Drake G.W.F. (editor) Handbook of Atomic, Molecular, and Optical Physics

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420 Part B Atoms

27.6 Fine Structure

The ﬁne structure of hydrogenic atoms in the non-recoil

limit is correctly described by the Dirac equation, which

gives for a state of principal quantum number n and

spin jE

= m

f(n, j), with

f(n, j) =



1 +

(Zα)

(n −β)



−1/2

. (27.48)

Here

β ≡ j +

−





j +



−(Zα)

, (27.49)

which gives the expansion

= m



1 −

(Zα)

−

(Zα)



j +

−



+···



(27.50)

With the use of the Bethe–Salpeter equation or NRQED

one can include recoil corrections exactly to order

m(Zα)

, and to this order

E = M +m

[ f(n, j) −1]−

[ f(n, j) −1]

(Zα)



j +

−

l +



(1 −δ

)

(27.51)

The last term in this expression leads to a slight break-

ing of the degeneracy of the 2s

1/2

and 2p

1/2

states: this

splitting is referred to as the Lamb shift. A larger break-

ing arises from the ﬁnite size of the nucleus, which in

the nonrelativistic limit shifts s-states by

∆E

(ﬁnite size) =

(Zα)





(27.52)

A major issue for the Lamb shift in hydrogen is the dis-

agreement between measurements at Stanford [27.28]

and Mainz [27.29] of the charge radius of the proton.

As discussed in Mohr’s chapter, the 18 kHz difference

creates difﬁculties in interpreting the experimental sta-

tus of the Lamb shift. While assuming the validity of

QED and completing all calculations that enter at the

few kHz level may allow one to determine the proton

charge radius independently, it is clearly highly desir-

able that a deﬁnitive electron-scattering experiment be

carried out. A promising alternative resolution to the

problem may come from measurements on muonic hy-

drogen undergoing at PSI. Because the scale of such

atoms is two orders of magnitude smaller than hy-

drogen, the effect of proton size is greatly enhanced,

and a measurement of the Lamb shift would then al-

low a very precise measurement of the proton charge

radius.

The largest correction to the Lamb shift comes

from the self-energy and vacuum polarization graphs

previously introduced in Sect. 27.2, with the understand-

ing that the free electron propagators are replaced by

Dirac–Coulomb propagators. The self-energy diagram

in Feynman gauge gives an energy shift

∆E

(SE) =−ie



r d





(2π)

ik·(r−r



)

+i

(r)

× γ



r, r



; E

−k









(27.53)

and the vacuum polarization a shift

∆E

(VP) =−ie







(2π)



2π

(r)

× γ

(r)

ik·(r−r



)







, r



; E



(27.54)

In both cases the counterterms discussed above are un-

derstood to be added. It is conventional to pull out the

overall behavior in Z,α, and the principal quantum

number n as follows:

∆E

(SE) +∆E

(VP) ≡ mα

(Zα)

πn

(Zα) .

(27.55)

The evaluation of the function F

(Zα) is quite difﬁcult.

At low Z, a perturbative expansion to order (Zα)

available, and is expressed as

(Zα) = A

+ A

ln(Zα)

−2

+(Zα)A

+(Zα)



(Zα)

−2

+ A

ln(Zα)

−2

+ A



+···

(27.56)

The constants A can be found in Mohr’s Chapt. 28. A

contains a constant that must be obtained numerically,

Part B 27.6

Quantum Electrodynamics 27.7 Hyperﬁne Structure 421

known as the Bethe logarithm. It can be deﬁned from

a nonrelativistic limit of the Coulomb gauge version of

(27.53), in which one carries out the integral over k

with Cauchy’s theorem, keeping only the photon pole,

makes the dipole approximation, and replaces α with

p/m. In that limit one has

∆E



2ω(2π)



−ω − E

(27.57)

where p

≡n|p|m. Because negative energy states

are not present, this expression diverges linearly with Λ.

