(Springer tracts in modern physics 222) michael i eides, howard grotch, valery a shelyuto (auth ) theory of light hydrogenic bound states springer verlag berlin heidelberg (2007)

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Michael I. Eides Howard Grotch Valery A. Shelyuto

Theory of
Light Hydrogenic
Bound States
With 108 Figures

ABC
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Michael I. Eides
Howard Grotch

Valery A. Shelyuto
Mendeleev Institute for
Metrology
Moskovsky Pr. 19
190005 St. Petersburg
Russia
E-mail:

University of Kentucky
Department of Physics
and Astronomy
Lexington, KY 40506
U.S.A.
E-mail:

Library of Congress Control Number: 2006933610
Physics and Astronomy Classification Scheme (PACS):
11.10.St, 12.20.-m, 31.30.Jv, 32.10.Fn, 36.10.Dr
ISSN print edition: 0081-3869
ISSN electronic edition: 1615-0430
ISBN-10 3-540-45269-9 Springer Berlin Heidelberg New York
ISBN-13 978-3-540-45269-0 Springer Berlin Heidelberg New York
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543210


Preface

Light one-electron atoms are a classical subject of quantum physics. The very
discovery and further progress of quantum mechanics is intimately connected
to the explanation of the main features of hydrogen energy levels. Each step
in the development of quantum physics led to a better understanding of the
bound state physics. The Bohr quantization rules of the old quantum theory
were created in order to explain the existence of the stable discrete energy
levels. The nonrelativistic quantum mechanics of Heisenberg and Schră
odinger
provided a self-consistent scheme for description of bound states. The relativistic spin one half Dirac equation quantitatively described the main experimental features of the hydrogen spectrum. Discovery of the Lamb shift
[1], a subtle discrepancy between the predictions of the Dirac equation and
the experimental data, triggered development of modern relativistic quantum
electrodynamics, and subsequently the Standard Model of modern physics.
Despite its long and rich history the theory of atomic bound states is
still very much alive today. New importance to the bound state physics was
given by the development of quantum chromodynamics, the modern theory of
strong interactions. It was realized that all hadrons, once thought to be the
elementary building blocks of matter, are themselves atom-like bound states
of elementary quarks bound by the color forces. Hence, from a modern point
of view, the theory of atomic bound states could be considered as a theoretical laboratory and testing ground for exploration of the subtle properties of
the bound state physics, free from further complications connected with the
nonperturbative effects of quantum chromodynamics which play an especially
important role in the case of light hadrons. The quantum electrodynamics and
quantum chromodynamics bound state theories are so intimately intertwined
today that one often finds theoretical research where new results are obtained
simultaneously, say for positronium and also heavy quarkonium.

The other powerful stimulus for further development of the bound state
theory is provided by the spectacular experimental progress in precise measurements of atomic energy levels. It suffices to mention that in about a
decade the relative uncertainty of measurement of the frequency of the 1S −2S

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VI

Preface

transition in hydrogen was reduced by four orders of magnitude from 3 · 10−10
[2] to 1.8 × 10−14 [3]. The relative uncertainty in measurement of the muonium hyperfine splitting was reduced by the factor 3 from 3.6 × 10−8 [4] to
1.2 × 10−8 [5].
This experimental development was matched by rapid theoretical progress,
and the comparison and interplay between theory and experiment has been
important in the field of metrology, leading to higher precision in the determination of the fundamental constants. We feel that now is a good time to review
modern bound state theory. The theory of hydrogenic bound states is widely
described in the literature. The basics of nonrelativistic theory are contained
in any textbook on quantum mechanics, and the relativistic Dirac equation
and the Lamb shift are discussed in any textbook on quantum electrodynamics and quantum field theory. An excellent source for the early results is the
classic book by Bethe and Salpeter [6]. A number of excellent reviews contain
more recent theoretical results, and a representative, but far from exhaustive,
list of these reviews includes [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17].
This book is an attempt to present a coherent state of the art discussion
of the theory of the Lamb shift and hyperfine splitting in light hydrogenlike
atoms. It is based on our earlier review [14]. The spin independent corrections
are discussed below mainly as corrections to the hydrogen and/or muonic
hydrogen energy levels, and the theory of hyperfine splitting is discussed in
the context of the hyperfine splitting in the ground state of muonium. These

simple atomic systems are singled out for practical reasons, because high precision experimental data either exists or is expected in these cases, and the
most accurate theoretical results are also obtained for these bound states.
However, almost all formulae below are also valid for other light hydrogenlike
systems, and some of these other applications will be discussed as well. We
will try to present all theoretical results in the field, with emphasis on more
recent results. Our emphasis on the theory means that, besides presenting
an exhaustive compendium of theoretical results, we will also try to present
a qualitative discussion of the origin and magnitude of different corrections
to the energy levels, to give, when possible, semiquantitative estimates of
expected magnitudes, and to describe the main steps of the theoretical calculations and the new effective methods which were developed in recent years.
We will not attempt to present a detailed comparison of theory with the latest
experimental results, leaving this task to the experimentalists. We will use the
experimental results only for illustrative purposes.
The book is organized as follows. In the introductory part we briefly discuss
the main theoretical approaches to the physics of weakly bound two-particle
systems. A detailed discussion then follows of the nuclear spin independent
corrections to the energy levels. First, we discuss corrections which can be calculated in the external field approximation. Second, we turn to the essentially
two-particle recoil and radiative-recoil corrections. Consideration of the spinindependent corrections is completed with discussion of the nuclear size and
structure contributions. A special section is devoted to the spin-independent

