Testing fundamental interactions on the helium atom

We critically examine the current status of theoretical calculations of the energies, the fine structure, and the isotope shift of the lowest-lying states of helium, searching for unresolved discrepancies with experiments. Calculations are performed within the quantum electrodynamics expansion in powers of the fine structure constant $\alpha$ and the electron-to-nucleus mass ratio $m/M$. For energies, theoretical results are complete through orders ${\alpha }^{6}m$ and ${\alpha }^6{m}^2/M$, with the resulting accuracy ranging from 0.5 to 2 MHz for the $n=2$ states. The fine-structure splitting of the $2{}^{3}P$ state is predicted with a much better accuracy, 1.7 kHz, as a consequence of a calculation of the next-order ${\alpha }^{7}m$ effect. An excellent agreement of the theoretical predictions with the recent measurements of the fine structure provides one of the best tests of the bound-state QED in few-electron systems. The isotope shift between ${}^3\mathrm{He}$ and ${}^4\mathrm{He}$ is treated with a subkilohertz accuracy, which allows for a high-precision determination of the differences of the nuclear charge radii $\delta r^2$. Several such determinations, however, yield results that are in a $4\sigma$ disagreement with each other, which remains unexplained. Apart from this, we find no significant discrepancies between theory and experiment for the helium atom. A further calculation of the yet unknown ${\alpha }^{7}m$ correction to energy levels will provide a sensitive test of universality in electromagnetic interactions of leptons by comparison of nuclear charge radii obtained by the helium and muonic helium spectroscopy.

Figure 1

Comparison of the theoretical and experimental results for the $2{}^{3}P_0$–$2{}^{3}P_2$ and $2{}^{3}P_1$–$2{}^{3}P_2$ intervals of the helium fine structure based on figure from Ref. [46]. Abbreviations are as follows: 2017ZH, Ref. [46]; 2015FE, Ref. [47]; 2010SM, Ref. [48]; 2009BO, Ref. [49]; 2005ZE, Ref. [50]; 2001GE, Ref. [52]; 2000CA, Ref. [53]; and 2010PA, Ref. [22].