Linking Electromagnetic Moments to Nuclear Interactions with a Global Physics-Driven Machine-Learning Emulator

Abstract

Understanding how specific components of the nuclear interaction shape observable properties of atomic nuclei remains a central challenge in nuclear structure research. While previous studies have focused on bulk observables such as nuclear energies and charge radii, it is unclear how distinct operator components of nuclear interactions impact complementary observables such as nuclear electromagnetic moments. Here, we develop a global, physics-constrained emulator to establish a quantitative link between electromagnetic moments and components of chiral nuclear forces. Unlike traditional sensitivity analyses that vary low-energy constants independently, we quantify parameter contributions while accounting for correlations within the physically supported parameter manifold. We show that, unlike bulk observables, electromagnetic moments probe complementary spin and isospin sectors of the interaction and exhibit a pronounced isotope-dependent sensitivity. These developments enable a quantitative assessment of the importance of prospective measurements, providing predictions with quantified uncertainties for observables that may be beyond the current experimental reach.

Figures (21)

• Figure 1: Architecture of the Fidelity-Resolved Affine Matrix Emulator (FRAME). The emulator maps the LEC vector $\alpha$, nuclear identifiers $(Z, N)$, and model-space fidelity $f$ to nuclear energies and electromagnetic observables. Top: The nuclear identifiers are embedded into a continuous latent representation, while the LECs enter the operator core directly. Middle: For each nucleus and fidelity, the effective Hamiltonian is constructed as an affine combination $M_k(\alpha, N, Z, f) = P_0 + \sum_i \alpha_i P_i + \sum_j h_j P_j$. Here, $\{P_i\}$ are learned Hermitian basis matrices, preserving the affine dependence on the LECs dictated by the chiral interaction. Bottom left: Predictions at successive fidelities $(e_{max})$ are controlled hierarchically by increasing the size of the model space. Bottom right: The assembled matrix $M_T$ is spectrally decomposed to yield energies and eigenvectors $v_j$. Observable expectation values $O_q$ are then computed as bilinears $\langle v_j|S_q|v_j\rangle$, where $S_q$ is a Hermitian operator constructed with the same architecture.
• Figure 2: Contrasting information content of bulk and electromagnetic observables. Posterior-weighted sensitivity analysis ($S_T$) for (a) binding energy ($E$), (b) charge radius ($R_{ch}$), (c) magnetic ($\mu$), and (d) electric ($Q$) quadrupole moments across the calcium chain ($^{37}$Ca--$^{55}$Ca). Red and blue bars indicate average positive and negative LEC contributions to the observable value, respectively. While bulk observables (top row) exhibit an LEC importance share independent of the neutron number and dominated by specific scalar couplings, EM moments (bottom row) display a scattered and isotope-dependent importance share with high volatility, indicating the importance of spin-isospin interactions that change with neutron number. A zoomed view of the importance of the subleading two-pion exchange LEC $c_2$ across the isotopic chain is shown in panel a).
• Figure 3: Sector resolved importance for $\mu$. Posterior-integrated importance aggregated by physical sector for $\mu$ across the calcium chain. The shaded regions represent the standard deviation within each physical sector, not statistical uncertainty. Two cartoon representations of a nearly single neutron hole case ($^{39}$Ca) and a single particle neutron limit for ($^{49}$Ca).
• Figure 4: Change in the correlation matrix of chiral LECs induced by the magnetic-moment update. We show $\Delta\rho_{ij} = \rho^\text{post}_{ij} - \rho^\text{prior}_{ij}$. Red arcs identify parameter pairs where magnetic moments enforce tighter constraints (locking), while blue arcs indicate the resolution of prior degeneracies (decoupling). Only statistically significant ($\sigma>1.5$) differences are shown. Node sizes are proportional to the marginal posterior shrinkage ($1-\sigma_{\text{post}}/\sigma_{\text{prior}}$).
• Figure 5: Global evolution of EM observables in Ca. We compare experimental values (black squares) garcia2016unexpectedlygarcia2015groundMill19 against the propagated posterior of the likelihood $\mathcal{L}_{2-4 + \mu\text{Ca}}$ (purple circles, 68% Degree of Belief) and reference interactions $\Delta$NNLO$_{\rm{{GO}}}$(394) (diamonds) jiang2020, 1.8/2.0 (EM)(triangles) PhysRevC.83.031301. Top: Absolute charge radii $R_{ch}$. Middle: Magnetic dipole moments $\mu$. Bottom: Electric quadrupole moments $Q$.