Download 12 - KVI

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)
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



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).
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 (j=1/2 and 3/2 is the
angular momentum) Dirac
introduced a relativisitic
equation.
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 (j=1/2 and 3/2 is the
angular momentum) Dirac
introduced a relativisitic
equation, which predicted
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 (j=1/2 and 3/2 is the
angular momentum) Dirac
introduced a relativisitic
equation, which predicted

existence of positron;
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 (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;
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 (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).
Hydrogen atom & QED
Two of these three predictions
happened to be not
absolutely correct.
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.
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.
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).
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.
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.
Hydrogen energy levels
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
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
The 2002 value is
Ry = 10 973 731.568 525(73) m-1.
The progress of the last period
was possible because of twophoton Doppler free
spectrocsopy.
1998
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
The 2002 value is
Ry = 10 973 731.568 525(73) m-1.
The progress of the last period
was possible because of twophoton Doppler free
spectrocsopy.
CODATA
2002
Two-photon Doppler-free
spectroscopy of hydrogen atom
Two-photon spectroscopy
v
n, k
n, - k
is free of linear Doppler
effect.
That makes cooling
relatively not too
important problem.
Two-photon Doppler-free
spectroscopy of hydrogen atom
Two-photon spectroscopy
v
n, k
n, - k
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.
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?
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
D(2) = L1s – 23× L2s
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
D(2) = L1s – 23× L2s
which we understand
much better since any
short distance effect
vanishes for D(2).
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
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.
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
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.
Lamb shift (2s1/2 – 2p1/2)
in the hydrogen atom
theory vs. experiment
Lamb shift (2s1/2 – 2p1/2)
in the hydrogen atom
theory vs. experiment

LS: direct
measurements of the
2s1/2 – 2p1/2 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).
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.
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 2s4s
 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).
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  n=8-12
Lamb shift (2s1/2 – 2p1/2)
in the hydrogen atom
theory vs. experiment
Uncertainties:
 Experiment: 2 ppm
 QED: 2 ppm
 Proton size 10 ppm
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 twolook corrections.
An exact in (Za)
calculation is needed
but may be not possible
for now.
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).
Hyperfine structure in
hydrogen & proton structure
• Hyperfine structure is a
relativistic effect ~ v2/c2
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
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.
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.
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.
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:
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;
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
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;
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.
Hyperfine structure
in light atoms
QED and nuclear effects
Bound
State
QED
Nuclear
Structure
23 ppm
- 33 ppm
Deuterium 23 ppm
138 ppm
Tritium
23 ppm
- 36 ppm
3He+
108 ppm
- 213 ppm
Hydrogen
• 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
Bound
State
QED
Nuclear
Structure
23 ppm
- 33 ppm
Deuterium 23 ppm
138 ppm
Tritium
23 ppm
- 36 ppm
3He+
108 ppm
- 213 ppm
Hydrogen
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;
Hyperfine structure
in light atoms
QED and nuclear effects
Bound
State
QED
Nuclear
Structure
23 ppm
- 33 ppm
Deuterium 23 ppm
138 ppm
Tritium
23 ppm
- 36 ppm
3He+
108 ppm
- 213 ppm
Hydrogen
The nuclear structure
effects are known very
badly.
• deuterium - the
corrections was
calculated, but the
uncertainty was not
presented;
Hyperfine structure
in light atoms
QED and nuclear effects
Bound
State
QED
Nuclear
Structure
23 ppm
- 33 ppm
Deuterium 23 ppm
138 ppm
Tritium
23 ppm
- 36 ppm
3He+
108 ppm
- 213 ppm
Hydrogen
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
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
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.
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.
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
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
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
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?
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.
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).
Comparison of HFS in 1s and
2s states
Theory of D21 = 8 × EHFS(2s) – EHFS(1s) [kHz]
Hydrogen
QED3
48.937
Deuterium
Helium-3 ion
11.305 6
– 1 189.252
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.
Comparison of HFS in 1s and
2s states
Theory of D21 = 8 × EHFS(2s) – EHFS(1s) [kHz]
Hydrogen
Deuterium
Helium-3 ion
QED3
48.937
11.305 6
– 1 189.252
(Za)4
0.006
0.0013
– 0.543
The only known 4th order term was the (Za)4 term.
Comparison of HFS in 1s and
2s states
Theory of D21 = 8 × EHFS(2s) – EHFS(1s) [kHz]
Hydrogen
QED3
48.937
Deuterium
Helium-3 ion
11.305 6
– 1 189.252
(Za)4
0.006
0.0013
– 0.543
QED4
0.018(3)
0.004 3(5)
– 1.137(53)
However, the (Za)4 term is only a part of 4th contributions.
Comparison of HFS in 1s and
2s states
Theory of D21 = 8 × EHFS(2s) – EHFS(1s) [kHz]
Hydrogen
QED3
QED4
NS
Theo
48.937
0.018(3)
– 0.002
48.953(3)
Deuterium
Helium-3 ion
11.305 6
– 1 189.252
0.004 3(5)
– 1.137(53)
0.002 6(2)
0.317(36)
11.312 5(5) –1 190.067(63)
The new 4th order terms and recently found higher order nuclear
size contributions are not small.
Comparison of HFS in 1s and
2s states
Theory of D21 = 8 × EHFS(2s) – EHFS(1s) [kHz]
Hydrogen
QED3
QED4
NS
Theo
Exp unc
48.937
0.018(3)
– 0.002
48.953(3)
0.23
Deuterium
Helium-3 ion
11.305 6
– 1 189.252
0.004 3(5)
– 1.137(53)
0.002 6(2)
0.317(36)
11.312 5(5) –1 190.083(63)
0.16
0.073
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.
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)
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)
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
QED tests in microwave

