Model-QED-operator approach to relativistic calculations of the nuclear recoil effect in many-electron atoms and ions

A model-operator approach to fully relativistic calculations of the nuclear recoil effect on energy levels in many-electron atomic systems is worked out. The one-electron part of the model operator for treating the normal mass shift beyond the Breit approximation is represented by a sum of semilocal and nonlocal potentials. The latter ones are constructed by employing the diagonal and off-diagonal matrix elements rigorously evaluated for hydrogenlike ions to first order in the electron-to-nucleus mass ratio. The specific mass shift beyond the lowest-order relativistic approximation has a form which can be directly employed in calculations. The capabilities of the method are probed by comparison of its predictions with the results of ab initio QED calculations. The proposed operator can be easily incorporated into any relativistic calculation based on the Dirac-Coulomb-Breit Hamiltonian.

Figure 1

One-electron nuclear recoil diagrams: the (a) Coulomb; (b), (c) one-transverse; and (d) two-transverse contributions. See the text and Ref. [26] for the description of the Feynman rules.

Figure 2

Two-electron nuclear recoil diagrams: the (a) Coulomb; (b), (c) one-transverse; and (d) two-transverse contributions.

Figure 3

Self-energy diagram.

Figure 4

Vacuum-polarization diagram.

Figure 5

One-photon exchange diagram.

Figure 6

The poles and the branch cuts of the integrand in the operator $\mathrm{P}\left({\varepsilon }_x\right)$ for the one- and two-transverse-photon contributions. Integration contour: the original one oriented along the real axis (left panel); the rotated to the imaginary one (right panel).

Figure 7

The poles and the branch cuts of the integrand in the operator $\mathrm{P}\left({\varepsilon }_x\right)$ for the one- and two-transverse-photon contributions. The integration contour in the complex plane is chosen to avoid all the singularities of the integrand.