QED calculation of transition probabilities in two-electron ions

An accurate QED calculation of transition probabilities for the low-lying two-electron configurations of multicharged ions is presented. The calculation is performed for the nondegenerate states $\left(1s2s\right){{}^{3}S}_1$, $\left(1s2p_{3/2}\right){{}^{3}P}_2$ ($M1$ and $M2$ transitions, respectively) and for the quasidegenerate states $\left(1s2p\right){{}^{1}P}_1$, $\left(1s2p\right){{}^{3}P}_1$ ($E1$ transitions) decaying to the ground state $\left(1s1s\right){{}^{1}S}_0$. Two-electron ions with nuclear-charge numbers $Z=10–92$ are considered. The line profile approach is employed for the description of the process in multicharged ions within the framework of QED.

Figure 1

Feynman graph, describing the photon scattering on an atomic electron. The wavy lines with the arrows describe the absorption and emission of photons with momenta $k$, $k^{\prime }$ and polarizations $e$, $e^{\prime }$, respectively. The double solid line denotes the electron in the field of the nucleus; $a_0$ corresponds to the ground electron state.

Figure 2

Feynman graph corresponding to the electron self-energy insertion into the electron propagator in Fig. 1. The wavy line denotes the virtual photon. The other notations are the same as in Fig. 1.

Figure 3

The process of double-photon scattering on the artificial lower than ground state $a_{\star }$. Notations are the same as in Fig. 1.

Figure 4

The Feynman graph representing the process of elastic photon scattering on one-electron ion. The double solid line denotes the electron in the field of the nucleus (Furry picture of QED). The boxes with the wavy lines describe the absorption and emission of the photons by electron the ground state. The letter $I$ at the internal electron line implies the resonance approximation, where only the resonant state $I$ remains in the sum over the intermediate states in the electron propagator.

Figure 5

The insertion of the one-loop electron self-energy in the internal electron line in Fig. 4. The wavy line describes the virtual photon. The other notations are the same as in Fig. 4.

Figure 6

The Feynman graph, illustrating Eq. (31). The box with the letter $\Sigma$ inside corresponds to the infinite number of successive insertions of the type (Fig. 5).

Figure 7

The Feynman graph representing the process of the photon emission. The labels $I$ and $F$ correspond to the initial and final states.

Figure 8

The Feynman graphs representing the process of photon emission in the LPA. This graph incorporates the graph in Fig. 7. The upper, central, and down vertices are specified by corresponding subscripts $\left(u\right)$, $\left(c\right)$, and $\left(d\right)$, respectively.

Figure 9

The Feynman graphs representing the single-photon transition in a two-electron ion to lowest order in $\alpha$. The notations are the same as in Figs. 4, 5, 6, 7, 8. $I_i$ $\left(i=1,2\right)$ and $F_i$ $\left(i=1,2\right)$ denote the initial and final states for the two electrons.

Figure 10

The Feynman graphs representing the process of elastic photon scattering on the two-electron ion within the LPA. These graphs incorporate the graphs in Fig. 9. See in the text the notations $a_0,b_0$.

Figure 11

The Feynman graphs representing the process of elastic photon scattering on the two-electron ion. The boxes with the letter $A_0$ inside and wavy lines depict complicated vertices describing the absorption and emission of a photon by the two-electron ion. The photon line in the center denotes the emission of a photon with frequency ${\omega }_0=\left|{\mathit{k}}_0\right|$ corresponding to the transition energy from the initial to the final two-electron state.

Figure 12

The Feynman graph representing the process of elastic photon scattering on the two-electron ion. In addition to Fig. 11 the box with letter $\Xi$ inside and with an external photon indicates the complicated vertex for the emission of a photon ${\omega }_0$ corresponding to the transition from the initial to the final state $\left(I\to F\right)$.

Figure 13

The Feynman graph representing the process of elastic photon scattering on the two-electron ion. This graph includes the one-photon exchange correction and, accordingly, it represents the next order of the perturbation theory (photon exchange correction) compared to the graph (Fig. 11).

Figure 14

The Feynman graphs representing the process of elastic photon scattering on the two-electron ion. In the lowest order of the perturbation theory these graphs reduce to the graphs (Fig. 13).

Figure 15

The Feynman graphs representing the process of elastic photon scattering on the one-electron ion. Multiple insertions of the self-energy operator into the lower outer electron line are made in the framework of the adiabatic approximation. The break in the electron lines denotes the possible multiple insertions.