HFS was measured in 1s
state of hydrogen (1420
MHz)
1s HFS: 1420 MHz
1s1/2 (F=0)
1s1/2 (F=1)
QED tests in microwave

HFS was measured in 1s
state of hydrogen (1420
MHz) and 2s state (177
MHz).
2s1/2(F=0)
2s1/2(F=0)
2s HFS: 177 MHz
1s1/2 (F=0)
1s1/2 (F=1)
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.
2s1/2
1s1/2 (F=0)
1s1/2 (F=1)
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.
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.
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.
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.
Muonium hyperfine splitting [kHz]
EF
4 459 031.88(50)
(g-2)e
5170.93
QED2
– 873.15
QED3
– 26.41
QED4
Hadr
Weak
– 0.55(22)
0.24
– 0.07
Theo
4 463 302.73(55)
Exp
4 463 302.78(5)
Muonium hyperfine splitting [kHz]
EF
4 459 031.88(50)
(g-2)e
5170.93
QED2
– 873.15
QED3
– 26.41
QED4
Hadr
Weak
– 0.55(22)
0.24
– 0.07
Theo
4 463 302.73(55)
Exp
4 463 302.78(5)
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]
EF
4 459 031.88(50)
(g-2)e
5170.93
QED2
– 873.15
QED3
– 26.41
QED4
Hadr
Weak
– 0.55(22)
0.24
– 0.07
Theo
4 463 302.73(55)
Exp
4 463 302.78(5)
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]
EF
4 459 031.88(50)
(g-2)e
5170.93
QED2
– 873.15
QED3
– 26.41
QED4
Hadr
Weak
– 0.55(22)
0.24
– 0.07
Theo
4 463 302.73(55)
Exp
4 463 302.78(5)
The higher order QED terms
(QED4) are similar to those
for D21.
The uncertainty comes from
recoil effects.
Muonium hyperfine splitting [kHz]
EF
4 459 031.88(50)
(g-2)e
5170.93
QED2
– 873.15
QED3
– 26.41
QED4
Hadr
Weak
– 0.55(22)
0.24
– 0.07
Theo
4 463 302.73(55)
Exp
4 463 302.78(5)
Non-QED effects:


Hadronic contributions
are known with
appropriate accuracy.
Effects of the weak
interactions are well
under control.
Muonium hyperfine splitting [kHz]
EF
4 459 031.88(50)
(g-2)e
5170.93
QED2
– 873.15
QED3
– 26.41
QED4
Hadr
Weak
– 0.55(22)
0.24
– 0.07
Theo
4 463 302.73(55)
Exp
4 463 302.78(5)
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.
Positronium spectroscopy &
Recoil effects
Positronium offers a unique
opportunity:

recoil effects are
enhanced
Positronium spectroscopy &
Recoil effects
Positronium offers a unique
opportunity:

