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Introduction to Bound-State Quantum Electrodynamics | SpringerLink

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Introduction to Bound-State Quantum Electrodynamics

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Handbook of Relativistic Quantum Chemistry

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Abstract

In this chapter we describe the fundamental aspects of bound-state quantum electrodynamics (BSQED). We recall the principal features of the Dirac equation. Then we describe quantum electrodynamics as a field theory. We provide the basic elements about representations, evolution operators and the S matrix. We then proceed to describe perturbation expansion of the S-matrix, and its relations with bound-state energies. We express this expansion in terms of Feynman diagrams. Finally we illustrate on practical examples the concepts of regularization, using the method of Pauli and Villars, and renormalization in coordinate space. We describe in detail the practical ways of doing the calculations, using self-energy, vacuum polarization, self-energy screening and QED corrections to the ladder approximation. Finally we show the quality of the agreement between BSQED and experiment by showing comparison for two- and three-electron ions transitions.

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References

  1. Jauch JM, Rohrlich F (1976) The theory of photons and electrons: the relativistic quantum field theory of charged particles with spin one-half. Texts and monographs in physics, 2nd edn. Springer, New York

    Book  Google Scholar 

  2. Indelicato P, Shabaev VM, Volotka AV (2004) Interelectronic-interaction effect on the transition probability in high-Z He-like ions. Phys Rev A 69(6):062506. http://link.aps.org/abstract/PRA/v69/e062506

    Article  CAS  Google Scholar 

  3. Dirac PAM (1928) The quantum theory of the electron. Proc R Soc Lond Ser A 117:610–624

    Article  Google Scholar 

  4. Yerokhin VA, Shabaev VM (2015) Nuclear recoil effect in the lamb shift of light hydrogenlike atoms. J Phys Chem Ref. Data 44(3):033103. http://scitation.aip.org/content/aip/journal/jpcrd/44/3/10.1063/1.4927487

  5. Artemyev AN, Shabaev VM, Yerokhin VA, Plunien G, Soff G (2005) QED calculation of the n = 1 and n = 2 energy levels in He-like ions. Phys Rev A 71(6):062104. http://link.aps.org/abstract/PRA/v71/e062104

    Article  CAS  Google Scholar 

  6. Yerokhin VA, Artemyev AN, Shabaev VM (2007) QED treatment of electron correlation in Li-like ions. Phys Rev A 75:062501. http://journals.aps.org/pra/abstract/10.1103/PhysRevA.75.062501

    Article  CAS  Google Scholar 

  7. Yerokhin VA, Surzhykov A (2012) Relativistic configuration-interaction calculation of energy levels of core-excited states in lithiumlike ions: argon through krypton. Phys Rev A 86(4):042507. http://link.aps.org/doi/10.1103/PhysRevA.86.042507

    Article  CAS  Google Scholar 

  8. Malyshev AV, Volotka AV, Glazov DA, Tupitsyn II, Shabaev VM, Plunien G (2014) QED calculation of the ground-state energy of berylliumlike ions. Phys Rev A 90(6):062517. http://link.aps.org/doi/10.1103/PhysRevA.90.062517

    Article  CAS  Google Scholar 

  9. Yerokhin VA, Surzhykov A, Fritzsche S (2014) Relativistic configuration-interaction calculation of Ka transition energies in beryllium-like iron. Phys Rev A 90(2):022509. http://link.aps.org/doi/10.1103/PhysRevA.90.022509

    Article  CAS  Google Scholar 

  10. Yerokhin VA, Surzhykov A, Fritzsche S (2015) Relativistic configuration-interaction calculation of K? transition energies in beryllium-like argon. Phys Scr 90(5):054003. http://stacks.iop.org/1402-4896/90/i=5/a=054003

    Article  CAS  Google Scholar 

  11. Artemyev AN, Shabaev VM, Tupitsyn II, Plunien G, Surzhykov A, Fritzsche S (2013) Ab initio calculations of the 2p3/2-2p1/2 fine-structure splitting in boronlike ions. Phys Rev A 88(3):032518. http://link.aps.org/doi/10.1103/PhysRevA.88.032518

    Article  CAS  Google Scholar 

  12. Dyall KG, Grant IP, Johnson CT, Parpia FA, Plummer EP (1989) GRASP: a general-purpose relativistic atomic structure program. Comput Phys Commun 55:425

    Article  CAS  Google Scholar 

  13. Indelicato P, Gorceix O, Desclaux JP (1987) MCDF studies of two electron ions II: radiative corrections and comparison with experiment. J Phys B At Mol Opt Phys 20(4):651. http://dx.doi.org/10.1088/0022-3700/20/4/007

    Article  CAS  Google Scholar 

  14. Indelicato P, Desclaux JP (1990) Multiconfiguration Dirac-Fock calculations of transition energies with QED corrections in three-electron ions. Phys Rev A 42(9):5139. http://journals.aps.org/pra/abstract/10.1103/PhysRevA.42.5139

    Article  CAS  Google Scholar 

  15. Cheng KT, Johnson WR, Sapirstein J (1993) Lamb-shift calculations for non-Coulomb potentials. Phys Rev A 47(3):1817. http://link.aps.org/doi/10.1103/PhysRevA.47.1817

    Article  CAS  Google Scholar 

  16. Cheng KT, Chen H, Johnson WR, Sapirstein J (2008) High-precision relativistic atomic structure calculations and the EBIT: Tests of quantum electrodynamics in highly charged ions. Can J Phys 86(1):33

    Article  CAS  Google Scholar 

  17. Shabaev VM, Tupitsyn II, Yerokhin VA (2013) Model operator approach to the Lamb shift calculations in relativistic many-electron atom. Phys Rev A 88(1):012513. http://link.aps.org/doi/10.1103/PhysRevA.88.012513

    Article  CAS  Google Scholar 

  18. Shabaev VM, Tupitsyn II, Yerokhin VA (2015) QEDMOD: Fortran program for calculating the model Lamb-shift operator. Comput Phys Commun 189:175. http://www.sciencedirect.com/science/article/pii/S0010465514004081

    Article  CAS  Google Scholar 

  19. Pauli W (1941) Relativistic field theories of elementary particles. Rev Mod Phys 13(3):203

    Article  Google Scholar 

  20. Foldy LL (1952) The electromagnetic properties of Dirac particles. Phys Rev 87(5):688. http://journals.aps.org/pr/abstract/10.1103/PhysRev.87.688

    Article  CAS  Google Scholar 

  21. Cheng KT, Childs WJ (1985) Ab initio calculation of 4fn 6s2 hyperfine structure in neutral rare-earth atoms. Phys Rev A 31(5):2775

    Article  CAS  Google Scholar 

  22. Boucard S, Indelicato P (2000) Relativistic many-body and QED effects on the hyperfine structure of lithium-like ions. Eur Phys J D 8(1):59

    Article  CAS  Google Scholar 

  23. Martin PC, Glauber RJ (1958) Relativistic theory of radiative orbital electron capture. Phys Rev 109(4):1307

    Article  CAS  Google Scholar 

  24. Rose ME (1961) Relativistic electron theory. Wiley, New York

    Google Scholar 

  25. Baym G (1969) Lectures in quantum mechanics. Benjamin, Cumming

    Google Scholar 

  26. Johnson MH, Lippmann BA (1950) Relativistic keppler problem. Phys Rev (A) 78:329. http://journals.aps.org/pr/abstract/10.1103/PhysRev.78.313

    Google Scholar 

  27. Indelicato P, Mohr PJ (1992) Coordinate space approach to the one-electron self-energy. Phys Rev A 46(1):172

    Article  Google Scholar 

  28. Mohr PJ, Kim YK (1992) Self-energy of excited states in a strong Coulomb field. Phys Rev A 45(5):2727. http://link.aps.org/doi/10.1103/PhysRevA.45.2727

    Article  Google Scholar 

  29. Mohr PJ (1974) Self-energy radiative corrections in hydrogen-like systems. Ann Phys 88:26

    Article  CAS  Google Scholar 

  30. Mohr PJ (1982) Self-energy of the n=2 states in a strong Coulomb field. Phys Rev A 26(5):2338

    Article  CAS  Google Scholar 

  31. Lamb WE, Retherford RC (1947) Fine structure of the hydrogen atom by a microwave method. Phys Rev 72(3):241. http://link.aps.org/doi/10.1103/PhysRev.72.241

    Article  CAS  Google Scholar 

  32. Lamb WE Jr, Retherford RC (1950) Fine structure of the hydrogen atom. Part I. Phys Rev 79(4):549. http://journals.aps.org/pr/abstract/10.1103/PhysRev.79.549

    Google Scholar 

  33. Kusch P, Foley HM (1948) Phys. The magnetic moment of the electron. Rev. 74(3):250

    Google Scholar 

  34. Bethe HA (1947) The electromagnetic shift of energy levels. Phys Rev 72(4):339. http://journals.aps.org/pr/abstract/10.1103/PhysRev.72.339

