Nucleon Electric Dipole Moments in Paramagnetic Molecules through Effective Field Theory

TL;DR

The paper develops an effective field theory to relate paramagnetic molecular EDMs to nucleon EDMs, enabling constraints on hadronic CP violation from molecular experiments. It identifies the relevant CP-odd interactions, distinguishes three photon-momentum regions, and derives a nucleus-wide effective coupling ${\bar C}_{SP}^{\rm eff}$ that governs electron-nucleus CP violation, including RG running. Through a detailed shell-model calculation for BaF (Ba-138), it shows that ultrasoft and, more importantly, potential NMEs contribute to the EDM, with potential NMEs exhibiting strong coherence and scaling with $Z$/$N$, leading to finite bounds on $d_p$ and $d_n$ from current measurements. The work also outlines how ratios of EDMs across systems can help identify the underlying hadronic CP-violating sources and discusses the experimental prospects needed to tighten nucleon EDM constraints via molecular experiments.

Abstract

Electric dipole moment (EDM) measurements using paramagnetic molecules have significantly advanced over the last decade. Traditionally, these experiments have been analyzed in terms of the electron EDM. However, paramagnetic molecules are also sensitive to hadronic sources of charge-parity (CP) violation, highlighting the need for a new framework to interpret the experimental results. In this Letter, we introduce an effective field theory framework to relate molecular EDMs to the EDMs of neutrons and protons. We identify the dominant contributions through power counting and pinpoint the necessary nuclear matrix elements. As a practical application, we employ the nuclear shell model to calculate these nuclear matrix elements for the polar molecule BaF. Finally, we estimate the limits on the nucleon EDMs set by current molecular EDM experiments.

Figures (5)

• Figure 1: Contributions to $\bar{C}_\text{SP}$ arising from the nucleon EDMs. We denote electrons by single and nucleons by double straight lines, nuclei by gray ovals (in Fig. \ref{['fig:2Npot']}) or bars (in Fig. \ref{['fig:usoftbox']}-\ref{['fig:betav_and_Csp']}), and photons by wavy lines. The black circle stands for the nucleon MDM, while the yellow, magenta, and blue squares indicate the CP-violating vertices: nucleon EDM, $\beta_v$ and $\bar{C}_{SP}$ effective vertices (see Eq. \ref{['eq:CSPandbetav']}), respectively. Fig. \ref{['fig:2Npot']} shows the two-nucleon potential-region contribution, and Fig. \ref{['fig:usoftbox']} the ultrasoft one. Fig. \ref{['fig:betav_and_Csp']} shows the two diagrams relevant to the matching and running of diagram \ref{['fig:usoftbox']} in an EFT with the nuclear ground state as the remaining degree of freedom.
• Figure 2: The ratio between $d_e^\text{equiv}$ and $d_n$, induced by various possible underlying sources of CP violation: the $\bar{\theta}$ term (blue band), the up quark EDM (orange) or chromo-EDM (grey). The plots are based on Refs. Dekens:2018bciPospelov:2001ysBhattacharya:2025blb regarding QCD matrix elements connecting the CP-violating sources to CP-violating hadronic couplings.
• Figure 3: $\bar{C}^{\rm usoft}_{SP}$ as a function of the excitation energy of the intermediate states, for three shell-model Hamiltonians.
• Figure 4: Absolute value for the proton ($m^{\rm pot,p}_{SP}$) and neutron ($m^{\rm pot,n}_{SP}$) contributions to $M^{\rm pot}_{SP}$, divided by the corresponding nucleon MDM, as a function of the atomic or neutron number. The results cover all nuclei in Table \ref{['tab:calc_Msp_pot_A']} and also show the best linear fit (see text) and $95\%$CL prediction bands.
• Figure 5: Normalized $c^{\rm pot}_{SP}(r)$ in units of fm${}^{-1}$ for $^{20}$Ne, $^{48}$Ca, and $^{138}$Ba for the contributions of the proton and neutron EDMs in \ref{['eq:Osp_pot_dp_dn']}. The dashed lines represent the normalized distributions with no radial dependence.