Bayesian analysis of nuclear polarizability corrections to the Lamb shift of muonic H-atoms and He-ions

TL;DR

This paper uses Bayesian analysis to evaluate nuclear polarizability corrections in muonic atoms, confirming the smallness of the expansion parameter and assessing uncertainties in theoretical calculations of the Lamb shift.

Contribution

It provides a Bayesian assessment of the nuclear polarizability corrections and their uncertainties, validating the use of the $ extit{ ext{eta}}$-expansion in light-nuclear systems.

Findings

The $ extit{ ext{eta}}$ parameter is confirmed to be much less than 1 in these systems.

Truncation uncertainties are consistent with previous estimates for most systems.

Larger uncertainties are found in the $ extit{ ext{mu}}^3$He$^+$ ion.

Abstract

The extraction of nuclear charge radii from spectroscopy experiments in muonic atoms is currently limited by the large uncertainties associated with the theoretical evaluation of the nuclear polarizability effects. To facilitate calculations, these polarizability corrections are conventionally expressed as an expansion in a dimensionless parameter $\eta$, which has been argued in previous literature to hold an approximate value of 0.33 in light-nuclear systems. In this work, we check this claim by doing a Bayesian analysis of the nuclear-polarizability corrections to the Lamb shift in $\mu^2$H and $\mu^3$H atoms and in $\mu^3$He$^+$ and $\mu^4$He$^+$ ions at various orders in the $\eta$-expansion. Our analysis supports the claim that $\eta \ll 1$ in these systems and finds truncation uncertainties that are similar to the previous estimate, the only exception being the truncation uncertainties in the $\mu^3$He$^+$ ion, which are found to be larger.