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Preface

VII

corrections in muonic atoms, with the emphasis on the theoretical specifics of
an atom where the orbiting lepton is heavier than the electron. Next we turn
to a systematic discussion of the physics of hyperfine splitting. As in the case

of spin-independent corrections, this discussion consists of two parts. First,
we use the external field approximation, and then turn to the corrections
which require two-body approaches for their calculation. A special section is
devoted to the nuclear size, recoil, and structure contributions to hyperfine
structure in hydrogen. The last section of the book contains some notes on
the comparison between theoretical and experimental results.
In all our discussions, different corrections to the energy levels are ordered
with respect to the natural small parameters such as α, Zα, m/M and nonelectrodynamic parameters like the ratio of the nucleon size to the radius
of the first Bohr orbit. These parameters have a transparent physical origin
in the light hydrogenlike atoms. Powers of α describe the order of quantum
electrodynamic corrections to the energy levels, parameter Zα describes the
order of relativistic corrections to the energy levels, and the small mass ratio
of the light and heavy particles is responsible for the recoil effects beyond the
reduced mass parameter present in a relativistic bound state.1 Corrections
which depend both on the quantum electrodynamic parameter α and the relativistic parameter Zα are ordered in a series over α at fixed power of Zα,
contrary to the common practice accepted in the physics of highly charged
ions with large Z. This ordering is more natural from the point of view of the
nonrelativistic bound state physics, since all radiative corrections (different
orders in α) to a contribution of a definite order Zα in the nonrelativistic
expansion originate from the same distances and describe the same physics.
On the other hand, the radiative corrections of the same order in α to the different terms in the nonrelativistic expansion over Zα are generated at vastly
different distances and could have drastically different magnitudes.
A few remarks about our notation. All formulae below are written for the
energy shifts. However, not energies but frequencies are measured in the spectroscopic experiments. The formulae for the energy shifts are converted to
the respective expressions for the frequencies with the help of the De Broglie
relationship E = hν. We will ignore the difference between the energy and
frequency units in our theoretical discussion. Comparison of the theoretical
expressions with the experimental data will always be done in the frequency
units, since transition to the energy units leads to loss of accuracy. All numerous contributions to the energy levels are generically called ∆E and as a
rule do not carry any specific labels, but it is understood that they are all

different.
Let us mention briefly some of the closely related subjects which are not
considered in this review. The physics of the high Z ions is nowadays a vast
and well developed field of research, with its own problems, approaches and
1

We will return to a more detailed discussion of the role of different small parameters below.

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VIII

Preface

tools, which in many respects are quite different from the physics of low Z
systems. We discuss below the numerical results obtained in the high Z calculations only when they have a direct relevance for the low Z atoms. The reader
can find a detailed discussion of the high Z physics in a number of reviews
(see, e.g., [18]). In trying to preserve a reasonable size of this text we decided
to omit discussion of positronium, even though many theoretical expressions
below are written in such form that for the case of equal masses they turn
into respective corrections for the positronium energy levels. Positronium is
qualitatively different from hydrogen and muonium not only due to the equality of the masses of its constituents, but because unlike the other light atoms
there exists a whole new class of corrections to the positronium energy levels
generated by the annihilation channel which is absent in other cases.
For many years, numerous friends and colleagues have discussed with us
the bound state problem, have collaborated on different projects, and have
shared with us their vision and insight. We are especially deeply grateful
to the late D. Yennie and M. Samuel, to G. Adkins, E. Borie, M. Braun,
A. Czarnecki, M. Doncheski, G. Drake, R. Faustov, U. Jentschura, K. Jungmann, S. Karshenboim, I. Khriplovich, T. Kinoshita, L. Labzowsky, P. Lepage,

A. Martynenko, K. Melnikov, A. Milshtein, P. Mohr, D. Owen, K. Pachucki,
V. Pal’chikov, J. Sapirstein, V. Shabaev, B. Taylor, A. Yelkhovsky, and
V. Yerokhin. This work was supported by the NSF grants PHY-0138210 and
PHY-0456462.

References
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.

W. E. Lamb, Jr. and R. C. Retherford, Phys. Rev. 72, 339 (1947).
M. G. Boshier, P. E. G. Baird, C. J. Foot et al, Phys. Rev. A 40, 6169 (1989).
M. Niering, R. Holzwarth, J. Reichert et al, Phys. Rev. Lett. 84, 5496 (2000).
F. G. Mariam, W. Beer, P. R. Bolton et al, Phys. Rev. Lett. 49, 993 (1982).
W. Liu, M. G. Boshier, S. Dhawan et al, Phys. Rev. Lett. 82, 711 (1999).
H. A. Bethe and E. E. Salpeter, Quantum Mechanics of One- and Two-Electron
Atoms, Springer, Berlin, 1957.
J. R. Sapirstein and D. R. Yennie, in Quantum Electrodynamics, ed. T. Kinoshita

(World Scientific, Singapore, 1990), p. 560.
V. V. Dvoeglazov, Yu. N. Tyukhtyaev, and R. N. Faustov, Fiz. Elem. Chastits
At. Yadra 25 144 (1994) [Phys. Part. Nucl. 25, 58 (1994)].
T. Kinoshita, Rep. Prog. Phys. 59, 3803 (1996).
J. Sapirstein, in Atomic, Molecular and Optical Physics Handbook, ed. G. W. F.
Drake, AIP Press, 1996, p. 327.
P. J. Mohr, in Atomic, Molecular and Optical Physics Handbook, ed. G. W. F.
Drake, AIP Press, 1996, p. 341.
K. Pachucki, D. Leibfried, M. Weitz, A. Huber, W. Kă
onig, and T. W. Hă
anch,
J. Phys. B 29, 177 (1996); 29, 1573(E) (1996).
T. Kinoshita, hep-ph/9808351, Cornell preprint, 1998.
M. I. Eides, H. Grotch, and V. A. Shelyuto, Phys. Rep. C 342, 63 (2001).
H. Grotch and D. A. Owen, Found. Phys. 32, 1419 (2002).