The Bethe logarithm is deﬁned by subtracting −1/ω

from the above denominator and carrying out the d

integration, giving

∆E



−E

) ln

−E

≡

4mα(Zα)

3πn



(n)Ry



(27.58)

Elegant high accuracy determinations of these quanti-

ties using Schwinger’s Coulomb Green’s function have

been made [27.30, 31]. Reference [27.32] contains an

extensive tabulation.

We next discuss recoil corrections to the Lamb shift.

One recoil effect is the modiﬁcation of the A constants

by a factor (m

)

(although part of A

is modi-

ﬁed instead by (m

)

). Another correction, similar

to the Lamb shift but involving a photon exchanged

between the electron and nucleus was ﬁrst derived by

Salpeter [27.25], and is given by

∆E

(Zα)

πn





Zα



−

ln k

(n)

−

−m



−m



(27.59)

where a

=−3 −ln 4, a

=−9/2, and a

=−13/3.

Present research is concerned with corrections of order

Finally we discuss the two-loop Lamb shift. The

leading contribution is [27.33–35]

∆E

(two-loop) =

mα

(Zα)

(Zα) (27.60)

where for s-states

=−

4358

1296

−

ln 2 −

ζ(3)

(27.61)

and for non-s states



197

−π

ln 2 +

ζ(3)



2(2l +1)

(27.62)

where C

= 2( j −l)/( j +1/2).TheZα corrections to

this quantity have been carried out [27.36,37], and con-

siderable progress has been made in calculating terms

of order (Zα)

[27.38], though a complete calculation

of the constant in that order has not yet been carried out.

27.7 Hyperﬁne Structure

27.7.1 Muonium Hyperﬁne Splitting

After the electron g −2, the best test of QED is af-

forded by ground state muonium hyperﬁne splitting.

This is dominated by the Fermi splitting, which is given

∞

(27.63)

where m

is the reduced mass. The leading cor-

rection to the Fermi splitting involves the electron

and muon anomalous magnetic moment, which act

to increase the splitting by a factor (1 +a

)(1 +a

It is convenient to split the remaining corrections

into non-recoil and recoil parts. The non-recoil terms

are given by

∆ν

non-recoil

= (1 +a

)



1 +a

(Zα)

+α(Zα)



ln 2 −



−

8α(Zα)

3π

ln(Zα)



ln(Zα)−ln 4 +

281

480



α(Zα)

[14.88 (0.29)]

(Zα)



. (27.64)

Part B 27.7

422 Part B Atoms

A recent development has been the complete eval-

uation of D

, which is a binding correction to

the two-loop g −2 contribution, by Kinoshita and

Nio [27.39], who found D

= 0.82(4).Notealso

the constant 14.88 has changed from the previous

value of 15.38 because of a new calculation of the

vacuum polarization component, as discussed fur-

ther in Mohr’s chapter. Subtracting ∆ν

non-recoil

from

experiment leaves 795 kHz to be accounted for by re-

coil. The present state of recoil corrections is given

∆ν

recoil



−

3Zα

−m

(Zα)



−2ln(Zα) −8ln2+



α(Zα)



−2ln

ζ(3) +

+2.15(14)



−



(27.65)

The full power of modern forms of the Bethe–Salpeter

equation was needed for the evaluation of the second

term [27.40]. While these terms account for just the

needed 795 kHz mentioned above, the relatively large

uncertainty in the muon mass leads to an uncertainty in

the Fermi splitting of 1.3 kHz, and further progress in

muonium hyperﬁne splitting will need this uncertainty

to be reduced.

