1. Precision test of bound-state QED and the fine structure constant  Savely G Karshenboim D.I. Mendeleev Institute for Metrology (St. Petersburg) and Max-Planck-Institut für Quantenoptik (Garching)

2. Outline Lamb shift in the hydrogen atom Hyperfine structure in light atoms Problems of bound state QED & Uncertainty of theoretical calculations Determination of the fine structure constants Search for  variations

3. Hydrogen atom & quantum mechanics Search for interpretation of regularity in hydrogen spectrum leads to establishment of Old quantum mechanics (Bohr theory) “New” quantum mechanics of Schrödinger and Heisenberg. The energy levels are E n = – ½  2 mc 2 /n 2 – no dependence on momentum (j).

4. Hydrogen atom & quantum mechanics Search for interpretation of regularity in hydrogen spectrum leads to establishment of Old quantum mechanics (Bohr theory) “New” quantum mechanics of Schrödinger and Heisenberg. The energy levels are E n = – ½  2 mc 2 /n 2 – no dependence on momentum (j). On a way to explain fine structure of some hydrogen lines which was due to a splitting between 2p 1/2 and 2p 3/2 (j=1/2 and 3/2 is the angular momentum) Dirac introduced a relativisitic equation.

5. Hydrogen atom & quantum mechanics Search for interpretation of regularity in hydrogen spectrum leads to establishment of Old quantum mechanics (Bohr theory) “New” quantum mechanics of Schrödinger and Heisenberg. The energy levels are E n = – ½  2 mc 2 /n 2 – no dependence on momentum (j). On a way to explain fine structure of some hydrogen lines which was due to a splitting between 2p 1/2 and 2p 3/2 (j=1/2 and 3/2 is the angular momentum) Dirac introduced a relativisitic equation, which predicted

6. Hydrogen atom & quantum mechanics Search for interpretation of regularity in hydrogen spectrum leads to establishment of Old quantum mechanics (Bohr theory) “New” quantum mechanics of Schrödinger and Heisenberg. The energy levels are E n = – ½  2 mc 2 /n 2 – no dependence on momentum (j). On a way to explain fine structure of some hydrogen lines which was due to a splitting between 2p 1/2 and 2p 3/2 (j=1/2 and 3/2 is the angular momentum) Dirac introduced a relativisitic equation, which predicted existence of positron;

7. Hydrogen atom & quantum mechanics Search for interpretation of regularity in hydrogen spectrum leads to establishment of Old quantum mechanics (Bohr theory) “New” quantum mechanics of Schrödinger and Heisenberg. The energy levels are E n = – ½  2 mc 2 /n 2 – no dependence on momentum (j). On a way to explain fine structure of some hydrogen lines which was due to a splitting between 2p 1/2 and 2p 3/2 (j=1/2 and 3/2 is the angular momentum) Dirac introduced a relativisitic equation, which predicted existence of positron; fine structure for a number of levels;

8. Hydrogen atom & quantum mechanics Search for interpretation of regularity in hydrogen spectrum leads to establishment of Old quantum mechanics (Bohr theory) “New” quantum mechanics of Schrödinger and Heisenberg. The energy levels are E n = – ½  2 mc 2 /n 2 – no dependence on momentum (j). On a way to explain fine structure of some hydrogen lines which was due to a splitting between 2p 1/2 and 2p 3/2 (j=1/2 and 3/2 is the angular momentum) Dirac introduced a relativisitic equation, which predicted existence of positron; fine structure for a number of levels; electron g factor (g = 2).

9. Hydrogen atom & QED Two of these three predictions happened to be not absolutely correct.

10. Hydrogen atom & QED Two of these three predictions happened to be not absolutely correct: It was discovered (Lamb) that energy of 2s 1/2 and 2p 1/2 is not the same.

11. Hydrogen atom & QED Two of these three predictions happened to be not absolutely correct. It was discovered (Lamb) that energy of 2s 1/2 and 2p 1/2 is not the same. It was also discovered (Rabi & Kusch) that hyperfine splitting of the 1s state in hydrogen atom has an anomalous contribution.

12. Hydrogen atom & QED Two of these three predictions happened to be not absolutely correct. It was discovered (Lamb) that energy of 2s 1/2 and 2p 1/2 is not the same. It was also discovered (Rabi & Kusch) that hyperfine splitting of the 1s state in hydrogen atom has an anomalous contribution, which was latter understood as a correction to the electron g factor (g – 2  0).

13. Hydrogen atom & QED Two of these three predictions happened to be not absolutely correct. It was discovered (Lamb) that energy of 2s 1/2 and 2p 1/2 is not the same. It was also discovered (Rabi & Kusch) that hyperfine splitting of the 1s state in hydrogen atom has an anomalous contribution, which was latter understood as a correction to the electron g factor (g – 2  0). It was indeed expected that quantum mechanics with classical description of photons is not complete. However, all attempts to reach appropriate results were unsuccessful for a while.

14. Hydrogen atom & QED Two of these three predictions happened to be not absolutely correct. It was discovered (Lamb) that energy of 2s 1/2 and 2p 1/2 is not the same. It was also discovered (Rabi & Kusch) that hyperfine splitting of the 1s state in hydrogen atom has an anomalous contribution, which was latter understood as a correction to the electron g factor (g – 2  0). It was indeed expected that quantum mechanics with classical description of photons is not complete. However, all attempts to reach appropriate results were unsuccessful for a while. Trying to resolve problem of the Lamb shift and anomalous magnetic moments an effective QED approach was created.