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References

IX

16. S. G. Karshenboim, Phys. Rep. 422, 1 (2005).
17. P. J. Mohr and B. N. Taylor, Rev. Mod. Phys. 77, 1 (2005).
18. P. J. Mohr, G. Plunien, and G. Soff, Phys. Rep. C 293, 227 (1998).

Lexington, Kentucky, USA
& Saint-Petersburg, Russia
Lexington, Kentucky, USA

Saint-Petersburg, Russia
August 2006

Michael Eides
Howard Grotch
Valery Shelyuto

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Contents

1

Theoretical Approaches to the Energy Levels
of Loosely Bound Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Nonrelativistic Electron in the Coulomb Field . . . . . . . . . . . . . . . 1
1.2 Dirac Electron in the Coulomb Field . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Bethe-Salpeter Equation and the Effective Dirac Equation . . . . 5
1.4 Nonrelativistic Quantum Electrodynamics . . . . . . . . . . . . . . . . . . 10
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2

General Features of the Hydrogen Energy Levels . . . . . . . . . .
2.1 Classification of Corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Physical Origin of the Lamb Shift . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Natural Magnitudes of Corrections to the Lamb Shift . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13
13
15
17
18

3

External Field Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1 Leading Relativistic Corrections with Exact Mass Dependence
3.2 Radiative Corrections of Order αn (Zα)4 m . . . . . . . . . . . . . . . . . .
3.2.1 Leading Contribution to the Lamb Shift . . . . . . . . . . . . .
3.2.2 Radiative Corrections of Order α2 (Zα)4 m . . . . . . . . . . .
3.2.3 Corrections of Order α3 (Zα)4 m . . . . . . . . . . . . . . . . . . . .
3.2.4 Total Correction of Order αn (Zα)4 m . . . . . . . . . . . . . . .
3.2.5 Heavy Particle Polarization Contributions
of Order α(Zα)4 m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 Radiative Corrections of Order αn (Zα)5 m . . . . . . . . . . . . . . . . . .
3.3.1 Skeleton Integral Approach to Calculations
of Radiative Corrections . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.2 Radiative Corrections of Order α(Zα)5 m . . . . . . . . . . . .
3.3.3 Corrections of Order α2 (Zα)5 m . . . . . . . . . . . . . . . . . . . .
3.3.4 Corrections of Order α3 (Zα)5 m . . . . . . . . . . . . . . . . . . . .

19
19
22
22
27
29

31

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36
36
38
40
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4

Contents

3.4 Radiative Corrections of Order αn (Zα)6 m . . . . . . . . . . . . . . . . . .
3.4.1 Radiative Corrections of Order α(Zα)6 m . . . . . . . . . . . .
3.4.2 Corrections of Order α2 (Zα)6 m . . . . . . . . . . . . . . . . . . . .
3.5 Radiative Corrections of Order α(Zα)7 m and of Higher Orders
3.5.1 Corrections Induced by the Radiative Insertions
in the Electron Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5.2 Corrections Induced by the Radiative Insertions
in the Coulomb Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5.3 Corrections of Order α2 (Zα)7 m . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

48

48
58
68

Essentially Two-Particle Recoil Corrections . . . . . . . . . . . . . . . .
4.1 Recoil Corrections of Order (Zα)5 (m/M )m . . . . . . . . . . . . . . . . .
4.1.1 Coulomb-Coulomb Term . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.2 Transverse-Transverse Term . . . . . . . . . . . . . . . . . . . . . . .
4.1.3 Transverse-Coulomb Term . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Recoil Corrections of Order (Zα)6 (m/M )m . . . . . . . . . . . . . . . . .
4.2.1 The Braun Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.2 Lower Order Recoil Corrections and the Braun Formula
4.2.3 Recoil Correction of Order (Zα)6 (m/M )m
to the S Levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.4 Higher Order in Mass Ratio Recoil Correction
of Order (Zα)6 (m/M )n m to the S Levels . . . . . . . . . . .
4.2.5 Recoil Correction of Order (Zα)6 (m/M )m
to the Non-S Levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3 Recoil Correction of Order (Zα)7 (m/M ) . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

81
81
83
85
87
89
89
92

68
73
76
77

93
94
94
95
98

5

Radiative-Recoil Corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
5.1 Corrections of Order α(Zα)5 (m/M )m . . . . . . . . . . . . . . . . . . . . . . 99
5.1.1 Corrections Generated by the Radiative Insertions
in the Electron Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
5.1.2 Corrections Generated by the Polarization Insertions
in the Photon Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
5.1.3 Corrections Generated by the Radiative Insertions
in the Proton Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
5.2 Corrections of Order α(Zα)6 (m/M )m . . . . . . . . . . . . . . . . . . . . . . 105
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

6

Nuclear Size and Structure Corrections . . . . . . . . . . . . . . . . . . . . 109
6.1 Main Proton Size Contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
6.1.1 Spin One-Half Nuclei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
6.1.2 Nuclei with Other Spins . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

