Relativistic recoil as a key to the fine-structure puzzle in muonic $^{90}\text{Zr}$

TL;DR

The paper addresses the long-standing muonic fine-structure anomaly in muonic Zr-90 and Sn-120 by performing fully relativistic QED calculations with a finite-size nucleus and a rigorous treatment of the relativistic recoil. It fits ab initio QED corrections to precision muonic transition data using a two-parameter Fermi charge distribution, from which the rms radius is extracted. For Zr-90 the rms radius is $4.2732(7)$ fm, and for Sn-120 it is $4.6518(34)$ fm, with a good fit indicating the relativistic recoil resolves the anomaly. The results indicate underestimation of uncertainties in many published radii and highlight the need for higher transitions and more realistic charge distributions beyond the 2pF model in heavier nuclei. Overall, the work provides a rigorous QED-based resolution to the muonic fine-structure puzzle and establishes a robust muonic benchmark for nuclear radii and precision tests of QED.

Abstract

The long-standing fine-structure anomaly in muonic $^{90}$Zr is resolved through a rigorous treatment of the relativistic-recoil effect. From a fit of ab initio QED calculations of the muonic $^{90}$Zr spectrum to precision measurements performed four decades ago, we extract a significantly more precise root-mean-square (rms) charge radius with 6-fold improvement in quality of the fit. A 2-parameter Fermi (2pF) distribution is assumed to model the nuclear charge density and yields a best-fit value of rms charge radius of $r_\text{rms}[^{90}\text{Zr}]=4.2732(7)$ fm ($χ^2 /{\text{DoF}} = 0.995$), in agreement with the previous muonic spectroscopy value, but a factor $6$ more precise, and 3$σ$ larger than the accepted literature value. Additionally, the same analysis has been performed for $^{120}$Sn, where the extracted value of $r_\text{rms}[^{120}\text{Sn}]=4.6518(34)$ fm ($χ^2 /{\text{DoF}} = 0.88$) is consistent with the accepted value. These results confirm our assumption that the muonic fine-structure puzzle arose from an incomplete treatment of QED effects and their uncertainties.

Figures (4)

• Figure 1: The figure depicts the energy shift from RRecoil as calculated in Yerokhin2023recoil. The plots show a) the lower order correction, b) the higher order corrections, and c) the total RRecoil, see also Eqs. \ref{['eq:recoil_total']}-\ref{['eq:recoil_lo']}. The orange (blue) curve shows the correction for the $2p_{1/2}$ ($2p_{3/2}$) state. It is evident, that the hierarchy for high $Z$ elements, like Pb, is inverted compared to low and moderate $Z$ elements like Zr and Sn.
• Figure 2: This plot shows the various NP models for $^{90}$Zr considered for the ensemble average. Each model predicts a set of NP corrections to the various transitions. All those are plotted against the mean value here.
• Figure 3: The fit of the spectrum of $\mu-^{90}$Zr to a 2pF distribution. The colored regions show the $5\sigma$ likelihood for the 2pF parameters and the dotted contours highlight the $5$, $3$, and $1\sigma$ steps. Each colour corresponds to a fit of different groups of transitions: purple only fits Run I, green corresponds to Run II, blue to Run III, and orange is the combined fit of all the transitions. The insets show the resulting residual transition energies in the same colours, with measured transitions in darker shade and resulting fine-structure splittings in lighter colour. The dashed lines correspond to curves of constant radius. The coloured circles indicate the best-fit parameters and the red circle the final values for the overall fit. Note that the best-fit value for Run I lies outside of the plotted parameter range, as indicated by the red arrow.
• Figure 4: The fit of the spectrum of $\mu-^{120}$Sn to a 2pF distribution. The colored regions show the $5\sigma$ likelihood for the 2pF parameters and the dotted contours highlight the $5$, $3$, and $1\sigma$ steps. Each colour corresponds to a fit of different groups of transitions: purple only fits Run I, green corresponds to Run II, blue to Run III, and orange is the combined fit of all the transitions. The insets show the resulting residual transition energies in the same colours, with measured transitions in darker shade and resulting fine-structure splittings in lighter colour. The dashed lines correspond to curves of constant radius. The coloured circles indicate the best-fit parameters and the red circle the final values for the overall fit. Note that the best-fit value for Run I lies outside of the plotted parameter range, as indicated by the red arrow.