Precision test of bound-state QED and the fine structure constant a

1
Precision test of bound-state QED and the fine
structure constant a

• 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 a 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
• En ½ a2mc2/n2
• 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
• En ½ a2mc2/n2
• no dependence on momentum (j).

• On a way to explain fine structure of some
  hydrogen lines which was due to a splitting
  between 2p1/2 and 2p3/2 (j1/2 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
• En ½ a2mc2/n2
• no dependence on momentum (j).

• On a way to explain fine structure of some
  hydrogen lines which was due to a splitting
  between 2p1/2 and 2p3/2 (j1/2 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
• En ½ a2mc2/n2
• no dependence on momentum (j).

• On a way to explain fine structure of some
  hydrogen lines which was due to a splitting
  between 2p1/2 and 2p3/2 (j1/2 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
• En ½ a2mc2/n2
• no dependence on momentum (j).

• On a way to explain fine structure of some
  hydrogen lines which was due to a splitting
  between 2p1/2 and 2p3/2 (j1/2 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
• En ½ a2mc2/n2
• no dependence on momentum (j).

• On a way to explain fine structure of some
  hydrogen lines which was due to a splitting
  between 2p1/2 and 2p3/2 (j1/2 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 2s1/2 and
  2p1/2 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 2s1/2 and
  2p1/2 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 2s1/2 and
  2p1/2 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 2s1/2 and
  2p1/2 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 2s1/2 and
  2p1/2 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.

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.

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.

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
n, k
n, - 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
n, k
n, - 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 r0.
• 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 r0.
• Now we have more data and more unknown variable.

• The idea is based on theoretical study of
• D(2) L1s 23 L2s

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 r0.
• Now we have more data and more unknown variable.

• The idea is based on theoretical study of
• D(2) L1s 23 L2s
• which we understand much better since any
  short distance effect vanishes for D(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 r0.
• Now we have more data and more unknown variable.

• The idea is based on theoretical study of
• D(2) L1s 23 L2s
• which we understand much better since any
  short distance effect vanishes for D(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 r0.
• Now we have more data and more unknown variable.

• The idea is based on theoretical study of
• D(2) L1s 23 L2s
• which we understand much better since any
  short distance effect vanishes for D(2).
• Theory of p and d states is also simple.
• Eventually the only unknow QED variable is the 1s
  Lamb shift L1s.

26
Lamb shift (2s1/2 2p1/2) in the hydrogen atom

• theory vs. experiment

27
Lamb shift (2s1/2 2p1/2) in the hydrogen atom

• theory vs. experiment

• LS direct measurements of the 2s1/2 2p1/2
  splitting.
• Sokolov--Yakovlevs result (2 ppm) is excluded
  because of possible systematic effects.
• The best included result is from Lundeen and
  Pipkin (10 ppm).

28
Lamb shift (2s1/2 2p1/2) in the hydrogen atom

• theory vs. experiment

• FS measurement of the 2p3/2 2s1/2 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 (2s1/2 2p1/2) 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)-4f(2s-4s).

30
Lamb shift (2s1/2 2p1/2) in the hydrogen atom

• theory vs. experiment

• The most accurate result is a comparison of
  independent absolute measurements
• Garching 1s-2s
• Paris 2s ? n8-12

31
Lamb shift (2s1/2 2p1/2) in the hydrogen atom

• theory vs. experiment

• Uncertainties
• 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 (Za) expansion of two-look
  corrections.
• An exact in (Za) 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
  v2/c2 

35
Hyperfine structure in hydrogen proton structure

• Hyperfine structure is a relativistic effect
  v2/c2 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
  v2/c2 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
  v2/c2 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
  v2/c2 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
  v2/c2 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
  v2/c2 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
  v2/c2 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
  v2/c2 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
  v2/c2 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
QED and nuclear effects

• 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.

45
Hyperfine structure in light atoms
QED and nuclear effects

• 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 

46
Hyperfine structure in light atoms
QED and nuclear effects

• The nuclear structure effects are known very
  badly.
• deuterium - the corrections was calculated, but
  the uncertainty was not presented

47
Hyperfine structure in light atoms
QED and nuclear effects

• 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

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 ms)
  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 ms),
  serves as a nucleus. Muon mass is 1/9 of proton
  mass.
• Positronium
• Positron is a nucleus. The atom is unstable
  (below 1 ms). 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
• DE 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
• DE 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
• DE A ?nl(0)2

wave function at r 0
54
HFS without the nuclear structure

• The leading nuclear contributions are of the
  form
• DE 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 (Za)3m3/pn3.
• What can we change in ?nl?

55
HFS without the nuclear structure

• The leading nuclear contributions are of the
  form
• DE 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 (Za)3m3/pn3.
• 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
• DE 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 (Za)3m3/pn3.
• 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 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.
58
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.
59
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.
60
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.
61
Comparison of HFS in 1s and 2s states

• Theory of D21 8 EHFS(2s) EHFS(1s) kHz

62
QED tests in microwave

• Lamb shift used to be measured either as a
  splitting between 2s1/2 and 2p1/2 (1057 MHz) or a
  big contribution into the fine splitting 2p3/2
  2s1/2 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 2s1/2 and 2p1/2 (1057 MHz)

2p3/2
2s1/2
2p1/2
Lamb shift 1057 MHz (RF)
64
QED tests in microwave

• Lamb shift used to be measured either as a
  splitting between 2s1/2 and 2p1/2 (1057 MHz) or a
  big contribution into the fine splitting 2p3/2
  2s1/2 11 THz (fine structure).

2p3/2
2s1/2
2p1/2
Fine structure 11 050 MHz (RF)
65
QED tests in microwave optics

• Lamb shift used to be measured either as a
  splitting between 2s1/2 and 2p1/2 (1057 MHz) or a
  big contribution into the fine splitting 2p3/2
  2s1/2 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).