6.1.3 Empirical Nuclear Form Factor and the Contributions
to the Lamb Shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

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XIII

6.2 Nuclear Size and Structure Corrections of Order (Zα)5 m . . . . . 114
6.2.1 Nuclear Size Corrections of Order (Zα)5 m . . . . . . . . . . . 114
6.2.2 Nuclear Polarizability Contribution of Order (Zα)5 m
to S-Levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
6.3 Nuclear Size and Structure Corrections of Order (Zα)6 m . . . . . 121
6.3.1 Nuclear Polarizability Contribution to P -Levels . . . . . . 121
6.3.2 Nuclear Size Correction of Order (Zα)6 m . . . . . . . . . . . 122
6.4 Radiative Corrections to the Finite Size Effect . . . . . . . . . . . . . . 124
6.4.1 Radiative Correction of Order α(Zα)5 r2 m3r
to the Finite Size Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
6.4.2 Higher Order Radiative Corrections
to the Finite Size Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
6.5 Weak Interaction Contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
7

Lamb Shift in Light Muonic Atoms . . . . . . . . . . . . . . . . . . . . . . . . 131
7.1 Closed Electron-Loop Contributions of Order αn (Zα)2 m . . . . . 133
7.1.1 Diagrams with One External Coulomb Line . . . . . . . . . . 133
7.1.2 Diagrams with Two External Coulomb Lines . . . . . . . . . 137

7.2 Relativistic Corrections to the Leading Polarization
Contribution with Exact Mass Dependence . . . . . . . . . . . . . . . . . 138
7.3 Higher Order Electron-Loop Polarization Contributions . . . . . . 141
7.3.1 Wichmann-Kroll Electron-Loop Contribution
of Order α(Zα)4 m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
7.3.2 Light by Light Electron-Loop Contribution
of Order α2 (Zα)3 m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
7.3.3 Diagrams with Radiative Photon and Electron-Loop
Polarization Insertion in the Coulomb Photon.
Contribution of Order α2 (Zα)4 m . . . . . . . . . . . . . . . . . . . 144
7.3.4 Electron-Loop Polarization Insertion
in the Radiative Photon. Contribution of
Order α2 (Zα)4 m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
7.3.5 Insertion of One Electron and One Muon Loops
in the same Coulomb Photon. Contribution
of Order α2 (Zα)2 (me /m)2 m . . . . . . . . . . . . . . . . . . . . . . . 146
7.4 Hadron Loop Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
7.4.1 Hadronic Vacuum Polarization Contribution
of Order α(Zα)4 m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
7.4.2 Hadronic Vacuum Polarization Contribution
of Order α(Zα)5 m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
7.4.3 Contribution of Order α2 (Zα)4 m Induced
by Insertion of the Hadron Polarization
in the Radiative Photon . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
7.4.4 Insertion of One Electron and One Hadron Loops
in the Same Coulomb Photon . . . . . . . . . . . . . . . . . . . . . . 150

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7.5 Standard Radiative, Recoil and Radiative-Recoil Corrections . . 150
7.6 Nuclear Size and Structure Corrections . . . . . . . . . . . . . . . . . . . . . 151
7.6.1 Nuclear Size and Structure Corrections
of Order (Zα)5 m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
7.6.2
Nuclear Size and Structure Corrections of Order
(Zα)6 m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
7.6.3 Radiative Corrections to the Nuclear Finite Size Effect 153
7.6.4 Radiative Corrections to Nuclear Polarizability
Contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
8

Physical Origin of the Hyperfine Splitting
and the Main Nonrelativistic Contribution . . . . . . . . . . . . . . . . 161
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

9

Nonrecoil Corrections to HFS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
9.1 Relativistic (Binding) Corrections to HFS . . . . . . . . . . . . . . . . . . 165
9.2 Electron Anomalous Magnetic Moment Contributions
(Corrections of Order αn EF ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
9.3 Radiative Corrections of Order αn (Zα)EF . . . . . . . . . . . . . . . . . . 169
9.3.1 Corrections of Order α(Zα)EF . . . . . . . . . . . . . . . . . . . . . 169
9.3.2 Corrections of Order α2 (Zα)EF . . . . . . . . . . . . . . . . . . . . 173

9.3.3 Corrections of Order α3 (Zα)EF . . . . . . . . . . . . . . . . . . . . 179
9.4 Radiative Corrections of Order αn (Zα)2 EF . . . . . . . . . . . . . . . . . 180
9.4.1 Corrections of Order α(Zα)2 EF . . . . . . . . . . . . . . . . . . . . 180
9.4.2 Corrections of Order α2 (Zα)2 EF . . . . . . . . . . . . . . . . . . . 184
9.5 Radiative Corrections of Order α(Zα)3 EF
and of Higher Orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
9.5.1 Corrections of Order α(Zα)3 EF . . . . . . . . . . . . . . . . . . . . 187
9.5.2 Corrections of Order α2 (Zα)3 EF
and of Higher Orders in α . . . . . . . . . . . . . . . . . . . . . . . . . 190
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190

10 Essentially Two-Body Corrections to HFS . . . . . . . . . . . . . . . . . 193
10.1 Recoil Corrections to HFS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
10.1.1 Leading Recoil Correction . . . . . . . . . . . . . . . . . . . . . . . . . 193
10.1.2 Recoil Correction of Relative Order (Zα)2 (m/M ) . . . . 195
10.1.3 Higher Order in Mass Ratio Recoil Correction
of Relative Order (Zα)2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
10.1.4 Recoil Corrections of Order (Zα)3 (m/M )EF . . . . . . . . . 197
10.2 Radiative-Recoil Corrections to HFS . . . . . . . . . . . . . . . . . . . . . . . 198
10.2.1 Corrections of Order α(Zα)(m/M )EF
and (Z 2 α)(Zα)(m/M )EF . . . . . . . . . . . . . . . . . . . . . . . . . 198
10.2.2 Electron-Line Logarithmic Contributions
of Order α(Zα)(m/M )EF . . . . . . . . . . . . . . . . . . . . . . . . . 200

