Theory of Light Hydrogenic Bound States

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Michael I. Eides Howard Grotch Valery A. Shelyuto

Theory ofLight HydrogenicBound States

With 108 Figures

ABC

Michael I. EidesHoward GrotchUniversity of KentuckyDepartment of Physicsand AstronomyLexington, KY 40506U.S.A.E-mail: [email protected]

[email protected]@adelphia.net

Valery A. ShelyutoMendeleev Institute forMetrologyMoskovsky Pr. 19190005 St. PetersburgRussiaE-mail: [email protected]

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

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Preface

Light one-electron atoms are a classical subject of quantum physics. The verydiscovery and further progress of quantum mechanics is intimately connectedto the explanation of the main features of hydrogen energy levels. Each stepin the development of quantum physics led to a better understanding of thebound state physics. The Bohr quantization rules of the old quantum theorywere created in order to explain the existence of the stable discrete energylevels. The nonrelativistic quantum mechanics of Heisenberg and Schrodingerprovided a self-consistent scheme for description of bound states. The rela-tivistic spin one half Dirac equation quantitatively described the main ex-perimental features of the hydrogen spectrum. Discovery of the Lamb shift[1], a subtle discrepancy between the predictions of the Dirac equation andthe experimental data, triggered development of modern relativistic quantumelectrodynamics, and subsequently the Standard Model of modern physics.

Despite its long and rich history the theory of atomic bound states isstill very much alive today. New importance to the bound state physics wasgiven by the development of quantum chromodynamics, the modern theory ofstrong interactions. It was realized that all hadrons, once thought to be theelementary building blocks of matter, are themselves atom-like bound statesof elementary quarks bound by the color forces. Hence, from a modern pointof view, the theory of atomic bound states could be considered as a theoret-ical laboratory and testing ground for exploration of the subtle properties ofthe bound state physics, free from further complications connected with thenonperturbative effects of quantum chromodynamics which play an especiallyimportant role in the case of light hadrons. The quantum electrodynamics andquantum chromodynamics bound state theories are so intimately intertwinedtoday that one often finds theoretical research where new results are obtainedsimultaneously, say for positronium and also heavy quarkonium.

The other powerful stimulus for further development of the bound statetheory is provided by the spectacular experimental progress in precise mea-surements of atomic energy levels. It suffices to mention that in about adecade the relative uncertainty of measurement of the frequency of the 1S−2S

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 muo-nium hyperfine splitting was reduced by the factor 3 from 3.6 × 10−8 [4] to1.2 × 10−8 [5].

This experimental development was matched by rapid theoretical progress,and the comparison and interplay between theory and experiment has beenimportant in the field of metrology, leading to higher precision in the determi-nation of the fundamental constants. We feel that now is a good time to reviewmodern bound state theory. The theory of hydrogenic bound states is widelydescribed in the literature. The basics of nonrelativistic theory are containedin any textbook on quantum mechanics, and the relativistic Dirac equationand the Lamb shift are discussed in any textbook on quantum electrodynam-ics and quantum field theory. An excellent source for the early results is theclassic book by Bethe and Salpeter [6]. A number of excellent reviews containmore 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 discussionof the theory of the Lamb shift and hyperfine splitting in light hydrogenlikeatoms. It is based on our earlier review [14]. The spin independent correctionsare discussed below mainly as corrections to the hydrogen and/or muonichydrogen energy levels, and the theory of hyperfine splitting is discussed inthe context of the hyperfine splitting in the ground state of muonium. Thesesimple atomic systems are singled out for practical reasons, because high pre-cision experimental data either exists or is expected in these cases, and themost accurate theoretical results are also obtained for these bound states.However, almost all formulae below are also valid for other light hydrogenlikesystems, and some of these other applications will be discussed as well. Wewill try to present all theoretical results in the field, with emphasis on morerecent results. Our emphasis on the theory means that, besides presentingan exhaustive compendium of theoretical results, we will also try to presenta qualitative discussion of the origin and magnitude of different correctionsto the energy levels, to give, when possible, semiquantitative estimates ofexpected magnitudes, and to describe the main steps of the theoretical calcu-lations and the new effective methods which were developed in recent years.We will not attempt to present a detailed comparison of theory with the latestexperimental results, leaving this task to the experimentalists. We will use theexperimental results only for illustrative purposes.

The book is organized as follows. In the introductory part we briefly discussthe main theoretical approaches to the physics of weakly bound two-particlesystems. A detailed discussion then follows of the nuclear spin independentcorrections to the energy levels. First, we discuss corrections which can be cal-culated in the external field approximation. Second, we turn to the essentiallytwo-particle recoil and radiative-recoil corrections. Consideration of the spin-independent corrections is completed with discussion of the nuclear size andstructure contributions. A special section is devoted to the spin-independent

Preface VII

corrections in muonic atoms, with the emphasis on the theoretical specifics ofan atom where the orbiting lepton is heavier than the electron. Next we turnto a systematic discussion of the physics of hyperfine splitting. As in the caseof spin-independent corrections, this discussion consists of two parts. First,we use the external field approximation, and then turn to the correctionswhich require two-body approaches for their calculation. A special section isdevoted to the nuclear size, recoil, and structure contributions to hyperfinestructure in hydrogen. The last section of the book contains some notes onthe comparison between theoretical and experimental results.

In all our discussions, different corrections to the energy levels are orderedwith respect to the natural small parameters such as α, Zα, m/M and non-electrodynamic parameters like the ratio of the nucleon size to the radiusof the first Bohr orbit. These parameters have a transparent physical originin the light hydrogenlike atoms. Powers of α describe the order of quantumelectrodynamic corrections to the energy levels, parameter Zα describes theorder of relativistic corrections to the energy levels, and the small mass ratioof the light and heavy particles is responsible for the recoil effects beyond thereduced mass parameter present in a relativistic bound state.1 Correctionswhich depend both on the quantum electrodynamic parameter α and the rel-ativistic 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 chargedions with large Z. This ordering is more natural from the point of view of thenonrelativistic bound state physics, since all radiative corrections (differentorders in α) to a contribution of a definite order Zα in the nonrelativisticexpansion 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 dif-ferent terms in the nonrelativistic expansion over Zα are generated at vastlydifferent distances and could have drastically different magnitudes.

A few remarks about our notation. All formulae below are written for theenergy shifts. However, not energies but frequencies are measured in the spec-troscopic experiments. The formulae for the energy shifts are converted tothe respective expressions for the frequencies with the help of the De Broglierelationship E = hν. We will ignore the difference between the energy andfrequency units in our theoretical discussion. Comparison of the theoreticalexpressions with the experimental data will always be done in the frequencyunits, since transition to the energy units leads to loss of accuracy. All nu-merous contributions to the energy levels are generically called ∆E and as arule do not carry any specific labels, but it is understood that they are alldifferent.

Let us mention briefly some of the closely related subjects which are notconsidered in this review. The physics of the high Z ions is nowadays a vastand 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 para-meters below.

VIII Preface

tools, which in many respects are quite different from the physics of low Zsystems. We discuss below the numerical results obtained in the high Z calcu-lations only when they have a direct relevance for the low Z atoms. The readercan 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 decidedto omit discussion of positronium, even though many theoretical expressionsbelow are written in such form that for the case of equal masses they turninto respective corrections for the positronium energy levels. Positronium isqualitatively different from hydrogen and muonium not only due to the equal-ity of the masses of its constituents, but because unlike the other light atomsthere exists a whole new class of corrections to the positronium energy levelsgenerated by the annihilation channel which is absent in other cases.

For many years, numerous friends and colleagues have discussed with usthe bound state problem, have collaborated on different projects, and haveshared with us their vision and insight. We are especially deeply gratefulto 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. Jung-mann, 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, andV. Yerokhin. This work was supported by the NSF grants PHY-0138210 andPHY-0456462.

References

1. W. E. Lamb, Jr. and R. C. Retherford, Phys. Rev. 72, 339 (1947).2. M. G. Boshier, P. E. G. Baird, C. J. Foot et al, Phys. Rev. A 40, 6169 (1989).3. M. Niering, R. Holzwarth, J. Reichert et al, Phys. Rev. Lett. 84, 5496 (2000).4. F. G. Mariam, W. Beer, P. R. Bolton et al, Phys. Rev. Lett. 49, 993 (1982).5. W. Liu, M. G. Boshier, S. Dhawan et al, Phys. Rev. Lett. 82, 711 (1999).6. H. A. Bethe and E. E. Salpeter, Quantum Mechanics of One- and Two-Electron

Atoms, Springer, Berlin, 1957.7. J. R. Sapirstein and D. R. Yennie, in Quantum Electrodynamics, ed. T. Kinoshita

(World Scientific, Singapore, 1990), p. 560.8. V. V. Dvoeglazov, Yu. N. Tyukhtyaev, and R. N. Faustov, Fiz. Elem. Chastits

At. Yadra 25 144 (1994) [Phys. Part. Nucl. 25, 58 (1994)].9. T. Kinoshita, Rep. Prog. Phys. 59, 3803 (1996).

10. J. Sapirstein, in Atomic, Molecular and Optical Physics Handbook, ed. G. W. F.Drake, AIP Press, 1996, p. 327.

11. P. J. Mohr, in Atomic, Molecular and Optical Physics Handbook, ed. G. W. F.Drake, AIP Press, 1996, p. 341.

12. K. Pachucki, D. Leibfried, M. Weitz, A. Huber, W. Konig, and T. W. Hanch,J. Phys. B 29, 177 (1996); 29, 1573(E) (1996).

13. T. Kinoshita, hep-ph/9808351, Cornell preprint, 1998.14. M. I. Eides, H. Grotch, and V. A. Shelyuto, Phys. Rep. C 342, 63 (2001).15. H. Grotch and D. A. Owen, Found. Phys. 32, 1419 (2002).

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 Michael EidesLexington, Kentucky, USA Howard GrotchSaint-Petersburg, Russia Valery ShelyutoAugust 2006

Contents

1 Theoretical Approaches to the Energy Levelsof Loosely Bound Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Nonrelativistic Electron in the Coulomb Field . . . . . . . . . . . . . . . 11.2 Dirac Electron in the Coulomb Field . . . . . . . . . . . . . . . . . . . . . . . 31.3 Bethe-Salpeter Equation and the Effective Dirac Equation . . . . 51.4 Nonrelativistic Quantum Electrodynamics . . . . . . . . . . . . . . . . . . 10References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2 General Features of the Hydrogen Energy Levels . . . . . . . . . . 132.1 Classification of Corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.2 Physical Origin of the Lamb Shift . . . . . . . . . . . . . . . . . . . . . . . . . 152.3 Natural Magnitudes of Corrections to the Lamb Shift . . . . . . . . 17References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3 External Field Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.1 Leading Relativistic Corrections with Exact Mass Dependence 193.2 Radiative Corrections of Order αn(Zα)4m . . . . . . . . . . . . . . . . . . 22

3.2.1 Leading Contribution to the Lamb Shift . . . . . . . . . . . . . 223.2.2 Radiative Corrections of Order α2(Zα)4m . . . . . . . . . . . 273.2.3 Corrections of Order α3(Zα)4m . . . . . . . . . . . . . . . . . . . . 293.2.4 Total Correction of Order αn(Zα)4m . . . . . . . . . . . . . . . 313.2.5 Heavy Particle Polarization Contributions

of Order α(Zα)4m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.3 Radiative Corrections of Order αn(Zα)5m . . . . . . . . . . . . . . . . . . 36

3.3.1 Skeleton Integral Approach to Calculationsof Radiative Corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.3.2 Radiative Corrections of Order α(Zα)5m . . . . . . . . . . . . 383.3.3 Corrections of Order α2(Zα)5m . . . . . . . . . . . . . . . . . . . . 403.3.4 Corrections of Order α3(Zα)5m . . . . . . . . . . . . . . . . . . . . 47

XII Contents

3.4 Radiative Corrections of Order αn(Zα)6m . . . . . . . . . . . . . . . . . . 483.4.1 Radiative Corrections of Order α(Zα)6m . . . . . . . . . . . . 483.4.2 Corrections of Order α2(Zα)6m . . . . . . . . . . . . . . . . . . . . 58

3.5 Radiative Corrections of Order α(Zα)7m and of Higher Orders 683.5.1 Corrections Induced by the Radiative Insertions

in the Electron Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 683.5.2 Corrections Induced by the Radiative Insertions

in the Coulomb Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 733.5.3 Corrections of Order α2(Zα)7m . . . . . . . . . . . . . . . . . . . . 76

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

4 Essentially Two-Particle Recoil Corrections . . . . . . . . . . . . . . . . 814.1 Recoil Corrections of Order (Zα)5(m/M)m . . . . . . . . . . . . . . . . . 81

4.1.1 Coulomb-Coulomb Term . . . . . . . . . . . . . . . . . . . . . . . . . . 834.1.2 Transverse-Transverse Term . . . . . . . . . . . . . . . . . . . . . . . 854.1.3 Transverse-Coulomb Term . . . . . . . . . . . . . . . . . . . . . . . . . 87

4.2 Recoil Corrections of Order (Zα)6(m/M)m . . . . . . . . . . . . . . . . . 894.2.1 The Braun Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 894.2.2 Lower Order Recoil Corrections and the Braun Formula 924.2.3 Recoil Correction of Order (Zα)6(m/M)m

to the S Levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 934.2.4 Higher Order in Mass Ratio Recoil Correction

of Order (Zα)6(m/M)nm to the S Levels . . . . . . . . . . . 944.2.5 Recoil Correction of Order (Zα)6(m/M)m

to the Non-S Levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 944.3 Recoil Correction of Order (Zα)7(m/M) . . . . . . . . . . . . . . . . . . . 95References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

5 Radiative-Recoil Corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 995.1 Corrections of Order α(Zα)5(m/M)m . . . . . . . . . . . . . . . . . . . . . . 99

5.1.1 Corrections Generated by the Radiative Insertionsin the Electron Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

5.1.2 Corrections Generated by the Polarization Insertionsin the Photon Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

5.1.3 Corrections Generated by the Radiative Insertionsin the Proton Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

5.2 Corrections of Order α(Zα)6(m/M)m . . . . . . . . . . . . . . . . . . . . . . 105References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

6 Nuclear Size and Structure Corrections . . . . . . . . . . . . . . . . . . . . 1096.1 Main Proton Size Contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

6.1.1 Spin One-Half Nuclei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1116.1.2 Nuclei with Other Spins . . . . . . . . . . . . . . . . . . . . . . . . . . . 1126.1.3 Empirical Nuclear Form Factor and the Contributions

to the Lamb Shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

Contents XIII

6.2 Nuclear Size and Structure Corrections of Order (Zα)5m . . . . . 1146.2.1 Nuclear Size Corrections of Order (Zα)5m . . . . . . . . . . . 1146.2.2 Nuclear Polarizability Contribution of Order (Zα)5m

to S-Levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1176.3 Nuclear Size and Structure Corrections of Order (Zα)6m . . . . . 121

6.3.1 Nuclear Polarizability Contribution to P -Levels . . . . . . 1216.3.2 Nuclear Size Correction of Order (Zα)6m . . . . . . . . . . . 122

6.4 Radiative Corrections to the Finite Size Effect . . . . . . . . . . . . . . 1246.4.1 Radiative Correction of Order α(Zα)5〈r2〉m3

r

to the Finite Size Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . 1246.4.2 Higher Order Radiative Corrections

to the Finite Size Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . 1276.5 Weak Interaction Contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

7 Lamb Shift in Light Muonic Atoms . . . . . . . . . . . . . . . . . . . . . . . . 1317.1 Closed Electron-Loop Contributions of Order αn(Zα)2m . . . . . 133

7.1.1 Diagrams with One External Coulomb Line . . . . . . . . . . 1337.1.2 Diagrams with Two External Coulomb Lines . . . . . . . . . 137

7.2 Relativistic Corrections to the Leading PolarizationContribution with Exact Mass Dependence . . . . . . . . . . . . . . . . . 138

7.3 Higher Order Electron-Loop Polarization Contributions . . . . . . 1417.3.1 Wichmann-Kroll Electron-Loop Contribution

of Order α(Zα)4m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1417.3.2 Light by Light Electron-Loop Contribution

of Order α2(Zα)3m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1437.3.3 Diagrams with Radiative Photon and Electron-Loop

Polarization Insertion in the Coulomb Photon.Contribution of Order α2(Zα)4m . . . . . . . . . . . . . . . . . . . 144

7.3.4 Electron-Loop Polarization Insertionin the Radiative Photon. Contribution ofOrder α2(Zα)4m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

7.3.5 Insertion of One Electron and One Muon Loopsin the same Coulomb Photon. Contributionof Order α2(Zα)2(me/m)2m . . . . . . . . . . . . . . . . . . . . . . . 146

7.4 Hadron Loop Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1487.4.1 Hadronic Vacuum Polarization Contribution

of Order α(Zα)4m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1487.4.2 Hadronic Vacuum Polarization Contribution

of Order α(Zα)5m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1497.4.3 Contribution of Order α2(Zα)4m Induced

by Insertion of the Hadron Polarizationin the Radiative Photon . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

7.4.4 Insertion of One Electron and One Hadron Loopsin the Same Coulomb Photon . . . . . . . . . . . . . . . . . . . . . . 150

XIV Contents

7.5 Standard Radiative, Recoil and Radiative-Recoil Corrections . . 1507.6 Nuclear Size and Structure Corrections . . . . . . . . . . . . . . . . . . . . . 151

7.6.1 Nuclear Size and Structure Correctionsof Order (Zα)5m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

7.6.2 Nuclear Size and Structure Corrections of Order(Zα)6m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

7.6.3 Radiative Corrections to the Nuclear Finite Size Effect 1537.6.4 Radiative Corrections to Nuclear Polarizability

Contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

8 Physical Origin of the Hyperfine Splittingand the Main Nonrelativistic Contribution . . . . . . . . . . . . . . . . 161References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

9 Nonrecoil Corrections to HFS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1659.1 Relativistic (Binding) Corrections to HFS . . . . . . . . . . . . . . . . . . 1659.2 Electron Anomalous Magnetic Moment Contributions

(Corrections of Order αnEF ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1679.3 Radiative Corrections of Order αn(Zα)EF . . . . . . . . . . . . . . . . . . 169

9.3.1 Corrections of Order α(Zα)EF . . . . . . . . . . . . . . . . . . . . . 1699.3.2 Corrections of Order α2(Zα)EF . . . . . . . . . . . . . . . . . . . . 1739.3.3 Corrections of Order α3(Zα)EF . . . . . . . . . . . . . . . . . . . . 179

9.4 Radiative Corrections of Order αn(Zα)2EF . . . . . . . . . . . . . . . . . 1809.4.1 Corrections of Order α(Zα)2EF . . . . . . . . . . . . . . . . . . . . 1809.4.2 Corrections of Order α2(Zα)2EF . . . . . . . . . . . . . . . . . . . 184

9.5 Radiative Corrections of Order α(Zα)3EF

and of Higher Orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1859.5.1 Corrections of Order α(Zα)3EF . . . . . . . . . . . . . . . . . . . . 1879.5.2 Corrections of Order α2(Zα)3EF

and of Higher Orders in α . . . . . . . . . . . . . . . . . . . . . . . . . 190References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190

10 Essentially Two-Body Corrections to HFS . . . . . . . . . . . . . . . . . 19310.1 Recoil Corrections to HFS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193

10.1.1 Leading Recoil Correction . . . . . . . . . . . . . . . . . . . . . . . . . 19310.1.2 Recoil Correction of Relative Order (Zα)2(m/M) . . . . 19510.1.3 Higher Order in Mass Ratio Recoil Correction

of Relative Order (Zα)2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 19710.1.4 Recoil Corrections of Order (Zα)3(m/M)EF . . . . . . . . . 197

10.2 Radiative-Recoil Corrections to HFS . . . . . . . . . . . . . . . . . . . . . . . 19810.2.1 Corrections of Order α(Zα)(m/M)EF

and (Z2α)(Zα)(m/M)EF . . . . . . . . . . . . . . . . . . . . . . . . . 19810.2.2 Electron-Line Logarithmic Contributions

of Order α(Zα)(m/M)EF . . . . . . . . . . . . . . . . . . . . . . . . . 200

Contents XV

10.2.3 Electron-Line Nonlogarithmic Contributionsof Order α(Zα)(m/M)EF . . . . . . . . . . . . . . . . . . . . . . . . . 201

10.2.4 Muon-Line Contribution of Order (Z2α)(Zα)(m/M)EF 20210.2.5 Leading Photon-Line Double Logarithmic

Contribution of Order α(Zα)(m/M)EF . . . . . . . . . . . . . 20310.2.6 Photon-Line Single-Logarithmic and Nonlogarithmic

Contributions of Order α(Zα)(m/M)EF . . . . . . . . . . . . 20410.2.7 Heavy Particle Polarization Contributions

of Order α(Zα)EF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20510.2.8 Leading Logarithmic Contributions

of Order α2(Zα)(m/M)EF . . . . . . . . . . . . . . . . . . . . . . . . 20610.2.9 Leading Four-Loop Contribution

of Order α3(Zα)(m/M)EF . . . . . . . . . . . . . . . . . . . . . . . . 20910.2.10 Corrections of Order α(Zα)(m/M)nEF . . . . . . . . . . . . . 20910.2.11 Corrections of Orders α(Zα)2(m/M)EF

and Z2α(Zα)2(m/M)EF . . . . . . . . . . . . . . . . . . . . . . . . . . 21010.3 Weak Interaction Contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213

11 Hyperfine Splitting in Hydrogen . . . . . . . . . . . . . . . . . . . . . . . . . . . 21711.1 Nuclear Size, Recoil and Structure Corrections of Orders

(Zα)EF and (Zα)2EF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21811.1.1 Corrections of Order (Zα)EF . . . . . . . . . . . . . . . . . . . . . . 21811.1.2 Recoil Corrections of Order (Zα)2(m/M)EF . . . . . . . . . 22611.1.3 Correction of Order (Zα)2m2r2EF . . . . . . . . . . . . . . . . . 22611.1.4 Correction of Order (Zα)3(m/Λ)EF . . . . . . . . . . . . . . . . 227

11.2 Radiative Corrections to Nuclear Size and Recoil Effects . . . . . . 22711.2.1 Radiative-Recoil Corrections of Order α(Zα)(m/Λ)EF 22711.2.2 Radiative-Recoil Corrections of Order α(Zα)(m/M)EF 22811.2.3 Heavy Particle Polarization Contributions . . . . . . . . . . . 229

11.3 Weak Interaction Contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231

12 Notes on Phenomenology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23312.1 Lamb Shifts of the Energy Levels . . . . . . . . . . . . . . . . . . . . . . . . . . 233

12.1.1 Values of Some Physical Constants . . . . . . . . . . . . . . . . . 23312.1.2 Theoretical Accuracy of S-State Lamb Shifts . . . . . . . . 23412.1.3 Theoretical Accuracy of P -State Lamb Shifts . . . . . . . . 23512.1.4 Theoretical Accuracy of the Interval

∆n = n3L(nS) − L(1S) . . . . . . . . . . . . . . . . . . . . . . . . . . . 23512.1.5 Classic Lamb Shift 2S 1

2− 2P 1

2. . . . . . . . . . . . . . . . . . . . . 236

12.1.6 1S Lamb Shift and the Rydberg Constant . . . . . . . . . . . 23812.1.7 Isotope Shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24512.1.8 Lamb Shift in Helium Ion He+ . . . . . . . . . . . . . . . . . . . . . 24612.1.9 1S − 2S Transition in Muonium . . . . . . . . . . . . . . . . . . . . 247

XVI Contents

12.1.10 Light Muonic Atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24812.2 Hyperfine Splitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250

12.2.1 Hyperfine Splitting in Hydrogen . . . . . . . . . . . . . . . . . . . . 25012.2.2 Hyperfine Splitting in Deuterium . . . . . . . . . . . . . . . . . . . 25112.2.3 Hyperfine Splitting in Muonium . . . . . . . . . . . . . . . . . . . . 252

12.3 Theoretical Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259

1

Theoretical Approaches to the Energy Levelsof 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 Schrodinger equation for an electron inthe field of an infinitely heavy Coulomb center with charge Z in terms of theproton charge1

(

− ∆

2m− Zα

r

)

ψ(r) = Enψ(r) , (1.1)

ψnlm(r) = Rnl(r)Ylm

(r

r

)

,

En = −m(Zα)2

2n2, n = 1, 2, 3 . . . ,

where n is called the principal quantum number. Besides the principal quan-tum number n each state is described by the value of orbital angular mo-mentum l = 0, 1, . . . , n − 1, and projection of the orbital angular momentumm = 0,±1, . . . ,±l. In the nonrelativistic Coulomb problem all states withdifferent orbital angular momentum but the same principal quantum numbern have the same energy, and the energy levels of the Schrodinger equationin the Coulomb field are n-fold degenerate with respect to the total angularmomentum quantum number. As in any spherically symmetric problem, theenergy levels in the Coulomb field do not depend on the projection of theorbital angular momentum on an arbitrary axis, and each energy level withgiven 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)DOI 10.1007/3-540-45270-2 1 c© Springer-Verlag Berlin Heidelberg 2007

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|v2|n〉 =⟨

n

∣∣∣∣

p2

m2

∣∣∣∣n

⟩

=(Zα)2

n2, (1.2)

⟨

n

∣∣∣∣

Zα

r

∣∣∣∣n

⟩

=m(Zα)2

n2,

⟨

n

∣∣∣∣

p2

2m

∣∣∣∣n

⟩

=m(Zα)2

2n2.

We see that due to the smallness of the fine structure constant α a one-electron atom is a loosely bound nonrelativistic system2 and all relativisticeffects 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 non-relativistic systems, since an expansion in Zα is problematic.

1.2 Dirac Electron in the Coulomb Field 3

in the atom. The smallest is determined by the binding energy ∼m(Zα)2, thenext is determined by the characteristic electron momenta ∼mZα, and thelast one is of order of the electron mass m.

Even in the framework of nonrelativistic quantum mechanics one canachieve a much better description of the hydrogen spectrum by taking into ac-count the finite mass of the Coulomb center. Due to the nonrelativistic natureof the bound system under consideration, finiteness of the nucleus mass leadsto substitution of the reduced mass instead of the electron mass in the for-mulae above. The finiteness of the nucleus mass introduces the largest energyscale 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 itsmomentum is described by the relativistic square root

√

p2 + m2 ≈ m +p2

2m− p4

8m3+ · · · . (1.3)

The kinetic energy operator in the Schrodinger equation corresponds to thequadratic term in this nonrelativistic expansion, and thus the Schrodingerequation describes only the leading nonrelativistic approximation to the hy-drogen energy levels.

The classical nonrelativistic expansion goes over p2/m2. In the case of theloosely bound electron, the expansion in p2/m2 corresponds to expansion in(Zα)2; hence, relativistic corrections are given by the expansion over evenpowers of Zα. As we have seen above, from the explicit expressions for theenergy levels in the Coulomb field the same parameter Zα also characterizesthe binding energy. For this reason, parameter Zα is also often called thebinding parameter, and the relativistic corrections carry the second name ofbinding corrections.

Note that the series expansion for the relativistic corrections in the boundstate problem goes literally over the binding parameter Zα, unlike the caseof the scattering problem in quantum electrodynamics (QED), where the ex-pansion parameter always contains an additional factor π in the denominatorand the expansion typically goes over α/π. This absence of the extra factorπ in the denominator of the expansion parameter is a typical feature of theCoulomb problem. As we will see below, in the combined expansions over αand Zα, expansion over α at fixed power of the binding parameter Zα al-ways goes over α/π, as in the case of scattering. Loosely speaking one couldcall successive terms in the series over Zα the relativistic corrections, andsuccessive terms in the expansion over α/π the loop or radiative corrections.

For the bound electron, calculation of the relativistic corrections shouldalso take into account the contributions due to its spin one half. Account forthe spin one half does not change the fundamental fact that all relativistic


(binding) corrections are described by the expansion in even powers of Zα, asin the naive expansion of the classical relativistic square root in (1.1). Onlythe coefficients in this expansion change due to presence of spin. A properdescription of all relativistic corrections to the energy levels is given by theDirac equation with a Coulomb source. All relativistic corrections may easilybe obtained from the exact solution of the Dirac equation in the externalCoulomb field (see, e.g., [1, 2])

Enj = mf(n, j) , (1.4)

where

f(n, j) =

1 +

(Zα)2(√

(j + 12 )2 − (Zα)2 + n − j − 1

2

)2

− 12

≈ 1 − (Zα)2

2n2− (Zα)4

2n3

(1

j + 12

− 34n

)

− (Zα)6

8n3

[1

(j + 12 )3

+3

n(j + 12 )2

+5

2n3− 6

n2(j + 12 )

]

+ · · · , (1.5)

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 componentsof the fine structure, unlike the nonrelativistic Schrodinger spectrum whereall levels with the same n are degenerate. However, not all degeneracy islifted in the spectrum of the Dirac equation: the energy levels correspondingto 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 sizeof the Coulomb source, recoil contributions, and by the dominating QEDloop 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 ofdiscussion below. We would like to emphasize that the quantum mechanical(recoil and finite nuclear size) effects alone do not predict anything of thescale of the experimentally observed Lamb shift which is thus essentially aquantum electrodynamic (field-theoretical) effect.

One trivial improvement of the Dirac formula for the energy levels mayeasily 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 thereduced mass of the electron-nucleus nonrelativistic system rather than theelectron mass. Below we will consider the expression with the reduced massfactor

Enj = m + mr[f(n, j) − 1] , (1.6)

rather than the naive expression in (1.4), as a starting point for calculationof corrections to the electron energy levels. In order to provide a solid start-ing point for further calculations the Dirac spectrum with the reduced mass

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 Equationand the Effective Dirac Equation

Quantum field theory provides an unambiguous way to find energy levels ofany composite system. They are determined by the positions of the poles ofthe respective Green functions. This idea was first realized in the form of theBethe-Salpeter (BS) equation for the two-particle Green function (see Fig. 1.2)[3]

G = S0 + S0KBSG , (1.7)

where S0 is a free two-particle Green function, the kernel KBS is a sum of alltwo-particle irreducible diagrams in Fig. 1.3, and G is the total two-particleGreen function.

Fig. 1.2. Bethe-Salpeter equation

At first glance the field-theoretical BS equation has nothing in commonwith the quantum mechanical Schrodinger and Dirac equations discussedabove. However, it is not too difficult to demonstrate that with selection ofa certain subset of interaction kernels (ladder and crossed ladder), followedby some natural approximations, the BS eigenvalue equation reduces in theleading approximation, in the case of one light and one heavy constituent, tothe Schrodinger or Dirac eigenvalue equations for a light particle in a field ofa heavy Coulomb center. The basics of the BS equation are described in manytextbooks (see, e.g., [2, 4, 5]), and many important results were obtained inthe BS framework.

However, calculations beyond the leading order in the original BS frame-work tend to be rather complicated and nontransparent. The reasons for thesecomplications can be traced to the dependence of the BS wave function onthe unphysical relative energy (or relative time), absence of the exact solution

Fig. 1.3. Kernel of the Bethe-Salpeter equation


in the zero-order approximation, non-reducibility of the ladder approximationto 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 thebound state problem in quantum field theory, but also by the unphysical clas-sification of diagrams with the help of the notion of two-body reducibility. Asit was known from the very beginning [3], there is a tendency to cancellationbetween the contributions of the ladder graphs and the graphs with crossedphotons. However, in the original BS framework, these graphs are treated inprofoundly different ways. It is quite natural, therefore, to seek such a mod-ification of the BS equation, that the crossed and ladder graphs play a moresymmetrical role. One also would like to get rid of other drawbacks of theoriginal BS formulation, preserving nevertheless its rigorous field-theoreticalcontents.

The BS equation allows a wide range of modifications since one can freelymodify both the zero-order propagation function and the leading order kernel,as long as these modifications are consistently taken into account in the rulesfor construction of the higher order approximations, the latter being consistentwith (1.7) for the two-particle Green function. A number of variants of theoriginal 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 restructurethe 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 ofthe Schrodinger or Dirac equations, and more or less transparent and regularway for selection of the kernels relevant for calculation of the corrections ofany required order.

We will describe, in some detail, one such modification, an effective Diracequation (EDE) which was derived in a number of papers [7, 8, 9, 10]. Thisnew equation is more convenient in many applications than the original BSequation, and we will derive some general formulae connected with this equa-tion. The physical idea behind this approach is that in the case of a looselybound system of two particles of different masses, the heavy particle spendsalmost all its life not far from its own mass shell. In such case some kind ofDirac equation for the light particle in an external Coulomb field should be anexcellent starting point for the perturbation theory expansion. Then it is con-venient to choose the free two-particle propagator in the form of the productof the heavy particle mass shell projector Λ and the free electron propagator

ΛS(p, l, E) = 2πiδ(+)(p2 − M2)p/ + M

E/ − p/ − m(2π)4δ(4)(p − l) , (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 thechoice of the reference frame), and γ-matrices associated with the light andheavy particles act only on the indices of the respective particle.


The free propagator in (1.8) determines other building blocks and theform of a two-body equation equivalent to the BS equation, and the regularperturbation 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

G(p, l, E) = S0(p, l, E) +∫

d4k

(2π)4

∫d4q

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

(1.9)where

S0(p, k, E) =i

p/ − M

i

E/ − l/ − m(2π)4δ(4)(p − l) . (1.10)

The amputated two-particle Green function GT satisfies the equation

GT = KBS + KBSS0GT . (1.11)

A new kernel corresponding to the free two-particle propagator in (1.8) maybe 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 forthe new kernel K (see Fig. 1.4)

K(q, l, E) = [I − KBS(S0 − ΛS)]−1KBS

= KBS(q, l, E) +∫

d4r

(2π)4KBS(q, r, E)

{i

r/ − M

i

E/ − r/ − m

− 2πiδ(+)(r2 − M2)r/ + M

E/ − r/ − m

}

KBS(r, l, E) + · · · . (1.13)

The new bound state equation is constructed for the two-particle Greenfunction defined by the relationship

G = ΛS + ΛSGT ΛS . (1.14)

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

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


G = ΛS + ΛSKG , (1.15)

or, explicitly,

G(p, l, E) = 2πiδ(+)(p2 − M2)p/ + M

E/ − p/ − m(2π)4δ(4)(p − l) (1.16)

+ 2πiδ(+)(p2 − M2)p/ + M

E/ − p/ − m

∫d4q

(2π)4K(p, q, E)G(q, l, E) .

This last equation is completely equivalent to the original BS equation, andmay be easily written in a three-dimensional form

G(p, l, E) =p/ + M

E/ − p/ − m

×{

(2π)3δ(3)(p − l) +∫

d3q

(2π)32EqiK(p, q, E)G(q, l, E)

}

,

(1.17)

where all four-momenta are on the mass shell p2 = l2 = q2 = M2, Eq =√

q2 + M2, and the three-dimensional two-particle Green function G is de-fined as follows

G(p, l, E) = 2πiδ(+)(p2 − M2)G(p, l, E)2πiδ(+)(l2 − M2) . (1.18)

Taking the residue at the bound state pole with energy En we obtain a ho-mogeneous equation

(E/n − p/ − m)φ(p, En) = (p/ + M)∫

d3q

(2π)32EqiK(p, q, En)φ(q, En) . (1.19)

Due to the presence of the heavy particle mass shell projector on the righthand side the wave function in (1.19) satisfies a free Dirac equation withrespect 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) =√

2EnU(p)ψ(p, En) , (1.21)

where

U(p) =

(√Ep + M I

√Ep − M

p · σ|p|

)

. (1.22)

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


Fig. 1.5. Effective Dirac equation

spinor of the heavy particle) satisfies the effective Dirac equation (seeFig. 1.5)

(E/n − p/ − m)ψ(p, En) =∫

d3q

(2π)32EqiK(p, q, En)ψ(q, En) , (1.23)

where

K(p, q, En) =U(p)K(p, q, En)U(q)

√4EpEq

, (1.24)

k = (En − p0,−p) is the electron momentum, and the crosses on the heavyline in Fig. 1.5 mean that the heavy particle is on its mass shell.

The inhomogeneous equation (1.17) also fixes the normalization of thewave function.

Even though the total kernel in (1.23) is unambiguously defined, we stillhave freedom to choose the zero-order kernel K0 at our convenience, in orderto obtain a solvable lowest order approximation. It is not difficult to obtaina regular perturbation theory series for the corrections to the zero-order ap-proximation corresponding to the difference between the zero-order kernel K0

and the exact kernel K0 + δK

En = E0n + (n|iδK(E0

n)|n)(1 + (n|iδK ′(E0

n)|n))

+ (n|iδK(E0n)Gn0

× (E0n)iδK(E0

n)|n)(1 + (n|iδK ′(E0

n)|n))

+ · · · , (1.25)

where the summation of intermediate states goes with the weight d3p/[(2π)32Ep] and is realized with the help of the subtracted free Green func-tion of the EDE with the kernel K0

Gn0(E) = G0(E) − |n)(n|E − E0

n

, (1.26)

conjugation is understood in the Dirac sense, and δK ′(E0n) ≡ (dK/dE)|E=E0

n.

The only apparent difference of the EDE (1.23) from the regular Diracequation is connected with the dependence of the interaction kernels on en-ergy. Respectively the perturbation theory series in (1.25) contain, unlike theregular nonrelativistic perturbation series, derivatives of the interaction ker-nels over energy. The presence of these derivatives is crucial for cancellationof the ultraviolet divergences in the expressions for the energy eigenvalues.

A judicious choice of the zero-order kernel (sum of the Coulomb and Breitpotentials, for more detail see, e.g, [6, 7, 10]) generates a solvable unperturbed


Fig. 1.6. Effective Dirac equation in the external Coulomb field

EDE in the external Coulomb field in Fig. 1.6.3 The eigenfunctions of thisequation may be found exactly in the form of the Dirac-Coulomb wave func-tions (see, e.g, [10]). For practical purposes it is often sufficient to approximatethese exact wave functions by the product of the Schrodinger-Coulomb wavefunctions with the reduced mass and the free electron spinors which dependon the electron mass and not on the reduced mass. These functions are veryconvenient for calculation of the high order corrections, and while below wewill often skip some steps in the derivation of one or another high order con-tribution from the EDE, we advise the reader to keep in mind that almost allcalculations below are done with these unperturbed wave functions.

1.4 Nonrelativistic Quantum Electrodynamics

A weakly bound state is necessarily nonrelativistic, v ∼ Zα (see discussionof the electron in the field of a Coulomb center above). Hence, there are twosmall parameters in a weakly bound state, namely, the fine structure constantα and nonrelativistic velocity v ∼ Zα. In the leading approximation weaklybound states are essentially quantum mechanical systems, and do not requirequantum field theory for their description. But a nonrelativistic quantum me-chanical description does not provide an unambiguous way for calculation ofhigher order corrections, when recoil and many particle effects become im-portant. On the other hand the Bethe-Salpeter equation provides an explicitquantum field theory framework for discussion of bound states, both weaklyand strongly bound. Just due to generality of the Bethe-Salpeter formalismseparation of the basic nonrelativistic dynamics for weakly bound states be-comes difficult, and systematic extraction of high order corrections over α andv ∼ Zα becomes prohibitively complicated.

Nonrelativistic quantum electrodynamics (NRQED) [11] is an attempt tocombine the simplicity of the quantum mechanical description with the powerand rigor of field theory. The idea is to write ordinary relativistic quantumelectrodynamics in the form of a nonrelativistic expansion with a Lagrangiancontaining vertices with arbitrary powers of fields. This is useful if we want toconsider essentially nonrelativistic processes, like nonrelativistic bound statesand threshold phenomena. In such a physical situation the dominant dynam-ics is nonrelativistic, and the calculations could be in principle simplified if

3 Strictly speaking the external field in this equation is not exactly Coulomb butalso includes a transverse contribution.

References 11

we could organize our expansions not only in terms of α but also in termsof the nonrelativistic velocity v ≈ p/m. In the scattering processes this ve-locity is determined by kinematics and is in our hands. In the nonrelativisticbound states it is connected with the coupling constant if the perturbationtheory works, but even if the perturbation theory fails, velocity still remainsa useful small parameter. The original idea of [11] was to introduce a finitecutoff Λ, and expand in p/Λ. Since the cutoff is finite we can choose it forconvenience. For bound states it is natural to choose Λ ∼ m. Then we havetwo small parameters α and p/Λ ∼ p/m ∼ v. Next we write all vertices whichare compatible with the symmetries of relativistic quantum electrodynamics.Then we calculate coefficients before the nonrelativistic vertices, comparingresults for the complete relativistic QED and for the NRQED. It is crucialto realize that we can compare coefficients up to arbitrary order in α at thefixed v. There are no problems with ultraviolet convergence since we considera theory with an explicit finite cutoff. Hence, if we want to calculate up toa particular order in v we can do it with arbitrary accuracy in α. If, as inthe bound state applications, we need certain overall accuracy in α (Zα) wesimply take the required order in v as to achieve the desired accuracy.

There are now numerous implementations of these basic ideas in the vastliterature, in the framework of schemes with an explicit cutoff as well as inthe framework of dimensional regularization. We will skip technical details ofNRQED here, referring the reader to papers [12, 13, 14, 15, 16] and referencestherein. In our discussions of high order corrections to the energy levels we willact in the general spirit of nonrelativistic QED, and use effective operatorsof NRQED for calculations of different contributions to the energy levels. Inmany cases the form and origin of these effective operators are intuitively clearand are more transparent than lengthy formal derivations.

References

1. J. D. Bjorken and S. D. Drell, Relativistic Quantum Mechanics, McGraw-HillBook Co., NY, 1964.

2. V. B. Berestetskii, E. M. Lifshitz, and L. P. Pitaevskii, Quantum electrodynam-ics, 2nd Edition, Pergamon Press, Oxford, 1982.

3. E. E. Salpeter and H. A. Bethe, Phys. Rev. 84, 1232 (1951).4. C. Itzykson and J.-B. Zuber, Quantum Field Theory, McGraw-Hill Book Co.,

NY, 1980.5. F. Gross, Relativistic Quantum Mechanics and Field Theory, Wiley, NY, 1993.6. H. Grotch and D. R. Yennie, Zeitsch. Phys. 202, 425 (1967).7. H. Grotch and D. R. Yennie, Rev. Mod. Phys. 41, 350 (1969).8. F. Gross, Phys. Rev. 186, 1448 (1969).9. L. S. Dulyan and R. N. Faustov, Teor. Mat. Fiz. 22, 314 (1975) [Theor. Math.

Phys. 22, 220 (1975)].10. P. Lepage, Phys. Rev. A 16, 863 (1977).11. W. E. Caswell and G. P. Lepage, Phys. Lett. B 167, 437 (1986).


12. T. Kinoshita and M. Nio, Phys. Rev. D 53, 4909 (1996).13. G. P. Lepage, preprint nucl-th/9706029, February 1997.14. P. Labelle, Phys. Rev. D 58, 093013 (1998).15. A. Pineda and J. Soto, Phys. Rev. D 59, 016005 (1999).16. A. H. Hoang, preprint hep-ph/0204299, April 2002.