27.7.2 Hydrogen Hyperﬁne Splitting

The situation in hydrogen hyperﬁne splitting is quite

different from the muonium case because of the struc-

ture of the proton. The recoil corrections in (27.65)that

involve ln m

involve the offshell muon propaga-

tor. Because the muon is a pointlike particle, there is

no uncertainty in the calculation. However, when it is

replaced by a proton, two strong-interaction problems

arise that limit the theoretical accuracy that can be at-

tained. The ﬁrst arises from the fact that the charge

distribution of the proton modiﬁes the hyperﬁne split-

ting. The fractional correction can be expressed in terms

of the electric and magnetic form factors of the proton

as follows:

(Zemach) =

2αm







− p





− p



1 +κ

−1



(27.66)

where κ is the anomalous magnetic moment of the pro-

ton. It contributes −38.72 ppm, with an error of 0.5 ppm

coming from the uncertainty in the form factors. The sec-

ond correction, δ

recoil

, replaces the ﬁrst term in ∆ν

recoil

discussed in muonium hfs. The logarithm of the ratio of

the muon and electron masses in that expression arises

from high internal momenta. In the case of hydrogen

hfs, this is sensitive to details of the proton structure.

The most recent evaluation of this quantity [27.41]gives

a5.68 ppm effect, which is smaller than the Zemach

correction only because of cancellation of individual

terms of comparable size. When these corrections are

added to the the other QED corrections, theory agrees

with experiment at under the 1 ppm level, with hadronic

uncertainties of about 0.5 ppm. This limits the size of

hadronic effects arising from the polarizability of the

proton, which is known to be bounded in magnitude by

4 ppm from inelastic electron scattering data [27.42].

As with the case of the Lamb shift in hydrogen, further

progress will require better understanding from either

an experimental or theoretical approach of properties of

the proton.

27.8 Orthopositronium Decay Rate

The decay rate of orthopositronium has been measured

with increasing accuracy over the years, and has reached

the level where corrections of order α

to the lowest

order decay rate need to be considered. The most recent

experiment [27.43] has determined

experiment

= 7.0404(10)(8) µs

−1

. (27.67)

The lowest order rate is

= 2

−9

9π

mα

= 7.211 1670(1) µs

−1

, (27.68)

so that radiative corrections must be quite large, about

2.4%, to account for the difference. These radiative

Part B 27.8

Quantum Electrodynamics 27.9 Precision Tests of QED in Neutral Helium 423

corrections are conventionally written as

theory



1 + A

lnα +B





−

3α

2π

+ C

lnα +D





+...



(27.69)

The constant A was ﬁrst correctly calculated in [27.44].

Analytic and numeric improvements to this work [27.45]

have led to the present value of

A =−10.286 606 ±0.000 010 , (27.70)

which accounts for all but 0.1% of the difference be-

tween theory and experiment. The logarithmic term

of order α

was determined by Caswell and Lep-

age [27.46] and the leading logarithmic term of order α

by Karshenboim [27.47, 48]. The coefﬁcient of the non-

leading logarithmic term in order α

, C, has recently

been determined, but is numerically insigniﬁcant. For

quite some time a discrepancy appeared to be present

between theory and experiment, because a set of experi-

ments in both powders and vacuum appeared to require

a very large coefﬁcient B. However, when this constant

was ﬁnally calculated [27.49], it was found to be

B =45.06(26),

(27.71)

which is too small to explain the discrepancy. How-

ever, another experiment [27.50] was consistent with

theory, and further work at Michigan led to the above re-

sult, which is in complete agreement with theory: there

is at present no discrepancy in the decay rate of or-

thopositronium. We note also progress in the theory of

the decay rate of parapositronium [27.51, 52], which is

also consistent with experiment.