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Contents

XV

10.2.3 Electron-Line Nonlogarithmic Contributions
of Order α(Zα)(m/M )EF . . . . . . . . . . . . . . . . . . . . . . . . . 201
10.2.4 Muon-Line Contribution of Order (Z 2 α)(Zα)(m/M )EF 202
10.2.5 Leading Photon-Line Double Logarithmic
Contribution of Order α(Zα)(m/M )EF . . . . . . . . . . . . . 203
10.2.6 Photon-Line Single-Logarithmic and Nonlogarithmic
Contributions of Order α(Zα)(m/M )EF . . . . . . . . . . . . 204
10.2.7 Heavy Particle Polarization Contributions
of Order α(Zα)EF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
10.2.8 Leading Logarithmic Contributions
of Order α2 (Zα)(m/M )EF . . . . . . . . . . . . . . . . . . . . . . . . 206
10.2.9 Leading Four-Loop Contribution
of Order α3 (Zα)(m/M )EF . . . . . . . . . . . . . . . . . . . . . . . . 209
10.2.10 Corrections of Order α(Zα)(m/M )n EF . . . . . . . . . . . . . 209
10.2.11 Corrections of Orders α(Zα)2 (m/M )EF
and Z 2 α(Zα)2 (m/M )EF . . . . . . . . . . . . . . . . . . . . . . . . . . 210
10.3 Weak Interaction Contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
11 Hyperfine Splitting in Hydrogen . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
11.1 Nuclear Size, Recoil and Structure Corrections of Orders
(Zα)EF and (Zα)2 EF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218
11.1.1 Corrections of Order (Zα)EF . . . . . . . . . . . . . . . . . . . . . . 218
11.1.2 Recoil Corrections of Order (Zα)2 (m/M )EF . . . . . . . . . 226
11.1.3 Correction of Order (Zα)2 m2 r2 EF . . . . . . . . . . . . . . . . . 226
11.1.4 Correction of Order (Zα)3 (m/Λ)EF . . . . . . . . . . . . . . . . 227
11.2 Radiative Corrections to Nuclear Size and Recoil Effects . . . . . . 227
11.2.1 Radiative-Recoil Corrections of Order α(Zα)(m/Λ)EF 227
11.2.2 Radiative-Recoil Corrections of Order α(Zα)(m/M )EF 228
11.2.3 Heavy Particle Polarization Contributions . . . . . . . . . . . 229

11.3 Weak Interaction Contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231
12 Notes on Phenomenology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
12.1 Lamb Shifts of the Energy Levels . . . . . . . . . . . . . . . . . . . . . . . . . . 233
12.1.1 Values of Some Physical Constants . . . . . . . . . . . . . . . . . 233
12.1.2 Theoretical Accuracy of S-State Lamb Shifts . . . . . . . . 234
12.1.3 Theoretical Accuracy of P -State Lamb Shifts . . . . . . . . 235
12.1.4 Theoretical Accuracy of the Interval
∆n = n3 L(nS) − L(1S) . . . . . . . . . . . . . . . . . . . . . . . . . . . 235
12.1.5 Classic Lamb Shift 2S 12 − 2P 12 . . . . . . . . . . . . . . . . . . . . . 236
12.1.6 1S Lamb Shift and the Rydberg Constant . . . . . . . . . . . 238
12.1.7 Isotope Shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245
12.1.8 Lamb Shift in Helium Ion He+ . . . . . . . . . . . . . . . . . . . . . 246
12.1.9 1S − 2S Transition in Muonium . . . . . . . . . . . . . . . . . . . . 247

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Contents

12.1.10 Light Muonic Atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248
12.2 Hyperfine Splitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250
12.2.1 Hyperfine Splitting in Hydrogen . . . . . . . . . . . . . . . . . . . . 250
12.2.2 Hyperfine Splitting in Deuterium . . . . . . . . . . . . . . . . . . . 251
12.2.3 Hyperfine Splitting in Muonium . . . . . . . . . . . . . . . . . . . . 252
12.3 Theoretical Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259

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1
Theoretical Approaches to the Energy Levels
of Loosely Bound Systems

1.1 Nonrelativistic Electron in the Coulomb Field
In the first approximation, energy levels of one-electron atoms (see Fig. 1.1)
are described by the solutions of the Schră
odinger equation for an electron in
the eld of an infinitely heavy Coulomb center with charge Z in terms of the
proton charge1
−

Zα
∆
−
2m
r

ψ(r) = En ψ(r) ,

ψnlm (r) = Rnl (r)Ylm
En = −

m(Zα)2
,
2n2

(1.1)

r
,
r

n = 1, 2, 3 . . . ,

where n is called the principal quantum number. Besides the principal quantum number n each state is described by the value of orbital angular momentum l = 0, 1, . . . , n − 1, and projection of the orbital angular momentum
m = 0, ±1, . . . , ±l. In the nonrelativistic Coulomb problem all states with
different orbital angular momentum but the same principal quantum number
n have the same energy, and the energy levels of the Schrăodinger equation
in the Coulomb field are n-fold degenerate with respect to the total angular
momentum quantum number. As in any spherically symmetric problem, the
energy levels in the Coulomb field do not depend on the projection of the
orbital angular momentum on an arbitrary axis, and each energy level with
given l is additionally 2l + 1-fold degenerate.