    Article  CAS  Google Scholar 

  35. Kroll NM, Lamb JW (1949) On the self-energy a the bound electron. Phys Rev 75(3):388

    Article  CAS  Google Scholar 

  36. Feynman RP (1948) A relativistic cut-off for quantum electrodynamics. Phys Rev 74(10):1430. http://journals.aps.org/pr/abstract/10.1103/PhysRev.74.1430

    Article  Google Scholar 

  37. Feynman RP (1949) The theory of positrons. Phys Rev 76(6):749. http://journals.aps.org/pr/abstract/10.1103/PhysRev.76.749

    Article  CAS  Google Scholar 

  38. Feynman RP (1949) Space-time approach to quantum electrodynamics. Phys Rev 76(6):769. http://journals.aps.org/pr/abstract/10.1103/PhysRev.76.769

    Article  Google Scholar 

  39. Schwinger J (1948) On quantum-electrodynamics and the magnetic moment of the electron. Phys Rev 73:416. http://journals.aps.org/pr/abstract/10.1103/PhysRev.73.416

    Article  CAS  Google Scholar 

  40. Schwinger J (1948) Quantum electrodynamics. I. A covariant formulation. Phys Rev 74(10):1439. http://journals.aps.org/pr/abstract/10.1103/PhysRev.74.1439

    Google Scholar 

  41. Schwinger J (1949) Quantum electrodynamics. II. Vacuum polarization and self-energy. Phys Rev 75(4):651. http://link.aps.org/doi/10.1103/PhysRev.75.651

    Google Scholar 

  42. Schwinger J (1949) Quantum electrodynamics. III. The electromagnetic properties of the electron-radiative corrections to scattering. Phys Rev 76(6):790. http://journals.aps.org/pr/abstract/10.1103/PhysRev.76.790

    Google Scholar 

  43. Dyson FJ (1949) The S matrix in quantum electrodynamics. Phys Rev 75(11):1736. http://journals.aps.org/pr/abstract/10.1103/PhysRev.75.1736

    Article  Google Scholar 

  44. Dyson FJ (1949) The radiation theories of Tomonaga, Schwinger, and Feynman. Phys Rev 75(3):486. http://journals.aps.org/pr/abstract/10.1103/PhysRev.75.486

    Article  Google Scholar 

  45. Brown GE, Ravenhall DE (1951) On the interaction of two electrons. Proc R Soc Lond Ser A 208:552

    Article  CAS  Google Scholar 

  46. Sucher J (1980) Foundation of the relativistic theory of many-electron atoms. Phys Rev A 22(2):348

    Article  CAS  Google Scholar 

  47. Mittleman MH (1981) Theory of relativistic effects on atoms: configuration-space hamiltonian. Phys Rev A 24(3):1167

    Article  CAS  Google Scholar 

  48. Heully JL, Lindgren I, lindroth E, Mårtensson-Pendrill AM (1986) Comment on relativistic wave equation and negative-energy state. Phys Rev A 33(6):4426

    Google Scholar 

  49. Grant IP (1987) On the relativistic electron-electron interaction. J Phys B At Mol Phys 20:L735

    Article  CAS  Google Scholar 

  50. Indelicato P (1995) Projection operators in multiconfiguration Dirac-Fock calculations. Application to the ground state of heliumlike ions. Phys Rev A 51(2):1132

    Google Scholar 

  51. Mohr PJ (1989) Quantum electrodynamics of high-Z few-electron atoms. In: Marrus R (ed) Physics of Highly-Ionized Atoms. NATO ASI Series, vol 201. Plenum, New York, pp 111–141

    Chapter  Google Scholar 

  52. Mohr PJ (1989) Relativistic, quantum electrodynamic, and weak interaction effects in atoms: Santa Barbara, CA 1988. In: Johnson WR, Mohr PJ, Sucher J (eds) Proceeding of the Institute for Theoretical Physics (Santa Barbara). AIP Conference Proceedings, vol 189. American Institute of Physics, New York

    Google Scholar 

  53. Mohr PJ, Plunien G, Soff G (1998) QED corrections in heavy atoms. Phys Rep 293(5&6), 227

    Article  CAS  Google Scholar 

  54. Cohen-Tannoudji C, Diu B, Laloë F (1996) Mécanique quantique. Hermann, Paris

    Google Scholar 

  55. Furry WH (1951) On bound states and scattering in Positrons theory. Phys Rev 81(1):115. http://journals.aps.org/pr/abstract/10.1103/PhysRev.81.115

    Article  Google Scholar 

  56. Mohr PJ (1985) Quantum electrodynamics of high-Z few-electron atoms. Phys Rev A 32(4):1949

    Article  CAS  Google Scholar 

  57. Gell-Mann M, Low F (1951) Bound states in quantum field theory. Phys Rev 84(2):350

    Article  CAS  Google Scholar 

  58. Fetter AL, Walecka JD (1971) Quantum theory of many-particle systems. International series in pure and applied physics. Mc Graw-Hill, New-York

    Google Scholar 

  59. Sucher J (1957) S-matrix formalism for level-shift calculations. Phys Rev 107:1448

    Article  CAS  Google Scholar 

  60. Mohr PJ (1996) QED in one electron systems. In: Drake GWF (ed) Handbook of atomic physics. American Institute of Physics, New-York, p 125

    Google Scholar 

  61. Dyson FJ (1952) Divergence of perturbation theory in quantum electrodynamics. Phys Rev 85(4):631. http://journals.aps.org/pr/abstract/10.1103/PhysRev.85.631

    Article  Google Scholar 

  62. Wick GC (1950) The evaluation of the collision matrix. Phys Rev 80(2):268. http://link.aps.org/doi/10.1103/PhysRev.80.268

    Article  Google Scholar 

  63. Mathematica Edition: Version 10 (2015) Wolfram Research, Inc., Champaign. http://www.wolfram.com

  64. Helton JW, Miller RL, Stankus M (1996) Ncalgebra: a mathematica package for doing noncommuting algebra. http://math.ucsd.edu/~ncalg

  65. Lindgren I, Persson H, Salomonson S, Karasiev V, Labzowsky L, Mitrushenkov A, Tokman M (1993) Second-order QED corrections for few-electron heavy ions: reducible Breit-Coulomb correction and mixed self-energy- vacuum polarization correction. J Phys B: At Mol Opt Phys 26:L503

    Article  CAS  Google Scholar 

  66. Persson H, Lindgren I, Labzowsky LN, Plunien G, Beier T, Soff G (1996) Second-order self-energy-vacuum-polarization contributions to the Lamb shift in highly charged few-electron ions. Phys Rev A 54(4):2805

    Article  CAS  Google Scholar 

  67. Yerokhin VA, Indelicato P, Shabaev VM (2003) Two-loop self-energy correction in high-Z hydrogen-like ions. Phys Rev Lett 91(7):073001

    Article  CAS  Google Scholar 

  68. Yerokhin VA, Indelicato P, Shabaev VM (2003) Evaluation of the two-loop self-energy correction to the ground state energy of H-like ions to all orders in Z? Eur Phys J D 25(3):203

    Article  CAS  Google Scholar 

  69. Yerokhin VA, Indelicato P, Shabaev VM (2005) Two-loop self-energy contribution to the Lamb shift in H-like ions. Phys Rev A 71(4):040101(R). http://link.aps.org/abstract/PRA/v71/e040101

  70. Yerokhin VA, Indelicato P, Shabaev VM (2005) Two-loop self-energy correction in a strong Coulomb nuclear field. J Exp Theor Phys 101(2):280–293

    Article  CAS  Google Scholar 

  71. Yerokhin VA, Indelicato P, Shabaev VM (2006) Nonperturbative calculation of the two-loop Lamb shift in Li-like ions. Phys Rev Lett 97(25):253004. http://link.aps.org/abstract/PRL/v97/e253004

    Article  CAS  Google Scholar 

  72. Yerokhin VA, Indelicato P, Shabaev VM (2007) Two-loop QED corrections in few-electron ions. Can J Phys 85(5):521. http://dx.doi.org/10.1140/epjd/e2006-00064-8

    Article  CAS  Google Scholar 

  73. Yerokhin VA, Indelicato P, Shabaev VM (2008) Two-loop QED corrections with closed fermion loops. Phys Rev A 77(6):062510 (12)

    Google Scholar 

  74. Yerokhin VA (2009) Two-loop self-energy for the ground state of medium-Z hydrogenlike ion. Phys Rev A 80(4):040501. http://link.aps.org/abstract/PRA/v80/e040501

    Article  CAS  Google Scholar 

  75. Yerokhin VA (2010) The two-loop self-energy: diagrams in the coordinate-momentum representation. Eur Phys J D 58(1):57. http://dx.doi.org/10.1140/epjd/e2010-00089-4