15. Hydrogen energy levels

16. Rydberg constant The Rydberg constant that is the most accurately measured fundamental constant. The improvement of accuracy has been nearly 4 orders of magnitude in 30 years. 1973 7.5 × 10 -8 1986 1.2 × 10 -9 1998 7.6 × 10 -12 2002 6.6 × 10 -12

17. Rydberg constant The Rydberg constant that is the most accurately measured fundamental constant. The improvement of accuracy has been nearly 4 orders of magnitude in 30 years. The 2002 value is Ry = 10 973 731.568 525(73) m -1. The progress of the last period was possible because of two- photon Doppler free spectrocsopy. 1973 7.5 × 10 -8 1986 1.2 × 10 -9 1998 7.6 × 10 -12 2002 6.6 × 10 -12 1998

18. Rydberg constant The Rydberg constant that is the most accurately measured fundamental constant. The improvement of accuracy has been nearly 4 orders of magnitude in 30 years. The 2002 value is Ry = 10 973 731.568 525(73) m -1. The progress of the last period was possible because of two- photon Doppler free spectrocsopy. 1973 7.5 × 10 -8 1986 1.2 × 10 -9 1998 7.6 × 10 -12 2002 6.6 × 10 -12 CODATA 2002

19. Two-photon Doppler-free spectroscopy of hydrogen atom Two-photon spectroscopy is free of linear Doppler effect. That makes cooling relatively not too important problem. v, k, - k

20. Two-photon Doppler-free spectroscopy of hydrogen atom Two-photon spectroscopy is free of linear Doppler effect. That makes cooling relatively not too important problem. All states but 2s are broad because of the E1 decay. The widths decrease with increase of n. However, higher levels are badly accessible. Two-photon transitions double frequency and allow to go higher. v, k, - k

21. Doppler-free spectroscopy & Rydberg constant Two-photon spectroscopy involves a number of levels strongly affected by QED. In “old good time” we had to deal only with 2s Lamb shift. Theory for p states is simple since their wave functions vanish at r=0. Now we have more data and more unknown variable. How has one to deal with that?

22. Doppler-free spectroscopy & Rydberg constant Two-photon spectroscopy involves a number of levels strongly affected by QED. In “old good time” we had to deal only with 2s Lamb shift. Theory for p states is simple since their wave functions vanish at r=0. Now we have more data and more unknown variable. The idea is based on theoretical study of  (2) = L 1s – 2 3 × L 2s

23. Doppler-free spectroscopy & Rydberg constant Two-photon spectroscopy involves a number of levels strongly affected by QED. In “old good time” we had to deal only with 2s Lamb shift. Theory for p states is simple since their wave functions vanish at r=0. Now we have more data and more unknown variable. The idea is based on theoretical study of  (2) = L 1s – 2 3 × L 2s which we understand much better since any short distance effect vanishes for  (2).

24. Doppler-free spectroscopy & Rydberg constant Two-photon spectroscopy involves a number of levels strongly affected by QED. In “old good time” we had to deal only with 2s Lamb shift. Theory for p states is simple since their wave functions vanish at r=0. Now we have more data and more unknown variable. The idea is based on theoretical study of  (2) = L 1s – 2 3 × L 2s which we understand much better since any short distance effect vanishes for  (2). Theory of p and d states is also simple.

25. Doppler-free spectroscopy & Rydberg constant Two-photon spectroscopy involves a number of levels strongly affected by QED. In “old good time” we had to deal only with 2s Lamb shift. Theory for p states is simple since their wave functions vanish at r=0. Now we have more data and more unknown variable. The idea is based on theoretical study of  (2) = L 1s – 2 3 × L 2s which we understand much better since any short distance effect vanishes for  (2). Theory of p and d states is also simple. Eventually the only unknow QED variable is the 1s Lamb shift L 1s.

26. Lamb shift (2s 1/2 – 2p 1/2 ) in the hydrogen atom theory vs. experiment

27. Lamb shift (2s 1/2 – 2p 1/2 ) in the hydrogen atom theory vs. experiment LS: direct measurements of the 2s 1/2 – 2p 1/2 splitting. Sokolov-&-Yakovlev’s result (2 ppm) is excluded because of possible systematic effects. The best included result is from Lundeen and Pipkin (~10 ppm).

28. Lamb shift (2s 1/2 – 2p 1/2 ) in the hydrogen atom theory vs. experiment FS: measurement of the 2p 3/2 – 2s 1/2 splitting. The Lamb shift is about of 10% of this effects. The best result leads to uncertainty of ~ 10 ppm for the Lamb shift.

29. Lamb shift (2s 1/2 – 2p 1/2 ) in the hydrogen atom theory vs. experiment OBF: the first generation of optical measurements. They were a relative measurements with frequencies different by a nearly integer factor. Yale: 1s-2s and 2s-4p Garching: 1s-2s and 2s- 4s Paris: 1s-3s and 2s-6s The result was reached through measurement of a beat frequency such as f(1s-2s)-4×f(2s-4s).

