# Calculations of QED Effects with the Dirac Green Function

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Dirac Green Function

#### 2.1. Representation in Terms of Regular and Irregular Solutions

#### 2.2. Finite Basis Set Representations

#### 2.3. Discussion

## 3. General Formulas

## 4. Angular Integration

## 5. Choice of the Integration Contour

## 6. Infrared Divergencies

## 7. Radial Integration

## 8. Magnetically-Perturbed Green Function

## 9. Numerical Calculations

## 10. Summary

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. Relativistic Slater Radial Integral

## Appendix B. Infrared Divergent Integrals

## References

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**Figure 1.**The one-loop self-energy correction. The double line represents the electron propagating in the binding field of the nucleus. The wavy line denotes the virtual photon.

**Figure 2.**The magnetic-vertex self-energy correction. The wavy line terminated by a cross denotes the interaction with an external magnetic field.

**Figure 3.**The double-vertex self-energy correction. The wavy line terminated by a triangle denotes the hyperfine interaction.

**Figure 4.**The poles and the branch cuts of the integrand of the matrix element of the self-energy operator and the integration contour ${C}_{LH}$ in the complex $\omega $ plane. The dashed lines (green) show the branch cuts of the photon propagator. The poles and the branch cuts of the electron propagator are shown by dots and the dashed-dot line (blue). The solid line (red) shows the integration contour ${C}_{LH}$.

**Table 1.**Numerical results for the one-loop self-energy correction for the $2s$ state of hydrogen-like calcium ($Z=20$), for the point nucleus, in terms of the standard scaled function $F(Z\alpha )=\delta E/[(\alpha /\pi )\phantom{\rule{0.166667em}{0ex}}{(Z\alpha )}^{4}/{n}^{3}]$, where $\delta E$ is a contribution to the energy in relativistic units. ${S}_{l}$ denotes the sum of partial-wave expansion ${\sum}_{\left|\kappa \right|=1}^{l}$; $\delta {S}_{l}$ is the increment with respect to the previous line.

l | ${\mathit{S}}_{\mathit{l}}$ | $\mathit{\delta}{\mathit{S}}_{\mathit{l}}$ | |
---|---|---|---|

$\langle {\Sigma}^{(2+)}\rangle $ | 1 | 82.268 19 | |

2 | 85.541 56 | 3.273 37 | |

3 | 86.515 41 | 0.973 85 | |

4 | 86.967 68 | 0.452 26 | |

5 | 87.223 70 | 0.256 02 | |

10 | 87.675 13 | 0.451 44 | |

15 | 87.790 89 | 0.115 75 | |

20 | 87.836 31 | 0.045 42 | |

30 | 87.869 99 | 0.033 68 | |

40 | 87.881 51 | 0.011 52 | |

50 | 87.886 57 | 0.005 06 | |

60 | 87.889 17 | 0.002 61 | |

∞ | 87.894 34 (26) | 0.005 17 (26) | |

$\langle {\Sigma}^{(0+1)}\rangle $ | −84.387 704 | ||

Total | 3.506 64 (26) | ||

P. J. Mohr [75] | 3.506 648 (2) | ||

Refs. [76,77] | 3.506 647 (5) |

**Table 2.**Numerical results for the self-energy correction to the g factor of the $2s$ state of hydrogen-like calcium ($Z=20$), for the point nucleus (in units of ${10}^{-6}$).

l | ${\mathit{S}}_{\mathit{l}}$ | $\mathit{\delta}{\mathit{S}}_{\mathit{l}}$ | |
---|---|---|---|

$\langle {\mathsf{\Lambda}}_{\mathrm{vr}}^{(2+)}\rangle $ | 1 | 36.130 52 | |

2 | 17.563 35 | −18.567 17 | |

3 | 14.605 25 | −2.958 10 | |

4 | 13.586 86 | −1.018 39 | |

5 | 13.115 22 | −0.471 64 | |

10 | 12.489 16 | −0.626 06 | |

15 | 12.379 38 | −0.109 78 | |

20 | 12.343 92 | −0.035 46 | |

25 | 12.328 78 | −0.015 15 | |

30 | 12.321 11 | −0.007 67 | |

35 | 12.316 76 | −0.004 35 | |

∞ | 12.306 43 (50) | −0.010 33 (50) | |

$\langle {\mathsf{\Lambda}}_{\mathrm{vr}}^{(0+1)}\rangle $ | 2237.914 11 | ||

$\langle {\mathsf{\Lambda}}_{\mathrm{ir}}\rangle $ | 75.453 02 | ||

Total | 2325.673 56 (50) | ||

Ref. [14] | 2325.674 (5) |

**Table 3.**Numerical results for the self-energy correction to the nuclear magnetic shielding constant $\sigma $ of the $1s$ state of hydrogen-like calcium ($Z=20$), for the point nucleus, in units of the scaled function $D(Z\alpha )=\delta \sigma /\left[{\alpha}^{2}{(Z\alpha )}^{3}\right]$, where $\delta \sigma $ is a contribution to the shielding constant.

l | ${\mathit{S}}_{\mathit{l}}$ | $\mathit{\delta}{\mathit{S}}_{\mathit{l}}$ | |
---|---|---|---|

$\langle {\mathsf{\Lambda}}_{\mathrm{dvr}}\rangle $ | 1 | −3.409 2 | |

2 | −5.550 9 | −2.141 7 | |

3 | −6.559 8 | −1.008 9 | |

4 | −7.111 6 | −0.551 7 | |

5 | −7.438 5 | −0.327 0 | |

10 | −7.941 8 | −0.503 2 | |

15 | −7.986 1 | −0.044 3 | |

20 | −7.968 3 | 0.017 7 | |

25 | −7.945 0 | 0.023 3 | |

30 | −7.925 5 | 0.019 5 | |

35 | −7.910 4 | 0.015 1 | |

∞ | −7.846 7 (32) | 0.063 7 (32) | |

$\langle {\mathsf{\Lambda}}_{\mathrm{der}}\rangle $ | 7.782 4 | ||

$\langle {\mathsf{\Lambda}}_{\mathrm{vr},\mathrm{Zee}}\rangle $ | 1.760 7 | ||

$\langle {\mathsf{\Lambda}}_{\mathrm{vr},\mathrm{hfs}}\rangle $ | −0.404 9 | ||

$\langle {\mathsf{\Lambda}}_{\mathrm{po}}\rangle $ | −2.217 0 | ||

Total | −0.925 5 (32) |

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Yerokhin, V.A.; Maiorova, A.V.
Calculations of QED Effects with the Dirac Green Function. *Symmetry* **2020**, *12*, 800.
https://doi.org/10.3390/sym12050800

**AMA Style**

Yerokhin VA, Maiorova AV.
Calculations of QED Effects with the Dirac Green Function. *Symmetry*. 2020; 12(5):800.
https://doi.org/10.3390/sym12050800

**Chicago/Turabian Style**

Yerokhin, Vladimir A., and Anna V. Maiorova.
2020. "Calculations of QED Effects with the Dirac Green Function" *Symmetry* 12, no. 5: 800.
https://doi.org/10.3390/sym12050800