2

General Featuresof the Hydrogen Energy Levels

2.1 Classification of Corrections

The zero-order effective Dirac equation with a Coulomb source provides onlyan approximate description of loosely bound states in QED, but the spectrumof this Dirac equation may serve as a good starting point for obtaining moreprecise results.

The magnetic moment of the heavy nucleus is completely ignored in theDirac equation with a Coulomb source, and, hence, the hyperfine splitting(HFS) of the energy levels is missing in its spectrum. Notice that the magneticinteraction between the nucleus and the electron may be easily described evenin the framework of the nonrelativistic quantum mechanics, and the respectivecalculation of the leading contribution to the hyperfine splitting was done along time ago by Fermi [1].

Other corrections to the Dirac energy levels do not arise in the quantummechanical treatment with a potential, and for their calculation, as well asfor calculation of the corrections to the hyperfine splitting, field-theoreticalmethods are necessary. All electrodynamic corrections to the energy levelsmay be written in the form of the power series expansion over three small pa-rameters α, Zα and m/M which determine the properties of the bound state.Account for the additional corrections of nonelectromagnetic origin inducedby the strong and weak interactions introduces additional small parameters,namely, the ratio of the nuclear radius and the Bohr radius, the Fermi con-stant, etc. It should be noted that the coefficients in the power series for theenergy levels might themselves be slowly varying functions (logarithms) ofthese parameters.

Each of the small parameters above plays an important and unique role.In order to organize further discussion of different contributions to the energylevels it is convenient to classify corrections in accordance with the smallparameters on which they depend.

Corrections which depend only on the parameter Zα will be called rela-tivistic or binding corrections. Higher powers of Zα arise due to deviation of


14 2 General Features of the Hydrogen Energy Levels

the theory from a nonrelativistic limit, and thus represent a relativistic ex-pansion. All such contributions are contained in the spectrum of the effectiveDirac equation in the external Coulomb field.

Contributions to the energy which depend only on the small parameters αand Zα are called radiative corrections. Powers of α arise only from the quan-tum electrodynamics loops, and all associated corrections have a quantumfield theory nature. Radiative corrections do not depend on the recoil fac-tor m/M and thus may be calculated in the framework of QED for a boundelectron in an external field. In respective calculations one deals only withthe complications connected with the presence of quantized fields, but thetwo-particle nature of the bound state and all problems connected with thedescription of the bound states in relativistic quantum field theory still maybe ignored.

Corrections which depend on the mass ratio m/M of the light and heavyparticles reflect a deviation from the theory with an infinitely heavy nucleus.Corrections to the energy levels which depend on m/M and Zα are called re-coil corrections. They describe contributions to the energy levels which cannotbe taken into account with the help of the reduced mass factor. The presenceof these corrections signals that we are dealing with a truly two-body problem,rather than with a one-body problem.

Leading recoil corrections in Zα (of order (Zα)4(m/M)n) still may betaken into account with the help of the effective Dirac equation in the ex-ternal field since these corrections are induced by the one-photon exchange.This is impossible for the higher order recoil terms which reflect the truly rel-ativistic two-body nature of the bound state problem. Technically, respectivecontributions are induced by the Bethe-Salpeter kernels with at least two-photon exchanges and the whole machinery of relativistic QFT is necessaryfor their calculation. Calculation of the recoil corrections is simplified by theabsence of ultraviolet divergences, connected with the purely radiative loops.

Radiative-Recoil corrections are the expansion terms in the expressions forthe energy levels which depend simultaneously on the parameters α, m/Mand Zα. Their calculation requires application of all the heavy artillery ofQED, since we have to account both for the purely radiative loops and for therelativistic two-body nature of the bound states.

The last class of corrections contains nonelectromagnetic corrections, ef-fects of weak and strong interactions. The largest correction induced by thestrong interaction is connected with the finiteness of the nuclear size.

Let us emphasize once more that hyperfine structure, radiative, recoil,radiative-recoil, and nonelectromagnetic corrections are all missing in theDirac energy spectrum. Discussion of their calculations is our main topicbelow.

2.2 Physical Origin of the Lamb Shift 15

2.2 Physical Origin of the Lamb Shift

According to QED an electron continuously emits and absorbs virtual photons(see the leading order diagram in Fig. 2.1) and as a result its electric chargeis spread over a finite volume instead of being pointlike1

〈r2〉 = −6dF1(−k2)

dk2 |k2=0≈ 2α

πm2ln

m

ρ≈ 2α

πm2ln(Zα)−2 . (2.2)

In order to obtain this estimate of the electron radius we have taken intoaccount that the electron is slightly off mass shell in the bound state. Hence,the would be infrared divergence in the electron charge radius is cut off byits virtuality ρ = (m2 − p2)/m which is of order of the nonrelativistic bindingenergy ρ ≈ m(Zα)2.

Fig. 2.1. Leading order contribution to the electron radius

The finite radius of the electron generates a correction to the Coulombpotential (see, e.g., [2])

δV =16〈r2〉∆V =

2π

3Zα〈r2〉δ(r) , (2.3)

where V = −Zα/r is the Coulomb potential.The respective correction to the energy levels is simply given by the matrix

element of this perturbation. Thus we immediately discover that the finite sizeof the electron produced by the QED radiative corrections leads to a shift ofthe hydrogen energy levels. Moreover, since this perturbation is nonvanishingonly at the source of the Coulomb potential, it influences quite differentlythe energy levels with different orbital angular momenta, and, hence, leadsto splitting of the levels with the same total angular momenta but differentorbital momenta. This splitting lifts the degeneracy in the spectrum of theDirac equation in the Coulomb field, where the energy levels depend only onthe principal quantum number n and the total angular momentum j.1 The numerical factor in (2.2) arises due to the common relation between the

expansion of the form factor and the mean square root radius

F (−k2) = 1 − 1

6〈r2〉k2 . (2.1)


It is very easy to estimate this splitting (shift of the S level energy)

∆E = 〈nS|δV |nS〉 ≈ |Ψn(0)|2 2π(Zα)3

〈r2〉

≈ 4m(Zα)4

n3

α

3πln[(Zα)−2]δl0 |n=2 ≈ 1330 MHz . (2.4)

This result should be compared with the experimental number of about1040 MHz and the agreement is satisfactory for such a crude estimate. Thereare two qualitative features of this result to which we would like to attract thereader’s attention. First, the sign of the energy shift may be obtained withoutcalculation. Due to the finite radius of the electron its charge in the S state ison the average more spread out around the Coulomb source than in the caseof the pointlike electron. Hence, the binding is weaker than in the case ofthe pointlike electron and the energy of the level is higher. Second, despitethe presence of nonlogarithmic contributions missing in our crude calculation,their magnitude is comparatively small, and the logarithmic term above isresponsible for the main contribution to the Lamb shift. This property is dueto the would be infrared divergence of the considered contribution, which iscutoff by the small (in comparison with the electron mass) binding energy.As we will see below, whenever a correction is logarithmically enhanced, therespective logarithm gives a significant part of the correction, as is the caseabove.

Another obvious contribution to the Lamb shift of the same leading orderis connected with the polarization insertion in the photon propagator (seeFig. 2.2). This correction also induces a correction to the Coulomb potential

δV = −Π(−k2)k4 |k2=0

∆V =α

15πm2∆V = − 4

15α(Zα)

m2δ(r) , (2.5)

and the respective correction to the S-level energy is equal to

∆E = 〈nS|δV |nS〉 = −|Ψn(0)|2 4α(Zα)15m2

(2.6)

= −4m(Zα)4

n3

α

15πδl0 |n=2 ≈ −30 MHz .

Once again the sign of this correction is evident in advance. The polariza-tion correction may be thought of as a correction to the electric charge of

Fig. 2.2. Leading order polarization insertion

2.3 Natural Magnitudes of Corrections to the Lamb Shift 17

the nucleon induced by the fact that the electron sees the proton from a fi-nite distance.2 This means that the electron, which has penetrated in thepolarization cloud, sees effectively a larger charge and experiences a strongerbinding force, which lowers the energy level. Experimental observation of thiscontribution to the Lamb shift played an important role in the developmentof modern quantum electrodynamics since it explicitly confirmed the veryexistence of the closed electron loops. Numerically the vacuum polarizationcontribution is much less important than the contribution connected with theelectron spreading due to quantum corrections, and the total shift of the levelis positive.

2.3 Natural Magnitudes of Correctionsto the Lamb Shift

Let us emphasize that the main contribution to the Lamb shift is a radiativecorrection itself (compare (2.4), (2.6)) and contains a logarithmic enhance-ment factor. This is extremely important when one wants to get a qualitativeunderstanding of the magnitude of the higher order corrections to the Lambshift discussed below. Due to the presence of this accidental logarithmic en-hancement it is impossible to draw conclusions about the expected magnitudeof higher order corrections to the Lamb shift simply by comparing them to themagnitude of the leading order contribution. It is more reasonable to extractfrom this leading order contribution the term which can be called the skeletonfactor and to use it further as a normalization factor. Let us write the leadingorder contributions in (2.4), (2.6) obtained above in the form

4m(Zα)4

n3× radiative correction , (2.7)

where the radiative correction is either the slope of the Dirac form factor,roughly speaking equal to m2dF1(−k2)/dk2

|k2=0 = α/(3π) ln(Zα)−2, or thepolarization correction m2Π(−k2)/k4

|k2=0 = α/(15π).It is clear now that the scale setting factor which should be used for qual-

itative estimates of the high order corrections to the Lamb shift is equal to4m(Zα)4/n3. Note the characteristic dependence on the principal quantumnumber 1/n3 which originates from the square of the wave function at theorigin |ψ(0)|2 ∼ 1/n3. All corrections induced at small distances (or at highvirtual momenta) have this characteristic dependence and are called state-independent. Even the coefficients before the leading powers of the low en-ergy logarithms ln(Zα)2 are state-independent since these leading logarithmsoriginate from integration over the wide intermediate momenta region fromm(Zα)2 to m, and the respective factor before the logarithm is determined2 We remind the reader that according to the common renormalization procedure

the electric charge is defined as a charge observed at a very large distance.


by the high momenta part of the integration region. Estimating higher ordercorrections to the Lamb shift it is necessary to remember, as mentioned above,that unlike the case of radiative corrections to the scattering amplitudes, inthe bound state problem factors Zα are not accompanied by an extra fac-tor π in the denominator. This well known feature of the Coulomb problemprovides one more reason to preserve Z in analytic expressions (even whenZ = 1), since in this way one may easily separate powers of Zα not accompa-nied by π from powers of α which always enter formulae in the combinationα/π.

References

1. E. Fermi, Z. Phys. 60, 320 (1930).2. J. D. Bjorken and S. D. Drell, Relativistic Quantum Mechanics, McGraw-Hill

Book Co., NY, 1964.

3

External Field Approximation

We will first discuss corrections to the basic Dirac energy levels which arisein the external field approximation. These are leading relativistic correctionswith exact mass dependence and radiative corrections.

3.1 Leading Relativistic Correctionswith Exact Mass Dependence

We are considering a loosely bound two-particle system. Due to the nonrela-tivistic nature of this bound state it is clear that the main (Zα)2 contributionto the binding energy depends only on one mass parameter, namely, on thenonrelativistic reduced mass, and does not depend separately on the masses ofthe constituents. Relativistic corrections to the energy levels of order (Zα)4,describing the fine structure of the hydrogen spectrum, are missing in the non-relativistic Schrodinger equation approach. The correct description of the finestructure for an infinitely heavy Coulomb center is provided by the relativis-tic Dirac equation, but it tells us nothing about the proper mass dependenceof these corrections for the nucleus of finite mass. There are no reasons toexpect that in the case of a system of two particles with finite masses rela-tivistic corrections of order (Zα)4 will depend only on the reduced mass. Thedependence of these corrections on the masses of the constituents should bemore complicated.

The solution of the problem of the proper mass dependence of the rela-tivistic corrections of order (Zα)4 may be found in the effective Hamiltonianframework. In the center of mass system the nonrelativistic Hamiltonian fora system of two particles with Coulomb interaction has the form

H0 =p2

2m+

p2

2M− Zα

r. (3.1)

In a nonrelativistic loosely bound system expansion over (Zα)2 correspondsto the nonrelativistic expansion over v2/c2. Hence, we need an effective


20 3 External Field Approximation

Hamiltonian including the terms of the first order in v2/c2 for proper descrip-tion of the corrections of relative order (Zα)2 to the nonrelativistic energylevels. Such a Hamiltonian was first considered by Breit [1], who realized thatall corrections to the nonrelativistic two-particle Hamiltonian of the first orderin v2/c2 may be obtained from the sum of the free relativistic Hamiltonians ofeach of the particles and the relativistic one-photon exchange. This conjectureis intuitively obvious since extra exchange photons lead to at least one extrafactor of Zα, thus generating contributions to the binding energy of order(Zα)5 and higher.

An explicit expression for the Breit potential was derived in [2] from theone-photon exchange amplitude with the help of the Foldy-Wouthuysen trans-formation1

VBr =πZα

2

(1

m2+

1M2

)

δ3(r) − Zα

2mMr

(

p2 +r(r · p) · p

r2

)

+Zα

r3

(1

4m2+

12mM

)

[r × p] · σ . (3.2)

A simplified derivation of the Breit interaction potential may be found inmany textbooks (see, e.g., [3]).

All contributions to the energy levels up to order (Zα)4 may be calculatedfrom the total Breit Hamiltonian

HBr = H0 + VI , (3.3)

where the interaction potential is the sum of the instantaneous Coulomb andBreit potentials in Fig. 3.1.

Fig. 3.1. Sum of the Coulomb and Breit kernels

The corrections of order (Zα)4 are just the first order matrix elements ofthe Breit interaction between the Coulomb-Schrodinger eigenfunctions of theCoulomb Hamiltonian H0 in (3.1). The mass dependence of the Breit inter-action is known exactly, and the same is true for its matrix elements. Thesematrix elements and, hence, the exact mass dependence of the contributions tothe energy levels of order (Zα)4, beyond the reduced mass, were first obtaineda long time ago [2]

1 We do not consider hyperfine structure now and thus omit in (3.2) all terms inthe Breit potential which depend on the spin of the heavy particle.

3.1 Leading Relativistic Corrections with Exact Mass Dependence 21

Etotnj = (m + M) − mr(Zα)2

2n2− mr(Zα)4

2n3

(1

j + 12

− 34n

+mr

4n(m + M)

)

+(Zα)4m3

r

2n3M2

(1

j + 12

− 1l + 1

2

)

(1 − δl0) . (3.4)

Note the emergence of the last term in (3.4) which lifts the characteristicdegeneracy in the Dirac spectrum between levels with the same j and l =j ± 1/2. This means that the expression for the energy levels in (3.4) alreadypredicts a nonvanishing contribution to the classical Lamb shift E(2S 1

2) −

E(2P 12). Due to the smallness of the electron-proton mass ratio this extra

term is extremely small in hydrogen. The leading contribution to the Lambshift, induced by the QED radiative correction, is much larger.

In the Breit Hamiltonian in (3.2) we have omitted all terms which dependon spin variables of the heavy particle. As a result the corrections to theenergy levels in (3.4) do not depend on the relative orientation of the spins ofthe heavy and light particles (in other words they do not describe hyperfinesplitting). Moreover, almost all contributions in (3.4) are independent notonly of the mutual orientation of spins of the heavy and light particles butalso of the magnitude of the spin of the heavy particle. The only exception isthe small contribution proportional to the term δl0, called the Darwin-Foldycontribution. This term arises in the matrix element of the Breit Hamiltonianonly for the spin one-half nucleus and should be omitted for spinless or spin onenuclei. This contribution combines naturally with the nuclear size correction,and we postpone its discussion to Subsect. 6.1.2 dealing with the nuclear sizecontribution.

In the framework of the effective Dirac equation in the external Coulombfield2 (see Fig. 1.6) the result in (3.4) was first obtained in [4] (see also [5, 6])and rederived once again in [7], where it was presented in the form

Etotnj = (m + M) + mr[f(n, j) − 1] − m2

r

2(m + M)[f(n, j) − 1]2

+(Zα)4m3

r

2n3M2

(1

j + 12

− 1l + 1

2

)

(1 − δl0) . (3.5)

This equation has the same contributions of order (Zα)4 as in (3.4), butformally this expression also contains nonrecoil and recoil corrections of order(Zα)6 and higher. The nonrecoil part of these contributions is definitely cor-rect since the Dirac energy spectrum is the proper limit of the spectrum of atwo-particle system in the nonrecoil limit m/M = 0. As we will discuss laterthe first-order mass ratio contributions in (3.5) correctly reproduce recoil cor-rections of higher orders in Zα generated by the Coulomb and Breit exchangephotons. Additional first order mass ratio recoil contributions of order (Zα)6

2 We remind the reader that the external field in this equation also contains atransverse contribution.


will be calculated below. Recoil corrections of order (Zα)8 were never calcu-lated and at the present stage the mass dependence of these terms should beconsidered as completely unknown.

Recoil corrections depending on odd powers of Zα are also missing in (3.5),since as was explained above all corrections generated by the one-photon ex-change necessarily depend on the even powers of Zα. Hence, to calculate recoilcorrections of order (Zα)5 one has to consider the nontrivial contribution ofthe box diagram. We postpone discussion of these corrections until Sect. 4.1.

It is appropriate to give an exact definition of what is called the Lambshift. In the early days of the Lamb shift studies, experimentalists measurednot a shift but the classical Lamb splitting E(2S 1

2) − E(2P 1

2) between the

energy levels which are degenerate according to the naive Dirac equation inthe Coulomb field. This splitting is an experimental observable defined in-dependently of any theory. Modern experiments now produce high precisionexperimental data for the nondegenerate 1S energy level, and the very notionof the Lamb shift in this case, as well as in the case of an arbitrary energylevel, does not admit an unambiguous definition. It is most natural to callthe Lamb shift the sum of all contributions to the energy levels which lift thedouble degeneracy of the Dirac-Coulomb spectrum with respect to l = j±1/2(see Sect. 1.2). There emerged an almost universally adopted convention tocall the Lamb shift the sum of all contributions to the energy levels beyondthe first three terms in (3.5) and excluding, of course, all hyperfine splittingcontributions. This means that we define the Lamb shift by the relationship

Etotnjl = (m + M) + mr[f(n, j) − 1]

− m2r

2(m + M)[f(n, j) − 1]2 + Lnjl ≡ EDR

nj + Lnjl . (3.6)

We will adopt this definition below.

3.2 Radiative Corrections of Order αn(Zα)4m

Let us turn now to the discussion of radiative corrections which may be cal-culated in the external field approximation.

3.2.1 Leading Contribution to the Lamb Shift

3.2.1.1 Radiative Insertions in the Electron Lineand the Dirac Form Factor Contribution

The main contribution to the Lamb shift was first estimated in the nonrela-tivistic approximation by Bethe [8], and calculated by Kroll and Lamb [9], andby French and Weisskopf [10]. We have already discussed above qualitativelythe nature of this contribution. In the effective Dirac equation framework the

3.2 Radiative Corrections of Order αn(Zα)4m 23

Fig. 3.2. Kernels with many spanned Coulomb photons

apparent perturbation kernels to be taken into account are the diagrams withthe radiative photon spanning any number of the exchanged Coulomb pho-tons in Fig. 2.1 and Fig. 3.2. The dominant logarithmic contribution to theLamb shift is produced by the slope of the Dirac form factor F1, but superfi-cially all these kernels can lead to corrections of order α(Zα)4 and one cannotdiscard any of them. An additional problem is connected with the infrareddivergence of the kernels on the mass shell. There cannot be any true infrareddivergence in the bound state problem since all would be infrared divergencesare cut off either at the inverse Bohr radius or by the electron binding energy.Nevertheless such spurious on-shell infrared divergences can complicate thecalculations.

An important step which greatly facilitates treatment of all these problemsconsists in separation of the radiative photon integration region with the helpof auxiliary parameter σ in such way that m(Zα)2 � σ � m(Zα). It is easyto see that in the high momentum region each additional Coulomb photonproduces an extra factor Zα, so it is sufficient to consider only the kernelwith one Coulomb photon in this region. Moreover, the auxiliary parameter σprovides an infrared cutoff for the vertex graph and thus solves the problemof the would be infrared divergence. Due to the choice of the parameter σ �m(Zα)2 one may ignore the binding energy in the high momentum region.The main contribution to the Lamb shift is induced by the Dirac form factorF1(k2) − 1 which is proportional to the transferred momentum squared atsmall momentum transfer. This transferred momentum squared factor exactlycancels the Coulomb photon propagator attached to the Dirac form factor, andmomentum space integrations over wave function momenta factorize, therebyproducing the wave function squared at the origin in the coordinate space. It isclear that if one would take into account the small virtuality of either externalelectron line, expanding the integrand in this virtuality, it would lead to anextra factor of momentum squared in the integrand, and after integration withthe wave function would lead to an extra factor (Zα)2 in the contribution tothe energy shift. Hence, since we are interested in the contribution of orderα(Zα)4, we may freely ignore the virtuality of the electron line in the kernelin the high momentum region. It is also clear even at this stage, that the highmomentum region does not produce any contribution for the non-S statesbecause the wave function vanishes at the origin for such states, and, hence,the logarithmic contribution is missing for the non-S states.


Of course, all approximations made above are invalid in the low momen-tum integration region, where one cannot consider only the kernel with theradiative photon spanning only one Coulomb exchange. For soft radiative pho-tons, with characteristic momenta of order m(Zα)2, graphs with any numberof spanned exchanged photons in Fig. 3.2 have the same order of magnitudeand one has to take these graphs into account simultaneously. This meansthat one has to calculate the matrix element of the exact self-energy operatorin the external Coulomb field between Dirac-Coulomb wave functions. Thisproblem may seem formidable at first sight, but it can be readily solved withthe help of old-fashioned perturbation theory by inserting a complete set ofintermediate states and performing calculations in the dipole approximation,which is adequate to accuracy α(Zα)4m. It should be mentioned that themagnitude of the upper boundary of the interval for the auxiliary parameterσ was chosen in order to provide validity of the dipole approximation.

The most important fact about the auxiliary parameter σ is that one canuse different approximations for calculations of the high- and low-momentumcontributions. In the high-momentum region the factor m(Zα)2/k ≤ (m(Zα)2/σ � 1 plays the role of a small parameter and one can consider binding effectsas small corrections. In the low-momentum region k/(mZα) ≤ σ/(mZα) � 1one may use the nonrelativistic multipole expansion, and the main contribu-tion in this region is given by the dipole contribution. The crucial point is thatfor k ∼ σ both expansions are valid simultaneously and one can match themwithout loss in accuracy. Matching the high- and low-momentum contribu-tions one obtains the classical result for the shift of the energy level generatedby the slope of the Dirac form factor

∆E ={[

13

lnm(Zα)−2

mr+

1172

]

δl0 −13

ln k0(n, l)}

4α(Zα)4

πn3

(mr

m

)3

m ,

(3.7)where mr = mM/(m + M) is the reduced mass and ln k0(n, l) is the so calledBethe logarithm. The factor m/mr arises in the argument of the would be in-frared divergent logarithm ln(m/λ) since in the nonrelativistic approximationthe energy levels of an atom depend only on the reduced mass and, hence, theinfrared divergence is cut off by the binding energy mr(Zα)2 [7].

The mass dependence of the correction of order α(Zα)4 beyond the re-duced mass factor is properly described by the expression in (3.7) as wasproved in [11, 12]. In the same way as for the case of the leading relativisticcorrection in (3.4), the result in (3.7) is exact in the small mass ratio m/M ,since in the framework of the effective Dirac equation all corrections of order(Zα)4 are generated by the kernels with one-photon exchange. In some earlierpapers the reduced mass factors in (3.7) were expanded up to first order inthe small mass ratio m/M . Nowadays it is important to preserve an exactmass dependence in (3.7) because current experiments may be able to detectquadratic mass corrections (about 2 kHz for the 1S level in hydrogen) to theleading nonrecoil Lamb shift contribution.


The Bethe logarithm is formally defined as a certain normalized infinitesum of matrix elements of the coordinate operator over the Schrodinger-Coulomb wave functions. It is a pure number which can in principle be calcu-lated with arbitrary accuracy, and high accuracy results for the Bethe loga-rithm can be found in the literature (see, e.g. [13, 14] and references therein).For convenience we have collected some values for the Bethe logarithms [14]in Table 3.1.

3.2.1.2 Pauli Form Factor Contribution

The Pauli form factor F2 also generates a small contribution to the Lambshift. This form factor does not produce any contribution if one neglects thelower components of the unperturbed wave functions, since the respectivematrix element is identically zero between the upper components in the stan-dard representation for the Dirac matrices which we use everywhere. Takinginto account lower components in the nonrelativistic approximation we easilyobtain an explicit expression for the respective perturbation

δV = − 14m2

[

∆V + 2 σ · m

mr[∇V × p]

]

F2(0) , (3.8)

where V = −Zα/r is the Coulomb potential. Note the appearance of anextra factor m/mr in the coefficient before the second term. This is read-ily obtained from an explicit consideration of the radiatively corrected onephoton exchange. In momentum space the term with the Laplacian of theCoulomb potential depends only on the exchanged momentum, while the sec-ond term contains explicit dependence on the electron momentum. Since thePauli form factor depends explicitly on the electron momentum and not onthe relative momentum of the electron-proton system, the transition to rel-ative momentum, which is the argument of the unperturbed wave functions,leads to emergence of an extra factor m/mr.

The interaction potential above generated by the Pauli form factor maybe written in terms of the spin-orbit interaction

δV =[Zαπ

m2δ3(r) +

Zαπ

r3mmr(s · l)

]

F2(0) , (3.9)

wheres =

σ

2, l = r × p . (3.10)

The respective contributions to the Lamb shift are given by

∆E|l=0 =(Zα)4m

n3F2(0)

(mr

m

)3

=α(Zα)4m

2πn3

(mr

m

)3

,

∆E|l �=0 =(Zα)4m

n3F2(0)

j(j + 1) − l(l + 1) − 3/4l(l + 1)(2l + 1)

(mr

m

)2

=α(Zα)4m

2πn3

j(j + 1) − l(l + 1) − 3/4l(l + 1)(2l + 1)

(mr

m

)2

, (3.11)


Table 3.1. Bethe Logarithms for Lower Levels [14]

ln k0(n, l) ∆E = − 43

ln k0(n, l)α(Zα)4

πn3 (mrm

)3m kHz

1S 2.984 128 555 765 498 −3 232 942.89

2S 2.811 769 893 120 563 −380 776.64

2P −0.030 016 708 630 213 4 064.93

3S 2.767 663 612 491 822 −111 052.94

3P −0.038 190 229 385 312 1 532.39

3D −0.005 232 148 140 883 209.94

4S 2.749 811 840 454 057 −46 548.27

4P −0.041 954 894 598 086 710.20

4D −0.006 740 938 876 975 114.11

4F −0.001 733 661 482 126 29.35

5S 2.740 823 727 854 572 −23 754.81

5P −0.044 034 695 591 878 381.65

5D −0.007 600 751 257 947 65.88

5F −0.002 202 168 381 486 19.09

5G −0.000 772 098 901 537 6.69

6S 2.735 664 206 935 105 −13 721.12

6P −0.045 312 197 688 974 227.27

6D −0.008 147 203 962 354 40.86

6F −0.002 502 179 760 279 12.55

6G −0.000 962 797 424 841 4.83

6H −0.000 407 926 168 297 2.05

7S 2.732 429 129 187 092 −8 630.49

7P −0.046 155 177 262 915 145.78

7D −0.008 519 223 293 658 26.91

7F −0.002 709 095 727 000 8.56

7G −0.001 094 472 739 370 3.46

7H −0.000 499 701 854 766 1.58

7I −0.000 240 908 258 717 0.76

8S 2.730 267 260 690 589 −5 777.18

8P −0.046 741 352 003 557 98.90

8D −0.008 785 042 984 125 18.59

8F −0.002 859 114 559 296 6.05

8G −0.001 190 432 043 318 2.52

8H −0.000 566 532 724 12 1.20

8I −0.000 290 426 172 391 0.61

8K −0.000 153 864 500 961 0.33


where we have used the Schwinger value [15] of the anomalous magnetic mo-ment F2(0) = α/(2π). Correct reduced mass factors have been retained in thisexpression instead of expanding in m/M .

3.2.1.3 Polarization Operator Contribution

The leading polarization operator contribution to the Lamb shift in Fig. 2.2was already calculated above in (2.6). Restoring the reduced mass factorswhich were omitted in that qualitative discussion, we easily obtain

∆E = 4π(Zα)|Ψn(0)|2 Π(−k2)k4 |k2=0

= −4α(Zα)4m15πn3

(mr

m

)3

δl0 . (3.12)

This result was originally obtained in [16] long before the advent of modernQED, and was the only known source for the 2S − 2P splitting. There is acertain historic irony that for many years it was the common wisdom “thatthis effect is . . . much too small and has, in addition, the wrong sign” [8] toexplain the 2S − 2P splitting.

3.2.2 Radiative Corrections of Order α2(Zα)4m

From the theoretical point of view, calculation of the corrections of orderα2(Zα)4 contains nothing fundamentally new in comparison with the correc-tions of order α(Zα)4. The scale for these corrections is provided by the factor4α2(Zα)4/(π2n3)m, as one may easily see from the respective discussion abovein Sect. 2.3. Corrections depend only on the values of the form factors andtheir derivatives at zero transferred momentum and the only challenge is tocalculate respective radiative corrections with sufficient accuracy.

3.2.2.1 Dirac Form Factor Contribution

Calculation of the contribution of order α2(Zα)4 induced by the radiativephoton insertions in the electron line is even simpler than the respective cal-culation of the leading order contribution. The point is that the second andhigher order contributions to the slope of the Dirac form factor are infraredfinite, and hence, the total contribution of order (Zα)4 to the Lamb shiftis given by the slope of the Dirac form factor. Hence, there is no need tosum an infinite number of diagrams. One readily obtains for the respectivecontribution

∆EF1 = −4π(Zα)|Ψn(0)|2 dF(2)1 (−k2)dk2 |k2=0

= 0.469 941 4 . . .4α2(Zα)4

π2n3

(mr

m

)3

m δl0 , (3.13)


where we have used the second order slope of the Dirac form factor generatedby the diagrams in Fig. 3.3

dF(2)1 (−k2)dk2 |k2=0

=[34ζ(3) − π2

2ln 2 +

49432

π2 +4 8195 184

]1

m2

(α

π

)2

≈ −0.469 941 4 . . .

m2

(α

π

)2

. (3.14)

The two-loop slope was considered in the early pioneer works [17, 18], andfor the first time the correct result was obtained numerically in [19]. This lastwork triggered a flurry of theoretical activity [20, 21, 22, 23], followed by thefirst completely analytical calculation in [24]. The same analytical result forthe slope of the Dirac form factor was derived in [25] from the total e+e−

cross section and the unitarity condition.


Calculation of the Pauli form factor contribution follows closely the one whichwas performed in order α(Zα)4, the only difference being that we have toemploy the second order contribution to the Pauli form factor (see Fig. 3.3)calculated a long time ago in [26, 27, 28] (the result of the first calculation[26] turned out to be in error)

F(2)2 (0) =

[34ζ(3) − π2

2ln 2 +

π2

12+

197144

](α

π

)2

≈ −0.328 478 9 . . .(α

π

)2

.

(3.15)Then one readily obtains for the Lamb shift contribution

Fig. 3.3. Two-loop electron formfactor


Fig. 3.4. Insertions of two-loop polarization operator

∆E|l=0 = −0.328 478 9 . . .α2(Zα)4m

π2n3

(mr

m

)3

,

∆E|l �=0 = −0.328 478 9 . . .α2(Zα)4m

π2n3

j(j + 1) − l(l + 1) − 3/4l(l + 1)(2l + 1)

(mr

m

)2

.

(3.16)


Here we use well known low momentum asymptotics of the second order po-larization operator [29, 30, 31] in Fig. 3.4

Π(−k2)k4 |k2=0

= − 41162m2

(α

π

)2

, (3.17)

and obtain [29]

∆E = −8281

α2(Zα)4mπ2n3

(mr

m

)3

δl0 . (3.18)

3.2.3 Corrections of Order α3(Zα)4m

3.2.3.1 Dirac Form Factor Contribution

Calculation of the corrections of order α3(Zα)4 is similar to calculation ofthe contributions of order α2(Zα)4. Respective corrections depend only onthe values of the three-loop form factors or their derivatives at vanishingtransferred momentum. The three-loop contribution to the slope of the Diracform factor (Fig. 3.5) was calculated analytically [32]

dF(2)1 (−k2)dk2 |k2=0

= −[258

ζ(5) − 1724

π2ζ(3) − 2929288

ζ(3) − 2179

a4 −217216

ln4 2

− 1031080

π2 ln2 2 +416712160

π2 ln 2 +389925920

π4

− 45497938880

π2 − 77513186624

]1

m2

(α

π

)3

≈ −0.171 72 . . .

m2

(α

π

)3

, (3.19)


where

a4 =∞∑

n=1

12nn4

. (3.20)

The respective contribution to the Lamb shift is equal to

∆EF1 = 0.171 72 . . .4α3(Zα)4

π3n3

(mr

m

)3

m δl0 . (3.21)


For calculation of the Pauli form factor contribution to the Lamb shift thethird order contribution to the Pauli form factor (Fig. 3.5), calculated numer-ically in [33], and analytically in [34] is used

F(3)2 (0) =

{8372

π2ζ(3) − 21524

ζ(5) +1003

[

(a4 +124

ln4 2) − 124

π2 ln2 2]

− 2392 160

π4 +13918

ζ(3) − 2989

π2 ln 2 +17 101

810π2 +

28 2595 184

}(α

π

)3

≈ 1.181 241 456 . . .(α

π

)3

. (3.22)

Then one obtains for the Lamb shift

∆E|l=0 = 1.181 241 456 . . .α3(Zα)4m

π3n3

(mr

m

)3

,

∆E|l �=0 = 1.181 241 456 . . .α3(Zα)4m

π3n3

j(j + 1) − l(l + 1) − 3/4l(l + 1)(2l + 1)

(mr

m

)2

.

(3.23)

Fig. 3.5. Examples of the three-loop contributions for the electron form factor


In this case the analytic result for the low frequency asymptotics of the thirdorder polarization operator (see Fig. 3.6) [35] is used


Fig. 3.6. Examples of the three-loop contributions to the polarization operator

Π(−k2)k4 |k2=0

= −(

8 1359 216

ζ(3) − π2 ln 215

+23π2

360− 325 805

373 248

)1

m2

(α

π

)3

≈ −0.362 654 440 . . .

m2

(α

π

)3

, (3.24)

and one obtains [36]

∆E = −1.450 617 763 . . .α3(Zα)4m

π3n3

(mr

m

)3

δl0 . (3.25)

3.2.4 Total Correction of Order αn(Zα)4m

The total contribution of order αn(Zα)4m is given by the sum of correctionsin (3.7), (3.11), (3.12), (3.13), (3.16), (3.18), (3.21), (3.23), (3.25). It is equalto

∆E|l=0 ={(

43

lnm(Zα)−2

mr− 4

3ln k0(n, 0) +

3845

)

+[

−94ζ(3) +

32π2 ln 2 − 10

27π2 − 2 179

648

](α

π

)

+[8524

ζ(5) − 12172

π2ζ(3) − 84 0712 304

ζ(3) − 7127

ln4 2 − 239135

π2 ln2 2

+4 787108

π2 ln 2 − 5689

a4 +15913 240

π4 − 252 2519 720

π2+679 44193 312

](α

π

)2}

× α(Zα)4mπn3

(mr

m

)3

={(

43

lnm(Zα)−2

mr− 4

3ln k0(n, 0) +

3845

)

+ 0.538 95 . . .(α

π

)

+ 0.417 504 . . .(α

π

)2}

α(Zα)4mπn3

(mr

m

)3

, (3.26)


for the S-states, and

∆E|l �=0 ={

−43

ln k0(n, l)(mr

m

)3

+[12

+(

34ζ(3) − π2

2ln 2 +

π2

12+

197144

)(α

π

)

+(

8372

π2ζ(3)− 21524

ζ(5) +1003

a4+2518

ln4 2 − 2518

π2 ln2 2 − 2392 160

π4

+13918

ζ(3) − 2989

π2 ln 2 +17 101

810π2 +

28 2595 184

)(α

π

)2]

×j(j + 1) − l(l + 1) − 3/4l(l + 1)(2l + 1)

(mr

m

)2}

α(Zα)4mπn3

={

−43

ln k0(n, l)(mr

m

)3

+[12− 0.328 478 9 . . .

(α

π

)

+ 1.181 241 456 . . .(α

π

)2]

×j(j + 1) − l(l + 1) − 3/4l(l + 1)(2l + 1)

(mr

m

)2}

α(Zα)4mπn3

(3.27)

for the non-S-states.Numerically corrections of order αn(Zα)4m for the lowest energy levels

give

∆E(1S) = 8 115 785.64 kHz ,

∆E(2S) = 1 037 814.43 kHz , (3.28)∆E(2P ) = −12 846.46 kHz .

Contributions of order α4(Zα)4m are suppressed by an extra factor α/πin comparison with the corrections of order α3(Zα)4m. Their expected mag-nitude is at the level of hundredths of kHz even for the 1S state in hydrogen,and they are too small to be of any phenomenological significance.

3.2.5 Heavy Particle Polarization Contributionsof Order α(Zα)4m

We have considered above only radiative corrections containing virtual pho-tons and electrons. However, at the current level of accuracy one has to con-sider also effects induced by the virtual muons and lightest strongly interact-ing particles. The respective corrections to the electron anomalous magneticmoment are well known [33] and are still too small to be of any practicalinterest for the Lamb shift calculations. Heavy particle contributions to thepolarization operator numerically have the same magnitude as polarizationcorrections of order α3(Zα)4. Corrections to the low-frequency asymptoticsof the polarization operator are generated by the diagrams in Fig. 3.7. Themuon loop contribution to the polarization operator


Fig. 3.7. Muon-loop and hadron contributions to the polarization operator

Π(−k2)k4 |k2=0

= − α

15πm2µ

(3.29)

immediately leads (compare (3.12)) to an additional contribution to the Lambshift [37, 38]

∆E = − 415

(mr

mµ

)2α(Zα)4

πn3mr δl0 . (3.30)

The hadronic polarization contribution to the Lamb shift was discussed ina number of papers [37, 38, 39]. The light hadron contribution to the polar-ization operator may easily be estimated with the help of vector dominance

Π(−k2)k4 |k2=0

= −∑

i

4πα

f2vi

m2vi

, (3.31)

where mviare the masses of the three lowest vector mesons and the vector

meson-photon vertex has the form em2vi

/fvi. Using the free quark loops for

the contribution of the heavy quark flavors one obtains the total hadronicvacuum polarization contribution to the Lamb shift in the form [38]

∆E = −4(

Σvi

4π2

f2vi

m2vi

+23

11 GeV2

)α(Zα)4

πn3m δl0 . (3.32)

Numerically this correction is −3.18 kHz for the 1S-state and −0.40 kHz forthe 2S-state in hydrogen.

Only the value of the leading coefficient in the low energy expansion ofthe hadronic vacuum polarization is needed for calculation of the hadroniccontribution to the Lamb shift (see the LHS of (3.32)). A model independentvalue of this coefficient may be obtained for the analysis of the experimentaldata on the low energy e+e− annihilation. Respective contribution to the 1SLamb shift [39] is −3.40(7) kHz. This value is compatible but more accuratethan the result in (3.32).3

All corrections of order αn(Zα)4 m are collected in Table 3.2.

3 It is not obvious that the hadronic vacuum polarization contribution should beincluded in the phenomenological analysis of the Lamb shift measurements, sinceexperimentally it is indistinguishable from an additional contribution to the pro-ton charge radius. We will return to this problem below in Sect. 6.1.3.

34

3E

xtern

alField

Approx

imatio

nTable 3.2. Contributions of Order αn(Zα)4m

4α(Zα)4

πn3 (mrm

)3m ≈ 3 250 137.65(4)

n3 kHz ∆E(1S) kHz ∆E(2S) kHz

Bethe (1947) [8]

French, Weisskopf (1949) [10][

13

ln m(Zα)−2

mr+ 11

72

]

δl0 − 13

ln k0(n, l) 7 925 175.34(8) 1 013 988.14(1)

Kroll, Lamb (1949) [9]

One-loopPauli FF l = 0 1

8406 267.21 50 783.40

One-loop

Pauli FF l �= 0 j(j+1)−l(l+1)−3/48l(l+1)(2l+1)

mmr

One-loopVacuum PolarizationUehling (1935) [16] − 1

15δl0 −216 675.85 −27 084.48

Two-loop Dirac FF

Appelquist, [− 34ζ(3) + π2

2ln 2 − 49

432π2 − 4 819

5 184]απδl0

Brodsky (1970) [19]Barbieri, Mignaco, ≈ 0.469 941 4 . . .α

πδl0 3 547.82 443.48

Remiddi (1971) [24]

Two-loop Pauli FF l = 0 ( 316

ζ(3) − π2

8ln 2 + π2

48+ 197

576)α

π

Sommerfield (1957) [28]Peterman (1957) [27] ≈ −0.082 119 7 . . . α

π−619.96 −77.50

Pauli FF l �= 0 ( 316

ζ(3) − π2

8ln 2 + π2

48+ 197

576)

Sommerfield (1957) [28] × j(j+1)−l(l+1)−3/4l(l+1)(2l+1)

απ

mmr

Peterman (1957) [27] ≈ −0.082 119 7 . . . j(j+1)−l(l+1)−3/4l(l+1)(2l+1)

mmr

απ

Two-loopVacuum PolarizationBaranger, Dyson, − 41

162(α

π)δl0 −1 910.67 −238.83

Salpeter (1952) [29]

3.2

Radia

tive

Correctio

ns

ofO

rder

αn(Z

α)4m

35

Table 3.2. (Continued)

Three-loop [ 258

ζ(5) − 1724

π2ζ(3) − 2929288

ζ(3) − 2179

a4

Dirac FF − 217216

ln4 2 − 1031080

π2 ln2 2 + 416712160

π2 ln 2Melnikov, + 3899

25920π4 − 454979

38880π2 − 77513

186624](α

π)2δl0

van Ritbergen (1999) [32] ≈ 0.171 72 . . .(απ)2δl0 3.01 0.38

Three-loop { 83288

π2ζ(3) − 21596

ζ(5) + 253

[(a4 + 124

ln4 2)Pauli FF l = 0 − 1

24π2 ln2 2] − 239

8 640π4 + 139

72ζ(3)

Kinoshita (1990) [33] − 14918

π2 ln 2 + 17 1013 240

π2 + 28 25920 736

}(απ)2

Laporta, Remiddi (1996) [34] ≈ 0.295 310 3 . . . (απ)2 5.18 0.65

Three-loop l �= 0 { 83288

π2ζ(3) − 21596

ζ(5) + 253

[(a4 + 124

ln4 2)Pauli FF − 1

24π2 ln2 2] − 239

8 640π4 + 139

72ζ(3) − 149

18π2 ln 2

Kinoshita (1990) [33] + 17 1013 240

π2 + 28 25920 736

} j(j+1)−l(l+1)−3/4l(l+1)(2l+1)

mmr

(απ)2

Laporta, Remiddi (1996) [34] ≈ 0.295 310 3 . . . j(j+1)−l(l+1)−3/4l(l+1)(2l+1)

mmr

(απ)2

Three-loop Vacuum Polarization

Baikov, Broadhurst (1995) [35] −( 8 1359 216

ζ(3) − π2 ln 215

+ 23π2

360− 325 805

373 248)

Eides, Grotch (1995) [36] ×(

απ

)2δl0 ≈ −0.362 654 4 . . .