30. Lamb shift (2s 1/2 – 2p 1/2 ) in the hydrogen atom theory vs. experiment The most accurate result is a comparison of independent absolute measurements: Garching: 1s-2s Paris: 2s  n=8-12

31. Lamb shift (2s 1/2 – 2p 1/2 ) in the hydrogen atom theory vs. experimentUncertainties: Experiment: 2 ppm QED: 2 ppm Proton size 10 ppm

32. Lamb shift in hydrogen: theoretical uncertainty Uncertainties: Experiment: 2 ppm QED: 2 ppm Proton size 10 ppm The QED uncertainty can be even higher because of bad convergence of (Z  ) expansion of two- look corrections. An exact in (Z  ) calculation is needed but may be not possible for now.

33. Lamb shift in hydrogen: theoretical uncertainty Uncertainties: Experiment: 2 ppm QED: 2 ppm Proton size 10 ppm The scattering data claimed a better accuracy (3 ppm), however, we should not completely trust them. It is likely that we need to have proton charge radius obtained in some other way (e.g. via the Lamb shift in muonic hydrogen – in the way at PSI).

34. Hyperfine structure in hydrogen & proton structure Hyperfine structure is a relativistic effect ~ v 2 /c 2

35. Hyperfine structure in hydrogen & proton structure Hyperfine structure is a relativistic effect ~ v 2 /c 2 and thus more sensitive to nuclear structure effects than the Lamb shift

36. Hyperfine structure in hydrogen & proton structure Hyperfine structure is a relativistic effect ~ v 2 /c 2 and thus more sensitive to nuclear structure effects than the Lamb shift, which involve for HFS relativistic momentum transfer.

37. Hyperfine structure in hydrogen & proton structure Hyperfine structure is a relativistic effect ~ v 2 /c 2 and thus more sensitive to nuclear structure effects than the Lamb shift, which involve for HFS relativistic momentum transfer. The bound state QED corrections to hydrogen HFS contributes 23 ppm.

38. Hyperfine structure in hydrogen & proton structure Hyperfine structure is a relativistic effect ~ v 2 /c 2 and thus more sensitive to nuclear structure effects than the Lamb shift, which involve for HFS relativistic momentum transfer. The bound state QED corrections to hydrogen HFS contributes 23 ppm. The nuclear structure term is about 40 ppm.

39. Hyperfine structure in hydrogen & proton structure Hyperfine structure is a relativistic effect ~ v 2 /c 2 and thus more sensitive to nuclear structure effects than the Lamb shift, which involve for HFS relativistic momentum transfer. The bound state QED corrections to hydrogen HFS contributes 23 ppm. The nuclear structure (NS) term is about 40 ppm. Three main NS efects:

40. Hyperfine structure in hydrogen & proton structure Hyperfine structure is a relativistic effect ~ v 2 /c 2 and thus more sensitive to nuclear structure effects than the Lamb shift, which involve for HFS relativistic momentum transfer. The bound state QED corrections to hydrogen HFS contributes 23 ppm. The nuclear structure (NS) term is about 40 ppm. Three main NS efects: nuclear recoil effects contribute 5 ppm and slightly depend on NS;

41. Hyperfine structure in hydrogen & proton structure Hyperfine structure is a relativistic effect ~ v 2 /c 2 and thus more sensitive to nuclear structure effects than the Lamb shift, which involve for HFS relativistic momentum transfer. The bound state QED corrections to hydrogen HFS contributes 23 ppm. The nuclear structure (NS) term is about 40 ppm. Three main NS efects: nuclear recoil effects contribute 5 ppm and slightly depend on NS; distributions of electric charge and magnetic moment (so called Zemach correction) is 40 ppm

42. Hyperfine structure in hydrogen & proton structure Hyperfine structure is a relativistic effect ~ v 2 /c 2 and thus more sensitive to nuclear structure effects than the Lamb shift, which involve for HFS relativistic momentum transfer. The bound state QED corrections to hydrogen HFS contributes 23 ppm. The nuclear structure (NS) term is about 40 ppm. Three main NS efects: nuclear recoil effects contribute 5 ppm and slightly depend on NS; distributions of electric charge and magnetic moment (so called Zemach correction) is 40 ppm and gives the biggest uncertainty of 6 ppm because of lack of magnetic radius;

43. Hyperfine structure in hydrogen & proton structure Hyperfine structure is a relativistic effect ~ v 2 /c 2 and thus more sensitive to nuclear structure effects than the Lamb shift, which involve for HFS relativistic momentum transfer. The bound state QED corrections to hydrogen HFS contributes 23 ppm. The nuclear structure (NS) term is about 40 ppm. Three main NS efects: nuclear recoil effects contribute 5 ppm and slightly depend on NS; distributions of electric charge and magnetic moment (so called Zemach correction) is 40 ppm and gives the biggest uncertainty of 6 ppm because of lack of magnetic radius; proton polarizability contributes below 4 ppm and is known badly.

