Higher-order recoil corrections for singlet states of the helium atom

We investigate the finite nuclear mass corrections in the helium atom in order to resolve a significant disagreement between the $2{}^{3}S-2{}^{3}P$ and $2{}^{3}S-2{}^{1}S$ transition isotope shifts. These two transitions lead to discrepant results for the nuclear charge radii difference between ${}^4\mathrm{He}$ and ${}^3\mathrm{He}$. The accurate treatment of the finite nuclear mass effects is quite complicated and requires the use of the quantum field theoretical approach. We derive the ${\alpha }^6{m}^2/M$ correction with the help of nonrelativistic QED and dimensional regularization of the three-body Coulombic system and present accurate numerical results for low-lying states. The previously reported $4\sigma$ discrepancy in the nuclear charge radius difference between ${}^3\mathrm{He}$ and ${}^4\mathrm{He}$ from two different atomic isotope shift transitions is confirmed, which calls for verification of experimental transition frequencies.