(απ

)2δl0 −6.36 −0.79

Muonic PolarizationKarshenboim (1995) [37]Eides, Shelyuto (1995) [38] − 1

15( m

mµ)2 −5.07 −0.63

Hadronic PolarizationFriar, Martorell, Sprung (1999) [39] −3.40 −0.40



3.3.1 Skeleton Integral Approach to Calculationsof Radiative Corrections

We have seen above that calculation of the corrections of order αn(Zα)4m(n > 1) reduces to calculation of higher order corrections to the propertiesof a free electron and to the photon propagator, namely to calculation of theslope of the electron Dirac form factor and anomalous magnetic moment, andto calculation of the leading term in the low-frequency expansion of the polar-ization operator. Hence, these contributions to the Lamb shift are independentof any features of the bound state. A nontrivial interplay between radiativecorrections and binding effects arises first in calculation of contributions oforder α(Zα)5m, and in calculations of higher order terms in the combinedexpansion over α and Zα.

Calculation of the contribution of order αn(Zα)5m to the energy shift iseven simpler than calculation of the leading order contribution to the Lambshift because the scattering approximation is sufficient in this case [40, 41, 42].Formally this correction is induced by kernels with at least two-photon ex-changes, and in analogy with the leading order contribution one could alsoanticipate the appearance of irreducible kernels with higher number of ex-changes. This does not happen, however, as can be proved formally, but infact no formal proof is needed. First one has to realize that for high exchangedmomenta expansion in Zα is valid, and addition of any extra exchanged pho-ton always produces an extra power of Zα. Hence, in the high-momentumregion only diagrams with two exchanged photons are relevant. Treatment ofthe low-momentum region is greatly facilitated by a very general feature ofthe Feynman diagrams, namely that the infrared behavior of any radiativelycorrected Feynman diagram (or more accurately any gauge invariant sumof Feynman diagrams) is milder than the behavior of the skeleton diagram.Consider the matrix element in momentum space of the diagram in Fig. 3.8with two exchanged Coulomb photons between the Schrodinger-Coulomb wavefunctions. We will take the external electron momenta to be on-shell and tohave vanishing space components. It is then easy to see that the contributionof such a diagram to the Lamb shift is given by the infrared divergent integral

− 16(Zα)5

πn3

(mr

m

)3

m

∫ ∞

0

dk

k4δl0 , (3.33)

Fig. 3.8. Skeleton diagram with two exchanged Coulomb photons


Fig. 3.9. Radiative insertions in the electron line

where k is the dimensionless momentum of the exchanged photon measuredin the units of the electron mass. This divergence has a simple physical in-terpretation. If we do not ignore small virtualities of the external electronlines and the external wave functions this two-Coulomb exchange adds oneextra rung to the Coulomb wave function and should simply reproduce it.The naive infrared divergence above would be regularized at the characteris-tic atomic scale mZα. Hence, it is evident that the kernel with two-photonexchange is already taken into account in the effective Dirac equation aboveand there is no need to try to consider it as a perturbation. Let us considernow radiative photon insertions in the electron line (see Fig. 3.9). Account ofthese corrections effectively leads to insertion of an additional factor L(k) inthe divergent integral above, and while this factor has at most a logarithmicasymptotic behavior at large momenta and does not spoil the ultraviolet con-vergence of the integral, in the low momentum region it behaves as L(k) ∼ k2

(again up to logarithmic factors), and improves the low frequency behaviorof the integrand. However, the integrand is still divergent even after inclusionof the radiative corrections because the two-photon-exchange box diagram,even with radiative corrections, contains a contribution of the previous or-der in Zα, namely the main contribution to the Lamb shift induced by theelectron form factor. This spurious contribution may be easily removed bysubtracting the leading low momentum term from L(k)/k4. The result of thesubtraction is a convergent integral which is responsible for the correction oforder α(Zα)5. As an additional bonus of this approach one does not need toworry about the ultraviolet divergence of the one-loop radiative corrections.The subtraction automatically eliminates any ultraviolet divergent terms andthe result is both ultraviolet and infrared finite.

Due to radiative insertions low integration momenta (of atomic ordermZα) are suppressed in the exchange loops and the effective integration mo-menta are of order m. Hence, one may neglect the small virtuality of externalfermion lines and calculate the above diagrams with on-mass-shell externalmomenta. Contributions to the Lamb shift are given by the product of thesquare of the Schrodinger-Coulomb wave function at the origin |ψ(0)|2 and thediagram. Under these conditions the diagrams in Fig. 3.9 comprise a gaugeinvariant set and may easily be calculated.

Contributions of the diagrams with more than two exchanged Coulombphotons are of higher order in Zα. This is obvious for the high exchangedmomenta integration region. It is not difficult to demonstrate that in theYennie gauge [43, 44, 45] contributions from the low exchanged momentum


region to the matrix element with the on-shell external electron lines remaininfrared finite, and hence, cannot produce any correction of order α(Zα)5.Since the sum of diagrams with the on-shell external electron lines is gaugeinvariant this is true in any gauge. It is also clear that small virtuality of theexternal electron lines would lead to an additional suppression of the matrixelement under consideration, and, hence, it is sufficient to consider only two-photon exchanges for calculation of all corrections of order α(Zα)5.

The magnitude of the correction of order α(Zα)5 may be easily estimatedbefore the calculation is carried out. We need to take into account the skeletonfactor 4m(Zα)4/n3 discussed above in Sect. 2.3, and multiply it by an extrafactor α(Zα). Naively, one could expect a somewhat smaller factor α(Zα)/π.However, it is well known that a convergent diagram with two external photonsalways produces an extra factor π in the numerator, thus compensating thefactor π in the denominator generated by the radiative correction. Hence,calculation of the correction of order α(Zα)5 should lead to a numerical factorof order unity multiplied by 4mα(Zα)5/n3.

3.3.2 Radiative Corrections of Order α(Zα)5m

3.3.2.1 Correction Induced by the Radiative Insertionsin the Electron Line

This correction is generated by the sum of all possible radiative insertions inthe electron line in Fig. 3.9. In the approach described above one has to calcu-late the electron factor corresponding to the sum of all radiative correctionsin the electron line, make the necessary subtraction of the leading infraredasymptote, insert the subtracted expression in the integrand in (3.33), andthen integrate over the exchanged momentum. This leads to the result

∆E = 4(

1 +11128

− 12

ln 2)

α(Zα)5

n3

(mr

m

)3

m δl0 , (3.34)

which was first obtained in [40, 41, 42] in other approaches. Note that numer-ically 1+11/128−1/2 ln 2 ≈ 0.739 in excellent agreement with the qualitativeconsiderations above.

3.3.2.2 Correction Induced by the Polarization Insertionsin the External Photons

The correction of order α(Zα)5 induced by the polarization operator insertionsin the external photon lines in Fig. 3.10 was obtained in [40, 41, 42] andmay again be calculated in the skeleton integral approach. We will use thesimplicity of the one-loop polarization operator, and perform this calculationin more detail in order to illustrate the general considerations above. Forcalculation of the respective contribution one has to insert the polarizationoperator in the skeleton integrand in (3.33)


Fig. 3.10. Polarization insertions in the Coulomb lines

1k2

→ α

πI1(k) , (3.35)

where

I1(k) =∫ 1

0

dvv2(1 − v2/3)

4 + (1 − v2)k2. (3.36)

Of course, the skeleton integral still diverges in the infrared after this substi-tution since

I1(0) =115

. (3.37)

This linear infrared divergence dk/k2 is effectively cut off at the characteristicatomic scale mZα, it lowers the power of the factor Zα, respective wouldbe divergent contribution turns out to be of order α(Zα)4, and correspondsto the polarization part of the leading order contribution to the Lamb shift.We carry out the subtraction of the leading low frequency asymptote of thepolarization operator insertion, which corresponds to the subtraction of theleading low frequency asymptote in the integrand for the contribution to theenergy shift

I1(k) ≡ I1(k) − I1(0) = −k2

4

∫ 1

0

dvv2(1 − v2)(1 − v2/3)

4 + (1 − v2)k2, (3.38)

and substitute the subtracted expression in the formula for the Lamb shiftin (3.33). We also insert an additional factor 2 in order to take into accountpossible insertions of the polarization operator in both photon lines. Then

∆E = − m(mr

m

)3 α(Zα)5

π2n3

32(1 − m2

M2 )

∫ ∞

0

dkI1(k)k2

δl0

= m(mr

m

)3 α(Zα)5

π2n3

8(1 − m2

M2 )

∫ 1

0

dv

∫ ∞

0

dkv2(1 − v2)(1 − v2/3)

4 + (1 − v2)k2δl0

=548

α(Zα)5

n3

(mr

m

)3

m δl0 . (3.39)

We have restored in (3.39) the characteristic factor 1/(1 − m2/M2) whichwas omitted in (3.33), but which naturally arises in the skeleton integral.However, it is easy to see that an error generated by the omission of thisfactor is only about 0.02 kHz even for the electron-line contribution to the 1Slevel shift, and, hence, this correction may be safely omitted at the presentlevel of experimental accuracy.


3.3.2.3 Total Correction of Order α(Zα)5m

The total correction of order α(Zα)5m is given by the sum of contributionsin (3.34), (3.39)

∆E = 4(

1 +11128

+5

192− 1

2ln 2

)α(Zα)5

n3

(mr

m

)3

m δl0

= 3.061 622 . . .α(Zα)5

n3

(mr

m

)3

m δl0

= 57 030.70 kHz|n=1 ,

= 7 128.84 kHz|n=2 . (3.40)


Corrections of order α2(Zα)5 have the same physical origin as correctionsof order α(Zα)5, and the scattering approximation is sufficient for their cal-culation [46]. We consider now corrections of higher order in α than in theprevious section and there is a larger variety of relevant graphs. All six gaugeinvariant sets of diagrams [46] which produce corrections of order α2(Zα)5 arepresented in Fig. 3.11. The blob called “2 loops” in Fig. 3.11 (f) means thegauge invariant sum of diagrams with all possible insertions of two radiativephotons in the electron line. All diagrams in Fig. 3.11 may be obtained fromthe skeleton diagram in Fig. 3.8 with the help of different two-loop radiativeinsertions. As in the case of the corrections of order α(Zα)5, corrections to theenergy shifts are given by the matrix elements of the diagrams in Fig. 3.11calculated between free electron spinors with all external electron lines onthe mass shell, projected on the respective spin states, and multiplied by thesquare of the Schrodinger-Coulomb wave function at the origin [46].

It should be mentioned that some of the diagrams under considerationcontain contributions of the previous order in Zα. These contributions areproduced by the terms proportional to the exchanged momentum squaredin the low-frequency asymptotic expansion of the radiative corrections, andare connected with integration over external photon momenta of characteris-tic atomic scale mZα. The scattering approximation is inadequate for theircalculation. In the skeleton integral approach these previous order contribu-tions arise as powerlike infrared divergences in the final integration over theexchanged momentum. We subtract leading low-frequency terms in the low-frequency asymptotic expansions of the integrands, when necessary, and thusremove the spurious previous order contributions.

3.3.3.1 One-Loop Polarization Insertions in the Coulomb Lines

The simplest correction is induced by the diagrams in Fig. 3.11 (a) with twoinsertions of the one-loop vacuum polarization in the external photon lines.


Fig. 3.11. Six gauge invariant sets of diagrams for corrections of order α2(Zα)5m

The contribution to the Lamb shift is given by the insertion of the one-looppolarization operator squared I2

1 (k) in the skeleton integral in (3.33), andtaking into account the multiplicity factor 3 one easily obtains [46, 47, 48]

∆E = −48α2(Zα)5

π3n3

(mr

m

)3

m

∫ ∞

0

dkI21 (k)δl0 = − 23

378α2(Zα)5

πn3

(mr

m

)3

mδl0 .

(3.41)

3.3.3.2 Insertions of the Irreducible Two-Loop Polarizationin the Coulomb Lines

The naive insertion 1/k2 → I2(k) of the irreducible two-loop vacuum polar-ization operator I2(k) [30, 31] in the skeleton integral in (3.33) would leadto an infrared divergent integral for the diagrams in Fig. 3.11 (b). This di-vergence reflects the existence of the correction of the previous order in Zαconnected with the two-loop irreducible polarization. This contribution of or-der α2(Zα)4m was discussed in Subsect. 3.2.2.3, and as we have seen therespective contribution to the Lamb shift is given simply by the product ofthe Schrodinger-Coulomb wave function squared at the origin and the leading


low-frequency term of the function I2(0). In terms of the loop momentumintegration this means that the relevant loop momenta are of the atomic scalemZα. Subtraction of the value I2(0) from the function I2(k) effectively re-moves the previous order contribution (the low momentum region) from theloop integral and one obtains the radiative correction of order α2(Zα)5m gen-erated by the irreducible two-loop polarization operator [46, 47, 48]

∆E = −32α2(Zα)5

π3n3

(mr

m

)3

m

∫ ∞

0

dk

k2[I2(k) − I2(0)] δl0

=(

5263

ln 2 − 2563

π +15 64713 230

)α2(Zα)5

πn3

(mr

m

)3

m δl0 . (3.42)

3.3.3.3 Insertion of One-Loop Electron Factor in the Electron Lineand of the One-Loop Polarization in the Coulomb Lines

The next correction of order α2(Zα)5 is generated by the gauge invariant setof diagrams in Fig. 3.11 (c). The respective analytic expression is obtainedfrom the skeleton integral by simultaneous insertion in the integrand of theone-loop polarization function I1(k) and of the expressions corresponding toall possible insertions of the radiative photon in the electron line. It is simplerfirst to obtain an explicit analytic expression for the sum of all these radiativeinsertions in the electron line, which we call the one-loop electron factor L(k)(explicit expression for the electron factor in different forms may be found in[12, 49, 50, 51]), and then to insert this electron factor in the skeleton inte-gral. It is easy to check explicitly that the resulting integral for the radiativecorrection is both ultraviolet and infrared finite. The infrared finiteness nicelycorrelates with the physical understanding that for these diagrams there isno correction of order α2(Zα)4 generated at the atomic scale. The respec-tive integral for the radiative correction was calculated both numerically andanalytically [49, 47, 51], and the result has the following elegant form

∆E = −32α2(Zα)5

π3n3

(mr

m

)3

m

∫ ∞

0

dkL(k)I1(k) δl0

=

(

83

ln2 1 +√

52

− 87263

√5 ln

1 +√

52

+62863

ln 2 − 2π2

9+

67 2826 615

)

× α2 (Zα)5

πn3

(mr

m

)3

m δl0 . (3.43)

3.3.3.4 One-Loop Polarization Insertionsin the Radiative Electron Factor

This correction is induced by the gauge invariant set of diagrams in Fig. 3.11(d) with the polarization operator insertions in the radiative photon. Therespective radiatively corrected electron factor is given by the expression [50]


L(k) =∫ 1

0

dvv2(1 − v2

3 )1 − v2

L(k, λ) , (3.44)

where L(k, λ) is just the one-loop electron factor used in (3.43) but with afinite photon mass λ2 = 4/(1 − v2).

Direct substitution of the radiatively corrected electron factor L(k) inthe skeleton integral in (3.33) would lead to an infrared divergence. Thisdivergence reflects existence in this case of the correction of the previous orderin Zα generated by the two-loop insertions in the electron line. The magnitudeof this previous order correction is determined by the nonvanishing value ofthe electron factor L(k) at zero

L(0) = −2F ′1(0) − 1

2F2(0) , (3.45)

which is simply a linear combination of the slope of the two-loop Dirac formfactor and the two-loop contribution to the electron anomalous magneticmoment.

Subtraction of the radiatively corrected electron factor removes this pre-vious order contribution which was already considered above, and leads to afinite integral for the correction of order α2(Zα)5 [50, 47]

∆E = −16α2(Zα)5

π3n3

(mr

m

)3

m

∫ ∞

0

dkL(k) − L(0)

k2δl0 (3.46)

= −0.072 90 . . .α2 (Zα)5

πn3

(mr

m

)3

m δl0 .

3.3.3.5 Light by Light Scattering Insertionsin the External Photons

The diagrams in Fig. 3.11 (e) with the light by light scattering insertions inthe external photons do not generate corrections of the previous order in Zα.They are both ultraviolet and infrared finite and respective calculations arein principle quite straightforward though technically involved. Only numericalresults were obtained for the contributions to the Lamb shift [47, 52]

∆E = −0.122 9 . . .α2 (Zα)5

πn3

(mr

m

)3

m δl0 . (3.47)

3.3.3.6 Diagrams with Insertions of Two Radiative Photonsin the Electron Line

As we have already seen, contributions of the diagrams with radiative in-sertions in the electron line always dominate over the contributions of thediagrams with radiative insertions in the external photon lines. This property


of the diagrams is due to the gauge invariance of QED. The diagrams (radia-tive insertions) with the external photon lines should be gauge invariant, andas a result transverse projectors correspond to each external photon. Theseprojectors are rational functions of external momenta, and they additionallysuppress low momentum integration regions in the integrals for energy shifts.Respective projectors are of course missing in the diagrams with insertionsin the electron line. The low momentum integration region is less suppressedin such diagrams, and hence they generate larger contributions to the energyshifts.

This general property of radiative corrections clearly manifests itself in thecase of six gauge invariant sets of diagrams in Fig. 3.11. By far the largestcontribution of order α2(Zα)5 to the Lamb shift is generated by the lastgauge invariant set of diagrams in Fig. 3.11 (f), which consists of nineteentopologically different diagrams [53] presented in Fig. 3.12. These nineteengraphs may be obtained from the three graphs for the two-loop electron self-energy by insertion of two external photons in all possible ways. Graphs inFig. 3.12 (a–c) are obtained from the two-loop reducible electron self-energydiagram, graphs in Fig. 3.12 (d − k) are the result of all possible insertionsof two external photons in the rainbow self-energy diagram, and diagramsin Fig. 3.12 (l–s) are connected with the overlapping two-loop self-energygraph. Calculation of the respective energy shift was initiated in [53, 54], wherecontributions induced by the diagrams in Fig. 3.12 (a–h) and in Fig. 3.12 (l)were obtained. Contribution of all nineteen diagrams to the Lamb shift wasfirst calculated in [55]. In the framework of the skeleton integral approach thecalculation was completed in [56, 38] with the result

∆E = −7.725(1) . . .α2 (Zα)5

πn3

(mr

m

)3

m δl0 , (3.48)

which confirmed the one in [55] but is about two orders of magnitude moreprecise than the result in [55, 57].

A few comments are due on the magnitude of this important result. It issometimes claimed in the literature that it has an unexpectedly large magni-tude. A brief glance at Table 3.3 is sufficient to convince oneself that this isnot the case. For the reader who followed closely the discussion of the scales ofdifferent contributions above, it should be clear that the natural scale for thecorrection under discussion is set by the factor 4α2(Zα)5/(πn3)m. The coef-ficient before this factor obtained in (3.48) is about −1.9 and there is nothingunusual in its magnitude for a numerical factor corresponding to a radiativecorrection. It should be compared with the respective coefficient 0.739 beforethe factor 4α(Zα)5/n3m in the case of the electron-line contribution of theprevious order in α.

The misunderstanding about the magnitude of the correction of orderα2(Zα)5m has its roots in the idea that the expansion of energy in a seriesover the parameter Zα at fixed power of α should have coefficients of orderone. As is clear from the numerous discussions above, however natural such


Fig. 3.12. Nineteen topologically different diagrams with two radiative photonsinsertions in the electron line

expansion might seem from the point of view of calculations performed with-out expansion over Zα, there are no real reasons to expect that the coefficientswould be of the same order of magnitude in an expansion of this kind. Wehave already seen that quite different physics is connected with the differentterms in expansion over Zα. The terms of order αn(Zα)4 (and αn(Zα)6, as wewill see below) are generated at large distances (exchanged momenta of orderof the atomic scale mZα) while terms of order αn(Zα)5 originate from thesmall distances (exchanged momenta of order of the electron mass m). Hence,it should not be concluded that there would be a simple way to figure out


Table 3.3. Radiative Corrections of Order αn(Zα)5m

4α(Zα)5

n3 ( mrm )3m ∆E(1S) ∆E(2S)

kHz kHz

One-Loop Electron-Line Insertions

Karplus, Klein, Schwinger (1951) [40, 41] (1 + 11128 − 1

2 ln 2)δl0 55 090.31 6 886.29

Baranger, Bethe, Feynman (1951) [42]

One-Loop Polarization Contribution

Karplus, Klein, Schwinger (1951) [40, 41]

Baranger, Bethe, Feynman (1951) [42] 5192 δl0 1 940.38 242.55

Two One-Loop Polarizations

Eides, Grotch, Owen (1992) [46]

Pachucki; Laporta (1993) [47, 48] − 231 512

απ δl0 −2.63 −0.33

Two-Loop Polarization

Eides, Grotch, Owen (1992) [46] ( 1363 ln 2 − 25

252 π + 15 64752 920 ) α

π δl0 21.99 2.75

Pachucki; Laporta (1993) [47, 48]

One-Loop Polarization

and Electron Factor

Eides, Grotch (1993) [49] ( 23 ln2 1+

√5

2 − 21863

√5 ln 1+

√5

2

Pachucki (1993) [47] + 15763 ln 2 − π2

18 + 33 64113 230 ) α

π δl0 26.45 3.31

Eides, Grotch, Shelyuto (1997) [51]

Polarization insertion

in the Electron Factor

Eides, Grotch (1993) [50]

−0.018 2 απ δl0 −3.15 −0.39

Pachucki (1993) [47]

Light by Light Scattering

Pachucki (1993) [47],

Eides, Grotch, Pebler (1994) [52]

−0.030 7 απ δl0 −5.31 −0.66

Insertions of Two Radiative

Photons in the Electron Line

Pachucki (1994) [55],

Eides, Shelyuto (1995) [56, 38] −1.931 2(3) απ δl0 −334.24(5) −41.78

Reducible Three-Loop

Polarization Insertions

Eides, Shelyuto (2003) [58] 2.651 9 (6)( απ )2δl0 0.27 0.03

Reducible Three-Loop

Radiative Insertions

Eides, Shelyuto (2004) [59] −5. 321 93 (1)( απ )2δl0 −0. 53 −0.07

the relative magnitude of the successive coefficients in an expansion over Zα.The situation is different for expansion over α at fixed power of Zα since thephysics is the same independent of the power of α, and respective coefficientsare all of order one, as in the series for the radiative corrections in scatteringproblems.


Fig. 3.13. Reducible three-loop diagrams

3.3.3.7 Total Correction of Order α2(Zα)5m

The total contribution of order α2(Zα)5 is given by the sum of contributionsin (3.41), (3.42), (3.43), (3.46), (3.47), (3.48) [51]

∆E =

(

83

ln2 1 +√

52

− 87263

√5 ln

1 +√

52

+68063

ln 2 − 2π2

9− 25π

63

+24 9012 205

− 7.921(1))

α2 (Zα)5

πn3

(mr

m

)3

m δl0

= −6.862(1)α2 (Zα)5

πn3

(mr

m

)3

m δl0 . (3.49)

∆E = −6.862(1)α2 (Zα)5

πn3

(mr

m

)3

m δl0

= −296.92 (4) kHz|n=1 ,

= −37.115 (5) kHz|n=2 . (3.50)


Corrections of order α3(Zα)5 are similar to the corrections of order α2(Zα)5,and can be calculated in the same approach. As should be clear from the dis-cussion above their natural scale is determined by the factor 4α3(Zα)5/(π2n3)m. Allowing for an additional numerical factor about 1 − 2 we conclude thata fair estimate of these corrections is about 1 kHz for the 1S-state and about0.1 kHz for the 2S-state.

Corrections of order α3(Zα)5 are generated by three-loop radiative in-sertions in the skeleton diagram in Fig. 3.8. All such corrections connectedwith the diagrams containing at least one one-loop or two-loop polarizationinsertion were obtained in [58]

∆E = 2.651 9 (6)α3(Zα)5

π2n3

(mr

m

)3

m δl0 . (3.51)

Relying on the experience with the corrections of order α2(Zα)5 we expectthat the numerically dominant part of the corrections of order α3(Zα)5 willbe generated by the gauge invariant set of diagrams with insertions of three


radiative photons in the electron line in the skeleton diagrams in Fig. 3.8.Only insertions of the three-loop one-particle reducible diagrams with radia-tive photons in the electron line in Fig. 3.13 were considered thus far. Thecontribution of these diagrams in the Yennie gauge is equal to [59]

∆E = −5. 321 93 (1)α3(Zα)5

π2n3

(mr

m

)3

m δl0 . (3.52)

Work on calculation of the remaining contributions of order α3(Zα)5 is inprogress now.


3.4.1 Radiative Corrections of Order α(Zα)6m

3.4.1.1 Logarithmic Contribution Induced by the RadiativeInsertions in the Electron Line

Unlike the corrections of order αn(Zα)5, corrections of order αn(Zα)6 dependon the large distance behavior of the wave functions. Roughly speaking thishappens because in order to produce a correction containing six factors ofZα one needs at least three exchange photons like in Fig. 3.14. The radiativephoton responsible for the additional factor of α does not suppress completelythe low-momentum region of the exchange integrals. As usual, long distancecontributions turn out to be state-dependent.

Fig. 3.14. Diagram with three spanned Coulomb photons

The leading correction of order α(Zα)6 contains a logarithm squared,which can be compared to the first power of logarithm in the leading ordercontribution to the Lamb shift. One can understand the appearance of thelogarithm squared factor qualitatively. In the leading order contribution tothe Lamb shift the logarithm was completely connected with the logarithmicinfrared singularity of the electron form factor. Now we have two exchangedloops and one should anticipate the emergence of a logarithm generated bythese loops. Note that the diagram with one exchange loop (e.g, relevant forthe correction of order α(Zα)5) cannot produce a logarithm, since in the ex-ternal field approximation the loop integration measure d3k is odd in the


exchanged momentum, while all other factors in the exchanged integral areeven in the exchanged momentum. Hence, in order to produce a logarithmwhich can only arise from the dimensionless integrand it is necessary to con-sider an even number of exchanged loops. These simple remarks may also beunderstood in another way if one recollects that in the relativistic correctionsto the Schrodinger-Coulomb wave function each power of logarithm is multi-plied by the factor (Zα)2 (this is evident if one expands the exact Dirac wavefunction near the origin).

The logarithm squared term is, of course, state-independent since the co-efficient before this term is determined by the high momentum integrationregion, where the dependence on the principal quantum number may enteronly via the value of the wave function at the origin squared. Terms linear inthe large logarithm are already state dependent. Logarithmic terms were firstcalculated in [60, 61, 62, 63]. For the S-states the logarithmic contribution isequal to

∆Elog|l=0 ={

−14

ln2

[m(Zα)−2

mr

]

+[43

ln 2 + ln2n

+ ψ(n + 1) − ψ(1)

− 601720

− 77180n2

]

ln[m(Zα)−2

mr

]}4α(Zα)6

πn3

(mr

m

)3

m , (3.53)

where

ψ(n) =n−1∑

1

1k

+ ψ(1) , (3.54)

is the logarithmic derivative of the Euler Γ -function ψ(x) = Γ ′(x)/Γ (x),ψ(1) = −γ.

For non-S-states the state-independent logarithm squared term disappearsand the single-logarithmic contribution has the form

∆Elog|l �=0 =[(

1 − 1n2

)(130

+112

δj, 12

)

δl1 +6 − 2l(l + 1)/n2

3(2l + 3)l(l + 1)(4l2 − 1)

]

× ln[m(Zα)−2

mr

]4α(Zα)6

πn3

(mr

m

)3

m . (3.55)

Calculation of the state-dependent nonlogarithmic contribution of orderα(Zα)6 is a difficult task, and has not been done for an arbitrary principalquantum number n. The first estimate of this contribution was made in [63].Next the problem was attacked from a different angle [64, 65]. Instead ofcalculating corrections of order α(Zα)6 an exact numerical calculation of allcontributions with one radiative photon, without expansion over Zα, wasperformed for comparatively large values of Z (n = 2), and then the result wasextrapolated to Z = 1. In this way an estimate of the sum of the contributionof order α(Zα)6 and higher order contributions α(Zα)7 was obtained (forn = 2 and Z = 1). We will postpone discussion of the results obtained in this


way up to Subsect. 3.5.1, dealing with corrections of order α(Zα)7, and willconsider here only the direct calculations of the contribution of order α(Zα)6.

An exact formula in Zα for all nonrecoil corrections of order α had theform

∆E = 〈n|Σ(2)|n〉 , (3.56)

where Σ(2) is an “exact” second-order self-energy operator for the electron inthe Coulomb field (see Fig. 3.15), and hence contains the unmanageable exactDirac-Coulomb Green function. The real problem with this formula is to ex-tract useful information from it despite the absence of a convenient expressionfor the Dirac-Coulomb Green function. Numerical calculation without expan-sion over Zα, mentioned in the previous paragraph, was performed directlywith the help of this formula.

Fig. 3.15. Exact second order self-energy operator

A more precise value than in [63] of the nonlogarithmic correction of orderα(Zα)6 for the 1S-state was obtained in [66, 67], with the help of a speciallydeveloped ”perturbation theory” for the Dirac-Coulomb Green function whichexpressed this function in terms of the nonrelativistic Schrodinger-CoulombGreen function [68, 69]. But the real breakthrough was achieved in [70, 71],where a new very effective method of calculation was suggested and veryprecise values of the nonlogarithmic corrections of order α(Zα)6 for the 1S-and 2S-states were obtained. We will briefly discuss the approach of papers[70, 71] in the next subsection.

3.4.1.2 Separation of the High- and Low-MomentumContributions. Nonlogarithmic Corrections

Starting with the very first nonrelativistic consideration of the main con-tribution to the Lamb shift [8] separation of the contributions of high- andlow-frequency radiative photons became a characteristic feature of the Lambshift calculations. The main idea of this approach was already explained inSubsect. 3.2.1.1, but we skipped over two obstacles impeding effective im-plementation of this idea. Both problems are connected with the effectiverealization of the matching procedure. In real calculations it is not alwaysobvious how to separate the two integration regions in a consistent way, sincein the high-momenta region one uses explicitly relativistic expressions, whilethe starting point of the calculation in the low-momenta region is the nonrel-ativistic dipole approximation. The problem is aggravated by the inclinationto use different gauges in different regions, since the explicitly covariant Feyn-man gauge is the simplest one for explicitly relativistic expressions in the


high-momenta region, while the Coulomb gauge is the gauge of choice in thenonrelativistic region. In order to emphasize the seriousness of these prob-lems it suffices to mention that incorrect matching of high- and low-frequencycontributions in the initial calculations of Feynman and Schwinger led to asignificant delay in the publication of the first fully relativistic Lamb shiftcalculation of French and Weisskopf [10]4! It was a strange irony of historythat due to these difficulties it became common wisdom in the sixties thatit is better to try to avoid the separation of the contributions coming fromdifferent momenta regions (or different distances) than to try to invent anaccurate matching procedure. A few citations are appropriate here. Bjorkenand Drell [73] wrote, having in mind the separation procedure: “The readermay understandably be unhappy with this procedure . . . we recommend therecent treatment of Erickson and Yennie [62, 63], which avoids the divisioninto soft and hard photons”. Schwinger [31] wrote: “. . . there is a moral herefor us. The artificial separation of high and low frequencies, which are han-dled in different ways, must be avoided.” All this was written even thoughit was understood that the separation of the large and small distances wasphysically quite natural and the contributions coming from large and smalldistances have a different physical nature. However, the distrust of the meth-ods used for separation of the small and large distances was well justifiedby the lack of a regular method of separation. Apparently different methodswere used for calculation of the high and low frequency contributions, high fre-quency contributions being commonly treated in a covariant four-dimensionalapproach, while old-fashioned nonrelativistic perturbation theory was used forcalculation of the low-frequency contributions. Matching these contributionsobtained in different frameworks was an ambiguous and far from obvious pro-cedure, more art than science. As a result, despite the fact that the methodsbased on separation of long and short distance contributions had led to somespectacular results (see, e.g., [74, 75]), their self-consistency remained suspect,especially when it was necessary to calculate the contributions of higher orderthan in the classic works. It seemed more or less obvious that in order to facil-itate such calculations one needed to develop uniform methods for treatmentof both small and large distances.

The actual development took, however, a different direction. Instead of re-jecting the separation of high and low frequencies, more elaborate methods ofmatching respective contributions were developed in the last decade, and thegeneral attitude to separation of small and large distances radically changed.Perhaps the first step to carefully separate the long and short distances wasdone in [5], where the authors had rearranged the old-fashioned perturbationtheory in such a way that one contribution emphasized the small momen-tum contributions and led to a Bethe logarithm, while in the other the smallmomentum integration region was naturally suppressed. Matching of bothcontributions in this approach was more natural and automatic. However, the

4 See fascinating description of this episode in [72].


price for this was perhaps too high, since the high momentum contributionwas to be calculated in a three-dimensional way, thus losing all advantages ofthe covariant four-dimensional methods.

Almost all new approaches, the skeleton integral approach described abovein Subsect. 3.3.1 ([38] and references there), ε-method described in this section[70, 71], nonrelativistic approach by Khriplovich and coworkers [76], nonrel-ativistic QED of Caswell and Lepage [77]) not only make separation of thesmall and large distances, but try to exploit it most effectively. In some cases,when the whole contribution comes only from the small distances, a rathersimple approach to this problem is appropriate (like in the calculation ofcorrections of order α2(Zα)4, α3(Zα)4, α(Zα)5 and α2(Zα)5 above, more ex-amples below) and the scattering approximation is often sufficient. In suchcases, would-be infrared divergences are powerlike. They simply indicate thepresence of the contributions of the previous order in Zα and may safely bethrown away. In other cases, when one encounters logarithms which get con-tributions both from the small and large distances, a more accurate approachis necessary such as the one described below. In any case “the separation oflow and high frequencies, which are handled in different ways” not only shouldnot be avoided but turns out to be a very convenient calculational tool andclarifies the physical nature of the corrections under consideration.

An effective method to separate contributions of low- and high-momentaavoiding at the same time the problems discussed above was suggested in[70, 71]. Consider in more detail the exact expression (3.56) for the sum ofall corrections of orders α(Zα)nm (n ≥ 4) generated by the insertion of oneradiative photon in the electron line

∆E = e2

∫d4k

(2π)4iDµν(k)〈n|γµG(p′ − k; p − k)γν |n〉 , (3.57)

where G(p′ − k; p − k) is the exact electron Green function in the externalCoulomb field. As was noted in [70, 71] one can rotate the integration contourover the frequency of the radiative photon in such a way that it encloses sin-gularities along the positive real axes in the ω (k0) plane. Then one considersseparately the region Re ω ≤ σ (region I) and Reω ≥ σ (region II), wherem(Zα)2 � σ � m(Zα). It is easy to see that due to the structure of thesingularities of the integrand, integration over k in the region I also goes onlyover the momenta smaller than σ (|k| ≤ σ), while in the region II the finalintegration over ω cuts off all would be infrared divergences of the integral.Hence, effective separation of high- and low-momenta integration regions isachieved in this way and, as was explained above, due to the choice of themagnitude of the parameter σ all would be divergences should exactly cancelin the sum of contributions of these regions. This cancellation provides anadditional effective method of control of the accuracy of all calculations. Itwas also shown in [71] that a change of gauge in the low-frequency regionchanges the result of the calculations by a term linear in σ. But anyway one


should discard such contributions matching high- and low-frequency contri-butions. The matrix element of the self-energy operator between the exactCoulomb-Dirac wave functions is gauge invariant with respect to changes ofgauge of the radiative photon [78]. Hence, it is possible to use the simpleFeynman gauge for calculation of the high-momenta contribution, and thephysical Coulomb gauge in the low-momenta part. It should be clear nowthat this method resolves all problems connected with the separation of thehigh- and low-momenta contributions and thus provides an effective tool forcalculation of all corrections with insertion of one radiative photon in theelectron line. The calculations originally performed in [70, 71, 79] success-fully reproduced all results of order α(Zα)4 and α(Zα)5 and produced a highprecision result for the constant of order α(Zα)6. Results for nonlogarithmiccontributions were later obtained also for excited S states in [80, 81], whereaccuracy of calculation was improved even further. The most recent resultsfor the 1S and 2S states are [80, 81]

∆Enon−log(1S) = −30.924 149 46 (1)α(Zα)6

π

(mr

m

)3

m , (3.58)

∆Enon−log(2S) = −31.840 465 09 (1)α(Zα)6

8π

(mr

m

)3

m .

We would like to emphasize two features of these results. First, the statedependence of the constant is very weak, and second, the scale of the constantis just of the magnitude one should expect. In order to make this last pointmore transparent let us write the total electron-line contribution of orderα(Zα)6 to the 1S energy shift in the form

∆E(1S) ≈{

− ln2

[m(Zα)−2

mr

]

+ 5.42 ln[m(Zα)−2

mr

]

− 30.92}

× α(Zα)6

π

(mr

m

)3

m . (3.59)

Now we see that the ratio of the nonlogarithmic term and the coefficient beforethe single-logarithmic term is about 31/5.4 ≈ 5.7 ≈ 0.6π2. It is well knownthat the logarithm squared terms in QED are always accompanied by thesingle-logarithmic and nonlogarithmic terms, and the nonlogarithmic termsare of order π2 (in relation with the current problem see, e.g., [62, 63]). Thisis just what happens in the present case.

Nonlogarithmic contributions of order α(Zα)6 to the energies of the non-S-states induced by the radiative photon insertions in the electron line wereobtained in the same framework in [82, 83, 84, 85]. The nonlogarithmic coef-ficients of order α(Zα)6 are similar to the nonlogarithmic coefficients of or-der α(Zα)4 in Table 3.1 (Bethe logarithms), and are called relativistic Bethelogarithms in [84, 85]. However, unlike the ordinary Bethe logarithms thenonlogarithmic coefficients of order α(Zα)6 depend not only on the principalquantum number n and the orbital momentum l, but also on the total angular


momentum j. We have collected the results for nonlogarithmic contributions[80, 81, 82, 83, 84, 85] in Table 3.4 in terms of the traditionally used coefficientA60 [62] which is defined by the relationship

∆E = A60α(Zα)6

πn3

(mr

m

)3

m . (3.60)

3.4.1.3 Correction Induced by the Radiative Insertionsin the External Photons

There are two kernels with radiative insertions in the external photon lineswhich produce corrections of order α(Zα)6 to the Lamb shift. First is our oldacquaintance – one-loop polarization insertion in the Coulomb line in Fig. 2.2.Its Fourier transform is called the Uehling potential [16, 86]. The second kernelcontains the light-by-light scattering diagrams in Fig. 3.16 with three externalphotons originating from the Coulomb source. The sum of all closed electronloops in Fig. 3.17 with one photon connected with the electron line and an ar-bitrary number of Coulomb photons originating from the Coulomb source maybe considered as a radiatively corrected Coulomb potential V . It generates ashift of the atomic energy levels

∆E = 〈n|V |n〉 . (3.61)

This potential and its effect on the energy levels were first considered in[87]. Since each external Coulomb line brings an extra factor Zα the energyshift generated by the Wichmann-Kroll potential increases for large Z. Forpractical reasons the effects of the Uehling and Wichmann-Kroll potentialswere investigated mainly numerically and without expansion in Zα, sinceonly such results could be compared with the experiments. Now there existmany numerical results for vacuum polarization contributions. In accordance

Fig. 3.16. Wichmann-Kroll potential

Fig. 3.17. Total one-loop polarization potential in the external field


Table 3.4. Nonlogarithmic Coefficient A60

α(Zα)6

πn3 (mrm

)3m kHz

1S 12

−30.924 149 46 (1) −1338.0434

2S 12

−31.840 465 09 (1) −172.2114

2P 12

−0.998 904 402 (1) −5.4026

2P 32

−0.503 373 465 (1) −2.7225

3S 12

−31.702 501 (1) −50.8045

3P 12

−1.148 189 956 (1) −1.8400

3P 32

−0.597 569 388 (1) −0.9576

3D 32

0.005 551 573 (1) 0.0089

3D 52

0.027 609 989 (1) 0.0442

4S 12

−31.561 922 (1) −21.3381

4P 12

−1.195 688 142 (1) −0.8083

4P 32

−0.630 945 795 (1) −0.4266

4D 32

0.005 585 985 (1) 0.0038

4D 52

0.031 411 862 (1) 0.0212

4F 52

0.002 326 988 (1) 0.0016

4F 72

0.007 074 961 (1) 0.0048

5S 12

−31.455 393 (1) −10.8882

5P 12

−1.216 224 512 (1) −0.4210

5P 32

−0.647 013 508 (1) −0.2240

5D 32

0.006 152 175 (1) 0.0021

5D 52

0.033 077 570 (1) 0.0114

5F 52

0.002 403 158 (1) 0.0008

5F 72

0.008 087 020 (1) 0.0028

5G 72

0.000 814 415 (1) 0.0003

5G 92

0.002 412 929 (1) 0.0008

6S 12

−31.375 130 (1) −6.2850

6P 12

−1.226 702 391 (1) −0.2457

6P 32

−0.656 154 893 (1) −0.1314

6D 32

0.006 749 745 (1) 0.0014

6D 52

0.033 908 493 (1) 0.0068

7S 12

−31.313 224 (1) −3.9501

7P 12

−1.232 715 957 (1) −0.1555

7P 32

−0.662 027 568 (1) −0.0835

7D 32

0.007 277 403 (1) 0.0009

7D 52

0.034 355 926 (1) 0.0043

8S 12

−31.264 257 (1) −2.6421

8D 32

0.007 723 850 (1) 0.0007

8D 52

0.034 607 492 (1) 0.0029


with our emphasis on the analytic results we will discuss here only ana-lytic contributions of order α(Zα)6, and will return to numerical results inSubsect. 3.5.2.