44. Hyperfine structure in light atoms Bound state QED term does not include anomalous magnetic moment of electron. The nuclear structure (NS) effects in all conventional light hydrogen-like atoms are bigger than BS QED term. NS terms are known very badly. Bound State QED Nuclear Structure Hydrogen23 ppm- 33 ppm Deuterium23 ppm138 ppm Tritium23 ppm- 36 ppm 3 He + 108 ppm- 213 ppm QED and nuclear effects

45. Hyperfine structure in light atoms The nuclear structure effects are known very badly. hydrogen - the uncertainty for the nuclear effects is about 15% being caused by a badly known distribution of the magnetic moment inside the proton and by proton polarizability effects; Bound State QED Nuclear Structure Hydrogen23 ppm- 33 ppm Deuterium23 ppm138 ppm Tritium23 ppm- 36 ppm 3 He + 108 ppm- 213 ppm QED and nuclear effects

46. Hyperfine structure in light atoms The nuclear structure effects are known very badly. deuterium - the corrections was calculated, but the uncertainty was not presented; Bound State QED Nuclear Structure Hydrogen23 ppm- 33 ppm Deuterium23 ppm138 ppm Tritium23 ppm- 36 ppm 3 He + 108 ppm- 213 ppm QED and nuclear effects

47. Hyperfine structure in light atoms The nuclear structure effects are known very badly. tritium - no result has been obtained up to date; helium-3 ion - no results has been obtained up to date Bound State QED Nuclear Structure Hydrogen23 ppm- 33 ppm Deuterium23 ppm138 ppm Tritium23 ppm- 36 ppm 3 He + 108 ppm- 213 ppm QED and nuclear effects

48. HFS without the nuclear structure There are few options to avoid nuclear structure effects: structure-free nucleus cancellation of the NS contributions combining two values

49. HFS without the nuclear structure There are few options to avoid nuclear structure effects: structure-free nucleus cancellation of the NS contributions combining two values Muonium: Muon, an unstable particle (lifetime ~ 2  s) serves as a nucleus. Muon mass is ~ 1/9 of proton mass.

50. HFS without the nuclear structure There are few options to avoid nuclear structure effects: structure-free nucleus cancellation of the NS contributions combining two values Muonium: Muon, an unstable particle (lifetime ~ 2  s), serves as a nucleus. Muon mass is ~ 1/9 of proton mass. Positronium: Positron is a nucleus. The atom is unstable (below 1  s). It is light and hard to cool, but the recoil effects are enhanced.

51. HFS without the nuclear structure There are few options to avoid nuclear structure effects: structure-free nucleus cancellation of the NS contributions combining two values The leading nuclear contributions are of the form:  E = A × |  nl (0)| 2

52. HFS without the nuclear structure There are few options to avoid nuclear structure effects: structure-free nucleus cancellation of the NS contributions combining two values The leading nuclear contributions are of the form:  E = A × |  nl (0)| 2 Coefficient determined by interaction with nucleus

53. HFS without the nuclear structure There are few options to avoid nuclear structure effects: structure-free nucleus cancellation of the NS contributions combining two values The leading nuclear contributions are of the form:  E = A × |  nl (0)| 2 wave function at r = 0

54. HFS without the nuclear structure The leading nuclear contributions are of the form:  E = A × |  nl (0)| 2. The coefficient A is nucleus-dependent and state-independent. The wave function is nucleus-independent state-dependent. For the s states: |  nl (0)| 2 = (Z  ) 3 m 3 /  n 3. What can we change in  nl ?

55. HFS without the nuclear structure The leading nuclear contributions are of the form:  E = A × |  nl (0)| 2. The coefficient A is nucleus-dependent and state-independent. The wave function is nucleus-independent state-dependent. For the s states: |  nl (0)| 2 = (Z  ) 3 m 3 /  n 3. m is the mass of orbiting particle: may be electron; muon.

56. HFS without the nuclear structure The leading nuclear contributions are of the form:  E = A × |  nl (0)| 2. The coefficient A is nucleus-dependent and state-independent. The wave function is nucleus-independent state-dependent. For the s states: |  nl (0)| 2 = (Z  ) 3 m 3 /  n 3. n is the principal quantum number; may be 1 (for the 1s state); 2 (for the 2s state).

57. Comparison of HFS in 1s and 2s states Theory of D 21 = 8 × E HFS (2s) – E HFS (1s) [kHz] HydrogenDeuteriumHelium-3 ion QED3 48.937 11.305 6– 1 189.252 QED3 is QED calculations up to the third order of expansion in any combinations of , (Z  ) or m/M – those are only corrections known for a while.

58. Comparison of HFS in 1s and 2s states Theory of D 21 = 8 × E HFS (2s) – E HFS (1s) [kHz] HydrogenDeuteriumHelium-3 ion QED3 48.937 11.305 6– 1 189.252 (Z  ) 4 0.006 0.0013 – 0.543 The only known 4 th order term was the (Z  ) 4 term.

59. Comparison of HFS in 1s and 2s states Theory of D 21 = 8 × E HFS (2s) – E HFS (1s) [kHz] HydrogenDeuteriumHelium-3 ion QED3 48.937 11.305 6– 1 189.252 (Z  ) 4 0.0060.0013– 0.543 QED40.018(3) 0.004 3(5) – 1.137(53) However, the (Z  ) 4 term is only a part of 4 th contributions.

60. Comparison of HFS in 1s and 2s states Theory of D 21 = 8 × E HFS (2s) – E HFS (1s) [kHz] HydrogenDeuteriumHelium-3 ion QED3 48.937 11.305 6– 1 189.252 QED40.018(3) 0.004 3(5) – 1.137(53) NS – 0.002 0.002 6(2) 0.317(36) Theo48.953(3)11.312 5(5)–1 190.067(63) The new 4 th order terms and recently found higher order nuclear size contributions are not small.

