Algebraic relations arising from mass polarization in one-photon and two-photon transitions of heliumlike ions*

Decay rates for two-photon transitions can be expressed as a power series in powers of μ/M for the case of finite nuclear mass M and reduced mass μ.  As shown previously [1], gauge invariance implies the existence of algebraic relationships which relate the expansion coefficients for the length and velocity forms.   These expansions provide a sensitive test of agreement between calculations in the length and velocity gauges at any desired order in μ/M.  In the present work, the algebraic relations are derived for arbitrary n-photon electric dipole (nE1) decay rates and tested in two-electron atoms up to the third-order terms powers of μ/M ≈ 10^-4. Both one-photon (singlet and triplet) and two-photon spontaneous decay rates for the heliumlike ions are shown numerically to satisfy the derived algebraic relations within the accuracy of the calculations. 

[1] A. T. Bondy, D. C. Morton, G. W. F. Drake, Phys. Rev. A 102, 052087 (2020).