precision test of bound-state qed and the fine structure constant savely g karshenboim d.i....

Precision test of bound-state QED and the fine structure

constant

Savely G Karshenboim

D.I. Mendeleev Institute for Metrology (St. Petersburg)and Max-Planck-Institut für Quantenoptik (Garching)

Outline Lamb shift in the hydrogen atom Hyperfine structure in light atoms Problems of bound state QED &

Uncertainty of theoretical calculations Determination of the fine structure

constants Search for variations

Hydrogen atom & quantum mechanicsSearch 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 2mc2/n2

– no dependence on momentum (j).






En = – ½ 2mc2/n2


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.






En = – ½ 2mc2/n2


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






En = – ½ 2mc2/n2



existence of positron;






En = – ½ 2mc2/n2



existence of positron; fine structure for a

number of levels;






En = – ½ 2mc2/n2



existence of positron; fine structure for a

number of levels; electron g factor (g = 2).

Hydrogen atom & QEDTwo of these three

predictions happened to be not absolutely correct.


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.




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.





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 2002 value is Ry = 10 973 731.568 525(73)

m-1.

The progress of the last period was possible because of two-photon Doppler free spectrocsopy.

1973 7.5×10-8

1986 1.2×10-9

1998 7.6×10-

12

2002 6.6×10-

12

1998

Rydberg constant



The 2002 value is Ry = 10 973 731.568 525(73)

m-1.

The progress of the last period was possible because of two-photon Doppler free spectrocsopy.

1973 7.5×10-8

1986 1.2×10-9

1998 7.6×10-

12

2002 6.6×10-

12

CODATA2002

Two-photon Doppler-free spectroscopy of hydrogen atom

Two-photon spectroscopy

is free of linear Doppler effect.

That makes cooling relatively not too important problem.

v

, k , - k

Two-photon Doppler-free spectroscopy of hydrogen atom

Two-photon spectroscopy

is free of linear Doppler effect.

That makes cooling relatively not too important problem.

All states but 2s are broad because of the E1 decay.

The widths decrease with increase of n.

However, higher levels are badly accessible.

Two-photon transitions double frequency and allow to go higher.

v

, k , - k

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?






The idea is based on theoretical study of

(2) = L1s – 23× L2s







(2) = L1s – 23× L2s

which we understand much better since any short distance effect vanishes for (2).







(2) = L1s – 23× L2s

which we understand much better since any short distance effect vanishes for (2).

Theory of p and d states is also simple.







(2) = L1s – 23× L2s which we understand

much better since any short distance effect vanishes for (2).

Theory of p and d states is also simple.

Eventually the only unknow QED variable is the 1s Lamb shift L1s.

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).



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.



OBF: the first generation of optical measurements. They were a relative measurements with frequencies different by a nearly integer factor.

Yale: 1s-2s and 2s-4p Garching: 1s-2s and

2s-4s Paris: 1s-3s and 2s-6s

The result was reached through measurement of a beat frequency such as

f(1s-2s)-4×f(2s-4s).



The most accurate result is a comparison of independent absolute measurements: Garching: 1s-2s Paris: 2s n=8-

12



Uncertainties: Experiment: 2

ppm QED: 2 ppm Proton size 10

ppm

Lamb shift in hydrogen: theoretical uncertainty



ppm

The QED uncertainty can be even higher because of bad convergence of (Z) expansion of two-look corrections.

An exact in (Z) calculation is needed but may be not possible for now.

Lamb shift in hydrogen: theoretical uncertainty



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


relativistic effect ~ v2/c2 and thus more sensitive to nuclear structure effects than the Lamb shift


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.




• The nuclear structure (NS) term is about 40 ppm.

• Three main NS efects:





• 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







• 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;







• 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

• 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

Hydrogen

23 ppm - 33 ppm

Deuterium

23 ppm 138 ppm

Tritium 23 ppm - 36 ppm

3He+ 108 ppm

- 213 ppm

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;

Bound State QED

Nuclear Structure

Hydrogen

23 ppm - 33 ppm

Deuterium

23 ppm 138 ppm


3He+ 108 ppm

- 213 ppm




• deuterium - the corrections was calculated, but the uncertainty was not presented;

Bound State QED

Nuclear Structure

Hydrogen

23 ppm - 33 ppm

Deuterium

23 ppm 138 ppm


3He+ 108 ppm

- 213 ppm




• tritium - no result has been obtained up to date;

• helium-3 ion - no results has been obtained up to date

Bound State QED

Nuclear Structure

Hydrogen

23 ppm - 33 ppm

Deuterium

23 ppm 138 ppm


3He+ 108 ppm

- 213 ppm


HFS without the nuclear structure

There are few options to avoid nuclear structure effects:

structure-free nucleus

cancellation of the NS contributions combining two values





Muonium:Muon, an unstable

particle (lifetime ~ 2 s) serves as a nucleus. Muon mass is ~ 1/9 of proton mass.





Muonium:Muon, an unstable particle

(lifetime ~ 2 s), serves as a nucleus. Muon mass is ~ 1/9 of proton mass.

