QED theory of the normal mass shift in few-electron atoms

The electron-electron interaction correction of first order in $1/Z$ to the one-electron part of the nuclear recoil effect on binding energies in atoms and ions is considered within the framework of the rigorous quantum electrodynamics (QED) approach. The calculations to all orders in $\alpha Z$ are performed for the $1s^2$ state in heliumlike ions and the $1s^{2}2s$ and $1s^{2}2p_{1/2}$ states in lithiumlike ions in the range $Z=5–100$. The results obtained are compared with the Breit-approximation values. The performed calculations complete a systematic treatment of the QED nuclear recoil effect up to the first order in $1/Z$. The correction obtained is combined with the previously studied two-electron part as well as the higher-order electron-correlation corrections evaluated within the Breit approximation to get the total theoretical predictions for the mass shifts.

Figure 1

One-electron nuclear recoil diagrams to zeroth order in $1/Z$: the Coulomb (a), one-transverse (b) and (c), and two-transverse (d) contributions. See the text and Ref. [21] for the description of the Feynman rules.

Figure 2

The second-order diagrams describing the electron-electron interaction correction to the one-electron two-transverse-photon contribution to the nuclear recoil effect. The analogous diagrams with the Coulomb and one-transverse photon recoil interactions have to be taken into account as well. See the text and Ref. [21] for the description of the diagram technique.

Figure 3

The one-photon exchange diagram which contributes to the second “disconnected” term in Eq. (26) along with the first-order diagrams in Fig. 1.

Figure 4

The interelectronic-interaction correction of first order in $1/Z$ to the one-electron part of the nuclear recoil effect on the binding energy of the $1s^2$ state expressed in terms of the dimensionless function $B\left(\alpha Z\right)$ defined by Eq. (39). The solid line represents the results of the QED calculations to all orders in $\alpha Z$, whereas the dashed line corresponds to the calculations based on the normal mass shift (NMS) operator given by Eq. (1).

Figure 5

The interelectronic-interaction correction of first order in $1/Z$ to the one-electron part of the nuclear recoil effect on the $2p_{1/2}\text{–}2s$ transition energy in Li-like ions expressed in terms of the dimensionless function $B\left(\alpha Z\right)$ defined by Eq. (39). Notations are the same as in Fig. 4.

Figure 6

Two-electron nuclear recoil diagrams to zeroth order in $1/Z$: the Coulomb (a), one-transverse (b) and (c), and two-transverse (d) contributions.

Figure 7

The interelectronic-interaction correction to the two-electron two-transverse-photon contribution to the nuclear recoil effect. The analogous diagrams with the Coulomb and one-transverse-photon recoil interactions have to be taken into account as well.

Figure 8

The nuclear recoil effect on the binding energy of the $1s^2$ state to first order in $1/Z$. The solid lines stand for the results of the QED calculations to all orders in $\alpha Z$, while the dashed lines correspond to the calculations based on the mass shift (MS) operator given by Eq. (3). The contributions of zeroth order in $1/Z$, $P_{\left[0\right]}\left(\alpha Z\right)=A\left(\alpha Z\right)$, and the sums of zeroth and first orders in $1/Z$, $P_{[0,1]}(\alpha Z,Z)=A\left(\alpha Z\right)+B\left(\alpha Z\right)/Z$, are shown with blue (circles) and red (squares) lines, respectively.

Figure 9

The nuclear recoil effect on the $2p_{1/2}\text{–}2s$ transition energy in Li-like ions to first order in $1/Z$. Notations are the same as in Fig. 8.

Figure 10

(a) The nuclear recoil effect on the $2p_{1/2}\text{–}2s$ transition energy in Li-like ions in terms of the function $P(\alpha Z,Z)$ defined by Eq. (41). The blue dashed line corresponds to the calculation performed by means of the mass shift operator (3) to all order in $1/Z$. The green dashed-dotted line includes the nontrivial QED contribution within the independent-electron approximation (${\mathrm{QED}}_{\left[0\right]}$). The violet dotted line accounts for the QED correction to first order in $1/Z$ (${\mathrm{QED}}_{[0,1]}$). The red solid line takes into account additionally the finite nuclear size (fns) correction $\delta A_{\mathrm{Breit}}^{\mathrm{fns},1\mathrm{el}}\left(\alpha Z\right)$. The corresponding data from Ref. [11] are shown with magenta circles and black diamonds. The error bars are not indicated. (b) The zoomed region for $Z=68–94$. The uncertainties of the present calculation and Ref. [11] are shown.