Uehling Potential Contribution

It is not difficult to present an exact formula containing all corrections pro-duced by the Uehling potential in Fig. 2.2 (compare with the respective ex-pression for the self-energy operator above)

∆E = 4π(Zα)⟨

n

∣∣∣∣

Π(k2)k4

∣∣∣∣n

⟩

. (3.62)

We have already seen that the matrix element of the first term of thelow-momentum expansion of the one-loop polarization operator between thenonrelativistic Schrodinger-Coulomb wave functions produces a correction oforder α(Zα)4. The next term in the low-momentum expansion of the polar-ization operator pushes characteristic momenta in the integrand to relativisticvalues, where the very nonrelativistic expansion is no longer valid, and evenmakes the integral divergent if one tries to calculate it between the nonrela-tivistic wave functions. Due to this effect we prefer to calculate the correctionof order α(Zα)5 induced by the one-loop polarization insertion (as well asthe correction of order α2(Zα)5 induced by the two-loop polarization) in theskeleton integral approach in Sect. 3.3. Note that all these corrections con-tribute only to the S-states. It is useful to realize that both these calculationsare, from another point of view, simply results of approximate calculation ofthe integral in (3.62) with accuracy (Zα)4 and (Zα)5. Our next task is tocalculate this integral with accuracy (Zα)6. In this order both small atomic(∼ mZα) and large relativistic (∼ m) momenta produce nonvanishing contri-butions to the integral, and as a result we get nonvanishing contributions tothe energy shifts also for states with nonvanishing angular momenta.

Consider first corrections to the energy levels with nonvanishing angularmomenta. The respective wave functions vanish at the origin in coordinatespace, hence only small photon momenta contribute to the integral, and onecan use the first two terms in the nonrelativistic expansion of the polarizationoperator

Π(k2)k4 ≈ − α

15πm2+

αk2

35m4(3.63)

for calculation of these contributions [88, 83] (compare (3.12) above). Cor-rections of order α(Zα)6 turn out to be nonvanishing only for l ≤ 1. For2P -states these corrections were first calculated in [65], and the result forarbitrary P -states [89] has the form

∆E(nPj) = − 415

(

1 − 1n2

)(114

+14δj 1

2

)α(Zα)6

πn3

(mr

m

)3

m . (3.64)


The respective correction to the energy levels of S-states originates bothfrom the large and small distances since the Schrodinger-Coulomb wave func-tion in the S-states does not vanish at small distances. Hence, one cannotimmediately apply low-momenta expansion of the polarization operator forcalculation of the matrix element in (3.62). The leading logarithmic state-independent contribution to the energy shift is still completely determined bythe first term in the low-momentum expansion of the polarization operatorin (3.63), but one has to consider the exact expression for the polarizationoperator in order to obtain the nonlogarithmic contribution.

The logarithmic term originates from the logarithmic correction to theSchrodinger-Coulomb wave function which arises when one takes into accountthe Darwin (δ-function) potential which arises in the nonrelativistic expansionof the Dirac Hamiltonian in the Coulomb field (see, e.g., [73, 3] and (3.96)below). Of course, the same logarithm arises if, instead of calculating correc-tions to the Schrodinger-Coulomb wave function, one expands the singularfactor in the Dirac-Coulomb wave function over Zα. The correction to theSchrodinger-Coulomb wave function Ψ at the distances of order 1/m has theform [44]

δΨ = −12(Zα)2 ln(Zα) Ψ , (3.65)

and substituting this correction in (3.62) one easily obtains the leading loga-rithmic contribution to the energy shift [60, 62, 63]

∆Elog(nS) = − 215

α(Zα)6

πn3ln

[m(Zα)−2

mr

](mr

m

)3

m . (3.66)

Note that the numerical factor before the leading logarithm here is simplythe product of the respective numerical factors in the correction to the wavefunction in (3.65), the low-frequency asymptote of the one-loop polarization−1/(15π) in (2.5), the factor 4π(Zα) in (3.62), and factor 2 which reflects thatboth wave functions in the matrix element in (3.62) have to be corrected.

Calculation of the nonlogarithmic contributions requires more effort. Com-plete analytic results for the lowest states were first obtained by P. Mohr [65]

∆E(1S) =115

[

−12

ln[m(Zα)−2

mr

]

+ ln 2 − 1289420

]4α(Zα)6

π

(mr

m

)3

m ,

∆E(2S) =115

[

−12

ln[m(Zα)−2

mr

]

− 743240

]4α(Zα)6

8π

(mr

m

)3

m . (3.67)

As was mentioned above, short distance contributions are state indepen-dent and always cancel in the differences of the form ∆n ≡ n3∆E(nS) −∆E(1S). This means that such state-dependent differences of energies con-tain only contributions of large distances and are much easier to calculate,


since one may employ a nonrelativistic approximation.5 The Uehling poten-tial contributions to the difference of level shifts n3∆E(nS) − ∆E(1S) werecalculated in [88] with the help of the nonrelativistic expansion of the polariza-tion operator in (3.63). The result of this calculation, in conjunction with theMohr result in (3.67), leads to an analytic expression for the Uehling potentialcontribution to the Lamb shift

∆E(nS) =115

[

−12

ln[m(Zα)−2

mr

]

− 431105

+ (ψ(n + 1) − ψ(1)) − 2(n − 1)n2

+1

28n2− ln

n

2

]4α(Zα)6

πn3

(mr

m

)3

m , (3.68)

where ψ(x) is the logarithmic derivative of the Euler Γ -function, see (3.54).

Wichmann-Kroll Potential Contribution

The only other contribution of order α(Zα)6 connected with the radiativeinsertions in the external photons is produced by the term trilinear in Zα inthe Wichmann-Kroll potential in Fig. 3.16. One may easily check that the firstterm in the small momentum expansion of the Wichmann-Kroll potential hasthe form [87, 92]

VWK(k) =(

1945

− π2

27

)α(Zα)3

m2. (3.69)

This potential generates the energy shift [93, 92]

∆E =(

1945

− π2

27

)α(Zα)6

πn3

(mr

m

)3

m δl0 , (3.70)

which is nonvanishing only for the S-states.


Corrections of order α2(Zα)6 are similar to the corrections of order α(Zα)6,and originate both from large and small distances. In principle they can be cal-culated in the same way. The total contribution of order α2(Zα)6 is a polyno-mial in ln(Zα)−2, starting with the cube of the large logarithm. The factor be-fore the leading logarithm cubed and the contribution of the logarithm squaredterms to the difference ∆EL(1S) − 8∆EL(2S) were obtained first. Calcula-tion of these contributions is relatively simple because large logarithms alwaysoriginate from the wide region of large virtual momenta (mZα � k � m) andthe respective matrix elements of the perturbation potentials depend only onthe value of the Schrodinger-Coulomb wave function or its derivative at the5 This well known feature was often used in the past. For example, differences of

the hyperfine splittings n3∆E(nS)−∆E(1S) were calculated much earlier [90, 91]than the hyperfine splittings themselves.


origin, as we have already seen above in discussion of the main contributionto the Lamb shift and corrections of order α(Zα)6. The other contributionsof order α2(Zα)6 depend on all distances, and their calculation is much moreinvolved. All terms in the polynomial in ln(Zα)−2 besides the constant areknown now.

3.4.2.1 S-States Electron-Line Contributions

Let us start with the leading logarithm cubed contribution of order α2(Zα)6.It is not hard to check that the effective potential corresponding to the irre-ducible electron two-loop self-energy operator in Fig. 3.18 does not generate aln3(Zα)−2 contribution to the Lamb shift. The only way to obtain the highestpower of the logarithm is to consider effective potentials corresponding to theone-loop electron self-energy operator in the external Coulomb field, and touse the ordinary second-order perturbation theory expression (see Fig. 3.19)

∆E = 2∑

m,m �=n

〈n|V1|m〉〈m|V2|n〉En − Em

, (3.71)

where V1 and V2 are the perturbation operators, and the factor 2 is due to twopossible orders of the perturbation operators; it is not present when V1 = V2.Summation over the intermediate states above includes integration over thecontinuous spectrum with the weight

∫d3k/(2π)3.

Fig. 3.18. Two-loop self-energy operator

A logarithmic matrix element of the first-order electron self-energy oper-ator (like the one producing the leading contribution to the Lamb shift) maybe considered in the framework of perturbation theory as a matrix element ofan almost local (it depends on the momentum transfer only logarithmically)perturbation theory operator since it is induced by the diagram with relativis-tic virtual momenta. We may use such a local operator in order to calculatethe higher order perturbation theory contributions [76]. The maximum powerof the large logarithm arises if we consider as perturbation (V1 = V2 = V )the first order quasilocal effective potential which generates the leading con-tribution to the Lamb shift (compare (2.3)), which in momentum space hasthe form

V = −8α(Zα)3m2

lnk

m. (3.72)


Fig. 3.19. Second order perturbation theory contribution with two one-loop self-energy operators

It is evident that this potential leads to a logarithm squared contributionof order α2(Zα)6 after substitution in (3.71). One may obtain one more log-arithm from the continuous spectrum contribution in (3.71). Due to localityof the potential, matrix elements reduce to the products of the values of therespective wave functions at the origin and the potentials in (3.72). The valueof the continuous spectrum Coulomb wave function at the origin is well known(see, e.g., [94]), and

|ψk(0)|2 =2πγr

k(1 − e−2πγr

k )≈ 1 +

πγr

k+ O

((γr

k

)2)

, (3.73)

where γr = mrZα. The leading term in the large momentum expansion in(3.73) generates an apparently linearly ultraviolet divergent contribution tothe energy shift, but this ultraviolet divergence is due to our nonrelativisticapproximation, and it would be cut off at the electron mass in a truly rela-tivistic calculation. What is more important this correction is of order (Zα)5,and may be safely omitted in the discussion of the corrections of order (Zα)6.Logarithmic corrections of order (Zα)6 are generated by the second term inthe high momentum expansion in (3.73)

∆E = −2(mrZα)4mr

π2n2

∫

dk〈n|V1|m〉〈m|V2|n〉

k. (3.74)

With the help of this formula one immediately obtains [95]

∆E = − 827

α2(Zα)6

π2n3ln3

[m(Zα)−2

mr

](mr

m

)5

m . (3.75)

Note that the scale of this contribution is once again exactly of theexpected magnitude, namely, this contribution is suppressed by the factor(α/π) ln(Zα)−2 in comparison with the leading logarithm squared contribu-tion of order α(Zα)6. Of course, the additional numerical suppression factor8/27 could not be obtained without real calculation. The calculation in [95]was followed by a series of papers [96, 97, 98] where a subset of all diagramswith two-loop radiative insertions in the electron line were calculated nu-merically without expansion in Zα. The results of these considerations werecontroversial, and the authors of [96, 98] even came to the conclusion that theleading logarithm contribution should be different from the result in (3.75).


The leading perturbation theory contribution in (3.75) was reproduced withthe help of the renormalization group equations in the approach based onnonrelativistic QED [99]. But this result should be considered more as a testof the validity of the renormalization group equations in a specific version ofNRQED, than as an independent confirmation of the value of the coefficientin (3.75). On the other hand an additional logarithm cubed contribution wasfound in a later paper [100]. However, only an incomplete set of diagrams wasconsidered in that paper. All doubts about validity of (3.75) were put to restin [101], where among other results a complete calculation of the logarithmcubed in the Coulomb gauge was presented. Later the relationship betweenthe results obtained with the help of the effective potentials and the resultsof the purely diagrammatic approach was additionally clarified in [102].

Numerically, the correction in (3.75) is about −28 kHz for the 1S-state.This is larger than the experimental error of the Lamb shift measurement,and other terms in the expansion of the correction of order α2(Zα)6 over thepower of the large logarithm ln(Zα)−2 are needed for any meaningful com-parison with the experimental results. Even the effective potential calculationof logarithm squared contributions to the energy shift of S-levels is impededby the fact that such contributions arise both from the discrete and contin-uous spectrum intermediate states in (3.71), and a complicated interplay ofcontributions from the different regions occurs. Hence, in such a calculation itis necessary to consider the contributions of the one-loop electron self-energyoperators more accurately and the local approximation used above becomesinappropriate. It is easier to use other methods for calculation of the coeffi-cients before the nonleading logarithms.

All corrections of order α2(Zα)6 are usually written as an expansion overthe powers of the large logarithm

∆E =[B63 ln3(Zα)−2 + B62 ln2(Zα)−2 + B61 ln(Zα)−2 + B60

] α2(Zα)6

π2n3m .

(3.76)As we already mentioned a breakthrough in calculating the coefficients B6n

was achieved in [101]. In this work the method for separating low and highfrequency contributions developed earlier for calculation of the corrections oforder α(Zα)6, and described above in Subsect. 3.4.1.2 was generalized for thecase of corrections of order α2(Zα)6. This generalization is far from beingstraightforward because due to insertions of two radiative photons in the elec-tron line the corrections in the diagrams in Fig. 3.18 and Fig. 3.20 contain two

Fig. 3.20. Reducible two-loop radiative insertions in the electron line


independent photon integration momenta and a nontrivial interplay betweentheir magnitudes should be taken into account.

The contribution to the logarithm squared coefficient generated by thediagrams in Fig. 3.18 and Fig. 3.20 was obtained in [101]. We write it in theform (see also [103])

B62(nS) = B62(1S) − 169

[

ln n − ψ(n) + ψ(1) − n − 1n

+n2 − 14n2

]

, (3.77)

whereB62(1S) = −16

9ln 2 +

1627

. (3.78)

In this notation we have separated the state-independent contribution B62(1S)originating exclusively from the short distances, first time obtained in [101],from the other contributions to this coefficient. The short distance contri-butions cancel in the difference B62(1S) − B62(nS) which corresponds tothe energy shifts combination ∆n = n3∆E(nS) − ∆E(1S). This differenceB62(1S) − B62(nS) was independently calculated earlier in the frameworkof the effective potential approach [104, 105, 106, 107], and we will brieflydescribe this calculation.

Fig. 3.21. Logarithm squared contributions to ∆EL(1S) − n3∆EL(nS)

The logarithm squared contributions to the difference of energies are gen-erated in the second order of perturbation theory by two one-loop vertexoperators, and in the first order of perturbation theory by the one two-loopvertex (see diagrams in Fig. 3.21). Due to cancellation of the short-distancestate-independent terms in the difference of energy levels only intermediatecontinuous spectrum states with momenta of the atomic scale mZα give con-tributions in the second order of perturbation theory [106, 107]. Then thelocal approximation for the one-loop vertices and the nonrelativistic approx-imation for the wave functions is sufficient for calculation of the logarithmsquared contribution to the energy difference generated by the first diagramin Fig. 3.21. Calculation of the contribution induced by the second ordervertex operator (second diagram in Fig. 3.21) is quite straightforward. Bothcontributions were calculated in a series of papers [104, 105, 106, 107], wherethe result for B62(1S) − B62(nS) in (3.77) was obtained for the first time.

The coefficient before the single logarithm generated by the diagrams inFig. 3.18 and Fig. 3.20 has the form [101, 103]


B61(nS) = B61(1S)−(

8027

− 329

ln 2)[

ln n − ψ(n) + ψ(1) − n − 1n

+n2 − 14n2

]

+43[N(nS) − N(1S)] , (3.79)

where [101, 103, 81]

B61(1S) = ζ(3)− 23π2 ln 2+

409

ln2 2− 15227

ln 2+1039432

π2+37991364800

+43N(1S) ,

(3.80)and the additive terms N(nS) introduced in [101] and calculated with highaccuracy in [108, 81] are collected in Table 3.5.

Table 3.5. Functions N(nS) and N(nP )

nS N(nS) nP N(nP )

1S 17.855 672 03 (1)

2S 12.032 141 58 (1) 2P 0.003 300 635 (1)

3S 10.449 809 (1) 3P 0.003 572 084 (1)

4S 9.722 413 (1) 4P −0.000 394 332 (1)

5S 9.304 114 (1) 5P −0.004 303 806 (1)

6S 9.031 832 (1) 6P −0.007 496 998 (1)

7S 8.840 123 (1) 7P −0.010 014 614 (1)

8S 8.697 639 (1) 8P −0.011 999 223 (1)

Corrections of order α2(Zα)6 are also generated by the insertions of theone-loop polarization operator in the one-loop electron self-energy operatorin the external field in Fig. 3.22. It was shown in [101] that the respectivelogarithm squared contribution vanishes. There are two contributions to thesingle-logarithmic coefficient. One is due to the effective potential generatedby the insertions of the one-loop electron polarization operator in the one-loop electron form factors in the same way as we obtained above the leadinglogarithmic contribution of order α(Zα)6 generated by the Uehling potentialin (3.66). The respective contribution to the coefficient B61(nS) is equal to8F ′(2)

1 (0) + 2F (2)2 (0) [101]

B(1)61 (nS) =

5432

π2 − 7162

. (3.81)

Later a second contribution to the single-logarithmic coefficient generatedby the diagram in Fig. 3.22 which cannot be accounted for by the form factorswas discovered in [81]

B(2)61 (nS) = − 61

864π2 +

493648

. (3.82)

Finally the total single-logarithmic coefficient generated by the diagramin Fig. 3.22 is equal to


Fig. 3.22. Polarization insertion in the first order self-energy operator

B61(nS) = B(1)61 (nS) + B

(2)61 (nS) = − 17

288π2 +

155216

. (3.83)

Let us turn to the nonlogarithmic coefficients B60. These coefficients inthe case of corrections of order α2(Zα)6m are similar to the nonlogarithmicterms in the leading contribution to the Lamb shift of order α(Zα)4m in (3.7).The dominant contribution to the nonlogarithmic term in (3.7) is given by theBethe logarithm, which describes the effect of the emission and absorption ofsoft virtual photons in the dipole approximation. Similarly one should expectthat the dominant low-energy contributions to the nonlogarithmic coefficientsB60 is given by the emission and absorption of two soft virtual photons in thedipole approximation. Respective contributions to the coefficients B60 for the1S and 2S states were obtained in [109, 110], where they were called two-loopBethe logarithms. The authors of [109, 110] estimate the magnitude of otheruncalculated contributions to the coefficients B60 as about 15% of the two-loop Bethe logarithms. We collected the results of [109, 110] in Table 3.6. The15% uncertainty in the value of the coefficients B60 determines the theoreticaluncertainty of the total α2(Zα)6 contribution to the energy shift. Numericallyit constitutes 0.9 kHz for the 1S-state and 0.1 kHz for the 2S-state.

Table 3.6. Two-Loop Bethe Logarithms

nS B60(nS)

1S −61.6 (3) ± 15%

2S −53.2 (3) ± 15%

3S −51.9 (6) ± 15%

4S −51.0 (8) ± 15%

5S −50.3 (8) ± 15%

6S −49.8 (8) ± 15%

The values of the coefficients B6n obtained above lead to a quite peculiarbehavior of different contributions to the energy shifts. To get a better ideaabout relative magnitude of these electron line contributions of order α2(Zα)6

let us write explicitly contributions to 1S and 2S states

∆E(1S) =[−0.296 ln3(Zα)−2 − 0.640 ln2(Zα)−2 + 48.417 ln(Zα)−2 − 61.6

]

× α2(Zα)6

π2m ≈ [−282 − 62 + 476 − 61]

α2(Zα)6

π2m , (3.84)


Fig. 3.23. Leading polarization operator logarithm squared contributions of orderα2(Zα)6

∆E(2S) =[−0.296 ln3(Zα)−2 + 0.461 ln2(Zα)−2 + 40.961 ln(Zα)−2 − 53.2

]

× α2(Zα)6

8π2m ≈ [−282 + 45 + 403 − 53]

α2(Zα)6

8π2m . (3.85)

We observe that contrary to our expectations the contributions generated bythe consecutive terms in the expansion over the powers of the large logarithmdo not decrease. The term linear in the large logarithm numerically domi-nates and even reverses the sign of the leading logarithmic contribution. Thecorrection of order α2(Zα)6m plays an important role in the analysis of theexperimental results on the Lamb shift, and an independent calculation of thecoefficients in (3.76) is clearly warranted. We could also expect that, as inthe case of corrections of order α(Zα)6m and α(Zα)7m, contributions of or-der α2(Zα)7m may be phenomenologically relevant. Therefore calculation ofthe corrections generated by the diagrams in Fig. 3.18 and Fig. 3.20 withoutexpansion in Zα is of great interest (see discussion below in Subsect. 3.5.3).

3.4.2.2 S-States Polarization Operator Contributions

Consider now contributions of order α2(Zα)6 generated by the polarizationinsertions in the external photon lines in Fig. 3.23. The general expression forthese corrections may be written as in (3.76). It is easy to see that the leadingterm generated by the diagrams in Fig. 3.23 is quadratic in the large loga-rithm, and hence, respective B63 = 0. These logarithm squared contributionscorresponding to the diagrams with at least one polarization insertion maybe calculated with the help of the effective potentials. The leading logarithmsquared term in Fig. 3.23 is generated when we combine the perturbationpotential in (3.72) which corresponds to the one-loop electron vertex and theperturbation potential in (2.5) which corresponds to the polarization operatorcontribution to the Lamb shift. Then the standard second order perturbationtheory (3.71) immediately leads to the expression [104]

B62 =845

. (3.86)

The contribution to the single logarithm coefficient generated by the di-agram in Fig. 3.23 and other diagrams with polarization insertions in the


Coulomb photons was obtained in [101]. We write it in the form (see also[103])

B61(nS) = B61(1S)− 3245

[

− ln n + ψ(n) − ψ(1) +n − 1

n− n2 − 1

4n2

]

, (3.87)

whereB61(1S) =

1615

ln 2 − 4012025

. (3.88)

In this notation we have separated the state-independent contribution B61(1S)originating exclusively from the short distances, first obtained in [101] from theother contributions to this coefficient. The short distance contributions cancelin the difference B61(1S) − B61(nS) which corresponds to the energy shiftscombination ∆n = n3∆E(nS)−∆E(1S). This difference B61(1S)−B61(nS)was independently calculated earlier in the framework of the effective potentialapproach [104].

As in the previous section, to get a better idea about relative magnitudeof different polarization contributions of order α2(Zα)6 let us write explicitlycontributions to 1S and 2S states

∆E(1S) =[0.178 ln2(Zα)−2 + 0.541 ln(Zα)−2 + B60

] α2(Zα)6

π2m

≈ [17.2 + 5.33 + B60]α2(Zα)6

π2m , (3.89)

∆E(2S) =[0.178 ln2(Zα)−2 + 0.101 ln(Zα)−2 + B60

] α2(Zα)6

8π2m .

≈ [17.2 + 0.993 + B60]α2(Zα)6

8π2m . (3.90)

Unlike the case of radiative insertions in the electron line, this time the co-efficients before the logarithms do not demonstrate any peculiarities. Therespective correction to the energy shift is about 2 kHz for the 1S state inhydrogen.

Recent progress in calculation of nonlogarithmic coefficients B60 is re-ported in [80, 81]. The authors of these papers derived a general NRQEDexpression for the contribution of all diagrams (both with radiative inser-tions in the electron line and in the external photons) to the difference of thecoefficients B60(1S)−B60(nS), which is necessary for calculation of the phe-nomenologically important energy difference ∆n = n3∆E(nS)−∆E(1S). Theaccuracy of formulae obtained in [80, 81] allows one to calculate the α2(Zα)6

contribution to the energy difference with accuracy about 0.1 kHz.

3.4.2.3 Non S-States Contributions

Consideration of the logarithm squared contributions to the energy levels withnonvanishing angular momenta is much simpler than for the S levels [104, 105].


Fig. 3.24. Effective potential corresponding to two-loop vertex

The second order perturbation theory term with two one-loop self-energyoperators does not generate any logarithm squared contribution for the statewith nonzero angular momentum since the respective nonrelativistic wavefunction vanishes at the origin. Only the two-loop vertex in Fig. 3.24 producesa logarithm squared term in this case. The respective perturbation potentialdetermined by the second term in the low-momentum expansion of the two-loop Dirac form factor [111] has the form

V2 = −2α2(Zα)k2

9πm4ln2(Zα)−2 . (3.91)

Calculation of the matrix element of this effective perturbation with thenonrelativistic wave functions for the P -states yields [104, 105] (see also [102])

∆E(nP ) ≡ B62(nP )α2(Zα)6m

π2n3ln2(Zα)−2 =

4(n2 − 1)27n2

α2(Zα)6mπ2n3

ln2(Zα)−2 ,

(3.92)while for l > 1 there are no logarithm squared contributions.

Terms linear in the large logarithm were recently obtained in [80, 81]

B61(nP 12) =

n2 − 1n2

(

−8 ln 227

+166405

)

+45N(nP ) , (3.93)

B61(nP 32) =

n2 − 1n2

(

−8 ln 227

+31405

)

+45N(nP ) , (3.94)

where the additive terms N(nP ), calculated with high accuracy in [108], arecollected in Table 3.5.

It was shown in [80, 81] that all logarithmic terms of order α2(Zα)6 vanishfor states with orbital momenta l > 1. A general NRQED expression for thenonlogarithmic terms for P and D states also was obtained in [80, 81].

3.4.2.4 Corrections of Order α3(Zα)6m

The analysis of the contributions of order α2(Zα)n confirms once again, asalso emphasized in [96], that there is no regular rule for the magnitude of the


coefficients before the successive terms in the series over Zα at fixed α. Thishappens because the terms, say of relative order Zα and (Zα)2, correspondto completely different physics at small and large distances and, hence, thereis no reason to expect a regular law for the coefficients in these series. Thisshould be compared with the series over α at fixed Zα. As we have shownabove, different terms in these series correspond to the same physics and hencethe coefficients in these series change smoothly and may easily be estimated.This is why we have organized the discussion in terms of such series. Notethat the best way to estimate an unknown correction of order, say α2(Zα)6m,which corresponds to the long distance physics, is to compare it with thelong distance correction of order α(Zα)6m, and not with the correction oforder α2(Zα)5m which corresponds to the short distance physics. Of course,such logic contradicts the spirit of the numerical calculations made withoutexpansion over Zα but it reflects properly the physical nature of differentcontributions at small Z.

Corrections of order α3(Zα)6 were never considered in the literature. Theyare suppressed in comparison to contributions of order α2(Zα)6 by at leastan additional factor α/π and are too small to be of any phenomenologicalinterest now.

3.5 Radiative Corrections of Order α(Zα)7mand of Higher Orders

Only partial results are known for corrections of order α(Zα)7m. However,recent achievements [112] in the numerical calculations without expansion inZα completely solve the problem of the corrections of order α(Zα)7m and ofhigher orders in Zα.

3.5.1 Corrections Induced by the Radiative Insertionsin the Electron Line

Consider first corrections of order α(Zα)7 induced by the radiative photoninsertions in the electron line. Due to the Layzer theorem [60] the diagramwith the radiative photon spanning four Coulomb photons does not lead to alogarithmic contribution. Hence, all leading logarithmic contributions of thisorder may be calculated with the help of second order perturbation theoryin (3.71). It is easy to check that the leading contribution is linear in thelarge logarithm and arises when one takes as the first perturbation the localpotential corresponding to the order α(Zα)5m contribution to the Lamb shift(3.34)

V1 = 4(

1 +11128

− 12

ln 2)

πα(Zα)2

m2, (3.95)

and the second perturbation corresponds to the Darwin potential

3.5

Radia

tive

Correctio

ns

ofO

rder

α(Z

α)7m

and

ofH

igher

Ord

ers69


4α(Zα)6

πn3 (mrm

)3m ≈ 173.074n3 kHz ∆E(1S) kHz ∆E(2S) kHz

Logarithmic Electron-LineContribution (l = 0)Layzer (1960) [60] − 1

4ln2[ m

mr(Zα)−2] + [ 4

3ln 2 + ln 2

n

Fried, Yennie (1960) [61] +ψ(n + 1) − ψ(1) − 601720

− 77180n2 ]

Erickson, Yennie (1965) [62, 63] × ln[ mmr

(Zα)−2] −1 882.77 −208.16

Logarithmic Electron-LineContribution (l �= 0)

Erickson, Yennie (1965) [62, 63] [(1 − 1n2 )( 1

30+ 1

12δj, 1

2)

+ 6−2l(l+1)/n2

3(2l+3)l(l+1)(4l2−1)] ln[ m

mr(Zα)−2]

Nonlogarithmic Electron-LineContribution

Pachucki (1993) (1S, 2S) [70, 71] A604

−1 338.04 −172.21[79], Jentschura, Pachucki (1996)(2P 1

2, 2P 3

2) [82]

Jentschura, Soff, Mohr (1997)(3P 1

2, 3P 3

2, 4P 1

2, 4P 3

2) [83]

Logarithmic PolarizationOperator ContributionLayzer (1960) [60] − 1

30ln[ m

mr(Zα)−2]δl0 −56.77 −7.10

Erickson, Yennie (1965) [62, 63]

70

3E

xtern

alField

Approx

imatio

nTable 3.7. (Continued)

Nonlogarithmic PolarizationOperator Contribution (l = 0)Mohr (1975) [65] 1

15[− 431

105+ ψ(n + 1) − ψ(1)

Ivanov, Karshenboim (1997) [88] − 2(n−1)

n2 + 128n2 − ln n

2]δl0 −27.41 −4.47

Nonlogarithmic PolarizationOperator Contribution (l = 1)

Mohr (1975) [65] − 115

(1 − 1n2 )[ 1

14+ 1

4δj, 1

2]

Manakov, Nekipelov,Fainstein (1989) [89]

Wichmann-Kroll Contribution

Wichmann, Kroll (1956) [87] ( 19180

− π2

108)δl0 2.45 0.31

Mohr (1976) [93, 92]

Leading Logarithmic Two-LoopElectron-Line Contribution

Karshenboim (1993) [95] − 227

(απ) ln3(Zα)−2δl0 −28.38 −3.55

Electron-Line Two-LoopLog Squared Term ∆E(nS)

Pachucki (2001) [101]{

− 49

ln 2 + 427

− 49

[

ln n − ψ(n)

Karshenboim (1996) [104, 105] +ψ(1) − n−1n

+ n2−14n2

]}

(απ) ln2(Zα)−2 −6.23 0.56

[106, 107]

(Continued)

3.5

Radia

tive

Correctio

ns

ofO

rder

α(Z

α)7m

and

ofH

igher

Ord

ers71


Single Logarithmic Two-Loop

Electron-Line Term ∆E(nS){

14ζ(3) − 1

6π2 ln 2 + 10

9ln2 2 − 38

27ln 2

Pachucki (2001) [101]

Jentschura (2003) [108] + 20273456

π2 + 426413259200

−(

2027

− 89

ln 2)[

ln n − ψ(n)

Jentschura, Czarnecki

Pachucki (2005) [81] +ψ(1) − n−1n

+ n2−14n2

]

+ 13N(nS)

}

(απ) ln(Zα)−2 47.89 5.06

Nonlogarithmic Two-LoopTerm ∆E(nS)

Pachucki, Jentschura (2003) [109] B604

(απ) −6.2 (9) −0.67 (10)

Electron-Line Two-LoopLog squared term ∆E(nP )

Karshenboim (1996) [104, 105] n2−127n2 (α

π) ln2(Zα)−2

Log Squared Two-LoopPolarization Term ∆E(nS)Karshenboim (1996) [104] 2

45(α

π) ln2(Zα)−2 1.73 0.22

Single Logarithmic Two-LoopPolarization Term ∆E(nS)

Pachucki (2001) [101]{

415

ln 2 − 4018100

− 845

[

− ln n + ψ(n)

Karshenboim (1996) [104] −ψ(1) + n−1n

− n2−14n2

]}

(απ) ln(Zα)−2 0.54 0.01

Single LogarithmicTerm ∆E(nPj)Czarnecki, Jentschura,

Pachucki (2005) [80]B61(nPj)

4(α

π) ln(Zα)−2


Fig. 3.25. Leading logarithmic contribution of order α(Zα)7 induced by the radia-tive photon

V2 = −πZα

2m2r

, (3.96)

where both potentials are written in momentum space (see Fig. 3.25). Substi-tuting these potentials in (3.71) one easily obtains [107]

∆E =(

2 +1164

− ln 2)

ln[m(Zα)−2

mr

]α(Zα)7

n3

(mr

m

)3

m δl0 . (3.97)

The Darwin potential generates the logarithmic correction to the nonrela-tivistic Schrodinger-Coulomb wave function in (3.65), and the result in (3.97)could be obtained by taking into account this correction to the wave func-tion in calculation of the contribution to the Lamb shift of order α(Zα)5m.This logarithmic correction is numerically equal 14.43 kHz for the 1S-level inhydrogen, and 1.80 kHz for the 2S level.

The nonlogarithmic contributions of order α(Zα)7 were never calculateddirectly. These corrections together with contributions in (3.60), (3.97) andthe corrections of higher orders in Zα are usually parametrized with the helpof an auxiliary function GSE(Zα)

∆E = GSE(Zα)α(Zα)6

πn3

(mr

m

)3

m , (3.98)

where GSE(Zα) = A60 +A71(Zα) ln(Zα)−2+ · · · (compare (3.60) and (3.97)).The function GSE(Zα) may be calculated numerically for not too small Zwithout expansion in Zα. For years the values of the function GSE(Zα) forsmall Z were obtained from the numerical results for larger Z [65, 113, 114]with the help of extrapolation. Many such extrapolations made under slightlydifferent assumptions are known in the literature [115, 116, 117, 118, 82, 83].

A spectacular success was achieved when the function GSE(Zα) was di-rectly calculated numerically for low Z [112, 119, 120, 121]. Using pow-erful resummation and convergence acceleration techniques, the authors of[112, 119, 120, 121] performed numerical calculations directly for low Z =1, . . . , 5 avoiding extrapolation from high Z. The GSE(Zα) self-energy con-tribution was obtained in this work with the numerical uncertainty 0.8 Hzfor the 1S-level in hydrogen, and 1.0 Hz for the 2P -level in hydrogen. Fromthe practical point of view the results [112, 119, 120, 121] completely solve allproblems with calculation of the higher order corrections in Zα of the form

3.5 Radiative Corrections of Order α(Zα)7m and of Higher Orders 73

α(Zα)n in the foreseeable future. We will use these results collected in Table3.8 in the numerical calculations of the positions of energy levels in hydrogenand helium below. The values of the function GSE(Zα) in Table 3.8 werecalculated for α = 1/137.036, but it was explicitly checked that variation ofthe inverse fine structure constant in the interval from α = 1/137.035 999 5to α = 1/137.036 000 5 does not change the entries in Table 3.8 at the currentlevel of accuracy [112, 119, 120, 121].

However, a number of theoretical questions connected with the electron-line corrections of order α(Zα)n remain unresolved. Comparing the pertur-bative results in Tables 3.4 and 3.7 with the numerical results in Table 3.8we immediately see that the difference between the respective contributionsto the Lamb shift is about 13 kHz for the 1S state in hydrogen and about1600 kHz for He+. This means that consistency between the analytic andnumerical results requires existence of relatively large nonlogarithmic con-tributions of order α(Zα)7 and/or large logarithmic contributions of orderα(Zα)8. Clearly this problem deserves further investigation. An independentnumerical calculation without expansion in Zα as well as an analytic calcu-lation of nonlogarithmic corrections of order α(Zα)7 and corrections of orderα(Zα)8 is also warranted.

3.5.2 Corrections Induced by the Radiative Insertionsin the Coulomb Lines

There are two contributions of order α(Zα)7m to the energy shift induced bythe Uehling and the Wichmann-Kroll potentials (see Fig. 3.10 and Fig. 3.16,respectively). Respective calculations go along the same lines as in the caseof the Coulomb-line corrections of order α(Zα)6 considered above.

Uehling Potential Contribution

The logarithmic contribution is induced only by the Uehling potential inFig. 3.10, and may easily be calculated exactly in the same way as the log-arithmic contribution induced by the radiative photon in (3.97). The onlydifference is that now the role of the perturbation potential is played by thekernel which corresponds to the polarization contribution to the Lamb shiftof order α(Zα)5m

V1 =548

πα(Zα)2

m2. (3.99)

Then we immediately obtain [65]

∆E =596

ln[m(Zα)−2

mr

]α(Zα)7

n3

(mr

m

)3

m δl0 . (3.100)

It is not difficult to calculate analytically nonlogarithmic corrections of orderα(Zα)7 generated by the Uehling potential. Using the formulae from [65] oneobtains for a few lower levels (see also [122] for the case of 1S-state)

74

3E

xtern

alField

Approx

imatio

n

Table 3.8. Values of the Function GSE(Zα)

Z = 1 Z = 2 Z = 3 Z = 4 Z = 5

1S 12

GSE(Zα) −30.290 24 (2) −29.770 967 (5) −29.299 170 (2) −28.859 222 (1) −28.443 372 3 (8)

∆E kHz −1310.615 0 (9) −82 441.40 (1) −924 177.23 (6) −5 114 662.6 (2) −19 229 745.3 (5)

2S 12

GSE(Zα) −31.185 15 (9) −30.644 66 (5) −30.151 93 (2) −29.691 27 (1) −29.255 033 (8)

∆E kHz −168.667 1 (5) −10 607.60 (2) −118 884.46 (8) −657 765.6 (2) −2 472 310.7 (7)

2P 12

GSE(Zα) −0.973 5 (2) −0.949 40 (5) −0.926 37 (2) −0.904 12 (1) −0.882 478 (8)

∆E kHz −5.265 (1) −328.63 (2) −3 652.54 (8) −20 029.4 (2) −74 577.2 (7)

2P 32

GSE(Zα) −0.486 5 (2) −0.470 94 (5) −0.456 65 (2) −0.443 13 (1) −0.430 244 (8)

∆E kHz −2.631 (1) −163.02 (2) −1 800.50 (8) −9 816.9 (2) −36 359.4 (7)

3S 12

GSE(Zα) −31.047 7 (9) −30.512 6 (2) −30.022 7 (2) −29.564 53 (6) −29.130 61 (4)

∆E kHz −49.755 (1) −3129.45 (2) −35 074.1 (2) −194 061.6 (4) −729 420 (1)

3P 12

GSE(Zα) −1.118 5 (9) −1.089 7 (2) −1.062 5 (1) −1.036 30 (6) −1.010 84 (4)

∆E kHz −1.792 (1) −114.71 (9) −1307 (1) −7 342 (6) −28 007 (23)

3P 32

GSE(Zα) −0.577 5 (9) −0.559 3 (2) −0.542 1 (1) −0.526 06 (6) −0.510 76 (4)

∆E kHz −0.925 (1) −59.23 (9) −675 (1) −3 791 (6) −14 460 (23)

4S 12

GSE(Zα) −30.912 (4) −30.380 0 (9) −29.892 4 (4) −29.437 1 (2) −29.006 0 (2)

∆E kHz −20.899 (3) −1314.50 (4) −14 732.6 (2) −81 516.9 (6) −306 408 (2)

4P 12

GSE(Zα) −1.164 (2) −1.134 1 (5) −1.105 5 (2) −1.078 1 (1) −1.051 20 (8)

∆E kHz −0.787 (1) −49.07 (2) −544.8 (1) −2 985.5 (3) −11 104.5 (8)

4P 32

GSE(Zα) −0.609 (2) −0.590 4 (5) −0.572 9 (2) −0.555 5 (1) −0.539 41 (8)

∆E kHz −0.412 (1) −25.55 (2) −282.4 (1) −1 538.3 (3) −5 698.1 (8)

3.5 Radiative Corrections of Order α(Zα)7m and of Higher Orders 75

∆E(1S) =596

[

ln[m(Zα)−2

mr

]

+ 2 ln 2 +2315

]

α(Zα)7(mr

m

)3

m ,

∆E(2S) =596

[

ln[m(Zα)−2

mr

]

+ 4 ln 2 +841480

]α(Zα)7

23

(mr

m

)3

m ,

∆E(2P 1

2

)=

413 072

α(Zα)7

23

(mr

m

)3

m ,

∆E(2P 3

2

)=

71 024

α(Zα)7

23

(mr

m

)3

m . (3.101)

There are no obstacles to exact numerical calculation of the Uehling po-tential contribution to the energy shift without expansion over Zα and suchcalculations have been performed with high accuracy (see [65, 117] and refer-ences therein). The results of these calculations may be conveniently presentedwith the help of an auxiliary function GU,7(Zα) defined by the relationship

∆E(l, n) =[

596

ln[m(Zα)−2

mr

]

δl0 +1π

Gl,nU,7(Zα)

]α(Zα)7

n3

(mr

m

)3

m . (3.102)

For the case of atoms with low Z (hydrogen and helium), values of the functionGU,7(Zα) for the states with n = 1, 2, 4 are tabulated in [117] and respectivecontributions may easily be calculated for other states when needed. Thesenumerical results may be used for comparison of the theory and experimentinstead of the results of order α(Zα)7 given above. We may also use theresults of numerical calculations in order to make an estimate of uncalculatedcontributions of the Uehling potential of order α(Zα)8 and higher. Accordingto [117]

Gl=0,n=1U (α) = 0.428 052 . (3.103)

Comparing this value with the order α(Zα)7m result in (3.101) we see that thedifference between the exact numerical result and analytic calculation up toorder α(Zα)7 is about 0.015 kHz for the 1S-level in hydrogen, and, taking intoaccount the accuracy of experimental results, one may use analytic results forcomparison of the theory and experiment without loss of accuracy. A similarconclusion is valid for other hydrogen levels.

Wichmann-Kroll Potential Contribution

Contribution of the Wichmann-Kroll potential in Fig. 3.16 may be calculatedin the same way as the respective contribution of order α(Zα)6m in (3.70)by taking the next term in Zα in the small momentum expansion of theWichmann-Kroll potential in (3.69). One easily finds [93, 92]

∆E =(

116

− 31π2

2 880

)α(Zα)7

n3

(mr

m

)3

m δl0 . (3.104)

This contribution is very small and it is clear that at the present levelof experimental accuracy calculation of higher order contributions of theWichmann-Kroll potential is not necessary.



4α(Zα)7

n3 (mrm

)3m ∆E(1S) kHz ∆E(2S) kHz

Logarithmic Electron-LineContributionKarshenboim (1994) [107] ( 139

256− 1

4ln 2) ln(Zα)−2δl0 14.43 1.80

NonlogarithmicElectron-Line ContributionMohr (1992) [114]Karshenboim (1995) [118] −0.76(5) −0.09(1)

Logarithmic PolarizationOperator ContributionMohr (1975) [65] 5

384ln(Zα)−2δl0 0.51 0.06

Nonlogarithmic PolarizationOperator Contribution∆E(nS, P ), Mohr (1975) [65] 0.15 0.03

Wichmann, Kroll (1956) [87] ( 164

− 31π2

11 520)δl0 −0.04 −0.01

Mohr (1976) [93, 92]

Corrections of orderα2(Zα)7 (±)α

π±1 ±0.1


Corrections of order α2(Zα)7m were never considered in the literature. Theyshould be suppressed in comparison with the corrections of order α(Zα)7 by atleast the factor α/π. Taking into account possible logarithmic enhancements,one could expect that these corrections are as large as 1 kHz for the 1S-stateand 0.1 kHz for 2S-state in hydrogen. This estimate means that calculationof these corrections is not only a challenging theoretical problem but is alsointeresting from the phenomenological point of view. The first results of cal-culation of all corrections of order α2(Zα)nm without expansion in Zα werereported in [123]. According to the results of this paper the effective value ofthe coefficient B60 (see (3.76)) for the 1S level in hydrogen which takes intoaccount also all contributions α2(Zα)nm of higher order in Zα is

B60 = −127 ± 30% . (3.105)

This value is two times larger than the result cited above in Table 3.6, andtaken at face value shifts the value of the 1S Lamb shift in hydrogen by 7 kHz.However, the authors of [123] show that extrapolation of their numerical datato (Zα) → 0 leads to the value of B60 twice as large as the analytic result forB60 in [109, 110]. Clearly any phenomenological conclusions in this situationare premature, and the problem of corrections of order α2(Zα)nm requires

References 77

more work. In view of this discrepancy we will assume that uncertainty of thecontribution of α2(Zα)n is about 4 kHz for 1S state and about 0.5 kHz for2S state in hydrogen.