Positronium:Positron is a nucleus. The

atom is unstable (below 1 s). It is light and hard to cool, but the recoil effects are enhanced.





The leading nuclear contributions are of the form:

E = A × |nl(0)|2






E = A × |nl(0)|2

Coefficientdeterminedby interactionwith nucleus






E = A × |nl(0)|2

wave functionat r = 0



E = A × |nl(0)|2.

The coefficient A is nucleus-dependent and state-independent.

The wave function is nucleus-independent state-dependent.

For the s states:

|nl(0)|2 = (Z)3m3/n3.

What can we change in nl?



E = A × |nl(0)|2.



For the s states:

|nl(0)|2 = (Z)3m3/n3.

m is the mass of orbiting particle: may be

electron; muon.



E = A × |nl(0)|2.



For the s states:

|nl(0)|2 = (Z)3m3/n3.

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 Deuterium Helium-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.




QED3 48.937 11.305 6 – 1 189.252

(Z)4 0.006 0.0013 – 0.543

The only known 4th order term was the (Z)4 term.




QED3 48.937 11.305 6 – 1 189.252

(Z)4 0.006 0.0013 – 0.543

QED4 0.018(3) 0.004 3(5)

– 1.137(53)

However, the (Z)4 term is only a part of 4th

contributions.




QED3 48.937 11.305 6 – 1 189.252

QED4 0.018(3) 0.004 3(5)

– 1.137(53)

NS – 0.002

0.002 6(2)

0.317(36)

Theo 48.953(3) 11.312 5(5)

–1 190.067(63)The new 4th order terms and recently found higher order

nuclear size contributions are not small.




QED3 48.937 11.305 6 – 1 189.252

QED4 0.018(3) 0.004 3(5)

– 1.137(53)

NS – 0.002

0.002 6(2)

0.317(36)

Theo 48.953(3) 11.312 5(5)

–1 190.083(63)

Exp unc

0.23 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.


measured either as a splitting between 2s1/2 and 2p1/2 (1057 MHz)

2s1/2

2p3/2

2p1/2

Lamb shift:1057 MHz(RF)



2s1/2

2p3/2

2p1/2

Fine structure:11 050 MHz(RF)

QED tests in microwave & optics Lamb shift used to be


However, the best fesult for the Lamb shift has been obtained up to now from UV transitions (such as 1s – 2s).

2s1/2

2p3/2

2p1/2

1s1/2

RF

1s – 2s:UV

QED tests in microwave HFS was measured in 1s

state of hydrogen (1420 MHz)

1s1/2 (F=0) 1s1/2 (F=1)

1s HFS: 1420 MHz

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)

1s1/2 (F=0) 1s1/2 (F=1)

2s HFS: 177 MHz

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.

Muonium hyperfine splitting [kHz]

EF 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

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 2 × cRy ×

/B ×(mr/m)3

The uncertainty comes from

/B .

EF 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

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 , (Z) or m/M are well known.

EF 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

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.

EF 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

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.

EF 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

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.

EF 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

Theo 4 463 302.73(55)

Exp 4 463 302.78(5)

Positronium spectroscopy & Recoil effects

Positronium offers a unique opportunity:

recoil effects are enhanced




and relatively low accuracy is sufficient for crucial tests.





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 HFS [MHz]

That is the same kind of correctionsas QED4 for muonium HFS.





That allows to do QED tests without any determination of fundamental constants.

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 HFS [MHz]

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) 11.16(16)

D, D21 11.28(6)

Accuracy in H and D is still not high enough to test QED.

Units are kHz

Theory Experiment


Units are kHz

H, D21 48.953(3) 49.13(13)

H, D21 48.53(23)

H, D21 49.13(40)

D, D21 11.312 5(5) 11.16(16)

D, D21 11.28(6)3He+, D21 – 1

190.083(63)– 1 189.979(71)

3He+, D21 – 1 190.1(16)

Accuracy in helium ion is much higher.


Units are still kHz

H, D21 48.953(3) 49.13(13)

H, D21 48.53(23)

H, D21 49.13(40)

D, D21 11.312 5(5) 11.16(16)

D, D21 11.28(6)3He+, D21 – 1

190.083(63)– 1 189.979(71)

3He+, D21 – 1 190.1(16)

Mu, 1s HFS

4 463 302.88(6)

4 463 302.78(5)

Muonium HFS is also obtained with a high accuracy.


H, D21 48.953(3) 49.13(13)

H, D21 48.53(23)

H, D21 49.13(40)

D, D21 11.312 5(5) 11.16(16)

D, D21 11.28(6)3He+, D21 – 1

190.083(63)– 1 189.979(71)

3He+, D21 – 1 190.1(16)

Mu, 1s HFS

4 463 302.88(6)

4 463 302.78(5)

Ps, 1s HFS 203 391.7(5) 203 389.10(7)

Ps, 1s HFS 203 397.5(16)

Units are kHz

Units for positroniumare MHz


Units are kHz for all but positronium (MHz).

