We apply point-particle effective field theory to electronic and muonic ${}^4\mathrm{He}^{+}$ ions, and use it to identify linear combinations of spectroscopic measurements for which the theoretical uncertainties are much smaller than for any particular energy levels. The error is reduced because these combinations are independent of all short-range physics effects up to a given order in the expansion in the small parameters $R/a_B$ and $Z\alpha$ (where $R$ and $a_B$ are the ion's nuclear and Bohr radii). In particular, the theory error is not limited by the precision with which nuclear matrix elements can be computed, or compromised, by the existence of any novel short-range interactions, should these exist. These combinations of ${}^4\mathrm{He}$ measurements therefore provide particularly precise tests of quantum electrodynamics. The restriction to ${}^4\mathrm{He}$ arises because our analysis assumes a spherically symmetric nucleus, but the argument used is more general and extendable to both nuclei with spin, and to higher orders in $R/a_B$ and $Z\alpha$.

• Received 19 September 2017
• Revised 17 September 2018

DOI:https://doi.org/10.1103/PhysRevA.98.052510