Validity of r^4ρ(r) weighting for reducing shape sensitivity in Barrett-parameter fitting

Ascertain whether using the r^4ρ(r) weighting in the least-squares fit of the muonic potential difference V_μ^i(r) − V_μ^f(r) to the model B r^k e^{−α r} reduces residual charge-distribution shape sensitivity compared to r^2ρ(r) weighting when extracting Barrett parameters and root-mean-square nuclear charge radii for medium-mass nuclei, specifically the chlorine isotopes 35Cl and 37Cl.

Background

To minimize charge-distribution model dependence in muonic x-ray radius extraction, the Barrett-moment method fits the potential difference generated by the muon in the initial and final states to B r^k e^{−α r} and uses the fitted parameters (k, α) to define a model-independent Barrett radius. The weighting used in the least-squares fit can influence residual shape sensitivity.

Earlier literature claimed that weighting the fit with r^4ρ(r) (instead of r^2ρ(r), which matches the integrand of the Barrett moment) could reduce shape sensitivity. In this paper, the authors adopted r^2ρ(r) weighting and report that they could not reproduce the claimed benefit of r^4ρ(r) weighting, noting identical extracted RMS radii (though with different fitted k and α). Whether r^4ρ(r) weighting genuinely improves shape robustness in medium-mass nuclei remains unresolved.