H, D21 48.953(3) 49.13(13) 1.4 0.09

H, D21 48.53(23) – 1.8

0.16

H, D21 49.13(40) 0.4 0.28

D, D21 11.312 5(5) 11.16(16) – 1.0

0.49

D, D21 11.28(6) -0.63He+, D21 – 1

190.083(63)– 1 189.979(71)

1.10 0.01

3He+, D21 – 1 190.1(16) 0.0 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) 203 389.10(7) – 2.9

4.4

Ps, 1s HFS 203 397.5(16) – 2.5

8.2

Shift/sigma


Units are kHz for all but positronium (MHz).

H, D21 48.953(3) 49.13(13) 1.4 0.09

H, D21 48.53(23) – 1.8

0.16

H, D21 49.13(40) 0.4 0.28

D, D21 11.312 5(5) 11.16(16) – 1.0

0.49

D, D21 11.28(6) -0.6 0.293He+, D21 – 1

190.083(63)– 1 189.979(71)

1.10 0.01

3He+, D21 – 1 190.1(16) 0.0 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) 203 389.10(7) – 2.9

4.4

Ps, 1s HFS 203 397.5(16) – 2.5

8.2

Sigma/EF

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


Three parameters of bound state QED:

is a QED parameter. It shows how many QED loops are involved.



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.






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.






Hydrogen-like gold or bismuth are with Z ~ 1. That is not good.

However, Z «1 is also not good!







However, Z « 1 is also not good!

Limit is Z = 0 related to an unbound atom.







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.






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).







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).






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 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.




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.




They are the same since one of parameters in QED3 is mainly m/M and so these corrections are recoil corrections.


Uncertainty of D21 in He+

involves both: recoil QED4 and higher-order two-loop effects.

Uncertainty of theoretical calculations and further tests Uncertainty in muonium



They are the same since one of parameters in QED3 is mainly m/M and so these corrections are recoil corrections.


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

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



The muonium result



The muonium result is consistent with others such as from electron g-2 but less accurate.

How one can measure ? QED

(g-2)e – the best! Bound state QED

Mu HFS & m/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

p/KJRK ~ p/B × h/me × for proton (or helion)

p = 2p/ħ KJ = 2e/h RK = h/e2

Calculable capacitor: a direct measurement of RK

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



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 & variations



Now: clocks enter optics and because of more oscillations in a given period they are potentially more accurate.

That is possible because of frequency comb technology which offers precision comparisons optics to optics and optics to RF.

Absolute determinations of optical frequencies is a way of practical realization of meter.

Meantime comparing various optical transitions to cesium HFS we look for variation at the level of few parts in 10-15 yr-1. (The result is negaive.)

Progress in variations since ACFC meeting (June 2003) Method:

f = C0 × c Ry × F()


f = C0 × c Ry × F()

and thusd ln{f}/dt = d ln{cRy}/dt

+ A × d ln/dt.


f = C0 × c Ry × F()

d ln{f}/dt = d ln{cRy}/dt

+ A × d ln/dt. Measurements: Optical transitions in Hg+ (NIST),

H (MPQ), Yb+ (PTB) versus Cs HFS;

Calcium is coming (PTB, NIST)


f = C0 × c Ry × F()


+ A × d ln/dt. Measurements: Optical transitions in Hg+ (NIST),

H (MPQ), Yb+ (PTB) versus Cs HFS;

Calcium is coming (PTB, NIST) Calculation of relativistic

corrections (Flambaum, Dzuba):

A = d lnF()/d ln


f = C0 × c Ry × F()


+ A × d ln/dt. Measurements: Optical transitions in Hg+ (NIST), H

(MPQ), Yb+ (PTB) versus Cs HFS; Calcium is coming (PTB, NIST) Calculation of relativistic

corrections (Flambaum, Dzuba):

A = d lnF()/d ln

Current laboratory constraints on variations of constants

X Variation d lnX/dt Model

(-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

p/e (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


For more detail on variation of constants:


For more detail on variation of constants:

Will appear in August

Contributors

Theory: Muonium HFS

(hadrons) Simon Eidelman Valery Shelyuto

2s HFS Volodya Ivanov

Experiments: 2s H and D

Hänsch´s group: Marc Fischer Peter Fendel Nikolai Kolachevsky

Constraints: Ekkehard Peik

(PTB) Victor Flambaum

Contributors and support

Theory: Muonium HFS

(hadrons) Simon Eidelman Valery Shelyuto

2s HFS Volodya Ivanov

Experiments: 2s H and D

T.W. Hänsch´s group: Marc Fischer Peter Fendel Nikolai Kolachevsky

Constraints: Ekkehard Peik (PTB) Victor Flambaum

Supported by RFBR, DFG, DAAD, Heareus etc

Welcome to Mangaratiba !

precision test of bound-state qed and the fine structure constant savely g karshenboim d.i....

Documents

hydrogen spectrum

hydrogen lines

angular momentum dirac

energy levels

interpretation of regularity

relativisitic equation

number of levels

existence of positron