    Article  CAS  Google Scholar 

  76. Källén G, Sabry A (1955) Fourth order vacuum polarization. Det Kongelige Danske Videnskabernes Selskab Matematisk-Fysiske Meddelelser 29(17):3

    Google Scholar 

  77. Fullerton LW, Rinker GA (1976) Accurate and efficient methods for the evaluation of vacuum-polarization potentials of order Za and Za2. Phys Rev A 13(3):1283

    Article  Google Scholar 

  78. Artemyev AN, Shabaev VM, Yerokhin VA (1997) Vacuum polarization screening corrections to the ground-state energy of two-electron ions. Phys Rev A 56(5):3529. http://journals.aps.org/pra/abstract/10.1103/PhysRevA.56.3529

    Article  CAS  Google Scholar 

  79. Artemyev AN, Beier T, Plunien G, Shabaev VM, Soff G, Yerokhin VA (2000) Vacuum-polarization screening corrections to the energy levels of heliumlike ions. Phys Rev A 62(2):022116. http://journals.aps.org/pra/abstract/10.1103/PhysRevA.62.022116

    Article  Google Scholar 

  80. Artemyev AN, Beier T, Plunien G, Shabaev VM, Soff G, Yerokhin VA (1999) Vacuum-polarization screening corrections to the energy levels of lithiumlike ions. Phys Rev A 60(1):45. http://journals.aps.org/pra/abstract/10.1103/PhysRevA.60.45

    Article  CAS  Google Scholar 

  81. Indelicato P, Mohr PJ (2001) Coordinate-space approach to the bound-electron self-energy: Self-Energy screening calculation. Phys Rev A 63(4):052507

    Article  CAS  Google Scholar 

  82. Yerokhin VA, Artemyev AN, Beier T, Plunien G, Shabaev VM, Soff G (1999) Two-electron self-energy corrections to the 2p1/2-2s transition energy in Li-like ions. Phys Rev A 60(5):3522. http://journals.aps.org/pra/abstract/10.1103/PhysRevA.60.3522

    Article  CAS  Google Scholar 

  83. Blundell SA, Mohr PJ, Johnson WR, Sapirstein J (1993) Evaluation of two-photon exchange graphs for highly charged heliumlike ions. PRA 48(4):2615. http://journals.aps.org/pra/abstract/10.1103/PhysRevA.48.2615

    Article  CAS  Google Scholar 

  84. Lindgren I, Persson H, Salomonson S, Labzowsky L (1995) Full QED calculations of two-photon exchange for heliumlike systems: analysis in the coulomb and feynman gauge. Phys Rev A 51(2):1167

    Article  CAS  Google Scholar 

  85. Mohr PJ, Sapirstein J (2000) Evaluation of two-photon exchange graphs for excited states of highly charged Heliumlike ions. Phys Rev A 62(5):052501

    Article  Google Scholar 

  86. Yerokhin VA, Artemyev AN, Shabaev VM, Sysak MM, Zherebtsov OM, Soff G (2001) Evaluation of the two-photon exchange graphs for the 2p 1∕2 − 2s transition in Li-like ions. Phys Rev A 64(3):032109. http://link.aps.org/doi/10.1103/PhysRevA.64.032109

    Article  CAS  Google Scholar 

  87. øAsén B, Salomonson S, Lindgren I (2002) Two-photon-exchange QED effects in the 1s2s 1S and 3S states of heliumlike ions. Phys Rev A 65(3):032516 (16)

    Google Scholar 

  88. Artemyev AN, Shabaev VM, Sysak MM, Yerokhin VA, Beier T, Plunien G, Soff G (2003) Evaluation of the two-photon exchange diagrams for the (1s)22p3/2 electron configuration in Li-like ions. Phys Rev A 67(6):062506 (7). http://journals.aps.org/pra/abstract/10.1103/PhysRevA.67.062506

  89. Sapirstein J, Cheng KT (2011) S-matrix calculations of energy levels of the lithium isoelectronic sequence. Phys Rev A 83(1):012504. http://link.aps.org/doi/10.1103/PhysRevA.83.012504

    Article  CAS  Google Scholar 

  90. Sapirstein J, Cheng KT (2015) S-matrix calculations of energy levels of sodiumlike ions. Phys Rev A 91(6):062508. http://link.aps.org/doi/10.1103/PhysRevA.91.062508

    Article  CAS  Google Scholar 

  91. Brown GE, Langer JS, Schaefer GW (1959) Lamb shift of a tightly bound electron: I. method. Proc R Soc Lond Ser A 251:92

    Google Scholar 

  92. Desiderio AM, Johnson WR (1971) Lamb Shift and binding energies of K electron in heavy atoms. Phys Rev A 3(4):1267

    Article  Google Scholar 

  93. Snyderman NJ (1991) Electron radiatives self-energy of highly stripped heavy atoms. Ann Phys 211:43

    Article  CAS  Google Scholar 

  94. Blundell SA, Snyderman NJ (1991) Basis set approach to calculating the radiative self-energy in highly ionized atoms. Phys Rev A 44(3):R1427

    Article  Google Scholar 

  95. Cheng KT, Johnson WR, Sapirstein J (1991) Screened Lamb shift calculations for lithiumlike uranium, sodiumlike platinum and copperlike gold. Phys Rev Lett 66(23):2960

    Article  CAS  Google Scholar 

  96. Le Bigot EO, Indelicato P, Mohr PJ (2001) QED self-energy contribution to highly-excited atomic states. Phys Rev A 64(5):052508 (14)

    Google Scholar 

  97. Jentschura UD, Mohr PJ, Soff G (1999) Calculation of the electron self-energy for low nuclear charge. Phys Rev Lett 82(1):53

    Article  CAS  Google Scholar 

  98. Jentschura UD, Mohr PJ, Soff G (2001) Electron self-energy for the K and L shells at low nuclear charge. Phys Rev A 63(4):042512

    Article  CAS  Google Scholar 

  99. Jentschura UD, Mohr PJ, Soff G, Weniger EJ (1999) Convergence acceleration via combined nonlinear-condensation transformations. Comput Phys Commun 116(1):28. http://www.sciencedirect.com/science/article/B6TJ5-3W3FP53-3/2/42586ad49205db06e1b5c90230bd7150

    Article  CAS  Google Scholar 

  100. Indelicato P, Mohr PJ (1998) Coordinate-space approach to the bound-electron self energy: Coulomb field calculation. Phys Rev A 58(1):165

    Article  CAS  Google Scholar 

  101. Indelicato P, Mohr PJ, Sapirstein J (2014) Coordinate-space approach to vacuum polarization. Phys Rev A 89(4):042121 http://link.aps.org/doi/10.1103/PhysRevA.89.042121

    Article  CAS  Google Scholar 

  102. Fried HM, Yennie DR (1958) New techniques in the Lamb shift calculation. Phys Rev 112(4):1391. http://link.aps.org/doi/10.1103/PhysRev.112.1391

    Article  CAS  Google Scholar 

  103. Coquereaux R (1980) Renormalization schemes in QED. Ann Phys 125(2):401. http://www.sciencedirect.com/science/article/pii/0003491680901396

    Article  CAS  Google Scholar 

  104. Narison S (1982) Techniques of dimensional regularization and the two-point functions of QCD and QED. Phys Rep 84(4):263. http://www.sciencedirect.com/science/article/pii/0370157382900230

    Article  CAS  Google Scholar 

  105. Blundell SA, Cheng KT, Sapirstein J (1997) Radiative corrections in atomic physics in the presence of perturbing potentials. Phys Rev A 55(3):1857. http://journals.aps.org/pra/abstract/10.1103/PhysRevA.55.1857

    Article  CAS  Google Scholar 

  106. Pauli W, Villars F (1949) On the invariant regularization in relativistic quantum theory. Rev Mod Phys 21(3):434. http://journals.aps.org/rmp/abstract/10.1103/RevModPhys.21.434

    Article  Google Scholar 

  107. Mohr PJ, Sapirstein JR (1985) Quantum electrodynamic energy shifts of quarks bound in a cavity. Phys Rev Lett 54(6):514

    Article  CAS  Google Scholar 

  108. Mohr PJ (1987) In: Greiner W (ed) Physics of strong fields. Plenum, New York, p 17

    Chapter  Google Scholar 

  109. Wichmann EH, Kroll NM (1956) Vacuum polarization in a strong Coulomb field. Phys Rev 101(2):843

    Article  Google Scholar 

  110. Uehling EA (1935) Polarization effects in the positron theory. Phys Rev 48(1):55

    Article  CAS  Google Scholar 

  111. Serber R (1935) Linear modifications in the Maxwell field equations. Phys Rev 48(1):49. http://link.aps.org/doi/10.1103/PhysRev.48.49