27.9 Precision Tests of QED in Neutral Helium

The Bethe–Salpeter equation mentioned above was de-

veloped in terms of an expansion of free electron

and nuclear propagators. However, if these propaga-

tors are replaced with two electron propagators in

an external Coulomb ﬁeld, the formalism, while now

more complicated, allows a rigorous treatment of he-

lium in the non-recoil limit. This was carried out by

Araki [27.53]andSucher [27.54], and allowed the-

oretical predictions up to order mα

.However,the

actual calculations are far more difﬁcult because of the

more complicated propagators. Firstly, the Schrödinger

equation cannot be solved analytically, but rather nu-

merically. However, the use of sophisticated Hylleras

basis sets [27.55, 56], or the more recently introduced

method of random exponents [27.57], leads to accura-

cies far beyond experimental precision. Secondly, the

ﬁne structure cannot be calculated analytically, but in-

stead involves expectation values of operators that are

somewhat singular: again the high-quality basis sets

just mentioned allow the precision evaluation of this

quantity [27.58, 59]. A third difﬁculty is the evalua-

tion of the Lamb shift. In helium the analog of the

Bethe logarithm discussed in connection with hydrogen

is much more difﬁcult to evaluate. However, signiﬁ-

cant progress has been made in recent years, with one

approach using Schwartz’s [27.60] idea of using an

integral representation of the numerator of the Bethe

logarithm [27.61, 62]. An even more accurate and com-

putationally simpler approach has been developed by

Drake and Goldman [27.63].

Most of the leading QED corrections in helium can

be expressed in terms of expectation values of various

operators O,where

O≡



∗

, r

)O(r

, r

)φ

, r

(27.72)

Explicitly, one has

(4)

H

+

2π

H

soo

+

H





ln α −

·s



δ(r

)



−

ln 2α



[

Zδ(r

) + Zδ(r

)

]

−

Q −

2α

3πm

M . (27.73)

Here



·L

−

[σ

·(r

× p

)σ

·(r

× p

)]



(27.74)

Part B 27.9

424 Part B Atoms

soo





× p



+σ



× p



(27.75)

and



·σ

−3σ



−

8π

·σ

)



. (27.76)

In the above, units in which = c = 1 are used. Then

p and 1/r are of order mα,andH

(4)

is of order mα

The quantity Q is deﬁned by

Q =

4π

lim

a→0



(a)

+4πδ(r

)[γ +ln(a)]



(27.77)

where r

(a) vanishes when r

< a and M is given by

M =





−E





−E



mα

(27.78)

where



∗

, r

)( p

+ p

)φ

, r

(27.79)

M can be written in terms of an expectation value if we

deﬁne the Bethe logarithm ln k

through

M ≡





−E



mα

. (27.80)

Calculations of the next order, that is, those of order

mα

, were carried out for triplet P states by Dou-

glas and Kroll [27.64] some time ago, but extension

of the method to S states has only recently become

possible through the use of NRQED techniques: we

note in particular a calculation of the 2

state by

Pachucki [27.65] and of the ground state by Korobov

and Ye lkh ovsky [27.66]. An outstanding problem in QED

is the completion of the extension of the ﬁne struc-

ture calculation to order mα

. This latter calculation

should allow an extraction of the ﬁne structure constant

from helium competitive to that obtained from electron

g −2 [27.67]. Work on this problem using both a Bethe–

Salpeter formalism [27.68]andNRQED [27.69]is

ongoing.

27.10 QED in Highly Charged One-Electron Ions

Because the Lamb shift scales as Z

and energy levels

as Z

, the relative importance of this effect increases as

one goes out along an isoelectronic sequence. In addi-

tion, the approach used in the previous section, which

relies on an expansion in powers of Zα, becomes inap-

propriate, and requires methods that do not use such an

expansion. Such methods were introduced by Wichmann

and Kroll [27.70] for vacuum polarization. They were

extended to the more difﬁcult self-energy calculation by

Brown and others [27.71], and the ﬁrst correct calcula-

tions using the method were carried out by Desiderio and

Johnson [27.72]. The basic idea is ﬁrst to carry out the

k integration in (27.53) analytically, leaving an inte-

gration over d

r,d



,andk

of a product of the electron

and photon propagators in coordinate space. If one then

makes a partial wave expansion of these propagators, the

angle integrations can be carried out, and the self-energy

becomes a sum over partial waves of an integral over two

radial coordinates and the photon energy, which integral

is carried out numerically. While simple in principle,

this method is numerically awkward because the parts

of the self-energy that are ultraviolet diver gent are not

well described in coordinate space. To solve this prob-

lem, the parts of the self-energy that have this sensitivity,

which are associated with the part of the electron prop-

agator in which the electron propagates freely, or with

a single interaction, can be separated out. These subtrac-

tions allowed the ﬁrst high-accuracy calculation of the

self-energy in Coulomb potentials by Mohr [27.73–75].