Concluding our discussion of the purely radiative corrections to the Lambshift let us mention once more that the main sources of the theoretical uncer-tainty in these contributions is connected with the nonlogarithmic correctionsof order α2(Zα)n and uncalculated contributions of orders α3(Zα)5, whichmay be as large as a few kHz for 1S-state and a few tenths of kHz for the2S-state in hydrogen. All other unknown purely radiative contributions to theLamb shift are much smaller.

References

1. G. Breit, Phys. Rev. 34, 553 (1929); ibid 36, 383 (1930); ibid 39, 616 (1932).2. W. A. Barker and F. N. Glover, Phys. Rev. 99, 317 (1955).3. V. B. Berestetskii, E. M. Lifshitz, and L. P. Pitaevskii, Quantum electrody-

namics, 2nd Edition, Pergamon Press, Oxford, 1982.4. H. Grotch and D. R. Yennie, Zeitsch. Phys. 202, 425 (1967).5. H. Grotch and D. R. Yennie, Rev. Mod. Phys. 41, 350 (1969).6. P. Lepage, Phys. Rev. A 16, 863 (1977).7. J. R. Sapirstein and D. R. Yennie, in Quantum Electrodynamics, ed. T. Ki-

noshita (World Scientific, Singapore, 1990), p.560.8. H. A. Bethe, Phys. Rev. 72, 339 (1947).9. N. M. Kroll and W. E. Lamb, Phys. Rev. 75, 388 (1949).

10. J. B. French and V. F. Weisskopf, Phys. Rev. 75, 1240 (1949).11. G. Bhatt and H. Grotch, Phys. Rev. A 31, 2794 (1985).12. G. Bhatt and H. Grotch, Ann. Phys. (NY) 178, 1 (1987).13. S. E. Haywood and J. D. Morgan III, Phys. Rev. A 32, 3179 (1985).14. G. W. F. Drake and R. A. Swainson, Phys. Rev. A 41, 1243 (1990).15. J. Schwinger, Phys. Rev. 73, 416 (1948).16. E. A. Uehling, Phys. Rev. 48, 55 (1935).17. J. Weneser, R. Bersohn, and N. M. Kroll, Phys. Rev. 91, 1257 (1953).18. M. F. Soto, Phys. Rev. Lett. 17, 1153 (1966); Phys. Rev. A 2, 734 (1970).19. T. Appelquist and S. J. Brodsky, Phys. Rev. Lett. 24, 562 (1970); Phys. Rev.

A 2, 2293 (1970).20. B. E. Lautrup, A. Peterman, and E. de Rafael, Phys. Lett. B 31, 577 (1970).21. R. Barbieri, J. A. Mignaco and E. Remiddi, Lett. Nuovo Cimento 3, 588 (1970).22. A. Peterman, Phys. Lett. B 34, 507 (1971); ibid. 35, 325 (1971)23. J. A. Fox, Phys. Rev. D 3, 3228 (1971); ibid. 4, 3229 (1971); ibid. 5, 492 (1972).24. R. Barbieri, J. A. Mignaco and E. Remiddi, Nuovo Cimento A 6, 21 (1971).25. E. A. Kuraev, L. N. Lipatov, and N. P. Merenkov, preprint LNPI 46, June

1973.26. R. Karplus and N. M. Kroll, Phys. Rev. 77, 536 (1950).27. A. Peterman, Helv. Phys. Acta 30, 407 (1957); Nucl. Phys. 3, 689 (1957).28. C. M. Sommerfield, Phys. Rev. 107, 328 (1957); Ann. Phys. (NY) 5, 26 (1958).29. M. Baranger, F. J. Dyson, and E. E. Salpeter, Phys. Rev. 88, 680 (1952).


30. G. Kallen and A. Sabry, Kgl. Dan. Vidensk. Selsk. Mat.-Fis. Medd. 29 (1955)No.17.

31. J. Schwinger, Particles, Sources and Fields, Vol.2 (Addison-Wesley, Reading,MA, 1973).

32. K. Melnikov and T. van Ritbergen, Phys. Rev. Lett. 84, 1673 (2000).33. T. Kinoshita, in Quantum Electrodynamics, ed. T. Kinoshita (World Scientific,

Singapore, 1990), p.218.34. S. Laporta and E. Remiddi, Phys. Lett. B 379, 283 (1996).35. P. A. Baikov and D. J. Broadhurst, preprint OUT-4102-54, hep-ph 9504398,

April 1995, published in the proceedings New Computing Technique in PhysicsResearch IV, ed. B. Denby and D. Perret-Gallix, World Scientific, 1995.

36. M. I. Eides and H. Grotch, Phys. Rev. A 52, 3360 (1995).37. S. G. Karshenboim, J. Phys. B: At. Mol. Opt. Phys. 28, L77 (1995).38. M. I. Eides and V. A. Shelyuto, Phys. Rev. A 52, 954 (1995).39. J. L. Friar, J. Martorell, and D. W. L. Sprung, Phys. Rev. A 59, 4061 (1999).40. R. Karplus, A. Klein, and J. Schwinger, Phys. Rev. 84, 597 (1951).41. R. Karplus, A. Klein, and J. Schwinger, Phys. Rev. 86, 288 (1952).42. M. Baranger, Phys. Rev. 84, 866 (1951); M. Baranger, H. A. Bethe, and R.

Feynman, Phys. Rev. 92, 482 (1953).43. A. A. Abrikosov, Zh. Eksp. Teor. Fiz. 30, 96 (1956) [Sov. Phys.-JETP 3, 71

(1956)].44. H. M. Fried and D. R. Yennie, Phys. Rev. 112, 1391 (1958).45. S. G. Karshenboim, V. A. Shelyuto, and M. I. Eides, Yad. Fiz. 47, 454 (1988)

[Sov. J. Nucl. Phys. 47, 287 (1988)].46. M. I. Eides, H. Grotch, and D. A. Owen, Phys. Lett. B 294, 115 (1992).47. K. Pachucki, Phys. Rev. A 48, 2609 (1993).48. S. Laporta, as cited in [47].49. M. I. Eides and H. Grotch, Phys. Lett. B 301, 127 (1993).50. M. I. Eides and H. Grotch, Phys. Lett. B 308, 389 (1993).51. M. I. Eides, H. Grotch, and V. A. Shelyuto, Phys. Rev. A 55, 2447 (1997).52. M. I. Eides, H. Grotch, and P. Pebler, Phys. Lett. B 326, 197 (1994); Phys.

Rev. A 50, 144 (1994).53. M. I. Eides, S. G. Karshenboim, and V. A. Shelyuto, Phys. Lett. B 312, 358

(1993); Yad. Phys. 57, 1309 (1994) [Phys. Atom. Nuclei 57, 1240 (1994)].54. M. I. Eides, S. G. Karshenboim, and V. A. Shelyuto, Yad. Phys. 57, 2246

(1994) [Phys. Atom. Nuclei 57, 2158 (1994)].55. K. Pachucki, Phys. Rev. Lett. 72, 3154 (1994).56. M. I. Eides and V. A. Shelyuto, Pis’ma Zh. Eksp. Teor. Fiz. 61, 465 (1995)

[JETP Letters 61, 478 (1995)].57. K. Pachucki, D. Leibfried, M. Weitz, A. Huber, W. Konig, and T. W. Hanch,

J. Phys. B 29, 177 (1996); 29, 1573(E) (1996).58. M. I. Eides and V. A. Shelyuto, Phys. Rev. A 68, 042106 (2003).59. M. I. Eides and V. A. Shelyuto, Phys. Rev. A 70, 022506 (2004).60. A. J. Layzer, Phys. Rev. Lett. 4, 580 (1960); J. Math. Phys. 2, 292, 308 (1961).61. H. M. Fried and D. R. Yennie, Phys. Rev. Lett. 4, 583 (1960).62. G. W. Erickson and D. R. Yennie, Ann. Phys. (NY) 35, 271 (1965).63. G. W. Erickson and D. R. Yennie, Ann. Phys. (NY) 35, 447 (1965).64. G. W. Erickson, Phys. Rev. Lett. 27, 780 (1971).65. P. J. Mohr, Phys. Rev. Lett. 34, 1050 (1975); Phys. Rev. A 26, 2338 (1982).

References 79

66. J. R. Sapirstein, Phys. Rev. Lett. 47, 1723 (1981).67. V. G. Pal’chikov, Metrologia 10, 3 (1987) /in Russian/.68. L. Hostler, J. Math. Phys. 5, 1235 (1964).69. J. Schwinger, J. Math. Phys. 5, 1606 (1964).70. K. Pachucki, Phys. Rev. A 46, 648 (1992).71. K. Pachucki, Ann. Phys. (NY) 236, 1 (1993).72. S. S. Schweber, QED and the Men Who Made It, Princeton University Press,

1994.73. J. D. Bjorken and S. D. Drell, Relativistic Quantum Mechanics, McGraw-Hill

Book Co., NY, 1964.74. E. E. Salpeter, Phys. Rev. 87, 553 (1952).75. T. Fulton and P. C. Martin, Phys. Rev. 95, 811 (1954).76. I. B. Khriplovich, A. I. Milstein, and A. S. Yelkhovsky, Physica Scripta, T46,

252 (1993).77. W. E. Caswell and G. P. Lepage, Phys. Lett. B 167, 437 (1986).78. J. A. Fox and D. R. Yennie, Ann. Phys. (NY) 81, 438 (1973).79. K. Pachucki, as cited in [112].80. A. Czarnecki, U. D. Jentschura, and K. Pachucki, Phys. Rev. Lett. 95, 180404

(2005).81. U. D. Jentschura, A. Czarnecki, and K. Pachucki, Phys. Rev. A 72, 062102

(2005).82. U. Jentschura and K. Pachucki, Phys. Rev. A 54, 1853 (1996).83. U. Jentschura, G. Soff, and P. J. Mohr, Phys. Rev. A 56, 1739 (1997).84. U. D. Jentschura, E.-O. Le Bigot, P. J. Mohr, et al, Phys. Rev. Lett. 90, 163001

(2003).85. E.-O. Le Bigot, U. D. Jentschura, P. J. Mohr, et al, Phys. Rev. A 68, 042101

(2003).86. R. Serber, Phys. Rev. 48, 49 (1935).87. E. H. Wichmann and N. M. Kroll, Phys. Rev. 101, 843 (1956) .88. V. G. Ivanov and S. G. Karshenboim, Yad. Phys. 60, 333 (1997) [Phys. Atom.

Nuclei 60, 270 (1997)].89. N. L. Manakov, A. A. Nekipelov, and A. G. Fainstein, Zh. Eksp. Teor. Fiz. 95,

1167 (1989) [Sov.Phys.-JETP 68, 673 (1989)].90. M. H. Mittleman, Phys. Rev. 107, 1170 (1957).91. D. E. Zwanziger, Phys. Rev. 121, 1128 (1961).92. P. J. Mohr, At. Data and Nucl. Data Tables, 29, 453 (1983) .93. P. J. Mohr, in Beam-Foil Spectroscopy, ed. I. A. Sellin and D. J. Pegg (Plenum

Press, New York, 1976), Vol. 1, p. 89.94. L. D. Landau and E. M. Lifshitz, “Quantum Mechanics”, 3d Edition,

Butterworth-Heinemann, 1997.95. S. G. Karshenboim, Zh. Eksp. Teor. Fiz. 103, 1105 (1993) [JETP 76, 541

(1993)].96. S. Mallampalli and J. Sapirstein, Phys. Rev. Lett. 80, 5297 (1998).97. I. Goidenko, L. Labzowsky, A. Nefiodov et al, Phys. Rev. Lett. 83, 2312 (1999).98. V. A. Yerokhin, Phys. Rev. A 62, 012508 (2000).99. A. V. Manohar and I. W. Stewart, Phys. Rev. Lett. 85, 2248 (2000).

100. V. A. Yerokhin, Phys. Rev. Lett. 86, 1990 (2001).101. K. Pachucki, Phys. Rev. A 63, 042503 (2001).102. U. Jentschura and I. Nandori, Phys. Rev. A 66, 022114 (2002).


103. U. D. Jentschura, Phys. Lett. B 564, 225 (2003).104. S. G. Karshenboim, Zh. Eksp. Teor. Fiz. 109, 752 (1996) [JETP 82, 403

(1996)].105. S. G. Karshenboim, J. Phys. B: At. Mol. Opt. Phys, 29, L29 (1996).106. S. G. Karshenboim, Yad. Phys. 58, 707 (1995) [Phys. Atom. Nuclei 58, 649

(1995)].107. S. G. Karshenboim, Zh. Eksp. Teor. Fiz. 106, 414 (1994) [JETP 79, 230

(1994)].108. U. D. Jentschura, J. Phys. A: Math. Gen. 36, L229 (2003).109. K. Pachucki and U. Jentschura, Phys. Rev. Lett. 91, 113005 (2003).110. U. D. Jentschura, Phys. Rev. A 70, 052108 (2004).111. D. R. Yennie, S. C. Frautchi, and H. Suura, Ann. Phys. (NY) 13, 379 (1961).112. U. Jentschura, P. J. Mohr, and G. Soff, Phys. Rev. Lett. 82, 53 (1999).113. P. J. Mohr and Y.-K. Kim, Phys. Rev. A 45, 2727 (1992).114. P. J. Mohr, Phys. Rev. A 46, 4421 (1992).115. A. van Wijngaarden, J. Kwela, and G. W. F. Drake, Phys. Rev. A 43, 3325

(1991).116. P. J. Mohr, Phys. Rev. A 44, R4089 (1991); Errata A 51, 3390 (1995).117. P. J. Mohr, in Atomic, Molecular and Optical Physics Handbook, ed. G. W. F.

Drake, AIP Press, 1996, p. 341.118. S. G. Karshenboim, Yad. Phys. 58, 309 (1995) [Phys. Atom. Nuclei 58, 262

(1995)].119. U. D. Jentschura, P. J. Mohr, and G. Soff, Phys. Rev. A 63, 042512 (2001).120. U. D. Jentschura and P. J. Mohr, Phys. Rev. A 69, 064103 (2004).121. U. D. Jentschura and P. J. Mohr, Phys. Rev. A 72, 014103 (2005).122. S. G. Karshenboim, Can. J. Phys. 76, 169 (1998); Zh. Eksp. Teor. Fiz. 116,

1575 (1999) [JETP 89, 850 (1999)].123. V. A. Yerokhin, P. Indelicato, and V. M. Shabaev, Phys. Rev. A 71, 040101(R)

(2005).

4

Essentially Two-Particle Recoil Corrections

4.1 Recoil Corrections of Order (Zα)5(m/M)m

Leading relativistic corrections of order (Zα)4 and their mass dependencewere discussed above in Sect. 3.1 in the framework of the Breit equationand the effective Dirac equation in the external field in Fig. 1.6. The exactmass dependence of these corrections could be easily calculated because allthese corrections are induced by the one-photon exchange. The effective Diracequation in the external field produces leading relativistic corrections withcorrect mass dependence because the one-photon exchange kernel is properlytaken into account in this equation. Some other recoil corrections of higherorders in Zα are also partially generated by the effective Dirac equation withthe external source. All such corrections are necessarily of even order in Zαsince all expansions for the energy levels of the Dirac equation are effectivelynonrelativistic expansions over v2; they go over (Zα)2, and, hence, the nextrecoil correction produced by the effective Dirac equation in the external fieldis of order (Zα)6. The result for the recoil correction of order (Zα)6(m/M),obtained in this way, is incomplete and we will improve it below. First wewill consider the even larger recoil correction of order (Zα)5(m/M), which iscompletely missed in the spectrum of the Breit equation or of the effectiveDirac equation with the Coulomb potential, and which can be calculatedonly by taking into account the two-particle nature of the QED bound-stateproblem.

The external field approximation is clearly inadequate for calculation ofthe recoil corrections and, in principle, one needs the machinery of the rel-ativistic two-particle equations to deal with such contributions to the en-ergy levels. The first nontrivial recoil corrections are generated by kernelswith two-photon exchanges. Naively one might expect that all correctionsof order (Zα)5(m/M)m are generated only by the two-photon exchanges inFig. 4.1. However, the situation is more complicated. More detailed consid-eration shows that the two-photon kernels are not sufficient and irreduciblekernels in Fig. 4.2 with arbitrary number of the exchanged Coulomb pho-


82 4 Essentially Two-Particle Recoil Corrections

Fig. 4.2. Irreducible kernels with arbitrary number of the exchanged Coulombphotons

tons spanned by a transverse photon also generate contributions of order(Zα)5(m/M)m. This effect is similar to the case of the leading order radia-tive correction of order α(Zα)4 considered in Subsubsect. 3.2.1.1 when, dueto a would-be infrared divergence, diagrams in Fig. 3.2 with any number ofthe external Coulomb photons spanned by a radiative photon give contribu-tions of one and the same order since the apparent factor Zα accompanyingeach extra external photon is compensated by a small denominator connectedwith the small virtuality of the bound electron. Exactly the same effect arisesin the case of the leading recoil corrections. All kernels with any numberof exchanged Coulomb photons spanned by an exchanged transverse photongenerate contributions to the leading recoil correction.

Let us describe this similarity between the leading contribution to theLamb shift and the leading recoil correction in more detail following a nicephysical interpretation which was given in [1]. The leading contribution tothe Lamb shift in (2.4) is proportional to the mean square of the electronradius which may be understood as a result of smearing of the fluctuatingelectron coordinate due to its interaction with the fluctuating electromagneticfield [2]. In the considerations leading to (2.4) we considered the proton asan infinitely heavy source of the Coulomb field. If we take into account thefiniteness of the proton mass, then the factor 〈r2〉 in (2.4) will turn into〈(∆r1 − ∆r2)2〉 = 〈(∆r1)2〉 + 〈(∆r2)2〉 − 2〈(∆r1)(∆r2)〉, where ∆r1 and ∆r2

are fluctuations of the coordinates of the electron and the proton, respectively.Averaging the squares of the fluctuations of the coordinates of both particlesproceeds exactly as in the case of the electron in the Coulomb field in (2.2) andgenerates the leading contribution to the Lamb shift and recoil correction ofrelative order (m/M)2. This recoil factor arises because the average fluctuationof the coordinate squared equal to the average radius squared of the particle isinversely proportional to mass squared of this particle. Hence, it is clear thatthe average 〈∆r1∆r2〉 generates a recoil correction of the first order in therecoil factor m/M . Note that the correlator 〈∆r1∆r2〉 is different from zeroonly when averaging goes over distances larger than the scale of the atom

4.1 Recoil Corrections of Order (Zα)5(m/M)m 83

1/(mZα) or in momentum space over fluctuating momenta of order mZαand smaller. For smaller distances (or larger momenta) fluctuations of thecoordinates of two particles are completely uncorrelated and the correlator oftwo coordinates is equal to zero. Hence, the logarithmic contribution to therecoil correction originates from the momentum integration region m(Zα)2 �k � m(Zα), unlike the leading logarithmic contribution to the Lamb shiftwhich originates from a wider region m(Zα)2 � k � m. A new feature of theleading recoil correction is that the upper cutoff to the logarithmic integrationis determined by the inverse size of the atom. We will see below how all thesequalitative features are reproduced in the exact calculations.

Complete formal analysis of the recoil corrections in the framework ofthe relativistic two-particle equations, with derivation of all relevant kernels,perturbation theory contributions, and necessary subtraction terms may beperformed along the same lines as was done for hyperfine splitting in [3]. How-ever, these results may also be understood without a cumbersome formalismby starting with the simple scattering approximation. We will discuss recoilcorrections below using this less rigorous but more physically transparent ap-proach.

As we have already realized from the qualitative discussion above, theleading recoil correction is generated at large distances, and small exchangedmomenta are relevant for its calculation. The choice of gauge of the exchangedphotons is, in such a case, determined by the choice of gauge in the effectiveDirac equation with the one photon potential. This equation was writtenin the Coulomb gauge and, hence, we have to use the Coulomb gauge alsoin the kernels with more than one exchanged photon. Since the Coulomband transverse propagators have different form in the Coulomb gauge it isnatural to consider separately diagrams with Coulomb-Coulomb, transverse-transverse and Coulomb-transverse exchanges.

4.1.1 Coulomb-Coulomb Term

Coulomb exchange is already taken into account in the construction of thezero-order effective Dirac equation, where the Coulomb source plays the roleof the external potential. Hence, additional contributions of order (Zα)5 couldbe connected only with the high-momentum Coulomb exchanges. Let us startby calculating the contribution of the skeleton Coulomb-Coulomb diagramswith on-shell external electron lines in Fig. 4.3, with the usual hope that theintegrals would tell us themselves about any possible inadequacy of such anapproximation.

Direct calculation of the two Coulomb exchange photon contribution leadsto the integral

∆E = − 4(1 − µ2)

(Zα)5mπn3

(mr

m

)3∫ ∞

0

dk

k4[f(µk) − µf(k)] , (4.1)

where


Fig. 4.3. Coulomb-Coulomb two-photon exchanges

f(k) = 3√

1 + k2 +1√

1 + k2, (4.2)

and µ = m/M . The apparent asymmetry of the expression in (4.1) withrespect to masses of the heavy and light particle emerged because the dimen-sionless momentum k in this formula is measured in terms of the electronmass.

At small momenta the function f(k) behaves as

f(µk) − µf(k) ≈ 4(1 − µ) − µ(1 − µ)k2 + O(k6) , (4.3)

and the skeleton integral in (4.1) diverges as∫

dk/k4 in the infrared region.The physical meaning of this low-momenta infrared divergence is clear; it cor-responds to the Coulomb exchange contribution to the Schrodinger-Coulombwave function. The Coulomb wave function graphically includes a sum ofCoulomb ladders and the addition of an extra rung does not change the wavefunction. However, if one omits the binding energy, as we have effectively doneabove, one would end up with an infrared divergent integral instead of theself-reproducing Schrodinger-Coulomb wave function. A slightly different wayto understand the infrared divergence in (4.1) is to realize that the terms in(4.3) which generate the divergent contribution correspond to the residue ofthe heavy proton pole in the box diagram. Once again these heavy particlepole contributions build the Coulomb wave function and we have to subtractthem not only to avoid an apparent divergence in the approximation when weneglect the binding energy, but in order to avoid double counting.

We would like to emphasize here that, even if one would forget about thethreat of double counting, an emerging powerlike infrared divergence wouldremind us of its necessity. Any powerlike infrared divergence is cutoff by thebinding energy, and has a well defined order in the parameter Zα. It is mostimportant that the integral in (4.1) does not contain any logarithmic infrareddivergence at small momenta. In such a case one can unambiguously subtractin the integrand the powerlike infrared divergent terms and the remainingintegral will be completely convergent. Then only high intermediate momentaof the order of the electron mass contribute to the subtracted integral, therespective diagram is effectively local in coordinate space, and the contributionto the energy shift of order (Zα)5 is simply given by the product of thisintegral and the nonrelativistic Schrodinger-Coulomb wave function squaredat the origin. Any attempt to take into account small virtuality of the externalelectron lines (equivalent to taking into account nonlocality of the diagramin the coordinate space) would lead to additional factors of Zα, which we do


Fig. 4.4. Transverse-transverse two-photon exchanges

not consider yet. Direct calculation, after subtraction, of the first two leadinglow-frequency terms in the integrand in (4.1) immediately gives

∆Esub = − 4(1 − µ2)

(Zα)5mπn3

(mr

m

)3∫ ∞

0

dk

k4{f(µk) − µf(k) (4.4)

−[4(1 − µ) − µ(1 − µ)k2]}

= −4µ

3(Zα)5m

πn3

(mr

m

)3

,

reproducing the well known result [4, 5, 6].Let us emphasize once again that an exact calculation (in contrast to the

calculation with the logarithmic accuracy) of the Coulomb-Coulomb contribu-tion with the help of the skeleton integral turned out to be feasible due to theabsence of the low-frequency logarithmic divergence. For the logarithmicallydivergent integrals the low-frequency cutoff is supplied by the wave function,and in such a case it is impossible to calculate the constant on the backgroundof the logarithm in the skeleton approximation. In such cases more accuratetreatment of the low-frequency contributions is warranted.

4.1.2 Transverse-Transverse Term

The kernels with two transverse exchanges in Fig. 4.4 give the following con-tribution to the energy shift in the scattering approximation

∆E = − 2µ

1 − µ2

(Zα)5mπn3

(mr

m

)3∫ ∞

0

dk(f(k) − µ3f(µk)) , (4.5)

wheref(k) =

1k− 1√

1 + k2. (4.6)

This integral diverges only logarithmically at small momenta. Hence, thiscontribution does not contain either corrections of the previous order or thenon-recoil corrections. The main low-frequency logarithmic divergence pro-duces ln Zα and the factor before this logarithm may easily be calculated inthe scattering approximation. This approximation is insufficient for calcula-tion of the nonlogarithmic contribution, and the respective calculation requiresa more accurate consideration [4, 5]. A new feature of the integral in (4.5),as compared with the other integrals discussed so far, is that the exchangedmomenta higher than the electron mass produce a nonvanishing contribution.


This new integration region from the electron to the proton mass, which wasdiscovered in [5], arises here for the first time in the bound state problem.As we will see below, especially in discussion of the hyperfine splitting, thesehigh momenta are responsible for a number of important contributions to theenergy shifts.

The high-momentum contribution to the Lamb shift is suppressed by thesecond power of the recoil factor (m/M)2, and is rather small. Let us notethat the result we will obtain below in this section is literally valid only foran elementary proton, since for the integration momenta comparable with theproton mass one cannot ignore the composite nature of the proton and has totake into account its internal structure as it is described by the phenomeno-logical form factors. It is also necessary to take into account inelastic contribu-tions in the diagrams with the two exchanged photons. We will consider theseadditional contributions later in Sect. 6.2 dealing with the nonelectromagneticcontributions to the Lamb shift.

The state-independent high-frequency contribution as well as the low-frequency logarithmic term are different from zero only for the S-states andmay easily be calculated with the help of (4.5)

∆E =[

−2µ2 ln µ

1 − µ2− 2 ln(1 + µ) + (2 ln Zα + 2 ln 2)

]

µ(Zα)5m

πn3

(mr

m

)3

δl0 ,

(4.7)in complete accord with the well known result [4, 5, 6]. Note that despite itsappearance this result is symmetric under permutation of the heavy and lightparticles, as expected beforehand, since the diagrams with two transverse ex-changes are symmetric. In order to preserve this symmetry we cut the integralfrom below at momenta of order mrZα, calculating the contribution in (4.7).

In order to obtain the state-dependent low-frequency contribution of thedouble transverse exchange it is necessary to restore the dependence of thegraphs with two exchanged photons on the external momenta and calculatethe matrix elements of these diagrams between the momentum dependentwave functions. Respective momentum integrals should be cut off from aboveat mrZα. The wave function momenta provide an effective lower cutoff for theloop integrals and one may get rid of the upper cutoff by matching the low-and high-frequency contributions. The calculation for an arbitrary principalquantum number is rather straightforward but tedious [4, 5, 7, 6, 8, 9] andleads to the result

∆E =

{[

2 ln2Zα

n+ 2[ψ(n + 1) − ψ(1)] +

n − 1n

+8(1 − ln 2)

3

− 2M2 ln m

mr− m2 ln M

mr

M2 − m2

]

δl0 −1 − δl0

l(l + 1)(2l + 1)

}

m

M

(Zα)5mπn3

(mr

m

)3

.

(4.8)

Let us emphasize that the total contribution of the double transverse ex-change is given by the matrix element of the two-photon exchanges between


the Schrodinger-Coulomb wave functions, and no kernels with higher numberof exchanges arise in this case, unlike the case of the main contribution to theLamb shift discussed in Subsubsect. 3.2.1.1 and the case of the transverse-Coulomb recoil contribution which we will discuss next.

4.1.3 Transverse-Coulomb Term

One should expect that the contribution of the transverse-Coulomb diagramsin Fig. 4.5 would vanish in the scattering approximation because, in thisapproximation, there are no external vectors which are needed in order tocontract the transverse photon propagator. The only available vector in thescattering approximation is the exchanged momentum itself, which turns intozero after contraction with the transverse photon propagator. It is easy tocheck that this is just what happens, the electron and proton traces are pro-portional to the exchanged momentum ki in the scattering approximation andvanish, being dotted with the transverse photon propagator.

Fig. 4.5. Transverse-Coulomb two-photon exchanges

This does not mean, however, that the diagrams with one transverse ex-change do not contribute to the energy shift. We still have to explore if anycontributions could be generated by the exchange of the transverse photon,with a small momentum between mr(Zα) and mr(Zα)2, when one clearlycannot neglect the momenta of the external wave functions which are of thesame order of magnitude. Hence, we have to consider all kernels in Fig. 4.2with a transverse exchanged photon spanning an arbitrary number of Coulombexchanges. As we have already discussed in the beginning of this section onemight expect that when the momentum of the transverse photon is smallerthan the characteristic atomic momentum mrZα (in other words when thewavelength of the transverse quantum is larger than the size of the atom) thecontribution to the Lamb shift generated by such a photon would only differby an additional factor m/M from the leading contribution to the Lamb shiftof order α(Zα)4m, simply because such a photon cannot tell the electron fromthe proton. The extra factor m/M is due simply to the smaller velocity of theheavy particle in the atom (we remind the reader that the transverse photoninteraction vertex with a charged particle in the nonrelativistic approxima-tion is proportional to the velocity of the particle). Old-fashioned perturbationtheory is more suitable for exploration of such small intermediate momentacontributions. Correction due to the exchange of the transverse photon isdescribed in this framework simply as a second order perturbation theory


contribution where the role of the perturbation potential plays the transversephoton emission (absorption) vertex. In this framework the Coulomb poten-tial plays the role of the unperturbed potential, so the simple second ordercontribution which we just described takes into account all kernels of therelativistic two-body equation in Fig. 4.2 with any number of the Coulombexchanges spanned by the transverse photon. Summation over intermediatestates in the nonrelativistic perturbation theory in our case means integrationover all intermediate momenta. It is clear that for momenta larger than thecharacteristic atomic momentum mrZα integration over external wave func-tion momenta decouples (and we obtain instead of the wave functions theirvalue at the origin) and one may forget about the binding energies in the inter-mediate states. Then the contribution of this high (larger than mrZα) regionof momenta reduces to the matrix element of the Breit interaction (transversequanta exchange). As we have explained above, this matrix element does notgive any contribution to the Lamb shift (but it gives the main contribution tohyperfine splitting, see below). All this means that the total recoil correctionof order (Zα)5(m/M)m may be calculated in the nonrelativistic approxima-tion. Calculations go exactly in the same way as calculation of the leading lowenergy contribution to the Lamb shift in Subsubsect. 3.2.1.1. Due to validityof the nonrelativistic approximation the matrix elements of the Dirac matricescorresponding to the emission of transverse quanta by the constituent parti-cles reduce to the velocities αi → pi/mi, where pi and mi are the momentaof the constituents (of the same magnitude and with opposite directions inthe center of mass system) and their masses. Note that the recoil factor arisessimply as a result of kinematics. Again, as in the case of the main contributionto the Lamb shift in Subsubsect. 3.2.1.1 one introduces an auxiliary parame-ter σ (mr(Zα)2 � σ � mr(Zα)/n) in order to facilitate further calculations.Then the low-frequency contribution coincides up to an extra factor 2Zm/M(extra factor 2 arises due to two ways for emission of the transverse quanta,by the first and the second constituents), with the respective low-frequencycontribution of order α(Zα)4m, and we may simply borrow the result fromthat calculation (compare (3.7))

∆E< =83

[

ln2σ

mr(Zα)2δl0 − ln k0(n, l)

]m

M

(Zα)5mπn3

(mr

m

)3

. (4.9)

In the region where k ≥ σ we may safely neglect the binding energies in thedenominators of the second-order perturbation theory and thus simplify theintegrand. After integration one obtains

∆E> =83

[

lnmr(Zα)

nσ+ [ψ(n + 1) − ψ(1)] − 1

2n+

56

+ ln 2

− 1 − δl0

2l(l + 1)(2l + 1)

]m

M

(Zα)5mπn3

(mr

m

)3

δl0 . (4.10)

Let us emphasize once more that, as was discussed above, the “high fre-quencies” in this formula are effectively cut at the characteristic bound state


momenta mr(Zα)/n. This leads to two specific features of the formula above.First, this expression contains the characteristic logarithm of the principalquantum number n, and, second, the logarithm of the recoil factor (1+m/M)is missing, unlike the case of the nonrecoil correction of order α(Zα)4m in(3.7). The source of this difference is easy to realize. In the nonrecoil case theeffective upper cutoff was supplied by the electron mass m, while at low fre-quency only the reduced mass enters all expressions. This mismatch betweenmasses leads to appearance of the logarithm of the recoil factor. In the presentcase, the effective upper cutoff mr(Zα)/n also depends only on the reducedmass, and, hence, an extra factor under the logarithm does not arise.

Matching both contributions we obtain [4, 5, 7, 6, 8]

∆E =83

{[

ln2

nZα+ [ψ(n + 1) − ψ(1)] − 1

2n+

56

+ ln 2]

δl0 − ln k0(n, l)

− 1 − δl0

2l(l + 1)(2l + 1)

}m

M

(Zα)5mπn3

(mr

m

)3

. (4.11)

The total recoil correction of order (Zα)5(m/M)m is given by the sum of theexpressions in (4.4), (4.8), and (4.11)

∆E ={[

23

ln(Zα)−1 +143

(

ln2n

+ ψ(n + 1) − ψ(1) +2n − 1

2n

)

− 19

− 2M2 ln m

mr− m2 ln M

mr

M2 − m2

]

δl0 −83

ln k0(n, l) − 7(1 − δl0)3l(l + 1)(2l + 1)

}

× m

M

(Zα)5mπn3

(mr

m

)3

. (4.12)

4.2 Recoil Corrections of Order (Zα)6(m/M)m

4.2.1 The Braun Formula

Calculation of the recoil corrections of order (Zα)6(m/M)m requires consid-eration of the kernels with three exchanged photons. As in the case of recoilcorrections of order (Zα)5 low exchange momenta produce nonvanishing con-tributions, external wave functions do not decouple, and exact calculations inthe direct diagrammatic framework are rather tedious and cumbersome. Thereis a long and complicated history of theoretical investigation on this correction.The program of diagrammatic calculation was started in [9, 10, 11]. Correc-tions obtained in these works contained logarithm Zα as well as a constantterm. However, completely independent calculations [12, 13] of both recoiland nonrecoil logarithmic contributions of order (Zα)6 showed that somewhatmiraculously all logarithmic terms cancel in the final result. This observationrequired complete reconsideration of the whole problem. The breakthrough


was achieved in [14], where one and the same result was obtained in twoapparently different frameworks. The first, more traditional approach, usedearlier in [15, 9, 10, 11], starts with an effective Dirac equation in the externalfield. Corrections to the Dirac energy levels are calculated with the help ofa systematic diagrammatic procedure. The other logically independent calcu-lational framework, also used in [14], starts with an exact expression for allrecoil corrections of the first order in the mass ratio of the light and heavyparticles m/M . This remarkable expression, which is exact in Zα, was firstdiscovered by M. A. Braun [16], and rederived and refined later in a numberof papers [17, 18, 14]. A particularly transparent representation of the Braunformula was obtained in [18]

∆Erec = − 1M

Re

∫dωd3k

(2π)4i〈n|(p−D(ω,k))G(E+ω)(p−D(ω,k))|n〉 , (4.13)

where G(E + ω) is the total Green function of the Dirac electron in the ex-ternal Coulomb field, and D(ω,k) is the transverse photon propagator in theCoulomb gauge

D(ω,k) = −4πZααk

ω2 − k2 + i0, (4.14)

and

αk = α − k(α · k)k2 , α = γ0γ . (4.15)

Before returning to the recoil corrections of order (Zα)6 we will digressto the Braun formula. We will not give a detailed derivation of this formula,referring the reader instead to the original derivations [16, 17, 18, 14]. We willhowever present below some physically transparent semiquantative argumentswhich make the existence and even the exact appearance of the Braun formulavery natural.

Let us return to the original Bethe-Salpeter equation (see (1.7)). As wehave already discussed there are many ways to organize the Feynman dia-grams which comprise the kernel of this equation. However, in all commonperturbation theory considerations of this kernel the main emphasis is onpresenting the kernel in an approximate form sufficient for calculation of cor-rections to the energy levels of a definite order in the coupling constant α.The revolutionary idea first suggested in [19] and elaborated in [16], was toreject such an approach completely, and instead to organize the perturbationtheory with respect to another small parameter, namely, the mass ratio of thelight and heavy particles. To this end an expansion of the heavy particle prop-agator over 1/M was considered in [16]. It is well known (see, e.g., [16]) thatin the leading order of this expansion the Bethe-Salpeter equation reduces tothe Dirac equation for the light particle in the external Coulomb field cre-ated by the heavy particle. This is by itself nontrivial, but well understood bynow, since to restore the Dirac equation in the external field one has to takeinto account irreducible kernels of the Bethe-Salpeter equation with arbitrary


Fig. 4.6. Irreducible kernels with crossed exchange photon lines

number of crossed exchange photon lines (see Fig. 4.6). Unlike the solutionsof the effective Dirac equation, considered above in Sect. 1.3, the solutionsof the Dirac equation obtained in this way contain as a mass parameter themass of the light particle, and not the reduced mass of the system. This is theprice one has to pay in the Braun approach for summation of all correctionsin the expansion over Zα. The zero-order Green function in this approach issimply the Coulomb-Dirac Green function. The next step in the derivationof the Braun formula is to consider all kernels of the Bethe-Salpeter equa-tion which produce corrections of order m/M . The crucial observation whichimmediately leads to the closed expression for the recoil corrections of orderm/M , is that all corrections linear in the mass ratio are generated by the ker-nels where all but one of the heavy particle propagators are replaced by theleading terms in their large mass expansion, and this remaining propagator isreplaced by the next term in the large mass expansion of the heavy particlepropagator. Respective kernels with the minimum number of exchanged pho-tons are the box and the crossed box diagrams in Fig. 4.1 where the heavyparticle propagator is replaced by the second term in its large mass expan-sion. All diagrams obtained from these two by insertions of any number ofexternal Coulomb photons between the two exchanges in Fig. 4.1 and/or ofany number of the radiative photons in the electron line also generate linearin the mass ratio corrections. It is not difficult to figure out that these arethe only kernels which produce corrections linear in the small mass ratio, allother kernels generate corrections of higher order in m/M , and, hence, arenot interesting in this context.

Then the linear in the small mass ratio contribution to the energy shift isequal to the matrix element of the two graphs in Fig. 4.1 with the total elec-tron Green function in the external Coulomb field instead of the upper electronline. This matrix element which should be calculated between the unperturbedDirac-Coulomb wave functions reduces after simple algebraic transformationsto the Braun formula in (4.13). All terms in the Braun formula have a trans-parent physical sense. The term containing product pp (first obtained in [19])originates from the exchange of two Coulomb photons, the terms with pDand Dp correspond to the exchange of Coulomb and magnetic (transverse)quanta, and the term DD is connected with the double transverse exchange.

Another useful perspective on the Braun formula is provided by the idea,first suggested in the original work [16], and later used as a tool to rederive(4.13) in [18, 14], that the recoil corrections linear in the small mass ratiom/M are associated with the matrix element of the nonrelativistic proton


Hamiltonian

H =(p − eA)2

2M. (4.16)

There is a clear one-to one correspondence between the terms in this non-relativistic Hamiltonian and the respective terms in (4.13). The latter couldbe obtained as matrix elements of the operators which enter the Hamiltonianin (4.16) [18, 14].

4.2.2 Lower Order Recoil Corrections and the Braun Formula

Being exact in the parameter Zα and an expansion in the mass ratiom/M the Braun formula in (4.13) should reproduce with linear accuracy inthe small mass ratio all purely recoil corrections of orders (Zα)2(m/M)m,(Zα)4(m/M)m, (Zα)5(m/M)m in (3.5) which were discussed above.

Corrections of lower orders in Zα are generated by the simplified Coulomb-Coulomb and Coulomb-transverse entries in (4.13). The main part of theCoulomb-Coulomb contribution in Eq. (4.13) may be written in the form

∆E(1)Coul =

12M

〈n|p2|n〉 , (4.17)

while the Breit (nonretarded) part of the magnetic contribution has the form

∆EBr = − 12M

〈n|p · D(0, k) + D(0, k) · p|n〉 . (4.18)

Calculation of the matrix elements in (4.17) and (4.18) is greatly simplified bythe use of the virial relations (see, e.g., [20, 21]), and one obtains the sum ofthe contributions in (4.17) and (4.18) in a very nice form [17] (compare (3.5))

∆E(1)Coul + ∆EBr =

m2 − E2nj

2M=

{

− m

M[f(n, j) − 1] − m

2M[f(n, j) − 1]2

}

m ,

(4.19)where Enj and f(n, j) are defined in (1.4) and (1.5), respectively. This repre-sentation again emphasizes the simple physical idea behind the Braun formulathat the recoil corrections of the first order in the small mass ratio m/M aregiven by the matrix elements of the heavy particle kinetic energy.

The recoil correction in (4.19) is the leading order (Zα)4 relativistic con-tribution to the energy levels generated by the Braun formula. All other con-tributions to the energy levels produced by the remaining terms in the Braunformula start at least with the term of order (Zα)5 [17]. The expression in(4.19) exactly reproduces all contributions linear in the mass ratio in (3.5).This is just what should be expected since it is exactly Coulomb and Breit po-tentials which were taken in account in the construction of the effective Diracequation which produced (3.5). The exact mass dependence of the terms oforder (Zα)2(m/M)m and (Zα)4(m/M)m is contained in (3.5), and, hence,


terms linear in the mass ratio in (4.19) give nothing new. It is important torealize at this stage that the contributions of order (Zα)6(m/M)m in (3.5)and (4.19) coincide as well, so any corrections of this order obtained with thehelp of the entries in the Braun formula not taken care of in (4.17) and (4.18),should be added to the order (Zα)6(m/M)m contribution in (3.5).