    Article  CAS  Google Scholar 

  112. Huang KN (1976) Calculation of the vacuum-polarization potential. Phys Rev A 14(4):1311

    Article  CAS  Google Scholar 

  113. Klarsfeld S (1977) Analytical expression for the evaluation of vacuum-polarization potentials in muonic atoms. Phys Lett 66B(1), 86

    Article  CAS  Google Scholar 

  114. Blomqvist J (1972) Vacuum polarization in exotic atoms. Nucl Phys B 48:95

    Article  CAS  Google Scholar 

  115. Hylton DJ (1985) Finite-nuclear-size corrections to the Uehling potential. Phys Rev A 32(3):1303. http://journals.aps.org/pra/abstract/10.1103/PhysRevA.32.1303

    Article  CAS  Google Scholar 

  116. Soff G, Mohr PJ (1988) Vacuum polarization in a strong external field. Phys Rev A 38(10):5066

    Article  CAS  Google Scholar 

  117. Persson H, Lindgren I, Salomonson S, Sunnergren P (1993) Accurate vacuum-polarization calculations. Phys Rev A 48(4):2772. http://link.aps.org/doi/10.1103/PhysRevA.48.2772

    Article  CAS  Google Scholar 

  118. Karplus R, Neuman M (1950) Non-linear interactions between electromagnetic fields. Phys Rev 80(3):380. http://link.aps.org/doi/10.1103/PhysRev.80.380

    Article  Google Scholar 

  119. Hylton DJ (1984) The reduced Dirac-Green function for the Coulomb potential. J Math Phys 25(4):1125. http://scitation.aip.org/content/aip/journal/jmp/25/4/10.1063/1.526255

    Article  Google Scholar 

  120. Lindgren I, Morrison J (1982) Atomic many-body theory. Atoms and plasmas, 2nd edn. Springer, Berlin

    Google Scholar 

  121. Kelly HP (1963) Correlation effects in atoms. Phys Rev 131(2):684. http://journals.aps.org/pr/abstract/10.1103/PhysRev.131.684

    Article  Google Scholar 

  122. Grant IP (1965) Relativistic self-consistent fields. Proc Phys Soc Lond 86:523

    Article  Google Scholar 

  123. Grant IP (1970) Relativistic calculation of atomic structures. Adv Phys 19(82):747

    Article  CAS  Google Scholar 

  124. Desclaux JP, Mayers DF, O’Brien F (1971) Relativistic atomic wavefunction. J Phys B: At Mol Optic Phys 4:631

    Article  CAS  Google Scholar 

  125. Desclaux JP (1975) A multiconfiguration relativistic Dirac-Fock program. Comput Phys Commun 9:31

    Article  Google Scholar 

  126. Lindgren I (1978) A coupled-cluster approach to the many-body perturbation theory for open-shell systems. Int J Quantum Chem 14(S12), 33. http://dx.doi.org/10.1002/qua.560140804

    Article  Google Scholar 

  127. Johnson WR, Sapirstein J (1986) Computation of second-order many-body corrections in relativistic atomic systems. Phys Rev Lett 57(9):1126. http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.57.1126

    Article  CAS  Google Scholar 

  128. Pohl R, Antognini A, Nez F, Amaro FD, Biraben F, Cardoso JMR, Covita DS, Dax A, Dhawan S, Fernandes LMP, Giesen A, Graf T, Hänsch TW, Indelicato P, Julien L, Kao CY, Knowles P, Bigot EOL, Liu YW, Lopes JAM, Ludhova L, Monteiro CMB, Mulhauser F, Nebel T, Rabinowitz P, dos Santos JMF, Schaller LA, Schuhmann K, Schwob C, Taqqu D, Veloso JFCA, Kottmann F (2010) The size of the proton. Nature 466(7303):213. http://www.nature.com/doifinder/10.1038/nature09250

    Article  CAS  Google Scholar 

  129. Antognini A, Nez F, Schuhmann K, Amaro FD, Biraben F, Cardoso JMR, Covita DS, Dax A, Dhawan S, Diepold M, Fernandes LMP, Giesen A, Gouvea AL, Graf T, Hänsch TW, Indelicato P, Julien L, Kao CY, Knowles P, Kottmann F, Le Bigot EO, Liu YW, Lopes JAM, Ludhova L, Monteiro CMB, Mulhauser F, Nebel T, Rabinowitz P, dos Santos JMF, Schaller LA, Schwob C, Taqqu D, Veloso JFCA, Vogelsang J, Pohl R (2013) Proton structure from the measurement of 2S-2P transition frequencies of muonic hydrogen. Science 339(6118):417. http://www.sciencemag.org/content/339/6118/417.abstract

    Article  CAS  Google Scholar 

  130. Trassinelli M, Kumar A, Beyer HF, Indelicato P, Märtin R, Reuschl R, Kozhedu YS, Brandau C, Bräuning H, Geyer S, Gumberidze A, Hess S, Jagodzinski P, Kozhuharov C, Liesen D, Trotsenko S, Weber G, Winters DFA, Stöhlker T (2009) Observation of the 2p 3∕2 → 2s 1∕2 intra-shell transition in He-like uranium. Europhys Lett 87(6):63001 (6). http://www.iop.org/EJ/abstract/0295-5075/87/6/63001

  131. Trassinelli M, Kumar A, Beyer HF, Indelicato P, Märtin R, Reuschl R, Kozhedub YS, Brandau C, Bräuning H, Geyer S, Gumberidze A, Hess S, Jagodzinski P, Kozhuharov C, Liesen D, Spillmann U, Trotsenko S, Weber G, Winters DFA, Stöhlker T (2011) Differential energy measurement between He- and Li-like uranium intra-shell transitions. Phys Scr T144:014003. http://stacks.iop.org/1402-4896/2011/i=T144/a=014003

    Article  CAS  Google Scholar 

  132. Chantler CT, Kinnane MN, Gillaspy JD, Hudson LT, Payne AT, Smale LF, Henins A, Pomeroy JM, Tan JN, Kimpton JA, Takacs E, Makonyi K (2012) Testing three-body quantum electrodynamics with trapped Ti20+ ions: evidence for a Z-dependent divergence between experiment and calculation. Phys Rev Lett 109(15):153001. http://link.aps.org/doi/10.1103/PhysRevLett.109.153001

    Article  CAS  Google Scholar 

  133. Kim YK, Baik DH, Indelicato P, Desclaux JP (1991) Resonance transition energies of Li-, Na-, and Cu-like ions. Phys Rev A 44(1):148. http://link.aps.org/doi/10.1103/PhysRevA.44.148

    Article  CAS  Google Scholar 

  134. Blundell SA (1993) Calculations of the screened self-energy and vacuum polarization in Li-like, Na-like and Cu-like ions. Phys Rev A 47(3):1790. http://journals.aps.org/pra/abstract/10.1103/PhysRevA.47.1790

    Article  CAS  Google Scholar 

  135. Chen MH, Cheng KT, Johnson WR, Sapirstein J (1995) Relativistic configuration-interaction calculations for the n=2 states of lithiumlike ions. Phys Rev A 52(1):266. http://link.aps.org/doi/10.1103/PhysRevA.52.266

    Article  CAS  Google Scholar 

  136. Cheng KT, Chen MH, Sapirstein J (2000) Quantum electrodynamic corrections in high-Z Li-like and Be-like ions. Phys Rev A 62(3):054501. http://journals.aps.org/pra/abstract/10.1103/PhysRevA.62.054501

    Article  Google Scholar 

  137. Kozhedub YS, Volotka AV, Artemyev AN, Glazov DA, Plunien G, Shabaev VM, Tupitsyn II, Stöhlker T (2010) Relativistic recoil, electron-correlation, and QED effects on the 2pj-2s transition energies in Li-like ions. Phys Rev A 81(4):042513. http://link.aps.org/doi/10.1103/PhysRevA.81.042513

    Article  CAS  Google Scholar 

  138. Hölzer G, Förster E, Klöpfel D, Beiersdorfer P, Brown GV, Crespo López-Urrutia JR, Widman K (1998) Absolute wavelength measurement of the Lyman-a transition of hydrogenic Mg11+. PRA 57(2):945. http://journals.aps.org/pra/abstract/10.1103/PhysRevA.57.945

    Article  Google Scholar 

  139. Tschischgale J, Klöpfel D, Beiersdorfer P, Brown GV, Förster E, Schulte-Schrepping H, Utter SB (2002) Absolute wavelength measurement of the Lyman-? transition of hydrogen-like silicon. CJP 80(8):867. http://rparticle.web-p.cisti.nrc.ca/rparticle/AbstractTemplateServlet?calyLang=eng%26journal=cjp%26volume=80%26year=0%26issue=8%26msno=p02-011