Later Blundell and Snyderman [27.76] used such sub-

tractions in a purely numeric approach that allowed the

treatment of non-Coulomb potentials, which must be

used for the many-electron ions discussed in the next

section. While the most accurate calculations have been

carried out for the case of the Coulomb potential, the

ﬁnite size of the nucleus cannot be neglected, particu-

larly at high Z. This issue has recently been studied by

several groups [27.77, 78], and is now well understood.

The experimental status of QED in one-electron ions is

discussed by Mohr in this volume. An important recent

development has been the calculation of the two-loop

Lamb shift using exact electron propagators. While so

far carried out only for the ground state in the ranges

Z > 40 [27.79] it should be straightforward, though

computationally intensive, to extend the calculations

both to excited states and to lower values of Z.

Part B 27.10

Quantum Electrodynamics 27.11 QED in Highly Charged Many-Electron Ions 425

27.11 QED in Highly Charged Many-Electron Ions

The previous applications of QED have been con-

centrated on simple systems, the most complex being

helium. However, slight modiﬁcations of the S-matrix

approach to QED described in Sect. 27.1 allow the treat-

ment of ions with any number of electrons. In practice,

this does not necessarily allow progress, since diagrams

of arbitrarily high order contribute at the level of 1 a.u.

in the many-body problem. However, when the nuclear

Coulomb ﬁeld dominates the Coulomb ﬁelds of the elec-

trons, a small parameter is available, the quantity 1/Z.

Physically this simply reﬂects the fact the the energy

of repulsion of any pair of electrons is of the order

of Z a.u., while the energy of attraction to the nu-

cleus scales as Z

a.u.. This 1/Z expansion has been

studied empirically in the context of many-body pertur-

bation theory, and it can be seen that while diagrams

involving two-photon interactions are important, those

involving three photons are highly suppressed. Thus the

possibility exists of putting the QED of highly charged

ions on the same footing as, for example, the electron

g −2, in the sense that evaluation of a limited num-

ber of Feynman diagrams will allow precision tests of

QED. However, while in principle one can start with

the original Furry representation, in practice, particu-

larly when larger numbers of electrons are present, it is

better to start with non-Coulomb potentials that build

in a major part of the electron screening present in

the ion. For example, if one considers sodiumlike plat-

inum (Z = 78), the valence electron should see a nuclear

charge of around 68 because of the ten ‘core’ electrons.

One potential that builds in this property is the so-called

‘core-Hartree’ potential [27.80],whichisdeﬁnedso

that

(r) = V

nuc

(r) +



(2 j

+1)v

(a, a;r),

(27.81)

where

(a, a;r) ≡

∞









) + f



)



(27.82)

with g

and f

the upper and lower radial compo-

nents of the Dirac wavefunction for core state a.For

sodium-like systems, a ranges over the 1s, 2s, 2p

1/2

and 2p

3/2

states. These states are solved for self con-

sistently. The resulting potential gives results close to

a full Dirac–Fock potential, but has the advantage of

being local, which makes the connection with QED

transparent. Because as r →∞,v

(a, a;r) → 1/r,the

long range behavior of this potential has the physi-

cally expected limit. It is very simple to modify the

Furry representation to include this potential: one simply

writes



rψ

†

(r)[α · p+βm +V

(r)]ψ(r)

(27.83)

and



rψ

†

(r)[V

nuc

(r) −V

(r)]ψ(r)

− e



rψ

†

(r)α · A(r)ψ(r)



r d



|r−r



†

(r)ψ(r)ψ

†



)ψ(r



(27.84)

The only difference with the QED discussed in the previ-

ous section is the ﬁrst term of H

, which acts in a manner

similar to the self-mass counterterm, though it is of

course ﬁnite. However, when more than one electron

is present, new types of Feynman diagram are encoun-

tered: a representative set of them is given in Fig. 27.3.

