Radiative potential and calculations of QED radiative corrections to energy levels and electromagnetic amplitudes in many-electron atoms

We derive an approximate expression for a “radiative potential” which can be used to calculate QED strong Coulomb field radiative corrections to energies and electric dipole $\left(E1\right)$ transition amplitudes in many-electron atoms with an accuracy of a few percent. The expectation value of the radiative potential gives radiative corrections to the energies. Radiative corrections to $E1$ amplitudes can be expressed in terms of the radiative potential and its energy derivative (the low-energy theorem): the relative magnitude of the radiative potential contribution is $\sim {\alpha }^3{Z}^2\ln (1∕{\alpha }^2{Z}^2)$, while the sum of other QED contributions is $\sim {\alpha }^3(Z_i+1{)}^2$, where $Z_i$ is the ion charge; that is, for neutral atoms $(Z_i=0)$ the radiative potential contribution exceeds other contributions $\sim Z^2$ times. The advantage of the radiative potential method is that it is very simple and can be easily incorporated into many-body theory approaches: relativistic Hartree-Fock, configuration interaction, many-body perturbation theory, etc. As an application we have calculated the radiative corrections to the energy levels and $E1$ amplitudes as well as their contributions ($-0.34\%$ and $+0.43\%$, respectively) to the parity nonconserving (PNC) $6s$-$7s$ amplitude in neutral cesium $(Z=55)$. Combining these results with the QED correction to the weak matrix elements $(-0.41\%)$ we obtain the total QED correction to the PNC $6s$-$7s$ amplitude, $(-0.32\pm 0.03)\%$. The cesium weak charge $Q_W=-72.66\left(29{)}_{\exp }\right(36{)}_{\mathrm{theor}}$ agrees with the Standard Model value $Q_W^{\mathrm{SM}}=-73.19\left(13\right)$, the difference is 0.53(48).

Figure 1

Diagrams for radiative energy shifts $⟨\widehat{L}⟩$ corresponding to (a) the self-energy and (b) vacuum polarization. The double line denotes a bound electron; the wavy line denotes a photon.

Figure 2

Diagrams for radiative corrections to the $E1$ amplitude: (a),(b) self-energy; (c) vertex; (d),(e) vacuum polarization; and (f) external photon polarization operator. Notations are the same as in Fig. 1; the dashed line with the cross denotes the laser driven external photon field.