61. Comparison of HFS in 1s and 2s states Theory of D 21 = 8 × E HFS (2s) – E HFS (1s) [kHz] HydrogenDeuteriumHelium-3 ion QED3 48.937 11.305 6– 1 189.252 QED40.018(3) 0.004 3(5) – 1.137(53) NS – 0.002 0.002 6(2) 0.317(36) Theo48.953(3)11.312 5(5)–1 190.083(63) Exp unc 0.23 0.16 0.073

62. QED tests in microwave Lamb shift used to be measured either as a splitting between 2s 1/2 and 2p 1/2 (1057 MHz) or a big contribution into the fine splitting 2p 3/2 – 2s 1/2 11 THz (fine structure). HFS was measured in 1s state of hydrogen (1420 MHz) and 2s state (177 MHz). All four transitions are RF transitions.

63. QED tests in microwave Lamb shift used to be measured either as a splitting between 2s 1/2 and 2p 1/2 (1057 MHz) 2s 1/2 2p 3/2 2p 1/2 Lamb shift: 1057 MHz (RF)

64. QED tests in microwave Lamb shift used to be measured either as a splitting between 2s 1/2 and 2p 1/2 (1057 MHz) or a big contribution into the fine splitting 2p 3/2 – 2s 1/2 11 THz (fine structure). 2s 1/2 2p 3/2 2p 1/2 Fine structure: 11 050 MHz (RF)

65. QED tests in microwave & optics Lamb shift used to be measured either as a splitting between 2s 1/2 and 2p 1/2 (1057 MHz) or a big contribution into the fine splitting 2p 3/2 – 2s 1/2 11 THz (fine structure). However, the best fesult for the Lamb shift has been obtained up to now from UV transitions (such as 1s – 2s). 2s 1/2 2p 3/2 2p 1/2 1s 1/2 RF 1s – 2s: UV

66. QED tests in microwave HFS was measured in 1s state of hydrogen (1420 MHz) 1s 1/2 (F=0) 1s 1/2 (F=1) 1s HFS: 1420 MHz

67. QED tests in microwave HFS was measured in 1s state of hydrogen (1420 MHz) and 2s state (177 MHz). 2s 1/2 (F=0) 1s 1/2 (F=0) 1s 1/2 (F=1) 2s HFS: 177 MHz

68. QED tests in microwave & optics HFS was measured in 1s state of hydrogen (1420 MHz) and 2s state (177 MHz). However, the best result for the 2s HFS was achieved at MPQ from a comparison of two UV two-photon 1s-2s transitions: for singlet (F=0) and triplet (F=1). The best result for D atom comes also from optics. 2s 1/2 1s 1/2 (F=0) 1s 1/2 (F=1)

69. 2s HFS: theory vs experiment The 1s HFS interval was measured for a number of H-like atoms; the 2s HFS interval was done only for the hydrogen atom, the deuterium atom, the helium-3 ion.

70. 2s HFS: theory vs experiment The 1s HFS interval was measured for a number of H-like atoms; the 2s HFS interval was done only for the hydrogen atom, the deuterium atom, the helium-3 ion.

71. 2s HFS: theory vs experiment The 1s HFS interval was measured for a number of H-like atoms; the 2s HFS interval was done only for the hydrogen atom, the deuterium atom, the helium-3 ion.

72. 2s HFS: theory vs experiment The 1s HFS interval was measured for a number of H-like atoms; the 2s HFS interval was done only for the hydrogen atom, the deuterium atom, the helium-3 ion.

73. Muonium hyperfine splitting [kHz] EFEF 4 459 031.88(50) (g-2) e 5170.93 QED2 – 873.15 QED3 – 26.41 QED4– 0.55(22) Hadr 0.24 Weak – 0.07 Theo4 463 302.73(55) Exp 4 463 302.78(5)

74. Muonium hyperfine splitting [kHz] The leading term (Fermi energy) is defined as a result of a non-relativistic interaction of electron (g=2) and muon: E F = 16/3  2 × cRy ×   /  B ×(m r /m) 3 The uncertainty comes from   /  B. EFEF 4 459 031.88(50) (g-2) e 5170.93 QED2 – 873.15 QED3 – 26.41 QED4– 0.55(22) Hadr 0.24 Weak – 0.07 Theo4 463 302.73(55) Exp 4 463 302.78(5)

75. Muonium hyperfine splitting [kHz] QED contributions up to the 3 rd order of expansion in either of small parameters , (Z  ) or m/M are well known. EFEF 4 459 031.88(50) (g-2) e 5170.93 QED2 – 873.15 QED3 – 26.41 QED4– 0.55(22) Hadr 0.24 Weak – 0.07 Theo4 463 302.73(55) Exp 4 463 302.78(5)

76. Muonium hyperfine splitting [kHz] The higher order QED terms (QED4) are similar to those for D 21. The uncertainty comes from recoil effects. EFEF 4 459 031.88(50) (g-2) e 5170.93 QED2 – 873.15 QED3 – 26.41 QED4– 0.55(22) Hadr 0.24 Weak – 0.07 Theo4 463 302.73(55) Exp 4 463 302.78(5)

77. Muonium hyperfine splitting [kHz] Non-QED effects: Hadronic contributions are known with appropriate accuracy. Effects of the weak interactions are well under control. EFEF 4 459 031.88(50) (g-2) e 5170.93 QED2 – 873.15 QED3 – 26.41 QED4– 0.55(22) Hadr 0.24 Weak – 0.07 Theo4 463 302.73(55) Exp 4 463 302.78(5)

78. Muonium hyperfine splitting [kHz] Theory is in an agreement with experiment. The theoretical uncertainty budget is the leading term and muon magnetic moment – 0.50 kHz; the higher order QED corrections (4 th order) – 0.22 kHz. EFEF 4 459 031.88(50) (g-2) e 5170.93 QED2 – 873.15 QED3 – 26.41 QED4– 0.55(22) Hadr 0.24 Weak – 0.07 Theo4 463 302.73(55) Exp 4 463 302.78(5)

79. Positronium spectroscopy & Recoil effects Positronium offers a unique opportunity: recoil effects are enhanced

80. Positronium spectroscopy & Recoil effects Positronium offers a unique opportunity: recoil effects are enhanced and relatively low accuracy is sufficient for crucial tests.