There is an interesting connection between these dia-

grams and the many-body perturbation theory (MBPT)

method of solving the many-electron Schrödinger equa-

tion. In this latter method the N-electron Hamiltonian is

written H = H

,where

i=N



i=1



· p

+β

m −

Zα

+U(r

)



(27.85)

a) b)

c) d)

Fig. 27.3a–e Two-photon diagrams contributing to energy

levels of highly charged ions

Part B 27.11

426 Part B Atoms

and



−r

−

i=N



i=1

U(r

). (27.86)

Care must be taken when working with the relativistic

version of the N-electron Schrödinger equation because

of negative energy states, a point that has been em-

phasized by Sucher [27.81]. This problem shows up in

MBPT in second order, where the associated energy shift

for the ground state of an alkali atom with valence elec-

tron v in the case when U is the Hartree–Fock potential

(2)

=−



amn

vamn

mnva

−g

mnav

)



+

−



abm

abmv

mvab

−g

mvba

)



+

−

. (27.87)

Here a and b range over the occupied core states, and

m and n over excited states including in principle neg-

ative energy states. However, if they are included, the

energy denominator in the ﬁrst term of the above ex-

pression can vanish when m is a positive energy state

and n negative energy, and vice versa. This problem can

be avoided by restricting sums over excited states to pos-

itive energy states, but this procedure must be justiﬁed in

the framework of QED. This can indeed be done [27.82],

with the result that the use of MBPT with the positive

energy restriction is justiﬁed, but negative energy state

effects enter in a well-deﬁned way, giving effects on the

order of the Lamb shift.

To illustrate the present status of QED in highly

charged many-electron ions, we consider the spectrum

of sodium-like platinum. The transition energy be-

tween 3p

3/2

and 3s

1/2

has been measured [27.83]to

be 653.44( 7) eV . When MBPT is applied to this transi-

tion through second order, the answer depends slightly

on the starting potential. The Hartree–Fock potential

gives 659.56 eV, while the core-Hartree potential de-

scribed above gives 659.59 eV. A modiﬁcation of the

core-Hartree potential in which the factor 2 j

+1isre-

duced by 1 for the last core state give 659.57 eV. This

spread of 0.03 eV gives a measure of the convergence of

MBPT: it should be compared with a spread of 12 eV

in lowest order and 0.4 eV in ﬁrst order. Thus the differ-

ence with experiment of 6.1 eV that is to be explained

by QED is reliable at the 0.1eVlevel.

The QED calculations in highly charged ions are

not as advanced as in the QED tests in simple neu-

tral atoms. As discussed above, only in the last decade

have calculations of one-loop radiative corrections in

non-Coulomb potentials begun to be carried out. In

the case of the core-Hartree potential they give exactly

the 6.1 eV required to explain the difference between

MBPT and experiment. This is actually somewhat un-

fortunate, since the set of two-photon Feynman diagrams

comprising the two-loop Lamb shift mentioned above,

some of which are shown in Fig. 27.3, should be de-

tectable. Speciﬁcally, scaling arguments indicate that

they should contribute at the few tenths of an eV level,

and it is likely the good agreement with experiment

found above involves fortuitous calculations. The graph

of Fig. 27.3a plays a particularly interesting role, since

it can be shown to ‘contain’ E

(2)

. Speciﬁcally, when

both photons are Coulomb photons and the electron

propagators are written in terms of a spectral represen-

tation, the fourth component of the internal momentum

integration can be carried out with contour methods.

When both propagators involve positive energy states,

(2)

from MBPT is precisely reproduced, including of

course the previously introduced ad hoc rule of hav-

ing only positive energy states in intermediate sums.

When one is positive and the other negative the un-

deﬁned term discussed above could arise, but since

in this case both poles are on the same side of the

axis, the contribution vanishes. However, when both

are negative, a contribution outside of MBPT,partof

the QED effect, arises. It is of order Z

a.u., in

other words 1/Z of the leading Lamb shift. For sodium-

like platinum, this should enter at the 0.1eV level. For

this reason, to be sensitive to corrections beyond the

one-loop Lamb shift, ever more precise experiments

along with more sophisticated MBPT calculations are

required.