recoil effects are
enhanced

and relatively low
accuracy is sufficient
for crucial tests.
Positronium spectroscopy &
Recoil effects
Positronium HFS [MHz]
EF
204 386.6
QED1
– 1 005.5
QED2
11.8
QED3
– 1.2(5)
Theo
203 391.7(5)
Exp
203 389.1(7)
Positronium offers a unique
opportunity:

recoil effects are
enhanced

and relatively low
accuracy is sufficient
for crucial tests.
That is the same kind of corrections
as QED4 for muonium HFS.
Positronium spectroscopy &
Recoil effects
Positronium HFS [MHz]
EF
204 386.6
QED1
– 1 005.5
QED2
11.8
QED3
– 1.2(5)
Theo
203 391.7(5)
Exp
203 389.1(7)
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 spectrum:
theory vs experiment
1s hyperfine structure
1s-2s interval
Precision tests QED with the HFS
H, D21
48.953(3)
49.13(13)
H, D21
48.53(23)
H, D21
49.13(40)
D, D21
11.312 5(5)
D, D21
11.16(16)
11.28(6)
Theory
Experiment
Accuracy in H and D is still not high
enough to test QED.
Units are kHz
Precision tests QED with the HFS
H, D21
48.953(3)
49.13(13)
H, D21
48.53(23)
H, D21
49.13(40)
D, D21
D, D21
3He+,
D21
3He+,
D21
11.312 5(5)
11.16(16)
11.28(6)
– 1 190.083(63) – 1 189.979(71)
– 1 190.1(16)
Units are kHz
Accuracy in helium ion is much
higher.
Precision tests QED with the HFS
H, D21
48.953(3)
49.13(13)
H, D21
48.53(23)
H, D21
49.13(40)
D, D21
D, D21
3He+,
D21
3He+,
D21
Mu, 1s HFS
11.312 5(5)
11.16(16)
11.28(6)
– 1 190.083(63) – 1 189.979(71)
– 1 190.1(16)
4 463 302.88(6) 4 463 302.78(5)
Units are still kHz
Muonium HFS is also obtained with a high accuracy.
Precision tests QED with the HFS
H, D21
48.953(3)
49.13(13)
H, D21
48.53(23)
H, D21
49.13(40)
D, D21
11.312 5(5)
D, D21
3He+,
D21
3He+,
D21
11.16(16)
11.28(6)
– 1 190.083(63) – 1 189.979(71)
– 1 190.1(16)
Mu, 1s HFS
4 463 302.88(6) 4 463 302.78(5)
Ps, 1s HFS
203 391.7(5)
Ps, 1s HFS
203 389.10(7)
203 397.5(16)
Units are kHz
Units for positronium
are MHz
Precision tests QED with the HFS
H, D21
49.13(13)
1.4
H, D21
48.53(23)
– 1.8 0.16
H, D21
49.13(40)
0.4
11.16(16)
– 1.0 0.49
11.28(6)
-0.6
D, D21
48.953(3)
11.312 5(5)
D, D21
3He+,
D21
3He+,
D21
– 1 190.083(63) – 1 189.979(71) 1.10
– 1 190.1(16)
0.0
0.09
0.28
0.01
0.18
Mu, 1s HFS
4 463 302.88(6) 4 463 302.78(5) – 0.2 0.11
Ps, 1s HFS
203 391.7(5)
Ps, 1s HFS
203 389.10(7)
– 2.9 4.4
203 397.5(16)
– 2.5 8.2
Units are kHz for all but positronium (MHz).
Shift/sigma
Precision tests QED with the HFS
H, D21
49.13(13)
1.4
H, D21
48.53(23)
– 1.8 0.16
H, D21
49.13(40)
0.4
11.16(16)
– 1.0 0.49
11.28(6)
-0.6
0.29
– 1 190.083(63) – 1 189.979(71) 1.10
0.01
D, D21
48.953(3)
11.312 5(5)
D, D21
3He+,
D21
3He+,
D21
– 1 190.1(16)
0.0
0.09
0.28
0.18
Mu, 1s HFS
4 463 302.88(6) 4 463 302.78(5) – 0.2 0.11
Ps, 1s HFS
203 391.7(5)
Ps, 1s HFS
203 389.10(7)
– 2.9 4.4
203 397.5(16)
– 2.5 8.2
Units are kHz for all but positronium (MHz).
Sigma/EF
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
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 #1xx+1 is bigger
than #1xx.
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 #1xx+1 is bigger
than #1xx.
However, bound state QED
calculations used to be
only for one- and twoloop contributions.
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!
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.
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.