    Google Scholar 

  140. Kubiçek K, Mokler PH, Mäckel V, Ullrich J, López-Urrutia JRC (2014) Transition energy measurements in hydrogenlike and heliumlike ions strongly supporting bound-state QED calculations. Phys Rev A 90(3):032508. http://link.aps.org/doi/10.1103/PhysRevA.90.032508

    Article  CAS  Google Scholar 

  141. Briand JP, Mossé JP, Indelicato P, Chevallier P, Girard-Vernhet D, Chétioui A, Ramos MT, Desclaux JP (1983) Spectroscopy of hydrogenlike and heliumlike argon. Phys Rev A 28(3):1413. http://link.aps.org/doi/10.1103/PhysRevA.28.1413

    Article  CAS  Google Scholar 

  142. Beyer HF, Deslattes RD, Folkmann F, LaVilla RE (1985) Determination of the 1s Lamb shift in one-electron argon recoil ions. J Phys B: At Mol Opt Phys 18(2):207. http://dx.doi.org/10.1088/0022-3700/18/2/008

    Article  CAS  Google Scholar 

  143. Rice JE, Reinke ML, Ashbourn JMA, Gao C, Victora MM, Chilenski MA, Delgado-Aparicio L, Howard NT, Hubbard AE, Hughes JW, Irby JH (2014) X-ray observations of Ca19+, Ca18+ and satellites from Alcator C-Mod tokamak plasmas. J Phys B: At Mol Opt Phys 47(7):075701. http://stacks.iop.org/0953-4075/47/i=7/a=075701

    Article  CAS  Google Scholar 

  144. Tarbutt MR, Silver JD (2002) Measurement of the ground-state Lamb shift of hydrogen-like Ti 21+. J Phys B: At Mol Opt Phys 35(6):1467. http://stacks.iop.org/0953-4075/35/i=6/a=305

    Article  CAS  Google Scholar 

  145. Gillaspy JD, Chantler CT, Paterson D, Hudson LT, Serpa FG, Takács E (2010) First measurement of Lyman alpha x-ray lines in hydrogen-like vanadium: results and implications for precision wavelength metrology and tests of QED. J Phys B: At Mol Opt Phys 43(7):074021. http://stacks.iop.org/0953-4075/43/i=7/a=074021

    Article  CAS  Google Scholar 

  146. Briand JP, Tavernier M, Indelicato P, Marrus R, Gould H (1983) High precision spectroscopic studies of Lyman a lines of hydrogenlike iron: a measurement of the 1s Lamb-shift. Phys Rev Lett 50(11):832. http://link.aps.org/doi/10.1103/PhysRevLett.50.832

    Article  CAS  Google Scholar 

  147. Chantler CT, Laming JM, Dietrich DD, Hallett WA, McDonald R, Silver JD (2007) Hydrogenic Lamb shift in iron Fe25+ and fine-structure Lamb shift. Phys Rev A 76(4):042116. http://link.aps.org/doi/10.1103/PhysRevA.76.042116

    Article  CAS  Google Scholar 

  148. Beyer HF, Indelicato P, Finlayson KD, Liesen D, Deslattes RD (1991) Measurement of the 1s lamb-shift in hydrogenlike nickel. Phys Rev A 43(1):223. http://link.aps.org/doi/10.1103/PhysRevA.43.223

    Article  CAS  Google Scholar 

  149. Chantler CT, Laming JM, Silver JD, Dietrich DD, Mokler PH, Finch EC, Rosner SD (2009) Hydrogenic Lamb shift in Ge31+ and the fine-structure Lamb shift. Phys Rev A 80(2):022508. http://link.aps.org/doi/10.1103/PhysRevA.80.022508

    Article  CAS  Google Scholar 

  150. Tavernier M, Briand JP, Indelicato P, Liesen D, Richard P (1985) Measurement of the 1s Lamb shift of hydrogenlike krypton. J Phys B: At Mol Opt Phys 18(11):L327. http://dx.doi.org/10.1088/0022-3700/18/11/004

    Article  CAS  Google Scholar 

  151. Briand JP, Indelicato P, Simionovici A, San Vicente V, Liesen D, Dietrich D (1989) Spectroscopic study of hydrogenlike and heliumlike xenon ions. Europhys Lett 9(3):225. http://iopscience.iop.org/0295-5075/9/3/007

    Article  CAS  Google Scholar 

  152. Widmann K, Beiersdorfer P, Brown GV, Löpez Urrutia JRC, Osterheld AL, Reed KJ, Scofield JH, Utter SB (2000) High-resolution measurements of the K-shell spectral lines of hydrogenlike and heliumlike xenon. In: X-Ray and inner-shell processes, vol 506, Chicago, pp 444–66. http://scitation.aip.org/content/aip/proceeding/aipcp/10.1063/1.1302773

  153. Beyer HF, Finlayson KD, Liesen D, Indelicato P, Chantler CT, Deslattes RD, Schweppe J, Bosch F, Jung M, Klepper O, König W, Moshammer R, Beckert K, Heuckhoff H, Franzke B, Gruber A, Nolden F, Spädtke P, Steck M (1993) X-ray transitions associated with electron capture into bare dysprosium. J Phys B: At Mol Opt Phys 26:1557. http://iopscience.iop.org/article/10.1088/0953-4075/26/9/004/meta

    Article  CAS  Google Scholar 

  154. Beyer HF, Liesen D, Bosch F, Finlayson KD, Jung M, Klepper O, Moshammer R, Beckert K, Eickhoff H, Franzke B, Nolden F, Spädtke P, Steck M, Menzel G, Deslattes RD (1994) X rays from radiative electron capture of free cooling electrons for precise Lamb-shift measurements at high Z: Au78+. Phys Lett A 184:435. http://www.sciencedirect.com/science?_ob=Article%26_udi=B6TVM-46SPK2H-8S%26_user=870092%26_coverDate=01%2F24%2F1994%26_alid=1427368629%26_rdoc=1%26_fmt=high%26_orig=search%26_cdi=5538%26_sort=r%26_docanchor=%26view=c%26_ct=1%26_acct=C000046800%26_version=1%26_urlVersion=0%26_userid=870092%26md5=95fca9083bd953fcc051dc24ee02498b

    Google Scholar 

  155. Beyer HF, Menzel G, Liesen D, Gallus A, Bosch F, Deslattes RD, Indelicato P, Stöhlker T, Klepper O, Moshammer R, Nolden F, Eickhoff H, Franzke B, Steck M (1995) Measurement of the ground-state Lambshift of hydrogenlike uranium at the electron cooler of the ESR. Zeitschrift für Physik D 35:169. http://www.springerlink.com/content/h68671l823m02g75/?p=83d9bb408bf8475db4f9330d57005a48%26pi=1

  156. Stöhlker T, Mokler PH, Bosch F, Dunford RW, Franzke F, Klepper O, Kozhuharov C, Ludziejewski T, Nolden F, Reich H, Rymuza P, Stachura Z, Steck M, Swiat P, Warczak A (2000) 1s Lamb shift in hydrogenlike uranium measured on cooled, decelerated ion beams. Phys Rev A 85(15):3109

    Google Scholar 

  157. Gumberidze A, Stöhlker T, Banaś D, Beckert K, Beller P, Beyer HF, Bosch F, Hagmann S, Kozhuharov C, Liesen D, Nolden F, Ma X, Mokler PH, Steck M, Sierpowski D, Tashenov S (2005) Quantum electrodynamics at strong electric fields: the ground state Lamb shift in hydrogenlike uranium. Phys Rev Lett 94(22):223001. http://link.aps.org/doi/10.1103/PhysRevLett.94.223001

    Article  CAS  Google Scholar 

  158. Amaro P, Schlesser S, Guerra M, Le Bigot EO, Isac JM, Travers P, Santos JP, Szabo CI, Gumberidze A, Indelicato P (2012) Absolute measurement of the “relativistic M1” transition energy in heliumlike argon. Phys Rev Lett 109(4):043005. http://link.aps.org/doi/10.1103/PhysRevLett.109.043005

    Article  CAS  Google Scholar 

  159. Szabo CI, Amaro P, Guerra M, Schlesser S, Gumberidze A, Santos JP, Indelicato P (2013) Reference free, high-precision measurements of transition energies in few electron argon ions. In: CAARI. AIP Conference Proceedings, vol 1525. AIP, Fort Worth, pp 68–72

    Google Scholar 

  160. Beiersdorfer P, Bitter M, von Goeler S, Will KW (1989) Experimental study of the x-ray transitions in the heliumlike isoelectronic sequence. Phys Rev A 40(1):150. http://journals.aps.org/pra/abstract/10.1103/PhysRevA.40.150

