1. Precision tests of bound-state QED: Muonium HFS etc 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 the nuclear structure HFS without nuclear effects Muonium HFS theory: summary

3. Hydrogen energy levels

4. Lamb shift (2s 1/2 – 2p 1/2 ) in the hydrogen atom theory vs. experiment Uncertainties: Experiment: 2 ppm QED: 2 ppm Proton size: 10 ppm Progress: QED: calculation of higher order corrections Proton size: may be

5. 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 effects: nuclear recoil effects contribute 5 ppm and slightly depend on NS; magnetic momentdistribution of electric charge and magnetic moment (so called Zemach correction) is 40 ppm; proton polarizability.

6. 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

7. 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 wave function at r = 0 |  nl (0)| 2 = (Z  ) 3 m 3 /  n 3 n=1 (for the 1s state – the ground state) n=2 (for the 2s state – the metastable state)

8. 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.

9. 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.

10. 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.

11. 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.

12. 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

13. 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.

14. 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.

15. 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.

16. 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)

17. 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)

18. 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)

19. 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)

20. Muonium hyperfine splitting [kHz] Non-QED effects: Hadronic contributions are known with appropriate accuracy. Their accuracy sets an ultimate limit on ab inition QED tests. 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)

21. 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)

22. 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.

23. 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

24. 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.

25. 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.

26. 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

27. 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

28. 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

29. 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

30. 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.

31. 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.

32. 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.

33. 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).

34. 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).

35. 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.

36. 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.

37. 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.