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Precision tests of bound-state QED: Muonium HFS etc

Explore the latest advancements in investigating nuclear structure effects, hydrogen energy levels theory, and higher-order QED corrections in bound-state Quantum Electrodynamics. Learn about Muonium hyperfine splitting, HFS theory, and experimental uncertainties.

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Precision tests of bound-state QED: Muonium HFS etc

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  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. 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 Lamb shift (2s1/2 – 2p1/2) in the hydrogen atom

  5. Hyperfine structure is a relativistic effect~ v2/c2and thusmoresensitive to nuclear structure effects than the Lamb shift, which involve for HFS relativistic momentum transfer. Thebound state QEDcorrections 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; distribution of electric charge and magnetic moment (so called Zemach correction) is 40 ppm; proton polarizability. Hyperfine structure in hydrogen & proton structure

  6. 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. Hyperfine structure in light atoms QED and nuclear effects

  7. 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: DE = A×|nl(0)|2 HFS without the nuclear structure wave function at r=0 |nl(0)|2= (Za)3m3/pn3 coefficient determined by interaction with nucleus 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 D21 = 8 × EHFS(2s) – EHFS(1s) [kHz] QED3 is QED calculations up to the third order of expansion in any combinations of a,(Za) or m/M – those are only corrections known for a while.

  9. Comparison of HFS in 1s and 2s states Theory of D21 = 8 × EHFS(2s) – EHFS(1s) [kHz] The only known 4th order term was the (Za)4 term.

  10. Comparison of HFS in 1s and 2s states Theory of D21 = 8 × EHFS(2s) – EHFS(1s) [kHz] However, the (Za)4 term is only a part of 4th contributions.

  11. Comparison of HFS in 1s and 2s states Theory of D21 = 8 × EHFS(2s) – EHFS(1s) [kHz] The new 4th order terms and recently found higher order nuclear size contributions are not small.

  12. Comparison of HFS in 1s and 2s states Theory of D21 = 8 × EHFS(2s) – EHFS(1s) [kHz]

  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]

  17. The leading term (Fermi energy) is defined as a result of a non-relativistic interaction of electron (g=2) and muon: EF= 16/3 a2× cRy ×mm/mB×(mr/m)3 The uncertainty comes from mm/mB. Muonium hyperfine splitting [kHz]

  18. QED contributions up to the 3rd order of expansion in either of small parameters a,(Za) or m/M are well known. Muonium hyperfine splitting [kHz]

  19. The higher order QED terms (QED4) are similar to those for D21. The uncertainty comes from recoil effects. Muonium hyperfine splitting [kHz]

  20. 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. Muonium hyperfine splitting [kHz]

  21. 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 (4th order) – 0.22 kHz. Muonium hyperfine splitting [kHz]

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

  23. Precision tests QED with the HFS Units are kHz Experiment Theory Accuracy in H and D is still not high enough to test QED.

  24. Precision tests QED with the HFS Units are kHz Accuracy in helium ion is much higher.

  25. Precision tests QED with the HFS Units are still kHz Muonium HFS is also obtained with a high accuracy.

  26. Precision tests QED with the HFS Units are kHz Units for positronium are MHz

  27. Precision tests QED with the HFS Units are kHz for all but positronium (MHz). Shift/sigma

  28. Precision tests QED with the HFS Units are kHz for all but positronium (MHz). Sigma/EF

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

  30. Three parameters of bound state QED: a is a QED parameter. It shows how many QED loops are involved. Za 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 a 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. Problems of bound state QED:

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

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

  33. Three parameters of bound state QED: a is a QED parameter. It shows how many QED loops are involved. Za 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). Problems of bound state QED:

  34. Three parameters of bound state QED: a is a QED parameter. It shows how many QED loops are involved. Za 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). Problems of bound state QED:

  35. Three parameters of bound state QED: a is a QED parameter. It shows how many QED loops are involved. Za 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. Problems of bound state QED:

  36. 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 theoretical calculations Uncertainty ofD21 in He+involves both: recoil QED4 and higher-order two-loop effects.

  37. 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. Precision physics of simple atoms & QED

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