    Article  CAS  Google Scholar 

  161. Payne AT, Chantler CT, Kinnane MN, Gillaspy JD, Hudson LT, Smale LF, Henins A, Kimpton JA, Takacs E (2014) Helium-like titanium x-ray spectrum as a probe of QED computation. J Phys B: At Mol Opt Phys 47(18):185001. http://stacks.iop.org/0953-4075/47/i=18/a=185001

    Article  CAS  Google Scholar 

  162. Briand JP, Tavernier M, Marrus R, Desclaux JP (1984) High-precision spectroscopic study of heliumlike iron. Phys Rev A 29(6):3143. http://link.aps.org/doi/10.1103/PhysRevA.29.3143

    Article  CAS  Google Scholar 

  163. Rudolph JK, Bernitt S, Epp SW, Steinbrügge R, Beilmann C, Brown GV, Eberle S, Graf A, Harman Z, Hell N, Leutenegger M, Müller A, Schlage K, Wille HC, Yavaş H, Ullrich J, Crespo López-Urrutia JR (2013) X-Ray resonant photoexcitation: linewidths and energies of K alpha transitions in highly charged Fe ions. Phys Rev Lett 111(10):103002. http://link.aps.org/doi/10.1103/PhysRevLett.111.103002

    Article  CAS  Google Scholar 

  164. Beiersdorfer P, Brown GV (2015) Experimental study of the x-ray transitions in the heliumlike isoelectronic sequence: updated results. Phys Rev A 91(3):032514. http://link.aps.org/doi/10.1103/PhysRevA.91.032514

    Article  CAS  Google Scholar 

  165. MacLaren S, Beiersdorfer P, Vogel DA, Knapp D, Marrs RE, Wong K, Zasadzinski R (1992) Precision measurement of Ka transitions in heliumlike Ge30+. Phys Rev A 45(1):329. http://link.aps.org/doi/10.1103/PhysRevA.45.329

    Article  CAS  Google Scholar 

  166. Briand JP, Indelicato P, Tavernier M, Liesen D, Beyer HF, Liu B, Warczak A, Desclaux JP (1984) Observation of hydrogenlike and heliumlike krypton spectra. Zeitschrift für Physik A 318:1

    Article  CAS  Google Scholar 

  167. Indelicato P, Tavernier M, Briand JP, Liesen D (1986) Experimental study of relativistic correlations and QED effects in heliumlike krypton ions. Zeitschrift für Physik D 2:249

    Article  CAS  Google Scholar 

  168. Epp SW, Steinbrügge R, Bernitt S, Rudolph JK, Beilmann C, Bekker H, Müller A, Versolato OO, Wille HC, Yavaş H, Ullrich J, Crespo López-Urrutia JR (2015) Single-photon excitation of K α in heliumlike Kr34+: Results supporting quantum electrodynamics predictions. Phys Rev A 92(2):020502. http://link.aps.org/doi/10.1103/PhysRevA.92.020502

    Article  CAS  Google Scholar 

  169. Briand JP, Indelicato P, Simionovici A, San Vicente V, Liesen D, Dietrich D (1989) Spectroscopic study of hydrogenlike and heliumlike xenon ions. Europhys Lett 9(3):225. http://iopscience.iop.org/0295-5075/9/3/007

    Article  CAS  Google Scholar 

  170. Kramida A, Ralchenko Y, Reader J, NIST ASD Team (2015) NIST Atomic Spectra Database (ver. 5.3). National Institute of Standards and Technology, Gaithersburg. [Online]. Available: http://physics.nist.gov/asd 20 Dec 2015

  171. Bosselmann P, Staude U, Horn D, Schartner KH, Folkmann F, Livingston AE, Mokler PH (1999) Measurements of 2s 2 S 1∕2 − 2p 2 P 1∕2, 3∕2 transition energies in lithiumlike heavy ions. II. Experimental results for Ag44+ and discussion along the isoelectronic series. PRA 59(3):1874

    Google Scholar 

  172. Feili D, Bosselmann P, Schartner KH, Folkmann F, Livingston AE, Träbert E, Ma X, Mokler PH (2000) Measurements of 2s 2 S 1∕2 − 2p 2 P 1∕2, 3∕2 transition energies in lithiumlike heavy ions. III. Experimental results for Sn47+ and Xe51+. Phys Rev A 62(2):022501. http://journals.aps.org/pra/abstract/10.1103/PhysRevA.62.022501

  173. Büttner R, Kraus B, Schartner KH, Folkmann F, Mokler PH, Möller G (1992) EUV spectroscopy of beam-foil excited 14.25 MeV/u Xe52+?Xe49+ ions. Zeitschrift für Physics D 22:693. http://springerlink.bibliotecabuap.elogim.com/article/10.1007/BF01437250

  174. Bernhardt D, Brandau C, Harman Z, Kozhuharov C, Böhm S, Bosch F, Fritzsche S, Jacobi J, Kieslich S, Knopp H, Nolden F, Shi W, Stachura Z, Steck M, Stöhlker T, Schippers S, Müller A (2015) Electron-ion collision spectroscopy: Lithium-like xenon ions. Phys Rev A 91(1):012710. http://link.aps.org/doi/10.1103/PhysRevA.91.012710

    Article  CAS  Google Scholar 

  175. Podpaly Y, Clementson J, Beiersdorfer P, Williamson J, Brown GV, Gu MF (2009) Spectroscopy of 2s 1∕2 − 2p 3∕2 transitions in W65+ through W71+. Phys Rev A 80(5):052504. http://link.aps.org/doi/10.1103/PhysRevA.80.052504

    Article  CAS  Google Scholar 

  176. Zhang X, Nakamura N, Chen C, Andersson M, Liu Y, Ohtani S (2008) Measurement of the QED energy shift in the 1s[sup2]2p[sub3∕2] −−1s[sup2]2s[sub1∕2] x-ray transition in Li-like208Pb79+. Phys Rev A 78(3):032504 http://link.aps.org/abstract/PRA/v78/e032504http://dx.doi.org/10.1103/PhysRevA.78.032504

    Article  CAS  Google Scholar 

  177. Beiersdorfer P, Osterheld AL, Scofield JH, Crespo Löpez-Urrutia JR, Widmann K (1998) Measurement of QED and hyperfine splitting in the 2s 1∕2 − 2p 3∕2 X-ray transition in Li-like209Bi80+. Phys Rev Lett 80(14):3022. http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.80.3022

    Article  CAS  Google Scholar 

  178. Beiersdorfer P, Osterheld A, Elliott SR, Chen MH, Knapp D, Reed K (1995) Structure and Lamb shift of 2s1/2-3p3/2 levels in lithiumlike Th87+ through neonlike Th80+. Phys Rev A 52(4):2693. http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.71.3939

    Article  CAS  Google Scholar 

  179. Nakano Y, Takano Y, Ikeda T, Kanai Y, Suda S, Azuma T, Bräuning H, Bräuning-Demian A, Dauvergne D, Stöhlker T, Yamazaki Y (2013) Resonant coherent excitation of the lithiumlike uranium ion: a scheme for heavy-ion spectroscopy. PRA 87(6):060501. http://link.aps.org/doi/10.1103/PhysRevA.87.060501

    Article  CAS  Google Scholar 

  180. Brandau C, Kozhuharov C, Müller A, Shi W, Schippers S, Bartsch T, Böhm S, Böhme C, Hoffknecht A, Knopp H, Grün N, Scheid W, Steih T, Bosch F, Franzke B, Mokler PH, Nolden F, Steck M, Stöhlker T, Stachura Z (2003) Precise determination of the 2s1/2–2p1/2 splitting in very heavy lithiumlike ions utilizing dielectronic recombination. PRL 91(7):073202. http://link.aps.org/doi/10.1103/PhysRevLett.91.073202

    Article  CAS  Google Scholar 

  181. Beiersdorfer P, Chen H, Thorn DB, Träbert E (2005) Measurement of the two-loop Lamb shift in lithiumlike U89+. Phys Rev Lett 95(23):233003. http://link.aps.org/abstract/PRL/v95/e233003

    Article  CAS  Google Scholar 

  182. Mooney T, Lindroth E, Indelicato P, Kessler E, Deslattes RD (1992) Precision measurements of K and L transitions in xenon: experiment and theory for the K, L and M levels. Phys Rev A 45(3):1531. http://link.aps.org/doi/10.1103/PhysRevA.45.1531

    Article  CAS  Google Scholar 

  183. Indelicato P, Lindroth E (1992) Relativistic effects, correlation, and QED corrections on K? transitions in medium to very heavy atoms. Phys Rev A 46(5):2426

    Article  CAS  Google Scholar 

  184. Indelicato P, Boucard S, Lindroth E (1998) Relativistic and many-body effects in K, L, and M shell ionization energy for elements with 10 ≤ Z ≤ 100 and the determination of the 1s Lamb shift for heavy elements. Eur Phys J D 3(1):29

    Article  CAS  Google Scholar 

  185. Deslattes RD, Kessler EG Jr, Indelicato P, de Billy L, Lindroth E, Anton J (2003) X-ray transition energies: new approach to a comprehensive evaluation. Rev Mod Phys 75(1):35. http://www.physics.nist.gov/PhysRefData/XrayTrans/index.html, http://link.aps.org/doi/10.1103/RevModPhys.75.35

    Google Scholar 

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Acknowledgements

Laboratoire Kastler-Brossel http://www.lkb.upmc.fr is Unité Mixte de Recherche UMR #8552 of Sorbonne Université, UPMC Paris-6; PSL-ENS; Collège de France and CNRS. The author is a member of the ExtreMe Matter Institute (EMMI) https://www.gsi.de/work/wissenschaftliche_netzwerke/helmholtz_allianz_emmi.htm, an Helmholtz Association Alliance (HA216/EMMI). P.I. wishes to thank P.B. for her continuous support and encouragements during the redaction of this chapter.