4.2.3 Recoil Correction of Order (Zα)6(m/M)m to the S Levels

Calculation of the recoil contribution of order (Zα)6(m/M)m to the nS statesgenerated by the Braun formula was first performed in [14]. Separation of thehigh- and low-frequency contributions was made with the help of the ε-method[22] described above in Subsubsect. 3.4.1.2. Hence, not only were contributionsof order (Zα)6(m/M)m obtained in [14], but also parts of recoil correctionsof order (Zα)5 linear in m/M , discussed in Sect. 4.1, were reproduced. Theolder methods of Sect. 4.1 produce the recoil corrections of order (Zα)5 witha correct nonrelativistic dependence on the reduced mass. The Braun formulain principle cannot reproduce this reduced mass dependence because it isconstructed as a first term in a rigorous expansion over the light-heavy massratio. However, the results for the recoil corrections of order (Zα)5 obtainedin [14] served as an important check of self-consistency of calculations. On theother hand calculations in [14], while in principle very straightforward, turnedout to be rather lengthy just because all corrections of previous orders in Zαwere recovered.

The agreement on the magnitude of (Zα)6(m/M)m contribution for the1S and 2S states obtained in the diagrammatic approach and in the frame-work of the Braun formula achieved in [14] seemed to put an end to all prob-lems connected with this correction. However, it was claimed in a later work[23], that the result of [14] is in error. The discrepancy between the resultsof [14, 23] was even more confusing since the calculation in [23] was alsoperformed with the help of the Braun formula. It was observed in [23] thatdue to the absence of the logarithmic contributions of order (Zα)6(m/M)mproved earlier in [13], the calculations may be organized in a more compactway than in [14]. The main idea of [23] is that it is possible to make someapproximations which are inadequate for calculation of the contributions ofthe previous orders in Zα, significantly simplifying calculation of the correc-tion of order (Zα)6(m/M)m. Due to absence of the logarithmic contributionsof order (Zα)6(m/M)m proved in [13], infrared divergences connected withthese crude approximations would be powerlike and can be safely thrownaway. Next, absence of logarithmic corrections of order (Zα)6(m/M)m meansthat it is not necessary to worry too much about matching the low- and high-frequency (long- and short-distance in terms of [23]) contributions, since eachregion will produce only nonlogarithmic contributions and correction termswould be suppressed as powers of the separation parameter. Of course, suchan approach would be doomed if the logarithmic divergences were present,since in such a case one could not hope to calculate an additive constant to


the logarithm, since the exact value of the integration cutoff would not beknown.

Despite all these nice features of the approach of [23] its result (as well asa corrected result of a later paper [24]) contradicted the result in [14]. Thediscrepancy was resolved in [25], where a new logically independent calculationof the recoil corrections of order (Zα)6(m/M)m was performed. A subtleerror in dealing with cutoffs in [23] was discovered and the result of [14] wasconfirmed. The recoil correction of order (Zα)6(m/M)m for S states witharbitrary principal quantum number n beyond that which is already containedin (3.5) has the form [14, 25]

∆Etot =(

4 ln 2 − 72

)(Zα)6

n3

m

Mm . (4.20)

This result was additionally confirmed in [26] where the recoil correction ofthe first order in the mass ratio was calculated without expansion over Zα for1S and 2S states in hydrogen.

4.2.4 Higher Order in Mass Ratio Recoil Correctionof Order (Zα)6(m/M)nm to the S Levels

Corrections of order (Zα)6(m/M)nm, n ≥ 2, are generated by the diagramsin Fig. 4.6. As in the case of recoil corrections of relative order (Zα)5 bothhigh frequency and low frequency integration regions produce nonzero con-tributions for n ≥ 2. Quadratic in mass ratio high frequency contribution(together with higher order terms in the mass ratio expansion) was obtainedin [35] in the NRQED framework

∆E =[

−(

8π2

3− 4

)

lnM

m− 12ζ(3) +

8π2

3+ 3

](Zα)6

π2n3

m

M

m3r

mM. (4.21)

The respective low frequency contribution was never calculated, so one canuse the result in (4.21) only to get an idea of the scale of the correction of order(Zα)6(m/M)nm. Numerically the contribution in (4.21) is about −0.09 kHzfor the 1S level in hydrogen, and calculation of the respective low frequencypart is warranted.

4.2.5 Recoil Correction of Order (Zα)6(m/M)mto the Non-S Levels

Recoil corrections of order (Zα)6(m/M)m to the energy levels with non-vanishing orbital angular momentum may also be calculated with the helpof the Braun formula [27]. We would prefer to discuss briefly another ap-proach, which was used in the first calculation of the recoil corrections oforder (Zα)6(m/M)m to the P levels [30]. The idea of this approach (see, e.g.,

4.3 Recoil Correction of Order (Zα)7(m/M) 95

review in [12]) is to extend the standard Breit interaction Hamiltonian (see,e.g., [28]) which takes into account relativistic corrections of order v2/c2 tothe next order in the nonrelativistic expansion, and also take into accountthe corrections of order v4/c4. Contrary to the common wisdom, such an ap-proach turns out to be quite feasible and effective, and it was worked out ina number of papers [29, 12], and references therein. This nonrelativistic ap-proach is limited to the calculation of the large distance (small intermediatemomenta) contributions since any short distance correction leads effectively toan ultraviolet divergence in this framework. Powerlike ultraviolet divergencesdemonstrate the presence of the corrections of the lower order in Zα (in con-trast to the scattering approximation where the presence of such correctionsreveals itself in the form of powerlike infrared divergences), and are not undercontrol in this approach. However, the logarithmic ultraviolet divergences arewell under control and produce logarithms of the fine structure constant. Anumber of logarithmic contributions to the energy levels and decay widthswere calculated in this approach [29, 12].

In the case of states with nonvanishing angular momenta the small dis-tance contributions are effectively suppressed by the vanishing of the wavefunction at the origin, and the perturbation theory becomes convergent in thenonrelativistic region. Then this nonrelativistic approach leads to an exactresult for the recoil correction of order (Zα)6(m/M)m for the P states [30]

∆E(nP ) =25

(

1 − 23n2

)(Zα)6

n3

m

Mm . (4.22)

Again, this expression contains only corrections not taken into account in(3.5). The approach of [30] may be generalized for calculation of the recoilcorrections to the energy levels with even higher orbital angular momenta.

The general expression for the recoil corrections of order (Zα)6(m/M)mto the energy level with an arbitrary nonvanishing angular momentum wasobtained in [27]

∆E(nL) =3

4(l − 12 )(l + 1

2 )(l + 32 )

(

1 − l(l + 1)3n2

)(Zα)6

n3

m

Mm . (4.23)

4.3 Recoil Correction of Order (Zα)7(m/M)

Recoil corrections of order (Zα)7(m/M) are connected only with the diagramswith multiphoton exchanges similar to the diagrams in Figs. 4.1–4.5. Follow-ing the approach developed in [1] one can classify all these diagrams andextract their leading logarithmic terms in the same way as for the corrections(Zα)6(m/M)m to the P levels above.

The leading logarithm squared contribution to the recoil correction of order(Zα)7(m/M) was independently obtained with the help of these methodsin [32, 31]


Table 4.1. Recoil Corrections to Lamb Shift

(Zα)5

πn3m3

rmM

∆E(1S) ∆E(2S)

kHz kHz

Coulomb-Coulomb Term

Salpeter (1952) [4] − 43δl0 −590.03 −73.75

Fulton, Martin (1954) [5]

Transverse-Transverse Term

Bulk Contribution

Salpeter (1952) [4]

{

2 ln 2Zαn

+ 2[ψ(n + 1) − ψ(1)]

Fulton, Martin (1954) [5] +n−1n

+8(1−ln 2)

3

}

δl0 −2 494.01 −305.46

Erickson, Yennie (1965) [7] − 1−δl0l(l+1)(2l+1)

Grotch, Yennie (1969) [6]

Erickson (1977) [8]Erickson, Grotch (1988) [9]

Transverse-Transverse Term

Very high momentum

Contribution − 2M2−m2

(

M2 ln mmr

− −0.48 −0.06

Fulton, Martin (1954) [5]

Erickson (1977) [8] m2 ln Mmr

)

δl0

Coulomb-Transverse Term

Salpeter (1952) [4] 83

{[

ln 2nZα

+ [ψ(n + 1) − ψ(1)]

Fulton, Martin (1954) [5] − 12n

+ 56

+ ln 2

]

δl0

Erickson, Yennie (1965) [7] − ln k0(n, l) − 1−δl02l(l+1)(2l+1)

}

5 494.03 720.56

Grotch, Yennie (1969) [6]Erickson (1977) [8]Erickson, Grotch (1988) [9]

∆E(nS)

Pachucki, Grotch (1993) [14] (4 ln 2 − 72)(πZα)δl0 −7.38 −0.92

Eides, Grotch (1997) [25]

∆E(nS), second orderin mass ratio, high frequency

Blokland, Czarnecki,

[

−(

8π2

3− 4

)

ln Mm

− 12ζ(3)

Melnikov (2002) [35] + 8π2

3+ 3

]Zαπ

mM

δl0 −0.09 −0.01

∆E(nL)(l �= 0)

Golosov, Elkhovski, 34(l− 1

2 )(l+ 12 )(l+ 3

2 )

Milshtein, Khriplovich (1995) [30] ×(

1 − l(l+1)

3n2

)

(πZα)

Jentschura, Pachucki (1996) [27]

∆E(nS)

Pachucki, Karshenboim (1999) [32] − 1115

(Zα)2 ln2(Zα)δl0 −0.42 −0.05

Melnikov, Yelkhovsky (1999) [31]

4.3 Recoil Correction of Order (Zα)7(m/M) 97

∆E = −1115

(Zα)7mπn3

ln2(Zα)m

M

(mr

m

)3

δl0 . (4.24)

Note that this expression is symmetric with respect to the masses of theconstituents, this result was obtained without expansion in the small massratio, and it is valid for arbitrary mass ratio.

Numerically the contribution in (4.24) is below 1 kHz. Due to linear depen-dence of the recoil correction on the electron-nucleus mass ratio, the respectivecontribution to the hydrogen-deuterium isotope shift (see Subsect. 12.1.7 be-low) is phenomenologically much more important, it is larger than the exper-imental uncertainty, and should be taken into account in comparison betweentheory and experiment at the current level of experimental uncertainty.

Further progress was also achieved in numerical calculations of higher orderrecoil corrections without expansion over Zα in the framework of the approachbased on (4.13). To describe these results we following [33, 34] (compare (3.98)and (3.102)) write all recoil corrections of order (Zα)6 and higher in theform (note absence of the characteristic factor (mr/m)3 which cannot bereproduced in the Braun formula framework)

∆E = PSE(Zα)(Zα)5

πn3

( m

M

)

m , (4.25)

where PSE(Zα) = A60 Zα + A72 ln2(Zα)(Zα)2 + · · ·, and the coefficients A60

and A72 are known analytically and are defined by the expressions in (3.5),(4.20), (4.22), and (4.24). The function PSE(Zα) for the lowest energy levels inhydrogenlike atoms with Z = 1 ∼ 100 was calculated numerically in [33, 34].The results for hydrogen are as follows

PSE

(1S 1

2

)= −0.016 16 (3), PSE

(2S 1

2

)= −0.016 17 (5) ,

PSE

(2P 1

2

)= 0.007 72 . (4.26)

The difference between the contribution to the energy shift calculated us-ing the numerical result in (4.26) and the sum of analytic results in Table 4.1may be considered as an estimate of the single logarithmic and nonlogarith-mic recoil corrections of order (Zα)7. This difference is equal 0.64(1) kHz forthe 1S level in hydrogen, and exceeds total logarithm squared contribution.Hence, the situation with the recoil corrections of order (Zα)7 resembles thecase with the nonrecoil corrections of order α2(Zα)6. As in the case of thenonrecoil corrections the numerical result in (4.26) indicates that the nonlead-ing corrections of order (Zα)7 are larger than the leading logarithm squaredterm. Clearly, an analytic calculation of the nonleading contributions of order(Zα)7 is warranted. Meanwhile, it is preferable to use apparently more pre-cise numerical results from [33, 34] instead of analytic results from (4.20) and(4.24) for comparison of the experimental data on hydrogen with theory.

This concludes our discussion of purely recoil corrections. We collected allanalytic results for these corrections in Table 4.1.


References

1. I. B. Khriplovich, A. I. Milstein, and A. S. Yelkhovsky, Physica Scripta, T46,252 (1993).

2. T. Welton, Phys. Rev. 74, 1157 (1948).3. M. I. Eides, S. G. Karshenboim, and V. A. Shelyuto, Ann. Phys. (NY) 205,

231, 291 (1991).4. E. E. Salpeter, Phys. Rev. 87, 553 (1952).5. T. Fulton and P. C. Martin, Phys. Rev. 95, 811 (1954).6. H. Grotch and D. R. Yennie, Rev. Mod. Phys. 41, 350 (1969).7. G. W. Erickson and D. R. Yennie, Ann. Phys. (NY) 35, 271 (1965).8. G. W. Erickson, J. Phys. Chem. Ref. Data 6, 831 (1977).9. G. W. Erickson and H. Grotch, Phys. Rev. Lett. 25, 2611 (1988); 63, 1326(E)

(1989).10. M. Doncheski, H. Grotch and D. A. Owen, Phys. Rev. A 41, 2851 (1990).11. M. Doncheski, H. Grotch and G. W. Erickson, Phys. Rev. A 43, 2152 (1991).12. I. B. Khriplovich, A. I. Milstein, and A. S. Yelkhovsky, Phys. Scr. T 46, 252

(1993).13. R. N. Fell, I. B. Khriplovich, A. I. Milstein and A. S. Yelkohovsky, Phys. Lett.

A 181, 172 (1993).14. K. Pachucki and H. Grotch, Phys. Rev. A 51, 1854 (1995).15. M. I. Eides and H. Grotch, Phys. Lett. B 301, 127 (1993).16. M. A. Braun, Zh. Eksp. Teor. Fiz. 64, 413 (1973) [Sov. Phys.-JETP 37, 211

(1973)].17. V. M. Shabaev, Teor. Mat. Fiz. 63, 394 (1985) [Theor. Math. Phys. 63, 588

(1985)].18. A. S. Yelkhovsky, preprint Budker INP 94-27, hep-th/9403095 (1994).19. L. N. Labzowsky, Proceedings of the XVII All-Union Congress on Spectroscopy,

Moscow, 1972, Part 2, p. 89.20. J. H. Epstein and S. T. Epstein, Am. J. Phys. 30, 266 (1962).21. V. M. Shabaev, J.Phys. B: At. Mol. Opt. Phys B24, 4479 (1991).22. K. Pachucki, Ann. Phys. (NY) 236, 1 (1993).23. A. S. Elkhovskii, Zh. Eksp. Teor. Fiz. 110, 431 (1996); JETP 83, 230 (1996).24. A. S. Elkhovskii, Zh. Eksp. Teor. Fiz. 113, 865 (1998) [JETP 86, 472 (1998)].25. M. I. Eides and H. Grotch, Phys. Rev. A 55, 3351 (1997).26. V.M. Shabaev, A.N.Artemyev, T. Beier et al, Phys. Rev. A 57, 4235 (1998);

V.M. Shabaev, A.N.Artemyev, T. Beier, and G. Soff, J. Phys. B: At. Mol. Opt.Phys B31, L337 (1998).

27. U. Jentschura and K. Pachucki, Phys. Rev. A 54, 1853 (1996).28. V. B. Berestetskii, E. M. Lifshitz, and L. P. Pitaevskii, Quantum electrodynam-

ics, 2nd Edition, Pergamon Press, Oxford, 1982.29. I. B. Khriplovich and A. S. Yelkhovsky, Phys. Lett. B 246, 520 (1990).30. E. A. Golosov, A. S. Elkhovskii, A. I. Milshtein, and I. B. Khriplovich, Zh. Eksp.

Teor. Fiz. 107, 393 (1995) [JETP 80, 208 (1995)].31. K. Melnikov and A. S. Yelkhovsky, Phys. Lett. B 458, 143 (1999).32. K. Pachucki and S. G. Karshenboim, Phys. Rev. A 60, 2792 (1999).33. V. M. Shabaev, A. N. Artemyev, T. Beier et al, Phys. Rev. A 57, 4235 (1998).34. V. M. Shabaev, in Lecture Notes in Physics, v.570, ed. S. G. Karshenboim et

al, Springer-Verlag, Berlin, Heidelberg, 2001, p.714.35. I. Blokland, A. Czarnecki, and K. Melnikov, Phys. Rev. D 65, 073015 (2002).

5

Radiative-Recoil Corrections

In the standard nomenclature the name radiative-recoil is reserved for therecoil corrections to pure radiative effects, i.e., for corrections of the formαm(Zα)n(m/M)k.

Let us start systematic discussion of such corrections with the recoil cor-rections to the leading contribution to the Lamb shift. The most important ob-servation here is that the mass dependence of all corrections of order αm(Zα)4

obtained above is exact, as was proved in [1, 2], and there is no additionalmass dependence beyond the one already present in (3.7)–(3.24). This con-clusion resembles the similar conclusion about the exact mass dependence ofthe contributions to the energy levels of order (Zα)4m discussed above, andit is valid essentially for the same reason. The high frequency part of thesecorrections is generated only by the one photon exchanges, for which we knowthe exact mass dependence, and the only mass scale in the low frequency part,which depends also on multiphoton exchanges, is the reduced mass.

5.1 Corrections of Order α(Zα)5(m/M)m

The first nontrivial radiative-recoil correction is of order α(Zα)5. We havealready discussed the nonrecoil contribution of this order in Subsect. 3.3.2.Due to the wave function squared factor this correction naturally containedan explicit factor (mr/m)3. Below we will discuss radiative-recoil correctionsof order α(Zα)5 with mass ratio dependence beyond this factor (mr/m)3.

5.1.1 Corrections Generated by the Radiative Insertionsin the Electron Line

Over the years different methods were applied for calculation of the radiative-recoil correction of order α(Zα)5. It was first considered in the diagrammaticapproach [1, 3, 2]. Later it was reconsidered on the basis of the Braun for-mula [4]. The Braun formula depends on the total electron Green function


100 5 Radiative-Recoil Corrections

in the external Coulomb field and automatically includes all radiative correc-tions to the electron line. Only one-loop insertions of the radiative photon inthe electron line should be preserved in order to obtain corrections of orderα(Zα)5. At first sight calculation of these corrections with the help of theBraun formula may seem to be overcomplicated because, as we already men-tioned above, the Braun formula produces a total correction of the first orderin the mass ratio. Hence, exact calculation of the radiative-recoil correction oforder α(Zα)5 with its help would produce not only the contributions we calledradiative-recoil above, but also the first term in the expansion over the massratio of the purely radiative contribution in (3.34). This contribution shouldbe omitted in order to avoid double counting. It is not difficult to organizethe calculations based on the Braun formula in such a way that the reducedmass correction connected with the nonrecoil contribution would not show upand calculation of the remaining terms would be significantly simplified [4].The idea is as follows. Purely radiative corrections of order α(Zα)5, togetherwith the standard (mr/m)3 factor were connected with the nonrelativisticheavy particle pole in the two photon exchange diagrams which correspondsto the zero order term in the proton propagator expansion over 1/M . On theother hand, the Braun formula explicitly picks up the first order term in theproton propagator heavy mass expansion. This means that the Braun formulaproduces the term corresponding to the reduced mass dependence of the non-recoil contribution only when the high integration momentum goes throughone of the external wave functions. New radiative-recoil contributions, whichdo not reduce to the tail of the mass ratio cubed factor in (3.34) are generatedonly by the integration region where the high momentum goes through theloop described by an explicit operator in the Braun formula. For calculationof the matrix element in this regime, it is sufficient to ignore external virtu-alities and to approximate the external wave functions by their values at theorigin. The respective calculation reduces therefore to a variant of the scatter-ing approximation calculation, the only difference being that the form of theskeleton structure is now determined by the Braun formula. As usual, in thescattering approximation approach the integral under consideration containspowerlike infrared divergence, which corresponds to the recoil contribution ofthe previous order in Zα and should simply be subtracted. After an explicitcalculation of the Braun formula contribution one obtains [4]

∆E ≈ −1.324α(Zα)5

n3

m

Mm δl0 . (5.1)

This result contradicts the result of earlier calculations in the diagrammaticapproach [1, 3, 2].

Later two groups of authors [5, 6, 7] realized that the radiative-recoil cor-rection of order α(Zα)5 is generated by the high integration momenta regionin the two-photon exchange kernels. Hence, there is no need to use the Braunformula for its calculation, instead it is sufficient to calculate the contribu-tion of the Feynman diagrams in Fig. 5.1 in the scattering approximation.

5.1 Corrections of Order α(Zα)5(m/M)m 101

Fig. 5.1. Electron-line radiative-recoil corrections

Technically calculations of the diagrams in these two papers were organizedin completely different ways, but both groups obtained one and the sameanalytic expression for the radiative-recoil correction of order α(Zα)5 [5, 6, 7]

∆E =(

6ζ(3) − 2π2 ln 2 +3π2

4− 14

)α(Zα)5

π2n3

m

M

(mr

m

)3

m δl0 , (5.2)

in an excellent agreement with the earlier numerical result in (5.1) [4].The method of direct analysis of the integration regions applied to the

bound state problem in [5, 6] allowed these authors also to obtain quadraticin mass ratio radiative-recoil corrections of order α(Zα)5

∆E =(

8 ln 2 − 12732

)α(Zα)5

n3

( m

M

)2 (mr

m

)3

m δl0 , (5.3)

as well as corrections of even higher orders in the small mass ratio. Due to thesmall magnitude of electron-proton mass ratio these higher order correctionsare not phenomenologically relevant in the case of hydrogen, but could beimportant for other quantum electrodynamic bound systems with larger massratio of the constituents, like muonic hydrogen and muonium.

5.1.2 Corrections Generated by the Polarization Insertionsin the Photon Lines

Calculation of the radiative-recoil correction generated by the one-loop polar-ization insertions in the exchanged photon lines in Fig. 5.2 follows the samepath as calculation of the correction induced by the insertions in the electronline. The respective correction was independently calculated analytically bothin the skeleton integral approach [8] and with the help of the Braun formula[4].

Due to the simplicity of the photon polarization operator the calculationbased on the scattering approximation [8] is so straightforward that we can

Fig. 5.2. Photon-line radiative-recoil corrections


present here all relevant formulae without making the text too technical. Theskeleton integral for the recoil corrections corresponding to the diagrams withtwo exchanged photons in Fig. 4.1 has the form

∆E =16(Zα)2|ψ(0)|2

m2(1 − µ2)

∫ ∞

0

kdk

(k2 + λ2)2

{

µ

√

1 +k2

4

(1k

+k3

8

)

(5.4)

−√

1 +µ2k2

4

(1k

+µ4k3

8

)

− µk2

8

(

1 +k2

2

)

+µ3k2

8

(

1 +µ2k2

2

)

+1k

}

,

where µ = m/M .We have already subtracted in (5.4) the nonrecoil part of the skeleton

integral. This subtraction term is given by the nonrelativistic heavy particlepole contribution (3.33) in the two photon exchange. Next, we insert thepolarization operator in the integrand in (5.4) according to the rule in (3.35).The skeleton integrand in (5.4) behaves as µ/k4 at small momenta and naivesubstitution in (3.35) leads to a linear infrared divergence. This divergence∫

dk/k2 would be cut off at the atomic scale 1/(mZα) by the wave functionmomenta in an exact calculation. The low-momentum contribution wouldclearly be of order α(Zα)4 and we may simply omit it since we already knowthis correction. Thus, to obtain the recoil correction of order α(Zα)5m it issufficient to subtract the leading low-frequency asymptote in the radiativelycorrected skeleton integrand. The subtracted integral for the radiative-recoilcorrection (the integral in (5.4) with inserted polarization operator and thelow-frequency asymptote subtracted) has to be multiplied by an additionalfactor 2 needed in order to take into account that the polarization may beinserted in each of the two photon lines in the skeleton diagrams in Fig. 4.1.It can easily be calculated analytically if one neglects contributions of higherorder in the small mass ratio [8]

∆E =(

2π2

9− 70

27

)α(Zα)5

π2n3

m

M

(mr

m

)3

m δl0 . (5.5)

Calculation of the same contribution with the help of the Braun formulawas made in [4]. In the Braun formula approach one also makes the substitu-tion in (3.35) in the propagators of the exchange photons, factorizes externalwave functions as was explained above (see Subsect. 5.1.1), subtracts the in-frared divergent part of the integral corresponding to the correction of previousorder in Zα, and then calculates the integral. The result of this calculation[4] nicely coincides with the one in (5.5)1.

1 The radiative-recoil correction to the Lamb shift induced by the polarizationinsertions in the exchanged photons was also calculated in [9]. The result of thatwork contradicts the results in [8, 4]. The calculations in [9] are made in the sameway as the calculation of the recoil correction of order (Zα)6(m/M)m in [10], andlead to a wrong result for the same reason.


5.1.3 Corrections Generated by the Radiative Insertionsin the Proton Line

We have discussed above insertions of radiative corrections either in the elec-tron line or in the exchanged photon line in the skeleton diagrams in Fig. 4.1.One more option, namely, insertions of a radiative photon in the heavy parti-cle line also should be considered. The leading order correction generated bysuch insertions is a radiative-recoil contribution of order (Z2α)(Zα)4(m/M)m.Note that this correction contains one less power of the parameter α thanthe first nontrivial radiative-recoil correction of order α(Zα)5(m/M)m gen-erated by the radiative insertions in the electron line considered above. Thereis nothing enigmatic about this apparent asymmetry, since the mass depen-dence of the leading order contribution to the Lamb shift of order α(Zα)4min (3.7) is known exactly, and thus the would-be radiative correction of or-der α(Zα)4(m/M)m is hidden in the leading order contribution to the Lambshift.

No new calculation is needed to obtain the correction of order (Z2α)(Zα)4

(m/M)m generated by the radiative insertions in the proton line. It is almostobvious that to order (Zα)4 the contributions to the energy level generatedby the radiative insertions in the fermion lines are symmetric with respect tointerchange of the light and heavy lines [11]. Then in the case of an elementaryproton radiative-recoil correction generated by the radiative photon insertionsin the heavy line may be obtained from the leading order contributions to theLamb shift in (3.7) and (3.11) by the substitutions m → M and α → (Z2α).Both substitutions are obvious, the first one arises because the leading termin the infrared expansion of the first order radiative corrections to the fermionvertex contains the mass of the particle under consideration, and the secondsimply reminds us that the charge of the heavy particle is Ze. Hence, theDirac form factor contribution is equal to

∆E ={[

13

lnM(Zα)−2

mr+

1172

]

δl0 −13

ln k0(n, l)}

4(Z2α)(Zα)4

πn3

m3r

M2,

(5.6)and the Pauli form factor leads to the correction

δE|l=0 =(Z2α)(Zα)4

2πn3

m3r

M2,

δE|l �=0 =j(j + 1) − l(l + 1) − 3/4

l(l + 1)(2l + 1)(Z2α)(Zα)4

2πn3

m2r

M. (5.7)

These formulae are derived for an elementary heavy particle, and do nottake into account the composite nature of the proton. The presence of thelogarithm of the heavy particle mass M in (5.6) indicates that the logarithmicloop integration in the form factor integral goes up to the mass of the particlewhere one could no longer ignore the composite nature of the proton. For thecomposite proton the integration would be cut from above not by the proton


mass but by the size of the proton. The usual way to account for the protonstructure is to substitute the proton form factor in the loop integral. Aftercalculation we obtain instead of (5.6)

∆E =

{[

13

lnΛ(Zα)−2

mr+

1172

+

(

− 124

− 7π

32Λ2

4M2+

23

(Λ2

4M2

)2

+ · · ·)]

δl0

− 13

ln k0(n, l)

}

4(Z2α)(Zα)4

πn3

m3r

M2, (5.8)

where Λ2 = 0.71 GeV2 is the parameter in the proton dipole form factor.As we have expected it replaces the proton mass in the role of the uppercutoff for the logarithmic loop integration. Note also that we have obtainedan additional constant in (5.8).

The anomalous magnetic moment contribution in (5.7) also would be mod-ified by inclusion of a nontrivial form factor, but the contribution to the protonmagnetic moment should be considered together with the nonelectromagneticcontributions to the proton magnetic moment. The anomalous magnetic mo-ment of the nucleus determined experimentally includes the electromagneticcontribution and, hence, even modified by the nontrivial form factor contri-bution in (5.7) should be ignored in the phenomenological analysis. Usuallythe total contribution of the proton anomalous magnetic moment is hiddenin the main proton charge radius contribution defined via the Sachs electricform factor2.

From the practical point of view, the difference between the results in (5.6)and (5.8) is about 0.18 kHz for 1S level in hydrogen and at the current level ofexperimental precision the distinctions between the expressions in (5.6) and(5.8) may be ignored in the discussion of the Lamb shift measurements. Thesedistinctions should, however, be taken into account in the discussion of thehydrogen-deuterium isotope shift (see below Subsect. 12.1.7).

An alternative treatment of the correction of order (Z2α)(Zα)4(m/M)mwas given in [4]. The idea of this work was to modify the standard definitionof the proton charge radius, and include the first order quantum electrody-namic radiative correction into the proton radius determined by the stronginteractions. From the practical point of view for the nS levels in hydrogen therecipe of [4] reduces to elimination of the constant 11/72 in (5.6) and omissionof the Pauli correction in (5.7). Numerically such a modification reduces thecontribution to the 1S energy level in hydrogen by 0.14 kHz in comparisonwith the naive result in (5.6), and increases it by 0.03 kHz in comparisonwith the result in (5.8). Hence, for all practical needs at the current level ofexperimental precision there are no contradictions between our result abovein (5.8), and the result in [4].

However, the approach of [4] from our point of view is unattractive; weprefer to stick with the standard definition of the intrinsic characteristics2 This topic will be discussed in more detail below in Subsect. 6.1.1.


of the proton as determined by the strong interactions. Of course, in suchan approach one has to extract the values of the proton parameters fromthe experimental data, properly taking into account quantum electrodynamiccorrections. Another advantage of the standard approach advocated above isthat in the case of the absence of a nontrivial nuclear form factor (as, forexample, in the muonium atom with an elementary nucleus) the formula in(5.8) reduces to the classical expression in (5.6).

5.2 Corrections of Order α(Zα)6(m/M)m

The leading logarithm squared radiative-recoil contribution of order α(Zα)6

(m/M) is connected with the diagrams with radiative photon insertions in thegraphs with multiphoton exchanges. These graphs are similar to the graphsgenerating the leading logarithm squared purely recoil contribution of order(Zα)7(m/M), discussed in 4.3. It is therefore not by chance that both thesecorrections were first obtained by the same methods and in the same works[12, 13]. The approximations used in these works did not include expansionin the small mass ratio, and therefore the obtained results are more generaland are valid for arbitrary mass ratio

∆E =(

−mr +83

m3r

mM

)α(Zα)6

πn3ln2(Zα)−2 δl0 . (5.9)

The first term here is of nonrecoil nature and was discussed earlier (see thelogarithm squared term in (3.53)). Note however, that the logarithm squaredterm in (3.53) is accompanied by the factor m3

r/(mM) instead of the factor mr

in (5.9). Hence, only the leading terms in the expansions of these coefficientsover the small mass ratio m/M coincide.

One could insist that the mass dependence in (5.9) is “natural” becausethe calculations leading to it are done without expansion over the mass ratio,and are therefore exact. On the other hand, the factor m3

r/(mM) symmetricwith respect to the masses naturally arises in all apparently nonrecoil calcu-lations just from the Schrodinger-Coulomb wave function squared. Accordingto the tradition we preserve the coefficient before the logarithm squared termin the form given in (3.53). Then the new contribution contained in (5.9) hasthe form [12, 13]

∆E =23

α(Zα)6

πn3ln2(Zα)−2 m3

r

mMδl0 . (5.10)

Due to linear dependence on the electron-nucleus mass ratio, the double loga-rithm contribution in (5.10) is nowadays very important for phenomenologicalanalysis of the hydrogen-deuterium isotope shift (see Subsect. 12.1.7).

Single logarithmic and nonlogarithmic radiative-recoil contributions of or-der α(Zα)6(m/M)m may be estimated as one half of the leading logarithm

106

5R

adia

tive-R

ecoil

Correctio

ns

Table 5.1. Radiative-Recoil Corrections

∆E(1S) kHz ∆E(2S) kHz

Radiative insertionsin the electron linePachucki (1995) [4]

Czarnecki, Melnikov (2001) [5, 6](

6ζ(3) − 2π2 ln 2

Eides, Grotch, Shelyuto (2001) [7] + 3π2

4− 14

)α(Zα)5

π2n3m3

rmM

δl0 −13.43 −1.68

Radiative insertionsin the electron line,second order in mass ratio

Czarnecki, Melnikov (2001) [5, 6](8 ln 2 − 127

32

)α(Zα)5

n3mM

m3r

mMδl0 0.009 0.001

Polarization insertionsEides, Grotch (1995) [8]

Pachucki (1995) [4] ( 2π2

9− 70

27)α(Zα)5

π2n3m3

rmM

δl0 −0.41 −0.05

Dirac FF insertions

in the heavy line{[

13

ln Λ(Zα)−2

mr+ (− 1

24− 7π

32Λ2

4M2

+ 23( Λ2

4M2 )2 + · · ·) + 1172

]

δl0

− 13

ln k0(n)}

4(Z2α)(Zα)4

πn3m3

rM2 4.58 0.58

Radiative insertionsin the electron line

Pachucki, Karshenboim (1999) [12] 23

α(Zα)6

πn3 ln2(Zα)−2 m3r

mMδl0 1.52 0.19


References 107

squared contribution. This constitutes about 0.8 kHz and 0.1 kHz for the 1Sand 2S levels in hydrogen, respectively. However, experience with the largecontributions generated by the nonleading terms of order α2(Zα)6(m/M)mand of order (Zα)7(m/M)m shows that we probably underestimate the mag-nitude of the nonleading terms in this way. In view of the rapid experimentalprogress in the Lamb shift and isotope shift measurements (see Table 12.3below) calculation of these nonleading corrections of order α(Zα)6(m/M)mdeserves further theoretical efforts.

References

1. G. Bhatt and H. Grotch, Phys. Rev. A 31, 2794 (1985).2. G. Bhatt and H. Grotch, Ann. Phys. (NY) 178, 1 (1987).3. G. Bhatt and H. Grotch, Phys. Rev. Lett. 58, 471 (1987).4. K. Pachucki, Phys. Rev. A 52, 1079 (1995).5. A. Czarnecki and K. Melnikov, Phys. Rev. Lett. 87, 013001 (2001).6. I. Blokland, A. Czarnecki, and K. Melnikov, Phys. Rev. D 65, 073015 (2002).7. M. I. Eides, H. Grotch, and V. A. Shelyuto, Phys. Rev. A 63, 052509 (2001).8. M. I. Eides and H. Grotch, Phys. Rev. A 52, 1757 (1995).9. A. S. Yelkhovsky, preprint Budker INP 97-80, hep-ph/9710377 (1997).

10. A. S. Elkhovskii, Zh. Eksp. Teor. Fiz. 110, 431 (1996); JETP 83, 230 (1996).11. T. Fulton and P. C. Martin, Phys. Rev. 95, 811 (1954).12. K. Pachucki and S. G. Karshenboim, Phys. Rev. A 60, 2792 (1999).13. K. Melnikov and A. S. Yelkhovsky, Phys. Lett. B 458, 143 (1999).

6

Nuclear Size and Structure Corrections

The one-electron atom is a composite nonrelativistic system loosely boundby electromagnetic forces. The characteristic size of the atom is of the orderof the Bohr radius 1/(mZα), and this scale is too large for effects of otherinteractions (weak and strong, to say nothing about gravitational) to play asignificant role. Nevertheless, in high precision experiments effects connectedwith the composite nature of the nucleus can become observable. By far themost important nonelectomagnetic contributions are connected with the fi-nite size of the nucleus and its structure. Both the finite radius of the protonand its structure constants do not at present admit precise calculation fromfirst principles in the framework of quantum chromodynamics (QCD) – themodern theory of strong interactions. Fortunately, the main nuclear parame-ters affecting the atomic energy levels may be either measured directly, oradmit almost model independent calculation in terms of other experimentallymeasured parameters.

Besides the strong interaction effects connected with the nucleus, stronginteractions affect the energy levels of atoms via nonleptonic contributionsto the photon polarization operator. Once again, these contributions admitcalculation in terms of experimental data, as we have already discussed abovein Subsect. 3.2.5. A minor contribution to the energy shift is also generatedby the weak gauge boson exchange to be discussed below.

6.1 Main Proton Size Contribution

The nucleus is a bound system of strongly interacting particles. Unfortunately,modern QCD does not provide us with the tools to calculate the bound stateproperties of the proton (or other nuclei) from first principles, since the QCDperturbation expansion does not work at large (from the QCD point of view)distances which are characteristic for the proton structure, and the nonper-turbative methods are not mature enough to produce good results.

M.I.x Eides et al.: Theory of Light Hydrogenic Bound States, STMP 222, 109–130 (2007)DOI 10.1007/3-540-45270-2 6 c© Springer-Verlag Berlin Heidelberg 2007

110 6 Nuclear Size and Structure Corrections

Fig. 6.1. Proton radius contribution to the Lamb shift. Bold dot corresponds tothe form factor slope

Fortunately, the characteristic scales of the strong and electromagneticinteractions are vastly different, and at the large distances which are relevantfor the atomic problem the influence of the proton (or nuclear) structure maybe taken into account with the help of a few experimentally measurable protonproperties. The largest and by far the most important correction to the atomicenergy levels connected with the proton structure is induced by its finite size.

The leading nuclear structure contribution to the energy shift is completelydetermined by the slope of the nuclear form factor in Fig. 6.1 (compare (2.1))

F (−k2) ≈ 1 − k2

6〈r2〉 . (6.1)

The respective perturbation potential is given by the form factor slope inser-tion in the external Coulomb potential (see (2.3))

− 4π(Zα)k2

→ 2π(Zα)3

〈r2〉 , (6.2)

and we immediately obtain

∆E =2π(Zα)

3〈r2〉|ψ(0)|2 =

2(Zα)4

3n3m3

r〈r2〉δl0 . (6.3)

We see that the correction to the energy level induced by the finiteness ofthe proton charge radius shifts the energy level upwards, and is nonvanishingonly for the S states1 Physically the finite radius of the proton means that theproton charge is smeared over a finite volume, and the electron spends sometime inside the proton charge cloud and experiences a smaller attraction thanin the case of the pointlike nucleus (Compare similar arguments in relationwith the finite radius of the electron below (2.4)).

The result in (6.3) needs some clarification. In the derivation above itwas implicitly assumed that the photon-nucleus vertex is determined by theexpression in (6.1). However, for a nucleus with spin this interaction dependson more than one form factor, and the effective slope of the photon-nucleusvertex contains in the general case some additional terms besides the nucleusradius. We will consider the real situation for nuclei of different spins below.1 Note the similarity of this discussion to the consideration of the level shift in-

duced by the polarization insertion in the external Coulomb photon in Sect. 2.2.However, unlike the present case the polarization insertion leads to a negativecontribution to the energy levels since the polarization cloud screens the sourcecharge.

6.1 Main Proton Size Contribution 111

6.1.1 Spin One-Half Nuclei

The photon-nucleus interaction vertex is described by the Dirac (F1) andPauli (F2) form factors

γµF1(k2) − 12M

σµνkνF2(k2) , (6.4)

where at small momentum transfers

F1(−k2) ≈ 1 − 〈r2〉F6

k2 , (6.5)

F2(0) =g − 2

2.

Hence, at low momenta the photon-nucleus interaction vertex (after the Foldy-Wouthuysen transformation and transition to the two-component nuclearspinors) is described by the expression

1 − k2 1 + 8F ′1M

2 + 2F2(0)8M2

= 1 − k2

[1

8M2+

〈r2〉F6

+g − 28M2

]

. (6.6)

For an elementary proton 〈r2〉F = 0, g = 2, and only the first term in thesquare brackets survives. This term leads to the well known local Darwinterm in the electron-nuclear effective potential (see, e.g., [1]) and generatesthe contribution proportional to the factor δl0 in (3.4). As was pointed out in[2], in addition to this correction, there exists an additional contribution ofthe same order produced by the term proportional to the anomalous magneticmoment in (6.6).

However, this is not yet the end of the story, since the proton charge radiusis usually defined via the Sachs electric form factor GE , rather than the Diracform factor F1

GE(−k2) ≈ 1 − 〈r2〉G6

k2 . (6.7)

The Sachs electric and magnetic form factors are defined as (see, e.g. [3])

GE(−k2) = F1(−k2) − k2

4M2F2(−k2) ,

GM (−k2) = F1(−k2) + F2(−k2) . (6.8)

In terms of this new charge radius the photon-nucleus vertex above has theform

1 − k2

[1

8M2+

〈r2〉G6

]

. (6.9)

We now see that for a real proton the charge radius contribution has exactlythe form in (6.3), where the charge radius is defined in (6.7). The only otherterm linear in the momentum transfer in the photon-nucleus vertex in (6.9)


generates the δl0 term in (3.4). Hence, if one uses the proton charge radiusdefined via the Sachs form factor, the net contribution of order (Zα)4m3

r/M2

has exactly the same form as if the proton were an elementary particle withg = 2.

Numerically the proton size correction is about 1100 kHz for the 1S state inhydrogen and about 140 kHz for the 2S state, and is thus much larger than theexperimental accuracy of the Lamb shift measurements. Unfortunately, pre-cise experimental determination of the proton root mean square (rms) chargeradius is a tricky business. Early measurements produced contradictory results[4, 5]. For many years the value rp = 0.862 (12) fm [5] was widely acceptedas a standard value of the proton rms radius. All recent analysis indicatesthat the real value of the proton rms radius is larger. An analysis of the lowmomentum transfer electron scattering data with account of the Coulomb andrecoil corrections [6] resulted in the proton radius value rp = 0.880 (15) fm.Another recent analysis of the elastic e±p scattering data resulted in an evenlarger value of the proton charge radius [7] rp = 0.897 (2) (1) (3) fm, wherethe error in the first brackets is due to statistics, the second error is due tonormalization effects, and the third error reflects the model dependence. Com-paring results of these two analysis one has to remember that the Coulombcorrections which played the most important role in [6] were ignored in [7].The results of [7] depend also on specific parametrization of the nucleon formfactors. The larger values of the rms charge radius were also confirmed in acomprehensive analysis in [8] which produced rp = 0.895 (18) fm. We will usethis last value in our numerical calculations.