1

We are using the system of units where ¯
h = c = 1.

M.I. Eides et al.: Theory of Light Hydrogenic Bound States, STMP 222, 1–12 (2007)
c Springer-Verlag Berlin Heidelberg 2007
DOI 10.1007/3-540-45270-2 1

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2

1 Theoretical Approaches to the Energy Levels of Loosely Bound Systems

Fig. 1.1. Hydrogen energy levels

Straightforward calculation of the characteristic values of the velocity,
Coulomb potential and kinetic energy in the stationary states gives
n|v 2 |n =

n

p2
n
m2

=

(Zα)2
,
n2

n

Zα
n
r

=

m(Zα)2
,
n2

n

p2
n
2m

=

m(Zα)2
.
2n2

(1.2)

We see that due to the smallness of the fine structure constant α a oneelectron atom is a loosely bound nonrelativistic system2 and all relativistic
effects may be treated as perturbations. There are three characteristic scales
2

We are interested only in low-Z atoms. High-Z atoms cannot be treated as nonrelativistic systems, since an expansion in Zα is problematic.

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1.2 Dirac Electron in the Coulomb Field

3

in the atom. The smallest is determined by the binding energy ∼m(Zα)2 , the
next is determined by the characteristic electron momenta ∼mZα, and the
last one is of order of the electron mass m.
Even in the framework of nonrelativistic quantum mechanics one can
achieve a much better description of the hydrogen spectrum by taking into account the finite mass of the Coulomb center. Due to the nonrelativistic nature
of the bound system under consideration, finiteness of the nucleus mass leads
to substitution of the reduced mass instead of the electron mass in the formulae above. The finiteness of the nucleus mass introduces the largest energy
scale in the bound system problem – the heavy particle mass.

1.2 Dirac Electron in the Coulomb Field
The relativistic dependence of the energy of a free classical particle on its
momentum is described by the relativistic square root
p 2 + m2 ≈ m +

p4
p2
−
+ ··· .
2m 8m3

(1.3)

The kinetic energy operator in the Schră
odinger equation corresponds to the
quadratic term in this nonrelativistic expansion, and thus the Schră
odinger
equation describes only the leading nonrelativistic approximation to the hydrogen energy levels.
The classical nonrelativistic expansion goes over p2 /m2 . In the case of the

loosely bound electron, the expansion in p2 /m2 corresponds to expansion in
(Zα)2 ; hence, relativistic corrections are given by the expansion over even
powers of Zα. As we have seen above, from the explicit expressions for the
energy levels in the Coulomb field the same parameter Zα also characterizes
the binding energy. For this reason, parameter Zα is also often called the
binding parameter, and the relativistic corrections carry the second name of
binding corrections.
Note that the series expansion for the relativistic corrections in the bound
state problem goes literally over the binding parameter Zα, unlike the case
of the scattering problem in quantum electrodynamics (QED), where the expansion parameter always contains an additional factor π in the denominator
and the expansion typically goes over α/π. This absence of the extra factor
π in the denominator of the expansion parameter is a typical feature of the
Coulomb problem. As we will see below, in the combined expansions over α
and Zα, expansion over α at fixed power of the binding parameter Zα always goes over α/π, as in the case of scattering. Loosely speaking one could
call successive terms in the series over Zα the relativistic corrections, and
successive terms in the expansion over α/π the loop or radiative corrections.
For the bound electron, calculation of the relativistic corrections should
also take into account the contributions due to its spin one half. Account for
the spin one half does not change the fundamental fact that all relativistic

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4

1 Theoretical Approaches to the Energy Levels of Loosely Bound Systems

(binding) corrections are described by the expansion in even powers of Zα, as
in the naive expansion of the classical relativistic square root in (1.1). Only
the coefficients in this expansion change due to presence of spin. A proper

description of all relativistic corrections to the energy levels is given by the
Dirac equation with a Coulomb source. All relativistic corrections may easily
be obtained from the exact solution of the Dirac equation in the external
Coulomb field (see, e.g., [1, 2])
Enj = mf (n, j) ,
where

(1.4)
− 12


2


f (n, j) = 1 +
(j +
≈ 1−
−



(Zα)
1 2
2)

− (Zα)2 + n − j −

(Zα)2
(Zα)4
−

2n2
2n3

1
j+

1
2

−

1
2

2

3
4n

1
3
5
6
(Zα)6
+
+ 3− 2
+ · · · , (1.5)
1
1
3

3
2
8n
2n
(j + 2 )
n(j + 2 )
n (j + 12 )

and j = 1/2, 3/2, . . . , n − 1/2 is the total angular momentum of the state.
In the Dirac spectrum, energy levels with the same principal quantum
number n but different total angular momentum j are split into n components
of the fine structure, unlike the nonrelativistic Schră
odinger spectrum where
all levels with the same n are degenerate. However, not all degeneracy is
lifted in the spectrum of the Dirac equation: the energy levels corresponding
to the same n and j but different l = j ± 1/2 remain doubly degenerate.
This degeneracy is lifted by the corrections connected with the finite size
of the Coulomb source, recoil contributions, and by the dominating QED
loop contributions. The respective energy shifts are called the Lamb shifts
(see exact definition in Sect. 3.1) and will be one of the main subjects of
discussion below. We would like to emphasize that the quantum mechanical
(recoil and finite nuclear size) effects alone do not predict anything of the
scale of the experimentally observed Lamb shift which is thus essentially a
quantum electrodynamic (field-theoretical) effect.
One trivial improvement of the Dirac formula for the energy levels may
easily be achieved if we take into account that, as was already discussed above,
the electron motion in the Coulomb field is essentially nonrelativistic, and,
hence, all contributions to the binding energy should contain as a factor the
reduced mass of the electron-nucleus nonrelativistic system rather than the
electron mass. Below we will consider the expression with the reduced mass

factor
(1.6)
Enj = m + mr [f (n, j) − 1] ,
rather than the naive expression in (1.4), as a starting point for calculation
of corrections to the electron energy levels. In order to provide a solid starting point for further calculations the Dirac spectrum with the reduced mass

