleons coupled to spin zero, just as dictated by the Gamow-Teller operator.

We explore the scaling of the potential NMEs with the number of protons and neutrons by calculating \(M^{\rm pot}_{SP}\) for several nuclei in different mass regions. Table 3 lists the results, which suggest that the potential NME indeed increases linearly with \(Z\) and \(N\). Figure 4 highlights this linear relation. It represents, for all nuclei, \(m^{\rm pot,p}_{SP}\) and \(m^{\rm pot,n}_{SP}\), normalized by the proton and neutron magnetic moment, as a function of \(Z\) (\(m^{\rm pot,p}_{SP}/\mu_p\)) or \(N\) (\(m^{\rm pot,n}_{SP}/\mu_n\)). A good linear relation common to the proton and neutron parts of all nuclei emerges, best fitted to \(m^{\rm pot,X}_{SP}=-9.74\,X\,\mu_X (\mathrm{fm}/e)\), where \(X\) stands either for protons (\(Z, p\)) or neutrons (\(N, n\)). The linear relation confirms that the potential NMEs is largely dominated by spin-zero pairs of protons and neutrons, as it includes nuclei where all nucleons form pairs—such as \(^{16}\)O, \(^{36}\)S (neutrons) or \(^{48}\)Ca (protons)—because they fill angular-momentum-closed shells. The contribution of proton-neutron pairs is minor.

Using the scaling function for the NMEs, we write a master formula one can use to compute the equivalent electron EDM \(d_e^\text{equiv} \equiv r_\text{mol} \bar C_\text{SP}/A\) [57] for any system \[{d}^{\rm equiv}_e=\frac{\sqrt{2}e^4m_e}{18\pi G_Fm_N} \frac{r_\text{mol}}{A} (-9.74) \left[ \, Z \, \mu_p\, d_p+N \,\mu_n \,d_n \right] \frac{\mathrm{fm}}{e} \label{eq:master32formula}\tag{17}\] where \(A\) is the mass number of the heaviest nucleus in each system, and \(r_\text{mol}\) is a molecular matrix element. For several molecules of experimental interest it is given by [50], [58]: \[\begin{align} r_\text{BaF}&=&\,4.46[18]\cdot 10^{-21} \text{ e cm}\; ,\nonumber\\ r_\text{ThO}&=&\,1.51[9] \cdot 10^{-20} \text{ e cm}\; ,\nonumber\\ r_{\text{HfF}^+}&=&9.17[52] \cdot 10^{-21} \text{ e cm}\,. \end{align}\]

Additionally to the full potential operator, indicated by \(V(r)=r\), Table 3 also presents NMEs for the operator just keeping the spin and isospin degrees of freedom, but without radial dependence. These results, denoted by \(V(r)=1\), also indicate a linear dependence with the number of neutrons and protons. In fact, for nucleons in fully angular-momentum-closed shells the NMEs exactly fit to \(-3\,X\,\mu_X d_X\; (\mathrm{fm}/e)\), as expected. For all the nuclei in Table 3, the results for \(m^{\rm pot,p}_{SP}\) and \(m^{\rm pot,n}_{SP}\) share a similar relation with the ones obtained for \(V(r)=1\), with a proportionality constant \(\sim(3.4-3.9)\) fm. This common factor reveals the lack of additional scaling in the potential NMEs due to the radial part of the operator.

Figure 5 further analyzes this aspect, showing the normalized radial distribution \(c_{SP}^{\rm pot}(r)\) for \(^{20}\)Ne, \(^{48}\)Ca, and \(^{138}\)Ba, defined by \[c_{SP}^{\rm pot}(r)=\frac{\sum_{i\neq j}\mu^{(i)}D^{(j)}r_{ij}N^{(ij)}(r_{ij})\delta(r-r_{ij})}{|M^{\rm pot}_{SP}|}\;. \label{eq:Csp95pot95rad}\tag{18}\] which fulfills the relation \[1=\int_0^{\infty}c^{\rm pot}_{SP}(r){\rm d}r\;. \label{eq:Csp95pot95int}\tag{19}\] The radial distributions in Fig. 5 show that, regardless of the size of the nucleus, the dominant contribution to the potential NME, shown in solid curves, stems from nucleons relatively close to each other. This property is dictated by the spin part of the NME, indicated by the dashed curves in Fig. 5. Even though the full potential operator—including the radial part—gives more relevance to nucleons further apart, pairs of nucleons at short distances still dominate.

References↩︎