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 $2^3S

• 2^1S$ transition isotope shifts. These two transitions lead to discrepant results for the nuclear charge radii difference between ${}^4$He and ${}^3$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 $\alpha^6 m^2/M$ 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 ${}^3$He and ${}^4$He from two different atomic isotope shift transitions is confirmed, which calls for verification of experimental transition frequencies.