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1.3 Bethe-Salpeter Equation and the Effective Dirac Equation

5

dependence in (1.6) should be itself derived from QED (see Sect. 3.1 below),
and not simply postulated on physical grounds as is done here.

1.3 Bethe-Salpeter Equation
and the Effective Dirac Equation
Quantum field theory provides an unambiguous way to find energy levels of
any composite system. They are determined by the positions of the poles of
the respective Green functions. This idea was first realized in the form of the
Bethe-Salpeter (BS) equation for the two-particle Green function (see Fig. 1.2)
[3]
(1.7)
G = S0 + S0 KBS G ,
where S0 is a free two-particle Green function, the kernel KBS is a sum of all
two-particle irreducible diagrams in Fig. 1.3, and G is the total two-particle
Green function.

Fig. 1.2. Bethe-Salpeter equation

At first glance the field-theoretical BS equation has nothing in common
with the quantum mechanical Schră
odinger and Dirac equations discussed
above. However, it is not too difficult to demonstrate that with selection of
a certain subset of interaction kernels (ladder and crossed ladder), followed
by some natural approximations, the BS eigenvalue equation reduces in the
leading approximation, in the case of one light and one heavy constituent, to
the Schră
odinger or Dirac eigenvalue equations for a light particle in a field of
a heavy Coulomb center. The basics of the BS equation are described in many
textbooks (see, e.g., [2, 4, 5]), and many important results were obtained in
the BS framework.
However, calculations beyond the leading order in the original BS framework tend to be rather complicated and nontransparent. The reasons for these
complications can be traced to the dependence of the BS wave function on
the unphysical relative energy (or relative time), absence of the exact solution

Fig. 1.3. Kernel of the Bethe-Salpeter equation

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6

1 Theoretical Approaches to the Energy Levels of Loosely Bound Systems

in the zero-order approximation, non-reducibility of the ladder approximation
to the Dirac equation, when the mass of the heavy particle goes to infinity,
etc. These difficulties are generated not only by the nonpotential nature of the
bound state problem in quantum field theory, but also by the unphysical classification of diagrams with the help of the notion of two-body reducibility. As

it was known from the very beginning [3], there is a tendency to cancellation
between the contributions of the ladder graphs and the graphs with crossed
photons. However, in the original BS framework, these graphs are treated in
profoundly different ways. It is quite natural, therefore, to seek such a modification of the BS equation, that the crossed and ladder graphs play a more
symmetrical role. One also would like to get rid of other drawbacks of the
original BS formulation, preserving nevertheless its rigorous field-theoretical
contents.
The BS equation allows a wide range of modifications since one can freely
modify both the zero-order propagation function and the leading order kernel,
as long as these modifications are consistently taken into account in the rules
for construction of the higher order approximations, the latter being consistent
with (1.7) for the two-particle Green function. A number of variants of the
original BS equation were developed since its discovery (see, e.g., [6, 7, 8, 9,
10]). The guiding principle in almost all these approaches was to restructure
the BS equation in such a way, that it would acquire a three-dimensional form,
a soluble and physically natural leading order approximation in the form of
the Schră
odinger or Dirac equations, and more or less transparent and regular
way for selection of the kernels relevant for calculation of the corrections of
any required order.
We will describe, in some detail, one such modification, an effective Dirac
equation (EDE) which was derived in a number of papers [7, 8, 9, 10]. This
new equation is more convenient in many applications than the original BS
equation, and we will derive some general formulae connected with this equation. The physical idea behind this approach is that in the case of a loosely
bound system of two particles of different masses, the heavy particle spends
almost all its life not far from its own mass shell. In such case some kind of
Dirac equation for the light particle in an external Coulomb field should be an
excellent starting point for the perturbation theory expansion. Then it is convenient to choose the free two-particle propagator in the form of the product
of the heavy particle mass shell projector Λ and the free electron propagator
ΛS(p, l, E) = 2πiδ (+) (p2 − M 2 )

p/ + M
(2π)4 δ (4) (p − l) ,
E/ − p/ − m

(1.8)

where pµ and lµ are the momenta of the incoming and outgoing heavy particle,
Eµ − pµ is the momentum of the incoming electron (E = (E, 0) – this is the
choice of the reference frame), and γ-matrices associated with the light and
heavy particles act only on the indices of the respective particle.

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1.3 Bethe-Salpeter Equation and the Effective Dirac Equation

7

The free propagator in (1.8) determines other building blocks and the
form of a two-body equation equivalent to the BS equation, and the regular
perturbation theory formulae in this case were obtained in [9, 10].
In order to derive these formulae let us first write the BS equation in (1.7)
in an explicit form
d4 k
(2π)4

G(p, l, E) = S0 (p, l, E) +
where
S0 (p, k, E) =

d4 q
S0 (p, k, E)KBS (k, q, E)G(q, l, E) ,
(2π)4
(1.9)

i
i
(2π)4 δ (4) (p − l) .
p/ − M E/ − /l − m

(1.10)

The amputated two-particle Green function GT satisfies the equation
GT = KBS + KBS S0 GT .