6.1.2 Nuclei with Other Spins

The general result for the nuclear charge radius and the Darwin-Foldy contri-bution for a nucleus with arbitrary spin was obtained in [9]. It was shown therethat one may write a universal formula for the sum of these contributions irre-spective of the spin of the nucleus if the nuclear charge radius is defined withthe help of the same form factor for any spin. However, for historic reasons,the definitions of the nuclear charge radius are not universal, and respectiveformulae have different appearances for different spins. We will discuss hereonly the most interesting cases of the spin zero and spin one nuclei.

The case of the spin zero nucleus is the simplest one. For an elementaryscalar particle the low momentum nonrelativistic expansion of the photon-scalar vertex starts with the k4/M4 term, and, hence, the respective con-tribution to the energy shift is suppressed by an additional factor 1/M2 incomparison to the spin one-half case. Hence, in the case of the scalar nucleusthere is no Darwin term δl0 in (3.4) [10, 11]. Interaction of the composite scalarnucleus with photons is described only by one form factor, and the slope ofthis form factor is called the charge radius squared. Hence, in the scalar casethe charge radius contribution is described by (6.3), and the Darwin term isabsent in (3.4).

6.1 Main Proton Size Contribution 113

The spin one case is more complicated since for the vector nucleus itsinteraction with the photon is described in the general case by three formfactors. The nonrelativistic limit of the photon-nucleus vertex in this casewas considered in [12], where it was shown that with the standard definitionof the deuteron charge radius (the case of the deuteron is the only phenom-enologically interesting case of the spin one nucleus) the situation with theDarwin-Foldy and the charge radius contribution is exactly the same as in thecase of the scalar particle. Namely, the Darwin-Foldy contribution is missingin (3.4) and the charge radius contribution is given by (6.3). It would beappropriate to mention here recent work [13], where a special choice of de-finition of the nuclear charge radius is advocated, namely it is suggested toinclude the Darwin-Foldy contribution in the definition of the nuclear chargeradius. While one can use any consistent definition of the nuclear charge ra-dius, this particular choice seems to us to be unattractive since in this caseeven a truly pointlike particle in the sense of quantum field theory (say anelectron) would have a finite charge radius even in zero-order approximation.The phenomenological aspects of the deuteron charge radius contribution tothe hydrogen-deuterium isotope shift will be discussed later in Subsect. 12.1.7.

6.1.3 Empirical Nuclear Form Factor and the Contributionsto the Lamb Shift

In all considerations above we have assumed the most natural theoretical def-inition of nuclear form factors, namely, the form factor was assumed to be anintrinsic property of the nucleus. Therefore, the form factor is defined via theeffective nuclear-photon vertex in the absence of electromagnetic interaction.Such a form factor can in principle be calculated with the help of QCD. Theelectromagnetic corrections to the form factor defined in this way may be cal-culated in the framework of QED perturbation theory. Strictly speaking allformula above are valid with this definition of the form factor.

However, in practice, form factors are measured experimentally and thereis no way to switch off the electromagnetic interaction. Hence, in order todetermine the form factor experimentally one has in principle to calculateelectromagnetic corrections to the elastic electron-nucleus scattering which isusually used to measure the form factors [4, 5]. In the usual fit to the exper-imental data not all electromagnetic corrections to the scattering amplitudeare usually taken into account (see e.g., discussion in [14, 15]). First, all vac-uum polarization insertions, excluding the electron vacuum polarization, areusually ignored. This means that respective contributions to the energy shiftin (3.32) are swallowed by the empirical value of the nuclear charge radiussquared. They are effectively taken into account in the contribution to the en-ergy shift in (6.3), and should not be considered separately. Next there are the


corrections of order (Z2α)(Zα)4 to the energy shift2. The perturbative elec-tromagnetic contributions to the Pauli form factor should be ignored, sincethey definitely enter the empirical value of the nuclear g-factor. The situationis a bit more involved with respect to the electromagnetic contribution to theDirac nuclear form factor. The QED contribution to the slope of the Diracform factor is infrared divergent, and, hence, one cannot simply include it inthe empirical value of the nuclear charge radius. Of course, as is well known,there is no real infrared divergence in the proper description of the electron-nucleus scattering if the soft photon radiation is properly taken into account(see, e.g., [10, 3]). This means that the proper determination of the empiricalproton form factor, on the basis of the experimental data, requires accountof the electromagnetic radiative corrections, and the measured value of thenuclear charge radius squared does not include the electromagnetic contribu-tion. Hence, the radiative correction of order (Z2α)(Zα)4 in (5.6) should beincluded in the comparison of the theory with the experimental data on theenergy shifts.

6.2 Nuclear Size and Structure Correctionsof Order (Zα)5m

Corrections of relative order (Zα)5 connected with the nonelementarity ofthe nucleus are generated by the diagrams with two-photon exchanges. Asusual all corrections of order (Zα)5, originate from high (on the atomic scale)intermediate momenta. Due to the composite nature of the nucleus, besidesintermediate elastic nuclear states, we also have to consider the contributionof the diagrams with inelastic intermediate states.

6.2.1 Nuclear Size Corrections of Order (Zα)5m

Let us consider first the contribution generated only by the elastic intermedi-ate nuclear states. This means that we will treat the nucleus here as a particlewhich interacts with the photons via a nontrivial experimentally measurableform factor GE(k2), i.e. the electromagnetic interaction of our nucleus is non-local, but we will temporarily ignore its excited states.

As usual we start with the skeleton integral contribution in (3.33) cor-responding to the two-photon skeleton diagram in Fig. 3.8. Insertion of thefactor GE(−k2) − 1 in the proton vertex corresponds to the presence of anontrivial proton form factor3.2 We have already considered these corrections together with other radiative-recoil

corrections above, in Subsect. 5.1.3. This discussion will be partially reproducedhere in order to make the present section self-contained.

3 Subtraction is necessary in order to avoid double counting since the subtractedterm in the vertex corresponds to the pointlike proton contribution, already takeninto account in the effective Dirac equation.

6.2 Nuclear Size and Structure Corrections of Order (Zα)5m 115

Fig. 6.2. Diagrams for elastic nuclear size corrections of order (Zα)5m with oneform factor insertion. Empty dot corresponds to factor GE(−k2) − 1

Fig. 6.3. Diagrams for elastic nuclear size corrections of order (Zα)5m with twoform factor insertions. Empty dot corresponds to factor GE(−k2) − 1

We have to consider diagrams in Fig. 6.2 with insertion of one factorGE(−k2) − 1 in the proton vertex (there are two such diagrams, hence anextra factor two below)4

∆E = −32mm3r

(Zα)5

πn3

∫ ∞

0

dk

k4

(GE(−k2) − 1

), (6.10)

and the diagrams in Fig. 6.3 with insertion of two factors GE(−k2)−1 in twoproton vertices

∆E = −16mm3r

(Zα)5

πn3

∫ ∞

0

dk

k4

(GE(−k2) − 1

)2. (6.11)

The low momentum integration region in the integral in (6.10) produces alinearly divergent infrared contribution, which simply reflects the presence ofthe correction of order (Zα)4, calculated in Sect. 6.1. We need to subtractthis divergent contribution, connected with the low energy expansion of theform factor in (6.7). Technically subtraction reduces to insertion of the term〈r2〉k2/6 in the brackets in (6.10). After subtraction, the integral in (6.10)also contains the finite contribution induced by high intermediate momenta,which should be taken on a par with the contribution in (6.11).

The scale of the integration momenta in (6.11), and (6.10) is determined bythe form factor scale. High momenta in the present context means momenta

4 Dimensionless integration momentum in (3.33) was measured in electron mass.We return here to dimensionful integration momenta, which results in an extrafactor m3 in the numerators in (6.10), (6.11) and (6.12) in comparison with thefactor in the skeleton integral (3.33). Notice also the minus sign before the mo-mentum in the arguments of form factors; it arises because in the equations belowk = |k|.


of the form factor scale, to be distinguished from high momenta in otherchapters which often meant momenta of the scale of the electron mass. Thecharacteristic momenta in the present case are much higher.

The total contribution of order (Zα)5 is given by the sum of the subtractedexpression in (6.10) and the contribution in (6.11)

∆E = −16mmr3 (Zα)5

πn3

∫ ∞

0

dk

k4

(

G2E(−k2) − 1 +

〈r2〉3

k2

)

. (6.12)

This expression is usually written in the form [16, 17]5

∆E = −m(Zα)5

3n3m3

r〈r3〉(2) , (6.13)

where 〈r3〉(2) is called the third Zemach moment [18], and is defined as

〈r3〉(2) =48π

∫ ∞

0

dk

k4

(

G2E(−k2) − 1 +

〈r2〉3

k2

)

. (6.14)

Carrying out the Fourier transformation, one obtains the coordinate spacerepresentation for the third Zemach moment, in terms of the weighted convo-lution of two nuclear charge densities ρ(r) [18]

〈r3〉(2) ≡∫

d3r1d3r2ρ(r1)ρ(r2)|r1 − r2|3 . (6.15)

Parametrically the result in (6.13) is of order m(Zα)5(m/Λ)3, where Λ isthe form factor scale. Hence, this correction is suppressed in comparison withthe leading proton size contribution not only by an extra factor Zα but also bythe extra small factor m/Λ. This explains the smallness of this contribution,even in comparison with the proton size correction of order (Zα)6 (see belowSubsect. 6.3.2), since one factor m/Λ in (6.13) is traded for a much largerfactor Zα in that logarithmically enhanced contribution.

The result in (6.13) depends on the third Zemach moment, or in otherwords, on a nontrivial weighted integral of the product of two charge den-sities, and cannot be measured directly, like the rms proton charge radius.This means that the correction under consideration may only conditionallybe called the proton size contribution. It depends on the fine details of theform factor momentum dependence, and not only on the directly measurablelow-momentum behavior of the form factor. This feature of the result is quitenatural taking into account high intermediate momenta characteristic for theintegral in (6.12). In the practically important cases of hydrogen and deu-terium, model dependent results for this contribution may be obtained with

5 The result in [17] has the factor m4r instead of mm3

r before the integral in (6.13).This difference could become important only after calculation of a recoil correctionto the contribution in (6.13).


the help of model dependent proton form factors and/or model independentexperimental data on electron-proton scattering [17, 19]. The latest modelindependent analysis of the proton-electron scattering and application of theexpression in (6.14) [19] produced the value 〈r3〉(2) = 2.71 (13) fm3 for thethird Zemach moment of the proton. The respective nuclear radius correctionof order (Zα)5 is equal to −40.0 (1.9) Hz for the 1S state and −5.0 (2) Hz forthe 2S state in hydrogen. These corrections are rather small. A much largercontribution arises in (6.13) to the energy levels of deuterium. The deuteron,unlike the proton, is a loosely bound system, with a radius much larger thanthe proton radius, and the respective correction to the energy levels is alsolarger. The contribution of the correction under consideration [20] (see also[21])

∆E = 0.49 kHz (6.16)

to the 2S − 1S energy splitting in deuterium should be taken into account inthe discussion of the hydrogen-deuterium isotope shift.

We started consideration of the proton size correction of relative order(Zα)5 by inserting the factors (GE(−k2) − 1) in the external field skeletondiagrams in Fig. 3.8. Technically the external field diagrams correspond to theheavy particle pole contribution in the sum of all skeleton diagrams with twoexchanged photons in Fig. 4.3-Fig. 4.5. In the absence of form factors the non-pole contributions of the diagrams in Fig. 4.3-Fig. 4.5 were suppressed by therecoil factor m/M in comparison with the heavy pole contribution, and thisjustified their separate consideration. However, as we have seen, insertion ofthe form factors pushes the effective integration momenta in the high momentaregion ∼ Λ even in the external field diagrams. Then even the external fieldcontribution contains the recoil factor (m/Λ)3. We might expect that, afterinsertion of the proton form factors, the nonpole contribution of the skeletondiagrams with two exchanged photons in Fig. 4.3-Fig. 4.5 would contain therecoil factor (m/M)(m/Λ)2, and would not be parametrically suppressed incomparison with the pole contribution in (6.13). The total contribution of theskeleton diagrams with the proton form factor insertions was calculated in[22] for the 2S state, and the difference between this result and the nonrecoilresult in (6.13) turned out to be −0.25 Hz. At the current level of theoreticaland experimental accuracy we can safely ignore such tiny differences betweenthe pole and total proton size contributions of order (Zα)5.

6.2.2 Nuclear Polarizability Contribution of Order (Zα)5mto S-Levels

The description of nuclear structure corrections of order (Zα)5m in terms ofnuclear size and nuclear polarizability contributions is somewhat artificial. Aswe have seen above the nuclear size correction of this order depends not onthe charge radius of the nucleus but on the third Zemach moment in (6.15).One might expect the inelastic intermediate nuclear states in Fig. 6.4 would


Fig. 6.4. Diagrams for nuclear polarizability correction of order (Zα)5m

generate corrections which are even smaller than those connected with thethird Zemach moment, but this does not happen. In reality, the contributionof the inelastic intermediate states turns out to be even larger than the elasticcontribution since the powerlike decrease of the form factor is compensated inthis case by the summation over a large number of nuclear energy levels. Theinelastic contributions to the energy shift were a subject of intensive study fora long time, especially for muonic atoms (see, e.g., [23, 24, 25, 26, 27, 28, 29]).Corrections to the energy levels were obtained in these works in the form ofcertain integrals of the inelastic nuclear structure functions, and the dominantcontribution is produced by the nuclear electric and magnetic polarizabilities.

The main feature of the polarizability contribution to the energy shift isits logarithmic enhancement [26, 30]. The appearance of the large logarithmmay easily be understood with the help of the skeleton integral. The heavyparticle factor in the two-photon exchange diagrams is now described by thephoton-nucleus inelastic forward Compton amplitude [31]

M = α(ω,k2)E · E∗ + β(ω,k2)B · B∗ , (6.17)

where α(ω,k2) and β(ω,k2) are proton (nuclear) electric and magnetic polar-izabilities.

In terms of this Compton amplitude the two-photon contribution has theform

∆E = −4α(mrZα)3

n3

∫d4k

(2π)4iDimDjn

k4

Tr{γi((1 + γ0)m − k)γj}k2 − 2mk0

Mmn ,

(6.18)which reduces after elementary transformations to6

∆E = −α(Zα)3

4πn3mr

3m

∫ ∞

0

dk{[(k4 + 8)√

k2 + 4 − k(k4 + 2k2 + 6)]α(ω, k2)

+ [k3(k2 + 4)32 − k(k4 + 6k2 + 6)]β(ω, k2)} , (6.19)

and we remind the reader that an extra factor Z2α is hidden in the definitionof the polarizabilities.

Ignoring the momentum dependence of the polarizabilities, one immedi-ately comes to a logarithmically ultraviolet divergent integral [31]

∆E = −α(Zα)3

πn3mr

3m[5α(0, 0) − β(0, 0)

](

lnΛ

m+ O(1)

)

, (6.20)

6 Integration in (6.19) goes over dimensionless momentum k measured in electronmass.


where Λ is an ultraviolet cutoff. In real calculations the role of the cutoff isplayed by the characteristic excitation energy of the nucleus.

The sign of the energy shift is determined by the electric polarizability andhas a clear physical origin. The electron polarizes the nucleus, an additionalattraction between the induced dipole and the electron emerges, and shiftsthe energy level down.

In the case of hydrogen the characteristic excitation energy is about 300MeV, the logarithm is rather large, and the logarithmic approximation worksvery well. Using the proton polarizabilities [32] one easily obtains the polar-izability contribution for the proton nS state [26, 31, 33, 34]

∆E(nS) = −70 (11) (7)n3

Hz , (6.21)

where the error in the first parenthesis describes the accuracy of the loga-rithmic approximation, and the error in the second parenthesis is due to theexperimental data on the polarizabilities.

A slightly different numerical polarizability contribution

∆E(1S) = −95 (7)n3

Hz , (6.22)

was obtained in [35]. Discrepancy between the results in (6.21) and (6.22) ispreserved even when both groups of authors use one and the same data onproton polarizabilities from [36]. Technically the disagreement between theresults in [26, 31, 33, 34] and [35] is due to the expression for the polarizabil-ity energy shift in the form of an integral of the total photoabsorption crosssection, which was used in [35]. This expression was derived in [30] under theassumption that the invariant amplitudes for the forward Compton scatteringsatisfy the dispersion relations without subtractions. Without this subtractionterm the dominant logarithmic contribution to the energy shift becomes pro-portional to the sum of the electric and magnetic polarizabilities α + β, whilein [26, 31, 33, 34] this contribution is proportional to another linear combina-tion of polarizabilities (see (6.20)). Restoring this necessary subtraction termthe authors of [22, 37] obtained the polarizability contribution

∆E(1S) = −82 (9)n3

Hz , (6.23)

which is compatible both with (6.21) and (6.22).In the other experimentally interesting case of deuterium, nuclear exci-

tation energies are much lower and a more accurate account of the internalstructure of the deuteron is necessary. As is well known, due to smallness ofthe binding energy the model independent zero-range approximation providesa very accurate description of the deuteron. The polarizability contributionsto the energy shift in deuterium are again logarithmically enhanced and inthe zero-range approximation one obtains a model independent result [38, 39]


∆E = −α(Zα)3

πn3mr

3m

{

5αd

[

ln8I

m+

120

]

− βd

[

ln8I

m− 1.24

]}

, (6.24)

where κ = 45.7 MeV is the inverse deuteron size, I = κ2/mp is the absolutevalue of the deuteron binding energy, and the deuteron electric and magneticpolarizabilities are defined as

αd(ω) =23

(Z2α)∑

n

(En + I)|〈0|r|n〉|2ω2 − (En + I)2

,

βd(0) =α(µp − µn)2

8mpκ2

1 + κ13κ

1 + κ1κ

, (6.25)

where κ1 = 7.9 MeV determines the position of the virtual level in the neutron-proton 1S0 state.

Numerically the polarizability contribution to the deuterium 1S energyshift in the zero range approximation is equal to [39]

∆E(1S) = (−22.3 + 0.31) kHz , (6.26)

where the first number in the parenthesis is the contribution of the electricpolarizability, and the second is the contribution of the magnetic polarizability.This zero range contribution results in the correction

∆E(1S − 2S) = 19.3 kHz , (6.27)

to the 1S − 2S interval, and describes the total polarizability contributionwith an accuracy of about one percent.

Experimental data on the deuterium-hydrogen isotope shift (see Table 12.4below) have an accuracy of about 0.1 kHz and, hence, a more accurate theo-retical result for the polarizability contribution is required. In order to obtainsuch a result it is necessary to go beyond the zero range approximation, andtake the deuteron structure into account in more detail. Fortunately, thereexist a number of phenomenological potentials which describe the propertiesof the deuteron in all details. Some calculations with realistic proton-neutronpotentials were performed [40, 41, 42, 43]. The most precise results were ob-tained in [43]

∆E(1S − 2S) = 18.58 (7) kHz , (6.28)

which are consistent with the results of the other works [38, 41].The result in (6.28) is obtained neglecting the contributions connected with

the polarizability of the constituent nucleons in the deuterium atom, and thepolarizability contribution of the proton in the hydrogen atom. Meanwhile,as may be seen from (6.21), proton polarizability contributions are compa-rable to the accuracy of the polarizability contribution in (6.28), and cannotbe ignored. The deuteron is a weakly bound system and it is natural to as-sume that the deuteron polarizability is a sum of the polarizability due to


relative motion of the nucleons and their internal polarizabilities. The nu-cleon polarizabilities in the deuteron coincide with the polarizabilities of thefree nucleons well within the accuracy of the logarithmic approximation [33].Therefore the proton polarizability contribution to the hydrogen-deuteriumisotope shift cancels, and the contribution to this shift, which is due to theinternal polarizabilities of the nucleons, is completely determined by the neu-tron polarizability. This neutron polarizability contribution to the hydrogen-deuterium isotope shift was calculated in the logarithmic approximation in[33]

∆E(1S − 2S) = 53 (9) (11) Hz . (6.29)

A compatible result for this contribution to the isotope shift was obtainedfrom the analysis of the experimental data on the proton and deuteron struc-ture functions in [37].

6.3 Nuclear Size and Structure Correctionsof Order (Zα)6m

6.3.1 Nuclear Polarizability Contribution to P -Levels

The leading polarizability contribution of order (Zα)5 obtained above is pro-portional to the nonrelativistic wave function at the origin squared, and hence,exists only for the S states. The leading polarizability contribution to the nonS-states is of order (Zα)6 and may easily be calculated. Consider the Comp-ton amplitude in (6.17) as the contribution to the bound state energy inducedby the external field of the electron at the nuclear site. Then we calculatethe matrix element in (6.17) between the electron states, considering the fieldstrengths from the Coulomb field generated by the orbiting electron. We ob-tain [23] (the overall factor 1/2 is due to the induced nature of the nucleardipole)

∆E = −αα

2

⟨

n, l

∣∣∣∣

1r4

∣∣∣∣n, l

⟩

= − 3n2 − l(l + 1)2n5(l + 3

2 )(l + 1)(l + 12 )l(l − 1

2 )α(Zα)4αm4

r

2.

(6.30)This energy shift is negative because when the electron polarizes the nucleusthis leads to an additional attraction of the induced dipole to the electron.

The contribution induced by the magnetic susceptibility may also be eas-ily calculated [23], but it is even of higher order in Zα (order (Zα)8) sincethe magnetic field strength behaves as 1/r3. This additional suppression ofthe magnetic effect is quite reasonable, since the magnetic field itself is arelativistic effect.

Consideration of the P -state polarizability contribution provides us witha new perspective on the S-state polarizability contribution. One could tryto consider the matrix element in (6.30) between the S states. Due to non-vanishing of the S-state wave functions at the origin this matrix element is


linearly divergent at small distances, which once more demonstrates that theS-state contribution is of a lower order in Zα than for the P -state, and thatfor its calculation one has to treat small distances more accurately than wasdone in (6.30).

6.3.2 Nuclear Size Correction of Order (Zα)6m

The nuclear size and polarizability corrections of order (Zα)5 obtained in Sect.6.2 are very small. As was explained there, the suppression of this contribu-tion is due to the large magnitude of the characteristic momenta responsiblefor this correction. The nature of the suppression is especially clear in thecase of the Zemach radius contribution in (6.13), which contains the smallfactor m3

r〈r3〉(2). The nuclear size correction of order (Zα)6m contains anextra factor Zα in comparison with the nuclear size and polarizability correc-tions of order (Zα)5, but its main part is proportional to the proton chargeradius squared. Hence, we should expect that despite an extra power of Zαthis correction is numerically larger than the nuclear size and polarizabilitycontributions of the previous order in Zα. As we will see below, calculationsconfirm this expectation and, moreover, the contribution of order (Zα)6m isadditionally logarithmically enhanced.

Nuclear size corrections of order (Zα)6 may be obtained in a quite straight-forward way in the framework of the quantum mechanical third order pertur-bation theory. In this approach one considers the difference between the elec-tric field generated by the nonlocal charge density described by the nuclearform factor and the field of the pointlike charge as a perturbation operator[16, 17].

The main part of the nuclear size (Zα)6 contribution which is proportionalto the nuclear charge radius squared may also be easily obtained in a simplerway, which clearly demonstrates the source of the logarithmic enhancementof this contribution. We will first discuss in some detail this simple-mindedapproach, which essentially coincides with the arguments used above to obtainthe main contribution to the Lamb shift in (2.4), and the leading proton radiuscontribution in (6.3).

The potential of an extended nucleus is given by the expression

V (r) = −Zα

∫

d3r′ρ(r′)

|r − r′| , (6.31)

where ρ(r′) is the nuclear charge density.Due to the finite size of the nuclear charge distribution, the relative dis-

tance between the nucleus and the electron is not constant but is subject toadditional fluctuations with probability ρ(r). Hence, the energy levels experi-ence an additional shift

∆E =⟨

n

∣∣∣∣

∫

d3r′′ρ(r′′)[V (r + r′′) − V (r)]∣∣∣∣n

⟩

. (6.32)


Taking into account that the size of the nuclear charge distribution is muchsmaller than the atomic scale, we immediately obtain

∆E =2π

3(Zα)〈r2〉

∫

d3rρ(r)ψ(r)+ψ(r) . (6.33)

We will now discuss contributions contained in (6.33) for different specialcases.

6.3.2.1 Correction to the nS-Levels

In the Schrodinger-Coulomb approximation the expression in (6.33) reducesto the leading nuclear size correction in (6.3). New results arise if we takeinto account Dirac corrections to the Schrodinger-Coulomb wave functions ofrelative order (Zα)2. For the nS states the product of the wave functions in(6.33) has the form (see, e.g, [17])

|ψ(r)|2 = |ψSchr(r)|2{

1 − (Zα)2[

ln2mrZα

n+ ψ(n) + 2γ +

94n2

− 1n− 11

4

]}

,

(6.34)and the additional contribution to the energy shift is equal to

∆E = −2(Zα)6

3n3m3

r〈r2〉[⟨

ln2mrZα

n

⟩

+ ψ(n) + 2γ +9

4n2− 1

n− 11

4

]

.

(6.35)This expression nicely illustrates the main qualitative features of the order

(Zα)6 nuclear size contribution. First, we observe a logarithmic enhancementconnected with the singularity of the Dirac wave function at small distances.Due to the smallness of the nuclear size, the effective logarithm of the ratioof the atomic size and the nuclear size is a rather large number; it is equal toabout −10 for the 1S level in hydrogen and deuterium. The result in (6.35)contains all state-dependent contributions of order (Zα)6.

A tedious third order perturbation theory calculation [16, 17] producessome additional state-independent terms with the net result being a few per-cent different from the naive result above. The additional state-independentcontribution beyond the naive result above has the form [20]

∆E =2(Zα)6

3n3m3

r

{〈r2〉2

−〈r3〉〈1

r 〉3

+∫

d3rd3r′ρ(r)ρ(r′)θ(|r| − |r′|)

×[

(r2 + r′2) ln|r′||r| − r′3

3|r| +r3

3|r′| +r2 − r′2

3

]

+ 6∫

d3rd3r′d3r′′ρ(r)ρ(r′)ρ(r′′)θ(|r| − |r′|)θ(|r′| − |r′′|)

×[r′′2

3ln

|r′||r′′|e −

r′′4

45|r||r′| +r′′3

9

(1|r′| +

1|r|

)

+r′2r′′2

36r2− 2r′r′′2

9r+

r′′2

9

]}

. (6.36)


Note that, unlike the leading naive terms in (6.35), this additional contributiondepends on more detailed features of the nuclear charge distribution thansimply the charge radius squared.

Detailed numerical calculations in the interesting cases of hydrogen anddeuterium were performed in [20]. Nuclear size contributions of order (Zα)6 tothe energy shifts in hydrogen are given in Table 6.17 and, as discussed above,they are an order of magnitude larger than the nuclear size and polarizabilitycontributions of the previous order in Zα.

Respective corrections to the energy levels in deuterium are even muchlarger than in hydrogen due to the large radius of the deuteron. The nuclearsize contribution of order (Zα)6 to the 2S−1S splitting in deuterium is equalto (we have used in this calculation the value of the deuteron charge radiusobtained in [44] from the analysis of all available experimental data)

∆E = −3.43 kHz , (6.37)

and in hydrogen∆E = −0.61 kHz . (6.38)

We see that the difference of these corrections gives an important contributionto the hydrogen-deuterium isotope shift.

6.3.2.2 Correction to the nP -Levels

Corrections to the energies of P -levels may easily be obtained from (6.33).Since the P -state wave functions vanish at the origin there are no chargeradius squared contributions of lower order, unlike the case of S states, andwe immediately obtain [17]

∆E(nPj) =(n2 − 1)(Zα)6

6n5m3

r〈r2〉δj 12

. (6.39)

There exist also additional terms of order (Zα)6 proportional to 〈r4〉 [17]but they are suppressed by an additional factor m2〈r2〉 in comparison withthe result above and may safely be omitted.

6.4 Radiative Corrections to the Finite Size Effect

6.4.1 Radiative Correction of Order α(Zα)5〈r2〉m3r

to the Finite Size Effect

Due to the large magnitude of the leading nuclear size correction in (6.3) atthe current level of experimental accuracy one also has to take into account7 All numbers in Table 6.1 are calculated for the proton radius rp = 0.895 (18) fm,

see discussion on the status of the proton radius results in Subsects. 6.1.1 and12.1.6.

6.4 Radiative Corrections to the Finite Size Effect 125

radiative corrections to this effect. These radiative corrections were first dis-cussed and greatly overestimated in [45]. The problem was almost immediatelyclarified in [46], where it was shown that the contribution is generated by largeintermediate momenta states and is parametrically a small correction of orderα(Zα)5m3

r〈r2〉. On the basis of the estimate in [46] the authors of [14] expectedthe radiative correction to the leading nuclear charge radius contribution tobe of order 10 Hz for the 1S-state in hydrogen.

The large magnitude of the characteristic integration momenta [46] is quiteclear. As we have seen above, in the calculation of the main proton chargeradius contribution, the exchange momentum squared factor in the numeratorconnected with the proton radius cancels with a similar factor in the denom-inator supplied by the photon propagator. Any radiative correction behavesas k2 at small momenta, and the presence of such a correction additionallysuppresses small integration momenta and pushes the characteristic integra-tion momenta into the relativistic region of order of the electron mass. Hence,the corrections may be calculated with the help of the skeleton integrals inthe scattering approximation. The characteristic integration momenta in theskeleton integral are of order of the electron mass, and are still much smallerthan the scale of the proton form factor. As a result the respective contributionto the energy shift depends only on the slope of the form factor.

The actual calculation essentially coincides with the calculation of the cor-rections of order α2(Zα)5 to the Lamb shift in Subsect. 3.3.3 but is technicallysimpler due to the triviality of the proton form factor slope contribution in(6.1).

There are two sources of radiative corrections to the leading nuclear sizeeffect, namely, the diagrams with one-loop radiative insertions in the electronline in Fig. 6.5, and the diagrams with one-loop polarization insertions in oneof the external Coulomb lines in Fig. 6.6.

6.4.1.1 Electron-Line Correction

Inserting the electron line factor [47, 48], the proton slope contribution (6.1),and the combinatorial factor 2 in the skeleton integral in (3.33), one obtains anintegral for the electron-line contribution which does not depend on any para-meters, and can be easily calculated numerically with an arbitrary precision[49]. Like a more complicated integral for the corrections of order α2(Zα)5in[48] this integral also admits an analytic evaluation, and the analytic resultwas obtained in [50, 51]

∆Ee−line =(

8 ln 23

− 236

)α(Zα)5

n3m3

r〈r2〉δl0 . (6.40)

6.4.1.2 Polarization Correction

Calculation of the diagrams with the polarization operator insertion proceedsexactly as in the case of the electron factor insertion. The only difference is


Fig. 6.5. Electron-line radiative correction to the nuclear size effect. Bold dot cor-responds to proton form factor slope

Fig. 6.6. Coulomb-line radiative correction to the nuclear size effect. Bold dot cor-responds to proton form factor slope

that one inserts an additional factor 4 in the skeleton integral to take intoaccount all possible ways to insert the polarization operator and the slopeof the proton form factor in the Coulomb photons. After an easy analyticcalculation one obtains [49, 52, 53]

∆Epol =α(Zα)5

2n3m3

r〈r2〉δl0 . (6.41)

6.4.1.3 Total Radiative Correction

The total radiative correction to the proton size effect is given by the sum ofcontributions in (6.41) and (6.40)

∆E =(

8 ln 23

− 103

)α(Zα)5

n3m3

r〈r2〉δl0 . (6.42)

This correction without separation of the polarization operator and elec-tron factor contributions was also considered in [54]. Correcting an apparentmisprint in that work, one finds the value −1.43 for the numerical coefficient in(6.42), which is slightly different from the result above. The origin of this minordiscrepancy is unclear, but the result in (6.42) was independently confirmedby at least two different groups [49, 50, 51] and is now firmly established.

Numerically the total radiative contribution in (6.42) for hydrogen is equalto

∆E(1S) = −0.138 kHz , (6.43)

∆E(2S) = −0.017 kHz ,

and for deuterium∆E(1S) = −0.841 kHz , (6.44)

∆E(2S) = −0.105 kHz .

These contributions should be taken into account in discussion of thehydrogen-deuterium isotope shift.

6.5 Weak Interaction Contribution 127

6.4.2 Higher Order Radiative Correctionsto the Finite Size Effect

One can consider also higher order radiative corrections to the finite size effect.Such contribution originating from very small distances and enhanced by thelarge logarithm ln(1/(mr0)) (r0 here is the radius of the nucleus) was obtainedin [50, 51, 55]. Its relative magnitude with respect to the leading finite sizecontribution is α(Zα)2 and it is universal for S and P state.

There exist also nonuniversal corrections of order α(Zα)6 both for S andP states. These corrections originate from large distances and are enhanced bylarge infrared logarithms ln(Zα) squared. Radiative corrections for S statesare enhanced by the large logarithm ln(Zα) squared, and are like the techni-cally similar purely radiative correction to the Lamb shift of order α2(Zα)6mconsidered above. The coefficient before this logarithm squared can be easilyobtained.

Radiative corrections to the finite size effect for P states of order α(Zα)6

originating from the atomic scale were calculated in [50, 51, 55] analytically.These corrections are enhanced by the large infrared logarithm ln(Zα).

Due to smallness even of the leading order finite size effect it is hard toimagine that these nice results would find any phenomenological applicationsfor light atoms, and we do not reproduce their calculation here.

6.5 Weak Interaction Contribution

The weak interaction contribution to the Lamb shift is generated by the Z-boson exchange in Fig. 6.7, which may be described by the effective locallow-energy Hamiltonian

HZ(L) = − 16πα

sin2 θW cos2 θW

(14− sin2 θW

)2mM

M2Z

×∫

d3x(ψ+(x)ψ(x)

) (Ψ+(x)Ψ(x)

), (6.45)

where MZ is the Z-boson mass, θW is the Weinberg angle, and ψ and Ψ are thetwo-component wave functions of the light and heavy particles, respectively.

Then we easily obtain the weak interaction contribution to the Lamb shiftin hydrogen [56]

Fig. 6.7. Z-boson exchange diagram


Table 6.1. Nuclear Size and Structure Corrections

∆E(1S) kHz ∆E(2S) kHz

Leading nuclear size

contribution 23n3 (Zα)4m3

r〈r2〉δl0 1 253 (50) 157 (6)

Proton form factorcontribution

of order (Zα)5

Borisoglebsky,

Trofimenko (1979) [16]

Friar (1979) [17]

Friar, Sick (2005) [19] −m(Zα)5

3n3 m3r〈r3〉(2)δl0 −0.040 (19) −0.005

Polarizability contribution

∆E(nS)

Startsev, Petrun’kin,

Khomkin (1976) [26]

Khriplovich, −α(Zα)3

πn3 mr3m

Sen’kov (1997) [31, 33, 34] ×[5α(0, 0) − β(0, 0)] ln Λm

−0.070(11)(7) −0.009(1)(1)

Polarizability contribution

∆E(nP ) −α(Zα)4αm4r

2

Ericson, Hufner (1972) [23] × 3n2−l(l+1)

2n5(l+ 32 )(l+1)(l+ 1

2 )l(l− 12 )

Nuclear size correctionof order (Zα)6 ∆E(nS)

Borisoglebsky, − 2(Zα)6

3n3 m3r〈r2〉

Trofimenko (1979) [16] ×[⟨

ln 2mrZαn

⟩+ ψ(n) + 2γ

Friar (1979) [17]

Friar, Payne (1997) [20] + 94n2 − 1

n− 11

4

]+ δE 0.709 (20) 0.095 (3)

Nuclear size correctionof order (Zα)6 ∆E(nPj)

Friar (1979) [17](Zα)6(n2−1)

6n5 m3r〈r2〉δj 1

2

Electron-lineradiative correction

Pachucki (1993) [54]Eides, Grotch (1997) [49]Milstein, Sushkov,Terekhov (2002) [50, 51]

(8 ln 2

3− 23

6

)m3

r〈r2〉α(Zα)5

n3 δl0 −0.199 (8) −0.025 (1)

Polarization operator

radiative correction

Friar (1979) [52]

Hylton (1985) [53]Pachucki (1993) [54]Eides, Grotch (1997) [49] 1

2m3

r〈r2〉α(Zα)5

n3 δl0 0.050 (2) 0.006

References 129

∆EZ(L) = −α(Zα)3mr

πn3

8Gm2r√

2α

(14− sin2 θW

)2

δl0

≈ −7.7 · 10−13 α(Zα)3mr

πn3δl0 . (6.46)

This contribution is too small to be of any phenomenological significance.

References

1. W. A. Barker and F. N. Glover, Phys. Rev. 99, 317 (1955).2. L. L. Foldy, Phys. Rev. 83, 688 (1951).3. V. B. Berestetskii, E. M. Lifshitz, and L. P. Pitaevskii, Quantum electrodynam-

ics, 2nd Edition, Pergamon Press, Oxford, 1982.4. D. J. Drickey and L. N. Hand, Phys. Rev. Lett. 9, 521 (1962); L. N. Hand, D.

J. Miller, and R. Wilson, Rev. Mod. Phys. 35, 335 (1963).5. G. G. Simon, Ch. Schmidt, F. Borkowski, and V. H. Walther, Nucl. Phys. A

333, 381 (1980).6. R. Rosenfelder, Phys. Lett. B 479, 381 (2000).7. V. V. Ezhela and B. V. Polishcuk, Protvino preprint IHEP 99-48, hep-

ph/9912401.8. I. Sick, Phys. Lett. B 576, 62 (2003).9. I. B. Khriplovich, A. I. Milstein, and R. A. Sen’kov, Phys. Lett. A221, 370

(1996); Zh. Eksp. Teor. Fiz. 111, 1935 (1997) [JETP 84, 1054 (1997]).10. J. D. Bjorken and S. D. Drell, Relativistic Quantum Mechanics, McGraw-Hill

Book Co., NY, 1964.11. D. A. Owen, Found. of Phys. 24, 273 (1994).12. K. Pachucki and S. G. Karshenboim, J. Phys. B: At. Mol. Opt. Phys. 28, L221

(1995).13. J. L. Friar, J. Martorell, and D. W. L. Sprung, Phys. Rev. A 56, 4579 (1997).14. J. R. Sapirstein and D. R. Yennie, in Quantum Electrodynamics, ed. T. Kinoshita

(World Scientific, Singapore, 1990), p.560.15. J. L. Friar, J. Martorell, and D. W. L. Sprung, Phys. Rev. A 59, 4061 (1999).16. L. A. Borisoglebsky and E. E. Trofimenko, Phys. Lett. B 81, 175 (1979).17. J. L. Friar, Ann. Phys. (NY) 122, 151 (1979).18. C. Zemach, Phys. Rev. 104, 1771 (1956).19. J. L. Friar and I. Sick, Phys. Rev. A 72, 040503(R) (2005).20. J. L. Friar and G. L. Payne, Phys. Rev. A 56, 5173 (1997).21. R. N. Faustov and A. P. Martynenko, Phys. Rev. A 67, 052506 (2003); A. P.

Martynenko and R. N. Faustov, Yad. Phys. 67, 477 (2004) [Phys. Atom. Nuclei67, 457 (2004)].

22. R. N. Faustov and A. P. Martynenko, Yad. Phys. 63, 915 (2000) [Phys. Atom.Nuclei 63, 845 (2000)].

23. T. E. O. Erickson and J. Hufner, Nucl. Phys. B 47, 205 (1972).24. J. Bernabeu and C. Jarlskog, Nucl. Phys. B 60, 347 (1973).25. J. Bernabeu and C. Jarlskog, Nucl. Phys. B 75, 59 (1974).26. S. A. Startsev, V. A. Petrun’kin, and A. L. Khomkin, Yad. Fiz. 23, 1233 (1976)

[Sov. J. Nucl. Phys. 23, 656 (1976)].


27. J. Bernabeu and C. Jarlskog, Phys. Lett. B 60, 197 (1976).28. J. L. Friar, Phys. Rev. C 16, 1540 (1977).29. R. Rosenfelder, Nucl. Phys. A 393, 301 (1983).30. J. Bernabeu and T. E. O. Ericson, Z. Phys. A - Atoms and Nuclei 309, 213

(1983).31. I. B. Khriplovich and R. A. Sen’kov, Novosibirsk preprint, nucl-th/9704043,

April 1997.32. B. E. MacGibbon, G. Garino, M. A. Lucas et al, Phys. Rev. C 52, 2097 (1995).33. I. B. Khriplovich and R. A. Sen’kov, Phys. Lett. A 249, 474 (1998).34. I. B. Khriplovich and R. A. Sen’kov, Phys. Lett. B 481, 447 (2000).35. R. Rosenfelder, Phys. Lett. B 463, 317 (1999).36. D. Babusci, G. Giordano, and G. Matone, Phys. Rev. C 57, 291 (1998).37. R. N. Faustov and A. P. Martynenko, Mod. Phys. Lett. A 16, 507 (2001).38. J. Martorell, D. W. Sprung, and D. C. Zheng, Phys. Rev. C 51, 1127 (1995).39. A. I. Milshtein, I. B. Khriplovich, and S. S. Petrosyan, Zh. Eksp. Teor. Fiz. 109,

1146 (1996) [JETP 82, 616 (1996)].40. Y. Lu and R. Rosenfelder, Phys. Lett. B 319, 7 (1993); B 333, 564(E) (1994).41. W. Leidemann and R. Rosenfelder, Phys. Rev. C 51, 427 (1995).42. J. L. Friar and G. L. Payne, Phys. Rev. C 55, 2764 (1997).43. J. L. Friar and G. L. Payne, Phys. Rev. C 56, 619 (1997).44. I. Sick and D. Trautman, Nucl. Phys. A637, 559 (1998).45. E. Borie, Phys. Rev. Lett. 47, 568 (1981).46. G. P. Lepage, D. R. Yennie, and G. W. Erickson, Phys. Rev. Lett. 47, 1640

(1981).47. M. I. Eides and H. Grotch, Phys. Lett. B 308, 389 (1993).48. M. I. Eides, H. Grotch, and V. A. Shelyuto, Phys. Rev. A 55, 2447 (1997).49. M. I. Eides and H. Grotch, Phys. Rev. A56, R2507 (1997).50. A. I. Milstein, O. P. Sushkov, and I. S. Terekhov, Phys. Rev. Lett. 28, 283003

(2002).51. A. I. Milstein, O. P. Sushkov, and I. S. Terekhov, Phys. Rev. A 67, 062103

(2003).52. J. L. Friar, Zeit. f. Physik A 292, 1 (1979); ibid. 303, 84 (1981)53. D. J. Hylton, Phys. Rev. A 32, 1303 (1985).54. K. Pachucki, Phys. Rev. A 48, 120 (1993).55. A. I. Milstein, O. P. Sushkov, and I. S. Terekhov, Phys. Rev. A 67, 062111

(2003).56. M. I. Eides, Phys. Rev. A 53, 2953 (1996).

7

Lamb Shift in Light Muonic Atoms

Theoretically, light muonic atoms have two main special features as comparedwith the ordinary electronic hydrogenlike atoms, both of which are connectedwith the fact that the muon is about 200 times heavier than the electron1.First, the role of the radiative corrections generated by the closed electronloops is greatly enhanced, and second, the leading proton size contributionbecomes the second largest individual contribution to the energy shifts afterthe polarization correction.