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  1. Laboratoire Kastler Brossel, UPMC-Sorbonne Universités, Collège de France and CNRS, Paris, France

    Paul Indelicato

  2. Laboratoire Kastler-Brossel, École Normale Supérieure,, PSL Research University, Paris, France

    Paul Indelicato

  3. National Institute of Standards and Technology, Gaithersburg, MD, USA

    Peter J. Mohr

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    Wenjian Liu

Appendices

Appendix: Useful Properties

Properties of Dirac γ Matrices

Basic Relations

To recover the relativistic invariant

$$\displaystyle{ p^{\mu }p_{\mu } = E^{2}/c^{2} -\mathbf{p}^{2} = m^{2}c^{2}, }$$
(436)

the γ μ matrices must obey

$$\displaystyle{ (\gamma ^{\mu }\gamma ^{\nu } +\gamma ^{\nu }\gamma ^{\mu }) = 2g^{\mu \nu } }$$
(437)

which can be rewritten as an anticommutator (noted \(\left \{,\right \}\))

$$\displaystyle{ \left \{\gamma ^{\mu },\gamma ^{\nu }\right \} = 2g^{\mu \nu }, }$$
(438)

where g μ ν = (1, −1, −1, −1) is the metric tensor for special relativity. We first evaluate the determinant of the γ μ. Taking the determinant of γ μ γ ν, we must have using (437) for μ ≠ ν \(Det\left [\gamma ^{\mu }\gamma ^{\nu }\right ] = Det\left [-\gamma ^{\mu }\gamma ^{\nu }\right ] = (-1)^{N}Det\left [\gamma ^{\mu }\gamma ^{\nu }\right ]\), where N is the dimension of the matrix. But since \(Det\left [\gamma ^{\mu }\gamma ^{\nu }\right ] = Det\left [\gamma ^{\mu }\gamma ^{\nu }\right ]\), one must have (−1)N = 1, i.e., N even. The smallest value for which this can be realized with 4 linearly independent matrices is N = 4. One can express the γ in terms of the Pauli matrices

$$\displaystyle{ \begin{array}{ccc} \sigma _{1} = \left (\begin{array}{cc} 0&1\\ 1 &0 \end{array} \right ),&\sigma _{2} = \left (\begin{array}{cc} 0& - i\\ i & 0 \end{array} \right )\;et&\sigma _{3} = \left (\begin{array}{cc} 1& 0\\ 0 & -1 \end{array} \right ) \end{array}. }$$
(439)

There are many representations used for the Dirac matrices. In atomic physics and BSQED, it is more convenient to use them in the form:

$$\displaystyle{ \begin{array}{cccc} \gamma ^{0} =\beta = \left (\begin{array}{cc} I& 0\\ 0 & -I \end{array} \right ),&\boldsymbol{\gamma } =\beta \boldsymbol{\alpha }= \left (\begin{array}{cc} 0& \boldsymbol{\sigma }\\ -\boldsymbol{\sigma } &0 \end{array} \right ),&\boldsymbol{\alpha } =\beta \boldsymbol{\gamma }= \left (\begin{array}{cc} 0& \boldsymbol{\sigma }\\ \boldsymbol{\sigma } &0 \end{array} \right )\;\boldsymbol{\Sigma } = \left (\begin{array}{cc} \boldsymbol{\sigma } &0\\ 0 & \boldsymbol{\sigma } \end{array} \right )\end{array}. }$$
(440)

Among other interesting properties, we can deduce from (437) \(\left (\gamma ^{0}\right )^{2} =\beta ^{2} = 1\) and \(\left (\gamma ^{i}\right )^{2} = -1\).

More Properties of the Dirac Matrices

One can rewrite the tensor γ μ γ ν as a sum of a symmetric and antisymmetric tensor as

$$\displaystyle{ \gamma ^{\mu }\gamma ^{\nu } = g^{\mu \nu } +\sigma ^{\mu \nu }. }$$
(441)

The Dirac γ matrix commutators and anticommutators can be rewritten as

$$\displaystyle\begin{array}{rcl} \left [\gamma ^{\mu },\gamma ^{\nu }\right ] =\gamma ^{\mu }\gamma ^{\nu } -\gamma ^{\nu }\gamma ^{\mu } = 2\sigma ^{\mu \nu }.& &{}\end{array}$$
(442)
$$\displaystyle\begin{array}{rcl} \left \{\gamma ^{\mu },\gamma ^{\nu }\right \} =\gamma ^{\mu }\gamma ^{\nu } +\gamma ^{\nu }\gamma ^{\mu } = 2g^{\mu \nu }& &{}\end{array}$$
(443)

where with our conventions the antisymmetric tensors \(\sigma ^{\mu \nu }\) is given by

$$\displaystyle{ \sigma ^{\mu \nu } = \left (\boldsymbol{\alpha },i\boldsymbol{\Sigma }\right ), }$$
(444)

from which one can deduce the following contractions, using only Eq. (437):

$$\displaystyle{ \begin{array}{rcl} \gamma ^{\mu }\gamma _{\mu }& =&4 \\ \gamma ^{\mu }\gamma ^{\nu }\gamma _{\mu }& =& - 2\gamma ^{\nu } \\ \gamma ^{\mu }\gamma ^{\lambda }\gamma ^{\nu }\gamma _{\mu }& =&4g^{\lambda \nu } \\ \gamma ^{\mu }\gamma ^{\lambda }\gamma ^{\nu }\gamma ^{\rho }\gamma _{\mu }& =& - 2\gamma ^{\rho }\gamma ^{\nu }\gamma ^{\lambda } \\ \gamma ^{\mu }\gamma ^{\lambda }\gamma ^{\nu }\gamma ^{\rho }\gamma ^{\sigma }\gamma _{\mu }& =&2\left (\gamma ^{\sigma }\gamma ^{\lambda }\gamma ^{\nu }\gamma ^{\rho } +\gamma ^{\rho }\gamma ^{\nu }\gamma ^{\lambda }\gamma ^{\sigma }\right ).\end{array} }$$
(445)

The traces are found to be

$$\displaystyle{ \begin{array}{rcl} Tr[I]& =&4 \\ Tr[\gamma ^{\mu }]& =&0. \end{array} }$$
(446)

We will use the Feynman slash in QED expressions:

$$\displaystyle{ \not\!p =\gamma p \equiv \gamma ^{\mu }p_{\mu } =\beta p_{0} -\boldsymbol{\gamma }\mathbf{p}. }$$
(447)

Another important properties of the Dirac matrices are given by the adjoints. One can easily show that

$$\displaystyle{ \left (\gamma ^{\mu }\right )^{\dag } =\gamma ^{0}\gamma ^{\mu }\gamma ^{0}. }$$
(448)

Using the commutation rules, one can see easily that \(\left (\gamma ^{0}\right )^{\dag } =\gamma ^{0}\) and \(\left (\gamma ^{i}\right )^{\dag } = -\gamma ^{i}\).