81. Positronium spectroscopy & Recoil effects Positronium offers a unique opportunity: recoil effects are enhanced and relatively low accuracy is sufficient for crucial tests. EFEF 204 386.6 QED1 – 1 005.5 QED2 11.8 QED3 – 1.2(5) Theo203 391.7(5) Exp203 389.1(7) Positronium HFS [MHz] That is the same kind of corrections as QED4 for muonium HFS.

82. Positronium spectroscopy & Recoil effects Positronium offers a unique opportunity: recoil effects are enhanced and relatively low accuracy is sufficient for crucial tests. That allows to do QED tests without any determination of fundamental constants. EFEF 204 386.6 QED1 – 1 005.5 QED2 11.8 QED3 – 1.2(5) Theo203 391.7(5) Exp203 389.1(7) Positronium HFS [MHz]

83. Positronium spectrum: theory vs experiment 1s hyperfine structure1s-2s interval

84. Precision tests QED with the HFS H, D 21 48.953(3)49.13(13) H, D 21 48.53(23) H, D 21 49.13(40) D, D 21 11.312 5(5)11.16(16) D, D 21 11.28(6) Accuracy in H and D is still not high enough to test QED. Units are kHz Theory Experiment

85. Precision tests QED with the HFS Units are kHz H, D 21 48.953(3)49.13(13) H, D 21 48.53(23) H, D 21 49.13(40) D, D 21 11.312 5(5)11.16(16) D, D 21 11.28(6) 3 He +, D 21 – 1 190.083(63)– 1 189.979(71) 3 He +, D 21 – 1 190.1(16) Accuracy in helium ion is much higher.

86. Precision tests QED with the HFS Units are still kHz H, D 21 48.953(3)49.13(13) H, D 21 48.53(23) H, D 21 49.13(40) D, D 21 11.312 5(5)11.16(16) D, D 21 11.28(6) 3 He +, D 21 – 1 190.083(63)– 1 189.979(71) 3 He +, D 21 – 1 190.1(16) Mu, 1s HFS4 463 302.88(6)4 463 302.78(5) Muonium HFS is also obtained with a high accuracy.

87. Precision tests QED with the HFS H, D 21 48.953(3)49.13(13) H, D 21 48.53(23) H, D 21 49.13(40) D, D 21 11.312 5(5)11.16(16) D, D 21 11.28(6) 3 He +, D 21 – 1 190.083(63)– 1 189.979(71) 3 He +, D 21 – 1 190.1(16) Mu, 1s HFS4 463 302.88(6)4 463 302.78(5) Ps, 1s HFS203 391.7(5)203 389.10(7) Ps, 1s HFS203 397.5(16) Units are kHz Units for positronium are MHz

88. Precision tests QED with the HFS Units are kHz for all but positronium (MHz). H, D 21 48.953(3)49.13(13)1.40.09 H, D 21 48.53(23)– 1.80.16 H, D 21 49.13(40)0.40.28 D, D 21 11.312 5(5)11.16(16)– 1.00.49 D, D 21 11.28(6)-0.6 3 He +, D 21 – 1 190.083(63)– 1 189.979(71)1.100.01 3 He +, D 21 – 1 190.1(16)0.00.18 Mu, 1s HFS4 463 302.88(6)4 463 302.78(5)– 0.20.11 Ps, 1s HFS203 391.7(5)203 389.10(7)– 2.94.4 Ps, 1s HFS203 397.5(16)– 2.58.2 Shift/sigma

89. Precision tests QED with the HFS Units are kHz for all but positronium (MHz). H, D 21 48.953(3)49.13(13)1.40.09 H, D 21 48.53(23)– 1.80.16 H, D 21 49.13(40)0.40.28 D, D 21 11.312 5(5)11.16(16)– 1.00.49 D, D 21 11.28(6)-0.60.29 3 He +, D 21 – 1 190.083(63)– 1 189.979(71)1.100.01 3 He +, D 21 – 1 190.1(16)0.00.18 Mu, 1s HFS4 463 302.88(6)4 463 302.78(5)– 0.20.11 Ps, 1s HFS203 391.7(5)203 389.10(7)– 2.94.4 Ps, 1s HFS203 397.5(16)– 2.58.2 Sigma/E F

90. Problems of bound state QED: Three parameters  is a QED parameter. It shows how many QED loops are involved. Z  is strength of the Coulomb interaction which bounds the atom m/M is the recoil parameter

91. Problems of bound state QED: Three parameters of bound state QED:  is a QED parameter. It shows how many QED loops are involved. Z  is strength of the Coulomb interaction which bounds the atom m/M is the recoil parameter QED expansions are an asymptotic ones. They do not converge. That means that with real  after calculation of 1xx terms we will find that #1xx+1 is bigger than #1xx.

