Effective Field Theory Perspective On King Non-linearity

TL;DR

The paper develops an effective field theory framework that systematically separates nuclear (SM) and atomic (QED) physics in King non-linearity of isotope shifts. By matching the Standard Model onto scalar NRQED in the infinite-nuclear-mass limit and then onto quantum-mechanical potentials, it expresses nuclear effects through a small set of Wilson coefficients tied to moments ⟨r^2⟩, ⟨r^4⟩ and to α_E, β_M, with the long-range piece arising from nuclear polarizability as a 1/r^4 potential. It shows that the controversial ⟨r^2⟩^2 contribution originates only at second order in quantum-mechanical perturbation theory and requires the full spectrum, while polarizability provides the leading SM source of King NL for current precision. The formalism clarifies which SM corrections affect King NL in hydrogen-like atoms, enabling precision inference about heavy nuclei and setting a clean stage to search for fifth forces in isotope-shift data.

Abstract

Precision spectroscopic measurements of isotope shifts have recently reached a high level of accuracy. Tests of King non-linearity (NL) along isotope chains have been proposed as a tool to search for fifth-force mediators. At the same time, these tests can potentially teach us about the structure of heavy nuclei at unprecedented precision, where King NL has already been observed in several systems. A robust interpretation of the existing data, however, is hampered by incomplete control over the Standard Model (SM) contributions. We develop a systematic effective field theory framework, matching the SM onto scalar non-relativistic QED in the infinite nuclear mass limit and then onto quantum-mechanical potentials. This approach organizes all nuclear effects into a small set of Wilson coefficients and cleanly separates short- and long-distance physics. We show that the commonly used treatment of the $\langle r^2\rangle^2$ term needs to be reconsidered, as it arises only at second-order in perturbation theory, and we derive the long-range $1/r^4$ potential from nuclear polarizability. Applying the framework to hydrogen-like systems, we provide a transparent classification of SM sources of King NL relevant for current and future isotope-shift experiments. The formalism can be applied to learn about the shape of the heavy scalar nuclei at a higher level of precision and detail than what was previously attainable.

Figures (7)

• Figure 1: NRQED Feynman diagrams for one photon exchange contributing to the QM potential $V(r)$. The dot, cross, and diamond vertices represent the minimal coupling (from $D_t$), and from $\langle r^2 \rangle$, and $\langle r^4 \rangle$ terms, respectively.
• Figure 2: Contribution from electric and magnetic polarizabilities (represented as a shaded circle) to the energy shift in a hydrogen-like atom.
• Figure 3: One-photon interaction with a scalar vertex represented as shaded circle.
• Figure 4: Two-photon interaction with a scalar vertex represented as shaded "blob".
• Figure 5: Two-photon exchange with two minimal $A^0$ couplings ($\propto Z^2$). In pNRQED these graphs correspond to the iteration of the Coulomb potential.