In recent years considerable progress has been made

in the evaluation of ‘two-photon’ physics in lithium-

like ions. This physics involves not only the two-loop

Lamb shift mentioned in the previous section, but also

vertex corrections to one-photon exchange and a full

QED treatment of the correlation effects discussed in

the previous paragraph. These effects have been included

in recent calculations [27.80, 84], and the extension to

sodiumlike and other alkalilike ions should be straight-

forward. This latter development would be desirable,

as new data on sodiumlike and copperlike uranium of

extremely high accuracy has become available.

As this progress is made, the interesting prospect

arises of extending the high-accuracy QED tests in sim-

ple neutral systems discussed in this review to more

complicated many-electron ions, and eventually to all

the neutral atoms in the periodic table.

Part B 27.11

Quantum Electrodynamics References 427

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Part B 27

428 Part B Atoms

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Part B 27

429

Tests of Funda

28. Tests of Fundamental Physics

This chapter describes comparisons of precise

measurements and theoretical predictions that

provide tests of our knowledge of fundamental

physics. The focus is on several quantitative tests

of quantum electrodynamics (QED).

The basic formulation of the theory of QED and

calculational methods are discussed in Chapt. 27.

Here, only the end results of calculations are

collected, numerically evaluated, and compared

with the corresponding experiments. (All quoted

uncertainties are meant to be approximately at

the one standard deviation level.)

It should be remarked that QED theory and

the fundamental constants that are employed

in evaluating the theoretical expressions are

intimately linked. Fundamental constants are

discussed in Chapt. 1. Values of the constants

needed for comparison of theory and experiment

are generally determined by other comparisons

of theory and experiment, so that only the

consistency of a set of tests is checked. However,

the fact that this overall consistency is maintained

at a high level of precision and over a broad range

of phenomena provides conﬁdence that QED is

sound despite mathematical shortcomings in its

formulation.

28.1 Electron g-Factor Anomaly ................... 429

28.2 Electron g-Factor in

and

...... 432

28.3 Hydrogen and Deuterium Atoms ........... 437

28.3.1 Dirac Eigenvalue ....................... 437

28.3.2 Relativistic Recoil ...................... 438

28.3.3 Nuclear Polarization .................. 439

28.3.4 Self Energy ............................... 439

28.3.5 Vacuum Polarization.................. 440

28.3.6 Two-Photon Corrections............. 441

28.3.7 Three-Photon Corrections........... 442

28.3.8 Finite Nuclear Size..................... 443

28.3.9 Nuclear-Size Correction to Self

Energy and Vacuum Polarization . 443

28.3.10 Radiative-Recoil Corrections ....... 444

28.3.11 Nucleus Self Energy ................... 444

28.3.12 Total Energy and Uncertainty ...... 444

28.3.13 Transition Frequencies

Between Levels with n = 2 ......... 445

References .................................................. 445

Recent reviews that cover topics in this chapter

are given in [28.1, 2]. Much of the content in this

chapter is reprinted from [28.3]. Values of the

fundamental constants used in calculations in this

chapter are the 2002 CODATA recommended values

[28.3].

28.1 Electron g-Factor Anomaly

The magnetic moment of any of the three charged

leptons (e, µ, τ) is written as

µ = g

s ,

(28.1)

where g is the g-factor of the particle, m is its mass, and

s is its spin. In (28.1), e is the elementary charge and is

positive. For the negatively charged leptons (e

−

, µ

−

,and

−

) g is negative, and for the corresponding antiparticles

, µ

,andτ

) g is positive. CPT invariance implies

that the masses and absolute values of the g-factors are

the same for each particle–antiparticle pair.

These leptons have eigenvalues of spin projection

=± /2, and in the case of the electron and positron

it is conventional to write, based on (28.1),

, (28.2)

where µ

= e /(2m

) is the Bohr magneton.

For nucleons or nuclei with spin I, the magnetic

moment can be written as

µ = g

I , (28.3)

Part B 28