92. Problems of bound state QED: Three parameters of bound state QED:  is a QED parameter. It shows how many QED loops are involved. Z  is strength of the Coulomb interaction which bounds the atom m/M is the recoil parameter QED expansions are an asymptotic ones. They do not converge. That means that with real  after calculation of 1xx terms we will find that #1xx+1 is bigger than #1xx. However, bound state QED calculations used to be only for one- and two- loop contributions.

93. Problems of bound state QED: Three parameters of bound state QED:  is a QED parameter. It shows how many QED loops are involved. Z  is strength of the Coulomb interaction which bounds the atom m/M is the recoil parameter Hydrogen-like gold or bismuth are with Z  ~ 1. That is not good. However, Z  «1 is also not good!

94. Problems of bound state QED: Three parameters of bound state QED:  is a QED parameter. It shows how many QED loops are involved. Z  is strength of the Coulomb interaction which bounds the atom m/M is the recoil parameter Hydrogen-like gold or bismuth are with Z  ~ 1. That is not good. However, Z  « 1 is also not good! Limit is Z  = 0 related to an unbound atom.

95. Problems of bound state QED: Three parameters of bound state QED:  is a QED parameter. It shows how many QED loops are involved. Z  is strength of the Coulomb interaction which bounds the atom m/M is the recoil parameter Hydrogen-like gold or bismuth are with Z  ~ 1. That is not good. However, Z  « 1 is also not good! Limit is Z  = 0 related to an unbound atom. The results contain big logarithms (ln1/Z  ~ 5) and large numerical coefficients.

96. Problems of bound state QED: Three parameters of bound state QED:  is a QED parameter. It shows how many QED loops are involved. Z  is strength of the Coulomb interaction which bounds the atom m/M is the recoil parameter For positronium m/M = 1. Calculations should be done exactly in m/M.

97. Problems of bound state QED: Three parameters of bound state QED:  is a QED parameter. It shows how many QED loops are involved. Z  is strength of the Coulomb interaction which bounds the atom m/M is the recoil parameter For positronium m/M = 1. Calculations should be done exactly in m/M. Limit m/M «1 is a bad limit. It is related to a charged “neutrino” (m=0).

98. Problems of bound state QED: Three parameters of bound state QED:  is a QED parameter. It shows how many QED loops are involved. Z  is strength of the Coulomb interaction which bounds the atom m/M is the recoil parameter For positronium m/M = 1. Calculations should be done exactly in m/M. Limit m/M «1 is a bad limit. It is related to a charged “neutrino” (m=0). The problems in calculations: appearance of big logarithms (ln(M/m)~5 in muonium).

99. Problems of bound state QED: Three parameters of bound state QED:  is a QED parameter. It shows how many QED loops are involved. Z  is strength of the Coulomb interaction which bounds the atom m/M is the recoil parameter All three parameters are not good parameters. However, it is not possible to do calculations exact for even two of them. We have to expand. Any expansion contains some terms and leave the others unknown. The problem of accuracy is a proper estimation of unknown terms.

100. Uncertainty of theoretical calculations Uncertainty in muonium HFS is due to QED4 corrections.

101. Uncertainty of theoretical calculations Uncertainty in muonium HFS is due to QED4 corrections. Uncertainty of positronium HFS and 1s-2s interval are due to QED3.

102. Uncertainty of theoretical calculations Uncertainty in muonium HFS is due to QED4 corrections. Uncertainty of positronium HFS and 1s-2s interval are due to QED3. They are the same since one of parameters in QED4 is m/M and so these corrections are recoil corrections.

103. Uncertainty of theoretical calculations Uncertainty in muonium HFS is due to QED4 corrections. Uncertainty of positronium HFS and 1s-2s interval are due to QED3. They are the same since one of parameters in QED3 is m/M and so these corrections are recoil corrections. Uncertainty of the hydrogen Lamb shift is due to higher- order two-loop self energy.

104. Uncertainty of theoretical calculations Uncertainty in muonium HFS is due to QED4 corrections. Uncertainty of positronium HFS and 1s-2s interval are due to QED3. They are the same since one of parameters in QED3 is mainly m/M and so these corrections are recoil corrections. Uncertainty of the hydrogen Lamb shift is due to higher- order two-loop self energy. Uncertainty of D 21 in He + involves both: recoil QED4 and higher-order two-loop effects.

105. Uncertainty of theoretical calculations and further tests Uncertainty in muonium HFS is due to QED4 corrections. Uncertainty of positronium HFS and 1s-2s interval are due to QED3. They are the same since one of parameters in QED3 is mainly m/M and so these corrections are recoil corrections. Uncertainty of the hydrogen Lamb shift is due to higher- order two-loop self energy. We hope that accuracy of D 21 in H and D will be improved, the He + will be checked and may be an experiment of Li ++ will be done.

106. Precision physics of simple atoms & QED There are four basic sources of uncertainty: experiment; pure QED theory; nuclear structure and hadronic contributions; fundamental constants.

107. Precision physics of simple atoms & QED There are four basic sources of uncertainty: experiment; pure QED theory; nuclear structure and hadronic contributions; fundamental constants. For hydorgen-like atoms and free particles pure QED theory is never a limiting factor for a comparison of theory and experiment. For helium QED is still a limiting factor.