2p3/2
2s1/2
RF
2p1/2
1s 2s UV
1s1/2
66
QED tests in microwave

• HFS was measured in 1s state of hydrogen (1420
  MHz)

1s HFS 1420 MHz
1s1/2 (F1)
1s1/2 (F0)
67
QED tests in microwave

• HFS was measured in 1s state of hydrogen (1420
  MHz) and 2s state (177 MHz).

2s1/2(F0)
2s1/2(F0)
2s HFS 177 MHz
1s1/2 (F1)
1s1/2 (F0)
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 (F0)
  and triplet (F1).
• The best result for D atom comes also from optics.

2s1/2
1s1/2 (F1)
1s1/2 (F0)
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
74
Muonium hyperfine splitting kHz

• The leading term (Fermi energy) is defined as a
  result of a non-relativistic interaction of
  electron (g2) and muon
• EF 16/3 a2 cRy mm/mB (mr/m)3
• The uncertainty comes from mm/mB .

75
Muonium hyperfine splitting kHz

• QED contributions up to the 3rd order of
  expansion in either of small parameters a, (Za)
  or m/M are well known.

76
Muonium hyperfine splitting kHz

• The higher order QED terms (QED4) are similar to
  those for D21.
• The uncertainty comes from recoil effects.

77
Muonium hyperfine splitting kHz

• Non-QED effects
• Hadronic contributions are known with appropriate
  accuracy.
• Effects of the weak interactions are well under
  control.

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 (4th order)
  0.22 kHz.

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.

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.

Positronium HFS MHz
83
Positronium spectrumtheory vs experiment

• 1s hyperfine structure

• 1s-2s interval

84
Precision tests QED with the HFS
Units are kHz
Experiment
Theory
Accuracy in H and D is still not high enough to
test QED.
85
Precision tests QED with the HFS
Units are kHz
Accuracy in helium ion is much higher.
86
Precision tests QED with the HFS
Units are still kHz
Muonium HFS is also obtained with a high accuracy.
87
Precision tests QED with the HFS
Units are kHz
Units for positronium are MHz
88
Precision tests QED with the HFS
Units are kHz for all but positronium (MHz).
Shift/sigma
89
Precision tests QED with the HFS
Units are kHz for all but positronium (MHz).
Sigma/EF
90
Problems of bound state QED

• 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

91
Problems of bound state QED

• 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 1xx1 is bigger than
  1xx.

92
Problems of bound state QED

• 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 1xx1 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
• 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!

94
Problems of bound state QED

• 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 0 related to an unbound atom.

95
Problems of bound state QED

• 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 0 related to an unbound atom.
• The results contain big logarithms (ln1/Za 5)
  and large numerical coefficients.

96
Problems of bound state QED

• 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.

97
Problems of bound state QED

• 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 (m0).

98
Problems of bound state QED

• 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 (m0).
• 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
• 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.

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 D21 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 D21 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.

108
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109
Muonium hyperfine splitting the fine structure
constant a

• Instead of a comparison of theory and experiment
  we can check if a from is consistent with other
  results.

110
Muonium hyperfine splitting the fine structure
constant a

• Instead of a comparison of theory and experiment
  we can check if a from is consistent with other
  results.
• The muonium result

111
Muonium hyperfine splitting the fine structure
constant a

• Instead of a comparison of theory and experiment
  we can check if a 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 a ?

• QED
• (g-2)e the best!
• Bound state QED
• Mu HFS mm/me
• Helium FS (excluded)
• Atomic physics
• h/m (cesium) me/mp the second best!
• Avogadro project
• h/m (neutron) Si lattice spacing

• Electric standards
• gyromagnetic ratio at low field measured as
• gp/KJRK mp/mB h/me a
• for proton (or helion)
• gp 2mp/h
• KJ 2e/h
• RK h/e2
• Calculable capacitor a direct measurement of RK

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 a 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 a variation at the level
  of few parts in 10-15 yr-1. (The result is
  negaive.)

116
Progress in a variations since ACFC meeting (June
2003)

• Method
• f C0 c Ry F(a)

117
Progress in a variations since ACFC meeting (June
2003)

• Method
• f C0 c Ry F(a)
• and thus
• d lnf/dt d lncRy/dt
• A d lna/dt.

118
Progress in a variations since ACFC meeting (June
2003)

• Method
• f C0 c Ry F(a)
• d lnf/dt d lncRy/dt
• A d lna/dt.
• Measurements
• Optical transitions in Hg (NIST), H (MPQ), Yb
  (PTB) versus Cs HFS
• Calcium is coming (PTB, NIST)

119
Progress in a variations since ACFC meeting (June
2003)

• Method
• f C0 c Ry F(a)
• d lnf/dt d lncRy/dt
• A d lna/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(a)/d lna

120
Progress in a variations since ACFC meeting (June
2003)

• Method
• f C0 c Ry F(a)
• d lnf/dt d lncRy/dt
• A d lna/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(a)/d lna

121
Progress in a variations since ACFC meeting (June
2003)

• Method
• f C0 c Ry F(a)
• d lnf/dt d lncRy/dt
• A d lna/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(a)/d lna

122
Current laboratory constraints on variations of
constants
123
Optical frequency measurements a variations

• For more detail on variation of constants

124
Optical frequency measurements a 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
• Hänschs 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. Hänschs group
• Marc Fischer
• Peter Fendel
• Nikolai Kolachevsky
• Constraints
• Ekkehard Peik (PTB)
• Victor Flambaum

Supported by RFBR, DFG, DAAD, Heareus etc
127
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128
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