Evidence Against Nuclear Polarization as Source of Fine-Structure Anomalies in Muonic Atoms

A long-standing problem of fine-structure anomalies in muonic atoms is revisited by considering the splittings $\mathrm{\Delta }2p=E_{2p_{3/2}}-E_{2p_{1/2}}$ in muonic ${}^{90}\mathrm{Zr}$, ${}^{120}\mathrm{Sn}$, and ${}^{208}\mathrm{Pb}$ and $\mathrm{\Delta }3p=E_{3p_{3/2}}-E_{3p_{1/2}}$ in muonic ${}^{208}\mathrm{Pb}$. State-of-the-art techniques from both nuclear and atomic physics are brought together in order to perform the most comprehensive to date calculations of nuclear-polarization energy shifts. Barring the more subtle case of $\mu \text{−}{}^{208}\mathrm{Pb}$, the results suggest that the dominant calculation uncertainty is much smaller than the persisting discrepancies between theory and experiment. We conclude that the resolution to the anomalies is likely to be rooted in refined quantum-electrodynamics corrections or even some other previously unaccounted-for contributions.

Figure 1

Leading-order NP effect: (a) effective self-energy Goldstone diagram with a dressed photon propagator; (b) ladder, (c) cross, and (d) seagull Feynman diagrams. A bound muon is denoted by a double line, while a nucleus is denoted by a single solid line. The shaded blob represents the NP insertion.

Figure 2

Theoretical values of the NP corrections for $\mu \text{−}{}^{90}\mathrm{Zr}$ in relation to the experimentally allowed range for $\mathrm{\Delta }2p^{\mathrm{NP}}$ as a function of $|\mathrm{\Delta }{E}_{1s_{1/2}}^{\mathrm{NP}}|$. The graph was adapted from Ref. [3].

Figure 3

Theoretical values of the NP corrections for $\mu \text{−}{}^{208}\mathrm{Pb}$ in relation to the experimentally allowed ranges for $\mathrm{\Delta }2p^{\mathrm{NP}}$ (a) and $\mathrm{\Delta }3p^{\mathrm{NP}}$ (b) as functions of $|\mathrm{\Delta }{E}_{1s_{1/2}}^{\mathrm{NP}}|$. The graphs were adapted from Refs. [2, 11].