109. Muonium hyperfine splitting & the fine structure constant  Instead of a comparison of theory and experiment we can check if  from is consistent with other results.

110. Muonium hyperfine splitting & the fine structure constant  Instead of a comparison of theory and experiment we can check if  from is consistent with other results. The muonium result

111. Muonium hyperfine splitting & the fine structure constant  Instead of a comparison of theory and experiment we can check if  from is consistent with other results. The muonium result is consistent with others such as from electron g-2 but less accurate.

112. How one can measure  ? QED (g-2) e – the best! Bound state QED Mu HFS & m  /m e Helium FS (excluded) Atomic physics h/m (cesium) & m e /m p – the second best! Avogadro project h/m (neutron) & Si lattice spacing Electric standards gyromagnetic ratio at low field measured as  p /K J R K ~  p /  B × h/m e ×  for proton (or helion)  p = 2  p /ħ K J = 2e/h R K = h/e 2 Calculable capacitor: a direct measurement of R K

113. Optical frequency measurements Length measurements are related to optics since RF has too large wave lengths for accurate measurements. Clocks used to be related to RF because of accurate frequency comparisons.

114. Optical frequency measurements Length measurements are related to optics since RF has too large wave lengths for accurate measurements. Clocks used to be related to RF because of accurate frequency comparisons. Now: clocks enter optics and because of more oscillations in a given period they are potentially more accurate. That is possible because of frequency comb technology which offers precision comparisons optics to optics and optics to RF.

115. Optical frequency measurements &  variations Length measurements are related to optics since RF has too large wave lengths for accurate measurements. Clocks used to be related to RF because of accurate frequency comparisons. Now: clocks enter optics and because of more oscillations in a given period they are potentially more accurate. That is possible because of frequency comb technology which offers precision comparisons optics to optics and optics to RF. Absolute determinations of optical frequencies is a way of practical realization of meter. Meantime comparing various optical transitions to cesium HFS we look for  variation at the level of few parts in 10 -15 yr -1. (The result is negaive.)

116. Progress in  variations since ACFC meeting (June 2003) Method: f = C 0 × c Ry × F(  )

117. Progress in  variations since ACFC meeting (June 2003) Method: f = C 0 × c Ry × F(  ) and thus d ln{f}/dt = d ln{cRy}/dt + A × d ln  /dt.

118. Progress in  variations since ACFC meeting (June 2003) Method: f = C 0 × c Ry × F(  ) d ln{f}/dt = d ln{cRy}/dt + A × d ln  /dt. Measurements: Optical transitions in Hg + (NIST), H (MPQ), Yb + (PTB) versus Cs HFS; Calcium is coming (PTB, NIST)

119. Progress in  variations since ACFC meeting (June 2003) Method: f = C 0 × c Ry × F(  ) d ln{f}/dt = d ln{cRy}/dt + A × d ln  /dt. Measurements: Optical transitions in Hg + (NIST), H (MPQ), Yb + (PTB) versus Cs HFS; Calcium is coming (PTB, NIST) Calculation of relativistic corrections (Flambaum, Dzuba): A = d lnF(  )/d ln 

120. Progress in  variations since ACFC meeting (June 2003) Method: f = C 0 × c Ry × F(  ) d ln{f}/dt = d ln{cRy}/dt + A × d ln  /dt. Measurements: Optical transitions in Hg + (NIST), H (MPQ), Yb + (PTB) versus Cs HFS; Calcium is coming (PTB, NIST) Calculation of relativistic corrections (Flambaum, Dzuba): A = d lnF(  )/d ln 

121. Progress in  variations since ACFC meeting (June 2003) Method: f = C 0 × c Ry × F(  ) d ln{f}/dt = d ln{cRy}/dt + A × d ln  /dt. Measurements: Optical transitions in Hg + (NIST), H (MPQ), Yb + (PTB) versus Cs HFS; Calcium is coming (PTB, NIST) Calculation of relativistic corrections (Flambaum, Dzuba): A = d lnF(  )/d ln 

122. Current laboratory constraints on variations of constants XVariation d lnX/dtModel  (-0.3±2.0)×10 -15 yr -1 -- {c Ry}(-2.1±3.1)×10 -15 yr -1 -- m e /m p (2.9±6.2)×10 -15 yr -1 Schmidt model p/ep/e (2.9±5.8)×10 -15 yr -1 Schmidt model gpgp (-0.1±0.5)×10 -15 yr -1 Schmidt model gngn (3±3)×10 -14 yr -1 Schmidt model

123. Optical frequency measurements &  variations For more detail on variation of constants:

124. Optical frequency measurements &  variations For more detail on variation of constants: Will appear in August

125. Contributors Theory: Muonium HFS (hadrons) Simon Eidelman Valery Shelyuto 2s HFS Volodya Ivanov Experiments: 2s H and D Hänsch´s group: Marc Fischer Peter Fendel Nikolai Kolachevsky Constraints: Ekkehard Peik (PTB) Victor Flambaum

126. Contributors and support Theory: Muonium HFS (hadrons) Simon Eidelman Valery Shelyuto 2s HFS Volodya Ivanov Experiments: 2s H and D T.W. Hänsch´s group: Marc Fischer Peter Fendel Nikolai Kolachevsky Constraints: Ekkehard Peik (PTB) Victor Flambaum Supported by RFBR, DFG, DAAD, Heareus etc

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