(1.11)

A new kernel corresponding to the free two-particle propagator in (1.8) may
be defined via this amputated two-particle Green function
GT = K + KΛSGT .

(1.12)

Comparing (1.11) and (1.12) one easily obtains the diagrammatic series for
the new kernel K (see Fig. 1.4)
K(q, l, E) = [I − KBS (S0 − ΛS)]−1 KBS
i
i
d4 r

= KBS (q, l, E) +
KBS (q, r, E)
(2π)4
/r − M E/ − /r − m
/r + M
− 2πiδ (+) (r2 − M 2 )
KBS (r, l, E) + · · · . (1.13)
E/ − /r − m
The new bound state equation is constructed for the two-particle Green
function defined by the relationship
G = ΛS + ΛSGT ΛS .

(1.14)

The two-particle Green function G has the same poles as the initial Green
function G and satisfies the BS-like equation

Fig. 1.4. Series for the kernal of the effective Dirac equation

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8

1 Theoretical Approaches to the Energy Levels of Loosely Bound Systems

G = ΛS + ΛSKG ,

(1.15)

or, explicitly,
p/ + M
(2π)4 δ (4) (p − l)
(1.16)
E/ − p/ − m
p/ + M
d4 q
K(p, q, E)G(q, l, E) .
+ 2πiδ (+) (p2 − M 2 )
E/ − p/ − m
(2π)4

G(p, l, E) = 2πiδ (+) (p2 − M 2 )

This last equation is completely equivalent to the original BS equation, and
may be easily written in a three-dimensional form
G(p, l, E) =

p/ + M
E/ − p/ − m
×

(2π)3 δ (3) (p − l) +

d3 q
iK(p, q, E)G(q, l, E) ,
(2π)3 2Eq
(1.17)

where all four-momenta are on the mass shell p2 = l2 = q 2 = M 2 , Eq =

q 2 + M 2 , and the three-dimensional two-particle Green function G is defined as follows
G(p, l, E) = 2πiδ (+) (p2 − M 2 )G(p, l, E)2πiδ (+) (l2 − M 2 ) .

(1.18)

Taking the residue at the bound state pole with energy En we obtain a homogeneous equation
(E/n − p/ − m)φ(p, En ) = (p/ + M )

d3 q
iK(p, q, En )φ(q, En ) . (1.19)
(2π)3 2Eq

Due to the presence of the heavy particle mass shell projector on the right
hand side the wave function in (1.19) satisfies a free Dirac equation with
respect to the heavy particle indices
(p/ − M )φ(p, En ) = 0 .

(1.20)

Then one can extract a free heavy particle spinor from the wave function in
(1.19)
φ(p, En ) = 2En U (p)ψ(p, En ) ,
(1.21)
where
U (p) =

Ep + M I
p·σ
Ep − M |p|

.

(1.22)

Finally, the eight-component wave function ψ(p, En ) (four ordinary electron
spinor indices, and two extra indices corresponding to the two-component

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1.3 Bethe-Salpeter Equation and the Effective Dirac Equation

9

Fig. 1.5. Effective Dirac equation

spinor of the heavy particle) satisfies the effective Dirac equation (see
Fig. 1.5)
(E/n − p/ − m)ψ(p, En ) =
where
K(p, q, En ) =

d3 q
iK(p, q, En )ψ(q, En ) ,
(2π)3 2Eq

(1.23)

¯ (p)K(p, q, En )U (q)
U

,
4Ep Eq

(1.24)

k = (En − p0 , −p) is the electron momentum, and the crosses on the heavy
line in Fig. 1.5 mean that the heavy particle is on its mass shell.
The inhomogeneous equation (1.17) also fixes the normalization of the
wave function.
Even though the total kernel in (1.23) is unambiguously defined, we still
have freedom to choose the zero-order kernel K0 at our convenience, in order
to obtain a solvable lowest order approximation. It is not difficult to obtain
a regular perturbation theory series for the corrections to the zero-order approximation corresponding to the difference between the zero-order kernel K0
and the exact kernel K0 + δK
En = En0 + (n|iδK(En0 )|n) 1 + (n|iδK (En0 )|n) + (n|iδK(En0 )Gn0
× (En0 )iδK(En0 )|n) 1 + (n|iδK (En0 )|n) + · · · ,

(1.25)

where the summation of intermediate states goes with the weight d3 p/
[(2π)3 2Ep ] and is realized with the help of the subtracted free Green function of the EDE with the kernel K0
Gn0 (E) = G0 (E) −

|n)(n|
,
E − En0

(1.26)

conjugation is understood in the Dirac sense, and δK (En0 ) ≡ (dK/dE)|E=En0 .

The only apparent difference of the EDE (1.23) from the regular Dirac
equation is connected with the dependence of the interaction kernels on energy. Respectively the perturbation theory series in (1.25) contain, unlike the
regular nonrelativistic perturbation series, derivatives of the interaction kernels over energy. The presence of these derivatives is crucial for cancellation
of the ultraviolet divergences in the expressions for the energy eigenvalues.
A judicious choice of the zero-order kernel (sum of the Coulomb and Breit
potentials, for more detail see, e.g, [6, 7, 10]) generates a solvable unperturbed

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