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

TL;DR

This paper resolves the muonic $^{90}$Zr fine-structure anomaly by incorporating relativistic recoil effects, leading to a more precise and accurate measurement of the nuclear charge radius that aligns with muonic spectroscopy.

Contribution

It introduces a rigorous relativistic-recoil treatment in QED calculations, significantly improving the precision of nuclear charge radius measurements from muonic atom spectra.

Findings

Achieved a 6-fold improvement in fit quality for $^{90}$Zr.

Obtained a charge radius consistent with muonic spectroscopy but more precise.

Confirmed the anomaly was due to incomplete QED effects treatment.

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 ($\chi^2 /{\text{DoF}} = 0.995$), in agreement with the previous muonic spectroscopy value, but a factor $6$ more precise, and 3$\sigma$ 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 ($\chi^2 /{\text{DoF}} = 0.88$) 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.