Asymptotic Properties

This section gives an example that indicates that the power series expansion of the wave function discussed in section “Singular Terms: Renormalization in Coordinate Space” leads to an asymptotic expansion for large values of | z | of the integrand in Eq. (190). A simple expression with the essential features of (190) is

$$\displaystyle{ u(y) =\int d\mathbf{x}_{2}\int d\mathbf{x}_{1}f(\mathbf{x}_{2})\left [a + y\mathbf{b} \cdot (\mathbf{x}_{2} -\mathbf{ x}_{1})\right ] \frac{e^{-y\vert \mathbf{x}_{2}-\mathbf{x}_{1}\vert }} {4\pi \vert \mathbf{x}_{2} -\mathbf{ x}_{1}\vert ^{2}}\ g(\mathbf{x}_{1}). }$$
(449)

If \(g(\mathbf{x}_{1})\) is expanded about the point \(\mathbf{x}_{2}\), we have

$$\displaystyle{ g(\mathbf{x}_{1}) = g(\mathbf{x}_{2})+(\mathbf{x}_{1}-\mathbf{x}_{2})\cdot \boldsymbol{\nabla }_{2}\ g(\mathbf{x}_{2})+\frac{1} {2}(x_{1}^{l}-x_{ 2}^{l})(x_{ 1}^{m}-x_{ 2}^{m}) \frac{\partial } {\partial x_{2}^{l}} \frac{\partial } {\partial x_{2}^{m}}g(\mathbf{x}_{2})+\cdots \,. }$$
(450)

Term by term integration over \(\mathbf{x}_{1}\) in (449) is elementary and yields

$$\displaystyle{ u(y) = \frac{a} {y}\int d\mathbf{x}f(\mathbf{x})\ g(\mathbf{x}) - \frac{2} {3y^{2}}\int d\mathbf{x}f(\mathbf{x})\ \mathbf{b} \cdot \boldsymbol{\nabla }\ g(\mathbf{x}) + \frac{a} {3y^{3}}\int d\mathbf{x}f(\mathbf{x})\nabla ^{2}g(\mathbf{x}) + \cdots \,, }$$
(451)

where the three terms correspond to the three terms in (450). Since gradients of the wave function are proportional to the momentum, the series depicted above is a power series in \(\mathbf{p}\) or Z α, as well as an asymptotic series in y −1.

Integration of Singular Terms

In section “Evaluation of the Singular Terms,” the result of integration over z is given for a number of singular terms in the high-energy part. Here, we indicate a method of evaluation of the integrals for one example:

$$\displaystyle{ I = \frac{1} {i} \int _{C_{H}}dz\ z\left ( \frac{1} {b + c} - \frac{1} {b' + c}\right ). }$$
(452)

Changes of variables y = −iz on the positive imaginary axis and y = iz on the negative imaginary axis lead to

$$\displaystyle\begin{array}{rcl} I& =& 2\ \mathrm{Im}\int _{0}^{\infty }dy\ y\bigg[ \frac{1} {y - iE_{n} + (1 + y^{2})^{1/2}} \\ & & \qquad - \frac{1} {(\varLambda ^{2} + (y - iE_{n})^{2})^{1/2} + (1 + y^{2})^{1/2}}\bigg].{}\end{array}$$
(453)

In the second term in (453), we make the replacement \((1 + y^{2})^{1/2} \rightarrow y\) with a resulting change in the integral of order \(\varLambda ^{-1}\). To integrate each of the two terms in (453) separately, we introduce a temporary cutoff

$$\displaystyle\begin{array}{rcl} I& =& 2\ \mathrm{Im}\lim _{Y \rightarrow \infty }\int _{0}^{Y }dy\ y\bigg[ \frac{1} {y - iE_{n} + (1 + y^{2})^{1/2}} \\ & & \qquad - \frac{1} {(\varLambda ^{2} + (y - iE_{n})^{2})^{1/2} + y}\bigg] + \mathcal{O}(\varLambda ^{-1}){}\end{array}$$
(454)

where the limit \(Y \rightarrow \infty \) is taken before the limit \(\varLambda \rightarrow \infty \). Each integral can be evaluated analytically, with the result for large Y that

$$\displaystyle\begin{array}{rcl} 2\ \mathrm{Im}\int _{0}^{Y }dy\ \frac{y} {y - iE_{n} + (1 + y^{2})^{1/2}}& =& \frac{E_{n}} {4} \bigg[2\ \ln (2Y ) - \frac{1} {E_{n}^{2}} \\ & & +\frac{1 - E_{n}^{4}} {E_{n}^{4}} \ln (1 + E_{n}^{2})\bigg] \\ & & +\mathcal{O}(Y ^{-1}) {}\end{array}$$
(455)

and

$$\displaystyle\begin{array}{rcl} 2\ \mathrm{Im}\int _{0}^{Y }dy\ \frac{y} {(\varLambda ^{2} + (y - iE_{n})^{2})^{1/2} + y}& =& \frac{E_{n}} {4} \bigg[2\ \ln (2Y ) -\ln \varLambda ^{2} - \frac{\varLambda ^{2}} {E_{n}^{2}} \\ & & -\left ( \frac{\varLambda ^{2}} {E_{n}^{2}} - 1\right )^{2}\ln \left (1 -\frac{E_{n}^{2}} {\varLambda ^{2}} \right )\bigg] \\ & & +\qquad \mathcal{O}(Y ^{-1}). {}\end{array}$$
(456)

Taking the difference for large \(\varLambda\) yields

$$\displaystyle{ I = \frac{E_{n}} {4} \left [\ln (\varLambda ^{2}) + \frac{3E_{n}^{2} - 2} {2E_{n}^{2}} + \frac{1 - E_{n}^{4}} {E_{n}^{4}} \ln \left (1 + E_{n}^{2}\right )\right ] + \mathcal{O}(\varLambda ^{-1}). }$$
(457)

The remaining integrals in Section “Evaluation of the Singular Terms,” can be evaluated in this way.

Angular Integrations

Formulas pertaining to integration over d Ω 1 in section “Evaluation of the Subtraction Terms” are given here. The calculation is facilitated by expressing the integral in terms of spherical angles \(\theta\) and ϕ of \(\mathbf{x}_{1}\) relative to the direction of \(\mathbf{x}_{2}\). In particular, we write

$$\displaystyle{ \mathbf{x}_{1} = x_{1}\cos \phi \sin \theta \ \hat{a} + x_{1}\sin \phi \sin \theta \ \hat{b} + x_{1}\cos \theta \ \hat{x}_{2} }$$
(458)

where \(\hat{a}\) and \(\hat{b}\) are orthogonal unit vectors in the plane perpendicular to \(\hat{x}_{2}\). For \(\xi =\cos \theta\), \(R = (x_{2}^{2} - 2x_{2}x_{1}\xi + x_{1}^{2})^{1/2}\), and f a function of R, we have

$$\displaystyle\begin{array}{rcl} \int d\varOmega _{1}(\mathbf{x}_{1} -\mathbf{ x}_{2})\ f(R)& =& \int _{-1}^{1}d\xi \int _{ 0}^{2\pi }d\phi \ \big(x_{ 1}\cos \phi \sqrt{1 -\xi ^{2}}\ \hat{a} \\ & & \qquad + x_{1}\sin \phi \sqrt{1 -\xi ^{2}}\ \hat{b} + x_{1}\xi \ \hat{x}_{2} -\mathbf{ x}_{2}\big)\ f(R) \\ & =& 2\pi \int _{-1}^{1}d\xi (\xi x_{ 1} - x_{2})\hat{x}_{2}\ f(R) {}\end{array}$$
(459)

and

$$\displaystyle\begin{array}{rcl} \int d\varOmega _{1}(x_{1}^{l} - x_{ 2}^{l})(x_{ 1}^{m} - x_{ 2}^{m})\ f(R)& =& \int _{ -1}^{1}d\xi \int _{ 0}^{2\pi }d\phi \ \big(x_{ 1}\cos \phi \sqrt{1 -\xi ^{2}}\ \hat{a}^{l} \\ & & \qquad + x_{1}\sin \phi \sqrt{1 -\xi ^{2}}\ \hat{b}^{l} + x_{ 1}\xi \ \hat{x}_{2}^{l} - x_{ 2}^{l}\big) \\ & & \times \big(x_{1}\cos \phi \sqrt{1 -\xi ^{2}}\ \hat{a}^{m} \\ & & \qquad + x_{1}\sin \phi \sqrt{1 -\xi ^{2}}\ \hat{b}^{m} \\ & & \qquad + x_{1}\xi \ \hat{x}_{2}^{m} - x_{ 2}^{m}\big)\ f(R) \\ & =& 2\pi \int _{-1}^{1}d\xi \big[x_{ 1}^{2}\frac{1} {2}(1 -\xi ^{2})(\hat{a}^{l}\hat{a}^{m} +\hat{ b}^{l}\hat{b}^{m}) \\ & & \qquad \qquad + (x_{1}\xi - x_{2})^{2}\hat{x}_{ 2}^{l}\hat{x}_{ 2}^{m}\big]f(R) \\ & =& 2\pi \int _{-1}^{1}d\xi f(R)\bigg[R^{2}\hat{x}_{ 2}^{l}\hat{x}_{ 2}^{m} \\ & & \qquad + \frac{1} {2}(1 -\xi ^{2})x_{ 1}^{2}(\delta _{ lm} - 3\hat{x}_{2}^{l}\hat{x}_{ 2}^{m})\bigg].{}\end{array}$$
(460)

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Indelicato, P., Mohr, P. (2016). Introduction to Bound-State Quantum Electrodynamics. In: Liu, W. (eds) Handbook of Relativistic Quantum Chemistry. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-41611-8_36-1

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