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.
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”
(m=0).
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”
(m=0).
The problems in
calculations:
appearance of big
logarithms (ln(M/m)~5
in muonium).
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.
Uncertainty of theoretical
calculations

Uncertainty in muonium HFS
is due to QED4 corrections.
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.
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.
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 higherorder two-loop self energy.
Uncertainty of theoretical
calculations



Uncertainty in muonium HFS

Uncertainty of the hydrogen
is due to QED4 corrections.
Lamb shift is due to higherorder two-loop self energy.
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
Uncertainty of D21 in He+
corrections.
involves both: recoil QED4
and higher-order two-loop
effects.
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 higherorder 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.
Precision physics of simple
atoms & QED
There are four basic
sources of
uncertainty:
 experiment;
 pure QED theory;
 nuclear structure
and hadronic
contributions;
 fundamental
constants.
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.
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.
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
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.
How one can measure

QED


Bound state QED



Mu HFS & mm/me
Helium FS (excluded)
Electric standards

for proton (or helion)

h/m (cesium) & me/mp –
the second best!
h/m (neutron) & Si lattice
spacing

gp = 2mp/ħ
KJ = 2e/h
2
 RK = h/e
Calculable capacitor: a
direct measurement of RK

Avogadro project

gyromagnetic ratio at low
field measured as
gp/KJRK ~ mp/mB × h/me × a
Atomic physics


(g-2)e – the best!

a?
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.
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.

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.)

Progress in a variations since
ACFC meeting (June 2003)

Method:
f = C0 × c Ry × F(a)
Progress in a variations since
ACFC meeting (June 2003)

Method:
f = C0 × c Ry × F(a)
and thus
d ln{f}/dt = d ln{cRy}/dt
+ A × d lna/dt.
Progress in a variations since
ACFC meeting (June 2003)

Method:
f = C0 × c Ry × F(a)
d ln{f}/dt = d ln{cRy}/dt
+ A × d lna/dt.
 Measurements:
Optical transitions in Hg+ (NIST), H
(MPQ), Yb+ (PTB) versus Cs HFS;

Calcium is coming (PTB, NIST)
Progress in a variations since
ACFC meeting (June 2003)

Method:
f = C0 × c Ry × F(a)
d ln{f}/dt = d ln{cRy}/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
Progress in a variations since
ACFC meeting (June 2003)

Method:
f = C0 × c Ry × F(a)
d ln{f}/dt = d ln{cRy}/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
Progress in a variations since
ACFC meeting (June 2003)

Method:
f = C0 × c Ry × F(a)
d ln{f}/dt = d ln{cRy}/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
Current laboratory constraints
on variations of constants
X
Variation d lnX/dt
Model
a
(-0.3±2.0)×10-15 yr-1
--
{c Ry}
(-2.1±3.1)×10-15 yr-1
--
me/mp
(2.9±6.2)×10-15 yr-1
Schmidt model
mp/me
(2.9±5.8)×10-15 yr-1
Schmidt model
gp
(-0.1±0.5)×10-15 yr-1
Schmidt model
gn
(3±3)×10-14 yr-1
Schmidt model
Optical frequency measurements
& a variations

For more detail on
variation of
constants:
Optical frequency measurements
& a variations

For more detail on
variation of
constants:
Will appear in August
Contributors
Theory:
 Muonium HFS
(hadrons)



Hänsch´s group:
 Marc Fischer
 Peter Fendel
 Nikolai Kolachevsky
Simon Eidelman
Valery Shelyuto
2s HFS

Experiments:
 2s H and D
Volodya Ivanov

Constraints:


Ekkehard Peik (PTB)
Victor Flambaum
Contributors and support
Theory:
 Muonium HFS
(hadrons)



T.W. Hänsch´s group:
 Marc Fischer
 Peter Fendel
 Nikolai Kolachevsky
Simon Eidelman
Valery Shelyuto
2s HFS

Experiments:
 2s H and D
Volodya Ivanov
Supported by RFBR,
DFG, DAAD, Heareus etc

Constraints:


Ekkehard Peik (PTB)
Victor Flambaum
Welcome to
Mangaratiba !