The reason for an enhanced contribution of the radiatively correctedCoulomb potential in Fig. 2.2 may be easily explained. The characteristicdistance at which the Coulomb potential is distorted by the polarization in-sertion is determined by the electron Compton length 1/me and in the caseof electronic hydrogen it is about 137 times less than the average distancebetween the atomic electron and the Coulomb source 1/(meZα). This is thereason why even the leading polarization contribution to the Lamb shift in(2.5) is so small for ordinary hydrogen. The situation with muonic hydrogen iscompletely different. This time the average radius of the muon orbit is aboutrat ≈ 1/(mZα) and is of order of the electron Compton length rC ≈ 1/me, therespective ratio is about rat/rC ≈ me/(mZα) ≈ 0.7, and the muon spends asignificant part of its life inside the region of the strongly distorted Coulombpotential. Qualitatively one can say that the muon penetrates deep in thescreening polarization cloud of the Coulomb center, and sees a larger un-screened charge. As a result the binding becomes stronger, and for examplethe 2S-level in muonic hydrogen in Fig. 7.1 turns out to be lower than the2P -level [1], unlike the case of ordinary hydrogen where the order of levels isjust the opposite. In this situation the polarization correction becomes by farthe largest contribution to the Lamb shift in muonic hydrogen.

1 Discussing light muonic atoms we will often speak about muonic hydrogen butalmost all results below are valid also for another phenomenologically interestingcase, namely muonic helium. In the Sections on light muonic atoms, m is themuon mass, M is the proton mass, and me is the electron mass.


132 7 Lamb Shift in Light Muonic Atoms

The relative contribution of the leading proton size contribution to theLamb shift interval in electronic hydrogen is about 10−4. It is determinedmainly by the ratio of the proton size contribution to the leading logarithmi-cally enhanced Dirac electron form factor slope contribution in (3.7) (whichis much larger than the polarization contribution for electronic hydrogen).The relatively larger role of the leading proton size contribution in muonichydrogen may also be easily understood qualitatively. Technically the lead-ing proton radius contribution in (6.3) is of order (Zα)4m3〈r2〉, where m isthe mass of the light particle, electron or muon in the case of ordinary andmuonic hydrogen, respectively. We thus see that the relative weight of theleading proton charge radius contribution to the Lamb shift, in comparisonwith the standard nonrecoil contributions, is enhanced in muonic hydrogenby the factor (m/me)2 in comparison with the relative weight of the leadingproton charge radius contribution in ordinary hydrogen, and it becomes largerthan all other standard nonrecoil and recoil contributions. Overall the weightof the leading proton radius contribution in the total Lamb shift in muonichydrogen is determined by the ratio of the proton size contribution to theleading electron polarization contribution. In electronic hydrogen the ratioof the proton radius contribution and the leading polarization contribution isabout 5×10−3, and is much larger than the weight of the proton charge radiuscontribution in the total Lamb shift. In muonic hydrogen this ratio is 10−2,four times larger than the ratio of the leading proton size contribution andthe leading polarization correction in electronic hydrogen. Both the leadingproton size correction and the leading vacuum polarization contribution areparametrically enhanced in muonic hydrogen, and an extra factor four in theirratio is due to an additional accidental numerical enhancement.

Below we will discuss corrections to the Lamb shift in muonic hydrogen,with an emphasis on the classic 2P − 2S Lamb shift, having in mind the ex-periment on measurement of this interval which is now under way [2] (see alsoSubsect. 12.1.10). Being interested in theory, we will consider even those cor-rections to the Lamb shift which are an order of magnitude smaller than theexpected experimental precision 0.008 meV. Such corrections could becomephenomenologically relevant for muonic hydrogen in the future. Another rea-son to consider these small corrections is that many of them scale as powersof the parameter Z, and produce larger contributions for atoms with higherZ. Hence, even being too small for hydrogen they become phenomenologicallyrelevant for muonic helium where Z = 2. In the spirit of nonrelativistic QEDused in other parts of this work we start with the nonrelativistic approxima-tion for muonic hydrogen. Other treatments of muonic hydrogen which avoidnonrelativistic expansion in Zα and start with the Dirac-Coulomb wave func-tions when possible exist in the literature (see, e.g, [3] and references therein).We prefer nonrelativistic approach because it allows a more systematic treat-ment, separation, and estimates of numerous small contributions.

7.1 Closed Electron-Loop Contributions 133

Fig. 7.1. Muonic hydrogen energy levels

7.1 Closed Electron-Loop Contributionsof Order αn(Zα)2m

7.1.1 Diagrams with One External Coulomb Line

7.1.1.1 Leading Polarization Contribution of Order α(Zα)2m

The effects connected with the electron vacuum polarization contributions inmuonic atoms were first quantitatively discussed in [4]. In electronic hydrogenpolarization loops of other leptons and hadrons considered in Subsect. 3.2.5played a relatively minor role, because they were additionally suppressed bythe typical factors (me/m)2. In the case of muonic hydrogen we have to dealwith the polarization loops of the light electron, which are not suppressed atall. Moreover, characteristic exchange momenta mZα in muonic atoms arenot small in comparison with the electron mass me, which determines themomentum scale of the polarization insertions (m(Zα)/me ≈ 1.5). We seethat even in the simplest case the polarization loops cannot be expanded in theexchange momenta, and the radiative corrections in muonic atoms induced bythe electron loops should be calculated exactly in the parameter m(Zα)/me.


Electron polarization insertion in the photon propagator in Fig. 2.2 inducesa correction to the Coulomb potential, which may be easily written in the form[5]

δV CV P (r) = −Zα

r

2α

3π

∫ ∞

1

dζe−2merζ

(

1 +1

2ζ2

) √ζ2 − 1ζ2

. (7.1)

The respective correction to the energy levels is given by the expectation valueof this perturbation potential

∆Enl = 〈nlm|δV |nlm〉 =∫ ∞

0

drR2nl(r)δV (r)r2

= −2α2Z

3π

∫ ∞

0

rdr

∫ ∞

1

dζR2nl(r)e

−2merζ

(

1 +1

2ζ2

) √ζ2 − 1ζ2

, (7.2)

where

Rnl(r) = 2(

mrZα

n

) 32

√

(n − l − 1)!n[(n + l)!]3

(2mrZα

nr

)l

e−mrZα

n r

× L2l+1n−l−1

(2mrZα

nr

)

(7.3)

is the radial part of the Schrodinger-Coulomb wave function in (1.1) (butnow it depends on the reduced mass), and L2l+1

n−l−1 is the associated Laguerrepolynomial, defined as in [6, 7]

L2l+1n−l−1(x) =

n−l−1∑

i=0

(−1)i[(n + l)!]2

i!(n − l − i − 1)!(2l + i + 1)!xi . (7.4)

The radial wave functions depend on radius only via the combination ρ =rmrZα and it is convenient to write it explicitly as a function of this dimen-sionless variable

Rnl(r) = 2(

mrZα

n

) 32

fnl

( ρ

n

)

, (7.5)

where

fnl

( ρ

n

)

≡√

(n − l − 1)!n[(n + l)!]3

(2ρ

n

)l

e−ρn L2l+1

n−l−1

(2ρ

n

)

. (7.6)

Explicit dependence of the leading polarization correction on the parame-ters becomes more transparent after transition to the dimensionless integra-tion variable ρ [4]

∆E(1)nl = −8α(Zα)2

3πn3Q

(1)nl (β)mr , (7.7)


where

Q(1)nl (β) ≡

∫ ∞

0

ρdρ

∫ ∞

1

dζf2nl

( ρ

n

)

e−2ρζβ

(

1 +1

2ζ2

) √ζ2 − 1ζ2

, (7.8)

and β = me/(mrZα). The integral Qnl(β) may easily be calculated nu-merically for arbitrary n. It was calculated analytically for the lower levelsn = 1, 2, 3 in [8, 9], and later these results were confirmed numerically in [10].Analytic results for all states with n = l + 1 were obtained in [11].

The leading electron vacuum polarization contribution to the Lamb shiftin muonic hydrogen in (7.8) is of order α(Zα)2m. Recall that the leadingvacuum polarization contribution to the Lamb shift in electronic hydrogenin (2.6) is of order α(Zα)4m. Thus, the relative magnitude of the lead-ing polarization correction in muonic hydrogen is enhanced by the factor1/(Zα)2 ∼ (m/me)2. This means that the electronic vacuum polarizationgives by far the largest contribution to the Lamb shift in muonic hydrogen.The magnitude of the energy shift in (7.7) is determined also by the dimen-sionless integral Qnl(β). At the physical value of β = me/(mrZα) ≈ 0.7 thisintegral is small (Q(1)

10 (β) ≈ 0.061, Q(1)20 (β) ≈ 0.056, Q

(1)21 (β) ≈ 0.0037) and

suppresses somewhat the leading electron polarization contribution.The expression for Q

(1)nl (β) in (7.8) is valid for any β, in particular we can

consider the case when m = me. Then β = m/(mrZα) � 1, and it is easyto show that the leading term in the expansion of the result in (7.7) over 1/βcoincides with the leading polarization contribution in electronic hydrogen in(2.6).

Numerically, contribution to the 2P − 2S Lamb shift in muonic hydrogenis equal to

∆E(2P − 2S) = 205.0074 meV . (7.9)

7.1.1.2 Two-Loop Electron Polarization Contributionof Order α2(Zα)2m

In electronic hydrogen the leading contribution generated by the two-loop ir-reducible polarization operator in Fig. 3.4 is of order α2(Zα)4m (see (3.18)),and is determined by the leading low-frequency term in the polarization op-erator. The reducible diagram in Fig. 7.2 with two one-loop insertions in theCoulomb photon does not generate a correction of the same order in electronichydrogen because it vanishes at the characteristic atomic momenta, which aresmall in comparison with the electron mass. In the case of muonic hydrogenatomic momenta are of order of the electron mass and the two-loop irreducibleand reducible polarization insertions in Fig. 7.2 both generate contributionsof order α2(Zα)2m and should be considered simultaneously.

The two-loop electron polarization contribution to the Lamb shift may becalculated exactly like the one-loop contribution, the only difference is that onehas to use as a perturbation potential the two-loop correction to the Coulomb


Fig. 7.2. Two-loop polarization insertions in the Coulomb photon

potential from [12]. We use it in the form of the integral representation derivedin [13] (see also [14])

δV (2)(r) =Zα

r

(α

π

)2∫ ∞

1

dζe−2merζ

{(13

54ζ2+

7108ζ4

+2

9ζ6

)√

ζ2 − 1

+(

−449ζ

+2

3ζ3+

54ζ5

+2

9ζ7

)

ln[

ζ +√

ζ2 − 1]

+(

43ζ2

+2

3ζ4

)√

ζ2 − 1 ln[8ζ(ζ2 − 1)

]+

(

− 83ζ

+2

3ζ5

)

F (ζ)}

,

(7.10)

where

F (ζ) =∫ ∞

ζ

dx

[3x2 − 1

x(x2 − 1)

]

ln[x+√

x2 − 1]− 1√x2 − 1

ln[8x(x2−1)] . (7.11)

Then we easily obtain

∆E(2)nl =

4α2(Zα)2

π2n3Q

(2)nl (β)mr , (7.12)

where

Q(2)nl (β) ≡

∫ ∞

0

ρdρ

∫ ∞

1

dζf2nl

( ρ

n

)

e−2ρζβ

{(13

54ζ2+

7108ζ4

+2

9ζ6

)√

ζ2 − 1

+(

−449ζ

+2

3ζ3+

54ζ5

+2

9ζ7

)

ln[

ζ +√

ζ2 − 1]

+(

43ζ2

+2

3ζ4

)√

ζ2 − 1 ln[8ζ(ζ2 − 1)

]+

(

− 83ζ

+2

3ζ5

)

F (ζ)}

.

(7.13)

Numerically, this correction for the 2P −2S Lamb shift was first calculatedin [10]

∆E(2P − 2S) = 1.5079 meV. (7.14)

7.1.1.3 Three-Loop Electron Polarization of Order α3(Zα)2m

As in the case of the two-loop electron polarization insertions in the exter-nal Coulomb line, reducible and irreducible three-loop polarization insertions


Fig. 7.3. Three-loop polarization insertions in the Coulomb photon

enter on par in muonic hydrogen, and we have to consider all respective cor-rections to the Coulomb potential in Fig. 7.3. One-, two-, and three-looppolarization operators were in one form or another calculated in the literature[5, 12, 15, 16, 17]. Numerical calculation of the respective contribution the2P − 2S splitting in muonic hydrogen was performed in [18]

∆E(2P − 2S) = 0.083 53 (1)α3(Zα)2

π3mr ≈ 0.0053 meV . (7.15)

7.1.2 Diagrams with Two External Coulomb Lines

7.1.2.1 Reducible Diagrams. Contributions of Order α2(Zα)2m

In electronic hydrogen characteristic exchanged momenta in the diagram inFig. 7.4 were determined by the electron mass, and since this mass in elec-tronic hydrogen is large in comparison with the characteristic atomic momentawe could ignore binding and calculate this diagram in the scattering approxi-mation. As a result the respective contribution was suppressed in comparisonwith the leading polarization contribution, not only by an additional factor αbut also by an additional factor Zα. The situation is completely different inthe case of muonic hydrogen. This time atomic momenta are just of order ofthe electron mass, one cannot neglect binding, and the additional suppressionfactor Zα is missing. As a result the respective correction in muonic hydro-gen is of the same order α2(Zα)2 as the contributions of the diagrams withreducible and irreducible two-loop polarization insertions in one and the sameCoulomb line considered above.

Formally the contribution of diagram Fig. 7.4 is given by the standardquantum mechanical second order perturbation theory term. Summation overthe intermediate states, which accounts for binding, is realized with the help

Fig. 7.4. Perturbation theory contribution with two one-loop polarization insertions


Fig. 7.5. Perturbation theory contribution of order α3(Zα)2 with polarization in-sertions

of the reduced Green function. Convenient closed expressions for the reducedGreen function in the lower states were obtained in [19] and independentlyreproduced in [20]. Numerical calculation of the contribution to the 2P − 2Ssplitting leads to the result [20, 21]

∆E(2P − 2S) = 0.012444α2(Zα)2

9π2mr = 0.1509 meV . (7.16)

7.1.2.2 Reducible Diagrams. Contributions of order α3(Zα)2m

As in the case of corrections of order α2(Zα)2m, not only the diagrams inFig. 7.3 with insertions of polarization operators in one and the same exter-nal Coulomb line but also the reducible diagrams Fig. 7.5 with polarizationinsertions in different external Coulomb lines generate corrections of orderα3(Zα)2m. Respective contributions were calculated in [18] with the help ofthe subtracted Coulomb Green function from [20]

∆E(2P − 2S) = 0.036 506 (4)α3(Zα)2

π3mr ≈ 0.0023 meV . (7.17)

Total contribution of order α3(Zα)2m is a sum of the contributions in(7.15) and (7.17) [18]

∆E(2P − 2S) = 0.120 045 (12)α3(Zα)2

π3mr = 0.0076 meV . (7.18)

7.2 Relativistic Corrections to the Leading PolarizationContribution with Exact Mass Dependence

The leading electron polarization contribution in (7.7) was calculated in thenonrelativistic approximation between the Schrodinger-Coulomb wave func-tions. Relativistic corrections of relative order (Zα)2 to this contribution mayeasily be obtained in the nonrecoil limit. To this end one has to calculatethe expectation value of the radiatively corrected potential in (7.1) betweenthe relativistic Coulomb-Dirac wave functions instead of averaging it with thenonrelativistic Coulomb-Schrodinger wave functions.

7.2 Relativistic Corrections to the Leading Polarization Contribution 139

Numerical calculations of the Uehling potential contribution to the energyshift without expansion over Zα (and therefore with account of the leadingnonrecoil relativistic corrections of order α(Zα)4m) are abundant in the liter-ature, see. e.g., [22], and references in the review [23]. Analytic results withoutexpansion over Zα were obtained for the states with n = l + 1, j = l + 1/2[24]. All these results might be very useful for heavy muonic atoms. However,in the case of muonic hydrogen with a relatively large muon-proton mass ratiorecoil corrections to the nonrecoil relativistic corrections of order α(Zα)4mmay be rather large, while corrections of higher orders in Zα are expected tobe very small. In such conditions it is reasonable to adopt another approachto the relativistic corrections, and try to calculate them from the start in thenonrecoil approximation with exact dependence on the mass ratio.

In the leading nonrelativistic approximation the one-loop electron polariza-tion insertion in the Coulomb photon generates a nontrivial correction (7.1) tothe unperturbed Coulomb binding potential in muonic hydrogen, which maybe written as a weighted integral of a potential corresponding to an exchangeby a massive photon with continuous mass

√t′ ≡ 2meζ

δV CV P (r) =

2α

3π

∫ ∞

1

dζ

(

1 +1

2ζ2

) √ζ2 − 1ζ2

(

−Zα

re−2meζr

)

. (7.19)

This situation is radically different from the case of electronic hydrogenwhere inclusion of the electron loop in the photon propagator generates effec-tively a δ-function correction to the Coulomb potential (compare discussionin Sect. 2.2).

Calculation of the leading relativistic corrections to the nonrelativisticelectronic vacuum polarization contribution may be done in the framework ofthe Breit approach used in Sect. 3.1 to derive leading relativistic correctionsto the ordinary one-photon exchange. All we need to do now is to derivean analogue of the Breit potential, which is generated by the exchange of onephoton with electron-loop insertion in Fig. 7.6. This derivation is facilitated bythe well known observation that the dispersion relation for the polarizationoperator allows one to represent the one-loop radiatively corrected photonpropagator as an integral over continuous photon mass (see, e.g., [5]). Theneverything one has to do to derive the analogue of the Breit potential is toobtain an expression for the Breit potential corresponding to the exchangeby a massive photon and then to integrate over the effective photon mass.Derivation of the massive Breit potential proceeds exactly as for the massless

Fig. 7.6. One-photon exchange with one-loop polarization insertion


case, and one obtains [20] (as in (3.2) we omit below all terms in the massiveBreit potential which depend on the spin of the heavy particle since we donot consider hyperfine structure now)

VBrV P (

√t′ ≡ 2meζ) =

Zα

2

(1

m2+

1M2

)(

πδ3(r) − m2eζ

2

re−2meζr

)

− Zαm2eζ

2

mM

e−2meζr

r(1 − meζr)

− Zα

2mMpi e

−2meζr

r

(

δij +rirj

r2(1 + 2meζr)

)

pj

+Zα

r3

(1

4m2+

12mM

)

e−2meζr(1 + 2meζr)[r × p] · σ .

(7.20)

Then the analogue of the Breit potential induced by the electron vacuumpolarization insertion is given by the integral

V BrV P =

2α

3π

∫ ∞

1

dζ

(

1 +1

2ζ2

) √ζ2 − 1ζ2

VV P (2meζ) . (7.21)

Calculation of the leading recoil corrections of order α(Zα)4 becomes nowalmost trivial. One has to take into account that in our approximation theanalogue of the Breit Hamiltonian in (3.3) has the form [20]

H =p2

2m+

p2

2M− Zα

r+ VBr + V C

V P + V BrV P , (7.22)

where VBr was defined in (3.2).Then the leading relativistic corrections of order α(Zα)4 may be easily ob-

tained as a sum of the first and second order perturbation theory contributionscorresponding to the diagrams in Fig. 7.7 [20]

∆E = 〈V BrV P 〉 + 2〈VBrG

′(En)V CV P 〉 . (7.23)

Numerical calculation of the respective contribution to the 2P − 2S Lambshift leads to the result [21, 25]

∆E(2P − 2S) = 0.0169 meV . (7.24)

In another approach this result was confirmed in [3].

Fig. 7.7. Relativistic corrections to the leading electron polarization contribution

7.3 Higher Order Electron-Loop Polarization Contributions 141

7.3 Higher Order Electron-Loop PolarizationContributions

7.3.1 Wichmann-Kroll Electron-Loop Contributionof Order α(Zα)4m

Contribution of the Wichmann-Kroll diagram in Fig. 3.16 with three externalfields attached to the electron loop [26] may be considered in the same way asthe polarization insertions in the Coulomb potential, and as we will see belowit generates a correction to the Lamb shift of order α(Zα)4m.

A convenient representation for the Wichmann-Kroll polarization potentialwas obtained in [13]

δV WK(r) =Zα

r

α(Zα)2

π

∫ ∞

0

dζe−2meζr 1ζ4

[

−π2

12

√

ζ2 − 1θ(ζ − 1)

+∫ ζ

0

dx√

ζ2 − x2f(x)

]

, (7.25)

where

f(x) = −2xLi2(x2) − x ln2(1 − x2) +1 − x2

x2ln(1 − x2) ln

1 + x

1 − x

+1 − x2

4xln2 1 + x

1 − x+

2 − x2

x(1 − x2)ln(1 − x2)

+3 − 2x2

1 − x2ln

1 + x

1 − x− 3x (7.26)

for x < 1, and

f(x) =1x2

Li2

(1x2

)

− 3x2 + 12x

[

Li2

(1x

)

− Li2

(

− 1x

)]

−2x2 − 12x2

[

ln2

(

1 − 1x2

)

+ ln2 x + 1x − 1

]

− (2x − 1) ln(

1 − 1x2

)

lnx + 1x − 1

+3x2 + 1

4xln2 x + 1

x − 1− 2 ln x ln

(

1 − 1x2

)

− 3x2 + 12x

lnx lnx + 1x − 1

+[

5 − x(3x2 − 2)x2 − 1

]

ln(

1 − 1x2

)

+[3x2 + 2

x− 3x2 − 2

x2 − 1

]

lnx + 1x − 1

+ 3 ln x − 3 (7.27)

for x > 1.This representation allows us to calculate the correction to the Lamb shift

in the same way as we have done above for the Uehling and Kallen-Sabry


potentials in (7.7) and (7.12), respectively. Let us write the potential in theform

δV WK(r) =Zα

r

α(Zα)2

πg(mer) . (7.28)

Then the contribution to the energy shift is given by the expression

∆E(WK)nl =

4α(Zα)4

πn3Q

(WK)nl (β)mr , (7.29)

where

Q(WK)nl (β) ≡

∫ ∞

0

ρdρf2nl

(ρ

n

)

g(ρβ) . (7.30)

The Uehling and Kallen-Sabry potentials are attractive, and shift the en-ergy levels down. Physically this corresponds to the usual charge screeningin QED, and one can say that at finite distances the muon sees a largerunscreened charge of the nucleus. From this point of view the Uehling andKallen-Sabry potentials are just the attractive potentials corresponding tothe excess of the bare charge over physical charge.

The case of the Wichmann-Kroll potential is qualitatively different. Dueto current conservation the total charge which induces the Wichmann-Krollpotential is zero [26]. Spatially the induced charge distribution consists oftwo components: a delta-function induced charge at the origin with the signopposite to the sign of the nuclear charge, and a spatially distributed com-pensating charge of the same sign as the nucleus. The radius of this spatialdistribution is roughly equal to the electron Compton length. As a result themuon which sees the nucleus from a finite distance experiences net repulsion,the Wichmann-Kroll potential shifts the levels up, and gives a positive con-tribution to the level shift (the original calculation in [27] produced a resultwith a wrong sign and magnitude).

Practical calculations of the Wichmann-Kroll contribution are greatly fa-cilitated by convenient approximate interpolation formulae for the potentialin (7.25). One such formula was obtained in [14] fitting the results of thenumerical calculation of the potential from [28]

g(x) = 0.361 662 331 exp[0.372 807 9 x

−√

4.416 798 x2 + 11.399 11 x + 2.906 096] . (7.31)

This expression fits the exact potential in the interval 0.01 < x < 1.0 with anaccuracy of about 1%, and due to an exponential decrease of the wave func-tions and smallness of the potential at large distances, it may be safely usedfor calculations at all x. After numerical calculation with this interpolationformula we obtain for the 2P − 2S Lamb shift [29]

∆E(2P − 2S) = −0.00103 meV . (7.32)

A result two times smaller than this contribution was obtained earlier in [20].Calculations [3] with another convenient interpolation formula [23, 30] for the


potential in (7.25) confirm the result in (7.32). One more way to calculate theWichmann-Kroll contribution numerically, is to use for the small values ofthe argument an asymptotic expansion of the potential in (7.25), which wasobtained in [13], and for the large values of the argument the interpolationformula from [31]. Calculations in both these approaches reproduce the nu-merical value for the 2P − 2S Lamb shift in (7.32). Besides calculations withall three forms of the interpolation formulae, we also calculated the energyshift for muonic helium with Z = 2, and reproduced the well known old heliumresults [31, 32, 23]. Therefore we consider the Wichmann-Kroll contributionin (7.32) to be firmly established.

7.3.2 Light by Light Electron-Loop Contributionof Order α2(Zα)3m

Light by light electron-loop contribution to the Lamb shift in Fig. 3.11 (e) inmuonic atoms was considered in [3, 33, 34, 35, 36, 37]. This is a correction oforder α2(Zα)3 in muonic hydrogen. Characteristic momenta in the electronpolarization loop are of the order of the atomic momenta in muonic hydrogen,and hence, one cannot neglect the atomic momenta calculating the matrixelement of this kernel as it was done in the case of electronic hydrogen. Aninitial numerical estimate in [33] turned out to be far too large, and consistentmuch smaller numerical estimates were obtained in [34, 35, 36].

The momentum-space potential generated by the light by light diagramsand the respective contribution to the energy shifts in heavy atoms were cal-culated numerically in [36]. Certain approximate expressions for the effectivemomentum space potential were obtained in [23, 35, 36]. After extensive nu-merical work the electron-loop light by light scattering contribution was cal-culated for muonic helium [32, 23], and turned out to be equal to 0.02 meVfor the 2P −2S interval. This approach was applied to muonic hydrogen withthe result [3]

∆E(2P − 2S) = 0.00135 (15) meV . (7.33)

Unfortunately, momenta of the external wave functions and the loop mo-menta were decoupled in this calculation, and as we explained above this is anillegitimate approximation for muonic hydrogen. Therefore, the contributionin (7.33) is at best an order of magnitude estimate, and we do not include thisresult in the collection of the contributions to the Lamb shift in muonic hy-drogen in Table 7.1. Further investigation of the purely electrodynamic lightby light contribution to the 2P − 2S interval in muonic hydrogen is clearlywarranted.


Fig. 7.8. Diagram with radiative photon and electron-loop polarization insertionin the Coulomb photon

7.3.3 Diagrams with Radiative Photon and Electron-LoopPolarization Insertion in the Coulomb Photon. Contributionof Order α2(Zα)4m

In electronic hydrogen the leading contributions of diagrams such as Fig. 7.8were generated at the scale of the mass of the light constituent. The dia-grams effectively looked like Fig. 3.11(c), could be calculated in the scatteringapproximation, and produced the corrections of order α2(Zα)5m. In muonichydrogen electron polarization insertion in the Coulomb photon is not sup-pressed at characteristic atomic momenta, and respective contribution to theenergy shift is only α times smaller than the contribution of the diagramswith insertions of one radiative photon in the muon line (leading diagramsfor the Lamb shift in case of electronic hydrogen). One should expect that, inthe same way as the leading Lamb shift contribution in electronic hydrogen,this contribution is also logarithmically enhanced and is of order α2(Zα)4m.This contribution was never calculated completely, the leading logarithmiccontribution was obtained in [20].

The leading logarithmic contribution generated by the diagrams with theradiative photon spanning any number of the Coulomb photons and oneCoulomb photon with electron-loop polarization insertion, the simplest ofwhich is presented in Fig. 7.8, may be calculated by closely following theclassical Bethe calculation [38] of the leading logarithmic contribution to theLamb shift in electronic hydrogen. As is well known, in the dipole approxima-tion the standard subtracted logarithmically divergent (at high frequencies)expression for the Lamb shift may be written in the form [20] (compare, e.g.,[5, 39])

∆E =2α

3πm2

∫

dω

⟨

n

∣∣∣∣p

H − En

H − En + ωp

∣∣∣∣n

⟩

, (7.34)

where H is the nonrelativistic Hamiltonian for the muon in the external fieldequal to the sum of the Coulomb field and radiatively corrected Coulomb fieldfrom (7.1)

Vtot = −Zα

r+ V V P

C . (7.35)

To obtain the leading contribution generated by the integral in (7.34), it issufficient to integrate over the wide logarithmic region mr(Zα)2 � ω � m,


where one can neglect the terms H −En in the denominator. Then one easilyobtains [20]

∆E =2α

3πm2ln

m

mr(Zα)2〈n|p(H − En)p|n〉 . (7.36)

Using the trivial identity

〈n|p(H − En)p|n〉 =〈n|∆(VC + V V P

C )|n〉2

, (7.37)

throwing away the standard leading polarization independent logarithmic cor-rection to the Lamb shift, which is also contained in this expression, and ex-panding the state vectors up to first order in the potential VV P one easilyobtains

∆E =α

3πm2ln

m

mr(Zα)2{〈n|∆V V P

C |n〉 + 2〈n|V V PC G′(En)∆VC |n〉} . (7.38)

This contribution to 2P − 2S splitting was calculated numerically in [20, 21]

∆E(2P − 2S) = −0.005 (1) meV . (7.39)

The uncertainty here is due to the unknown nonlogarithmic terms. Calculationof these nonlogarithmic terms is one of the future tasks in the theory of muonichydrogen.

7.3.4 Electron-Loop Polarization Insertionin the Radiative Photon. Contribution of Order α2(Zα)4m

Contributions of order α2(Zα)4m in muonic hydrogen generated by the two-loop muon form factors have almost exactly the same form as the respectivecontributions in the case of electronic hydrogen. The only new feature is con-nected with the contribution to the muon form factors generated by insertionof one-loop electron polarization in the radiative photon in Fig. 7.9. Respectiveinsertion of the muon polarization in the electron form factors in electronichydrogen is suppressed as (me/m)2, but insertion of a light loop in the muoncase is logarithmically enhanced.

Fig. 7.9. Electron polarization insertion in the radiative photon


The graph in Fig. 7.9 is gauge invariant and generates a correction to theslope of the Dirac form factor, which was calculated in [40]

dF(2)1 (−k2)dk2 |k2=0

= −[19

ln2 m

me− 29

108ln

m

me+

π2

54+

3951296

+ O

(me

m

)]

× 1m2

(α

π

)2

≈ −(

2.216 56 + O

(me

m

))1

m2

(α

π

)2

. (7.40)

Then the contribution to the Lamb shift has the form [40]

∆EF1 = −4π(Zα)|Ψn(0)|2 dF(2)1 (−k2)dk2 |k2=0

=(

2.216 56 + O

(me

m

))4α2(Zα)4

π2n3

(mr

m

)3

m δl0 . (7.41)

We also have to consider the electron-loop contribution to the muon anom-alous magnetic moment

F(2)2 (0) =

[13

lnm

me− 25

36+

π2

4me

m− 4

(me

m

)2

lnm

me+ 3

(me

m

)2

+ O

((me

m

)3)](α

π

)2

≈(

1.094 26 + O

(me

m

))(α

π

)2

. (7.42)

The first two terms in this expression were obtained in [41, 42], and an exactanalytic result without expansion over me/m was calculated in [43, 44].Then one readily obtains for the Lamb shift contribution [23]

∆E|l=0 = 1.094 26α2(Zα)4m

π2n3

(mr

m

)3

,

∆E|l �=0 = 1.094 26α2(Zα)4m

π2n3

j(j + 1) − l(l + 1) − 3/4l(l + 1)(2l + 1)

(mr

m

)2

. (7.43)

Numerically for the 2P 12− 2S 1

2interval we obtain

∆E(

2P 12− 2S 1

2

)

= −0.0015 meV . (7.44)

7.3.5 Insertion of One Electron and One Muon Loops in the sameCoulomb Photon. Contribution of Order α2(Zα)2(me/m)2m

Contribution of the mixed polarization graph with one electron- and onemuon-loop insertions in the Coulomb photon in Fig. 7.10 may be easily calcu-lated by the same methods as the contributions of purely electron loops, and


Fig. 7.10. Electron- and muon-loop polarization insertions in the Coulomb photon

it was first considered in [45]. The momentum space perturbation potentialcorresponding to the mixed loop diagram is given by the expression (factor 2is due to two diagrams)

2Πµ(k2)Πe(k2)

k6, (7.45)

where Πµ(k2) and Πe(k2) are the muon- and electron-loop polarization oper-ators, respectively (compare (2.5)).

The characteristic integration momenta in the matrix element of this per-turbation potential between the Coulomb-Schrodinger wave functions are ofthe atomic scale mZα, and are small in comparison with the muon mass m.Hence, in the leading approximation the muon polarization may be approxi-mated by the first term in its low-frequency expansion

2α

15πm2

Πe(k2)k2

. (7.46)

This momentum space potential is similar to the momentum space potentialcorresponding to insertion of the electron-loop polarization in the Coulombphoton, considered in Subsubsect. 7.1.1.1. The only difference is in the overallmultiplicative constant, and that the respective expression in the case of theone electron polarization insertion contains k4 in the denominator insteadof k2 in (7.46). This means that the mixed loop contribution is suppressedin comparison with the purely electron loops by an additional recoil factor(me/m)2.

Similarly to (7.1) it is easy to write a coordinate space representation forthe perturbation potential corresponding to the diagram in Fig. 7.10

δV (r) =Zα

r

1645

(α

π

)2 (me

m

)2∫ ∞

1

dζe−2meζr

(

1 +1

2ζ2

)√

ζ2 − 1 . (7.47)

Then we easily obtain

∆E(3)nl =

64α2(Zα)2

45π2n3

(me

mµ

)2

Q(3)nl (β)mr , (7.48)

where

Q(3)nl (β) ≡

∫ ∞

0

ρdρ

∫ ∞

1

dζf2nl

( ρ

n

)

e−2ρζβ

(

1 +1

2ζ2

)√

ζ2 − 1 . (7.49)


Fig. 7.11. Hadron polarization insertion in the Coulomb photon

Due to the additional recoil factor (me/m)2 this contribution is suppressedby four orders of magnitude in comparison with the nonrecoil correctionsgenerated by insertion of two electron loops in the Coulomb photon (compare(7.12)). Numerically, for the 2P − 2S interval we obtain

∆E(2P − 2S) = 0.00007 meV . (7.50)

7.4 Hadron Loop Contributions

7.4.1 Hadronic Vacuum Polarization Contributionof Order α(Zα)4m

Masses of pions are only slightly larger than the muon mass, and we shouldexpect that the contribution of the diagram with insertion of the hadronicvacuum polarization in the Coulomb photon in Fig. 7.11 is of the same orderof magnitude as contribution of the respective diagrams with muon vacuumpolarization. The hadronic polarization correction is of order α(Zα)4m. Itdepends only on the leading low-momentum asymptotic term in the hadronicpolarization operator, and has the same form as in the case of electronichydrogen in Subsect. 3.2.5. It was considered in the literature many timesand consistent results were obtained in [18, 46, 47, 48, 49]. According to [50]

∆E(nS) = −0.671 (15) ∆Eµ , (7.51)

where ∆Eµ is the muon-loop polarization contribution to the Lamb shift in(3.30)2.

The latest treatment of this diagram in [51, 52] produced

∆E(nS) = −0.638 (22) ∆Eµ . (7.52)

Respective results for the 2P − 2S splitting are

∆E(2P − 2S) = 0.0113 (3) meV , (7.53)

and∆E(2P − 2S) = 0.0108 (4) meV . (7.54)

2 In the case of muonic hydrogen mr in (3.30) is the muon-proton reduced mass.

7.4 Hadron Loop Contributions 149

As in the case of electronic hydrogen this correction may be hidden inthe main proton radius contribution to the Lamb shift and we ignored it inthe phenomenological discussion of the Lamb shift in electronic hydrogen (seediscussion in Subsect. 6.1.3). However, we include the hadronic polarizationin the theoretical expression for the Lamb shift in muonic hydrogen havingin mind that in the future all radiative corrections should be properly takeninto account while extracting the value of the proton charge radius from thescattering and optical experimental data.

7.4.2 Hadronic Vacuum Polarization Contributionof Order α(Zα)5m

Due to the analogy between contributions of the diagrams with muon andhadron vacuum polarizations, it is easy to see that insertion of hadron vacuumpolarization in one of the exchanged photons in the skeleton diagrams withtwo-photon exchanges generates a correction of order α(Zα)5 (see Fig. 7.12).Calculation of this correction is straightforward. One may even take into ac-count the composite nature of the proton and include the proton form factorsin photon-proton vertices. Such a calculation was performed in [51, 52] andproduced a very small contribution

∆E(2P − 2S) = 0.000047 meV . (7.55)

Fig. 7.12. Hadron polarization contribution of order α(Zα)5

7.4.3 Contribution of Order α2(Zα)4m Induced by Insertionof the Hadron Polarization in the Radiative Photon

The muon mass is only slightly lower than the pion mass, and we shouldexpect that insertion of hadronic vacuum polarization in the radiative pho-ton in Fig. 7.13 will give a contribution to the anomalous magnetic momentcomparable with the contribution induced by insertion of the muon vacuumpolarization.

Respective corrections are written via the slope of the Dirac form factorand the anomalous magnetic moment exactly as in Subsect. 7.3.4. The onlydifference is that the contributions to the form factors are produced by thehadronic vacuum polarization.

Numerically this contribution to the 2P − 2S interval was calculated in[51, 52]


Fig. 7.13. Hadron polarization insertion in the radiative photon

∆E(2P − 2S) = −0.000015 meV , (7.56)

and is too small to be of any practical significance.In the case of electronic hydrogen this hadronic insertion in the radiative

photon is additionally suppressed in comparison with the contribution of theelectron vacuum polarization roughly speaking as (me/mπ)2.

7.4.4 Insertion of One Electron and One Hadron Loopsin the Same Coulomb Photon

Due to similarity between the muon and hadron polarizations, such a correc-tion generated by the diagram in Fig. 7.14 should be of the same order as therespective correction with the muon loop in (7.50) and thus is too small forany practical needs. It may be easily calculated.

Fig. 7.14. Hadron and electron polarization insertions in the Coulomb photon

7.5 Standard Radiative, Recoiland Radiative-Recoil Corrections

All corrections to the energy levels obtained above in the case of ordinaryhydrogen and collected in the Tables 3.2, 3.3, 3.7, 3.8, 3.9, 4.1, 5.1 are stillvalid for muonic hydrogen after an obvious substitution of the muon massinstead of the electron mass in all formulae. These contributions are includedin Table 7.1.

7.6 Nuclear Size and Structure Corrections 151

7.6 Nuclear Size and Structure Corrections

Nuclear size and structure corrections for the electronic hydrogen were con-sidered in Chap. 6 and are collected in Table 7.1. Below we will consider whathappens with these corrections in muonic hydrogen. The form of the mainproton size contribution of order (Zα)4m3

r〈r2〉 from (6.3) does not change

∆E(2P − 2S) = − (Zα)4

12m3

r〈r2〉 . (7.57)

Having in mind that the data from the muonic hydrogen Lamb shift ex-periment will be used for measurement of the rms proton charge radius [2] itis useful to write this correction in the form

∆E(2P − 2S) = −5.1975〈r2〉 meV , (7.58)

where 〈r2〉 is assumed to be measured in fermi squared. For rp ≡√〈r2〉 =

0.895 (18) fm, we obtain

∆E(2P − 2S) = −4.163 (188) meV . (7.59)

7.6.1 Nuclear Size and Structure Corrections of Order (Zα)5m

7.6.1.1 Nuclear Size Corrections of Order (Zα)5m

The nuclear size correction of order (Zα)5m in muonic hydrogen in the ex-ternal field approximation is given by (6.13). Unlike ordinary hydrogen, inmuonic hydrogen it makes a difference if we use m4

r or mm3r in this expression

(compare footnote after (6.13)). We will use the factor m4r as obtained in [53]

∆E(2P − 2S) =(Zα)5

24m4

r〈r3〉(2) , (7.60)

but the true check of this factor could be provided only by calculation of theleading recoil correction to this contribution. The total nuclear size correc-tion of order (Zα)5 with account for recoil is given by the sum of two-photondiagrams in Figs. 6.2 and 6.3. As in the case of electronic hydrogen, due tolarge effective integration momenta, it is sufficient to calculate these diagramsin the scattering approximation. Results of these calculations again may bewritten in terms of the third Zemach moment (6.15). The third Zemach mo-ment is a characteristic of electric charge density in the proton, and it cannotbe model independently connected with rms proton charge radius. However,using model dependent proton form factors one can obtain 〈r3〉(2) in terms ofthe rms radius rp [3, 21, 20, 53, 54]. The contribution to the Lamb shift withaccount for recoil and with the dipole form factor was calculated in [20, 21, 54]with the result

∆E(2P − 2S) = 0.0363 r3p meV , (7.61)


where rp is assumed to be measured in fermi. Comparing this result withthe result [55] calculated according to (7.60) we see that the additional recoilcontribution turns out to be about 0.0015 meV, which is negligible for thecurrent experimental perspective. This justifies formula (7.60) with the factorm4

r.The latest calculation in [3] with (7.60) and the dipole form factor pro-

duced∆E(2P − 2S) = 0.0347 r3

p meV , (7.62)

or numerically∆E(2P − 2S) = 0.0249 (20) meV . (7.63)

A model independent analysis of the proton-electron scattering and appli-cation of the expression in (6.14) produced the value 〈r3〉(2) = 2.71 (13) fm3

for the third Zemach moment of the proton [56]. Numerically, the respectivecontribution to the Lamb shift is

∆E(2P − 2S) = 0.0247 (12) meV . (7.64)

7.6.1.2 Nuclear Polarizability Contribution of Order (Zα)5mto S-Levels

Calculation of the nuclear structure corrections of order (Zα)5m generatedby the diagrams in Fig. 6.4 follows the same route as in the case of electronichydrogen in Subsect. 6.2.2 starting with the forward Compton scattering am-plitude. The only difference is that due to the relatively large mass of themuon the logarithmic approximation is not valid any more, and one has tocalculate the integrals more accurately. According to [21, 57]

∆E = −0.095 (18)n3

δl0 meV , (7.65)

while the result in [58] is

∆E = −0.136 (30)n3

δl0 meV . (7.66)

We think that the reasons for a minor discrepancy between these resultsare the same as for a similar discrepancy in the case of electronic hydrogen(see discussion in Subsect. 6.2.2). The improved result in [54] is

∆E = −0.129n3

δl0 meV . (7.67)

However, the experimental data on the proton form factors used in [54] con-tained some misprints. Corrections to this experimental data were taken intoaccount in [21].

We will adopt the result in (7.65) for further discussion. The respectivecontribution to the 2P − 2S splitting is [21]

∆E(2P − 2S) = 0.012 (2) meV . (7.68)


7.6.2 Nuclear Size and Structure Corrections of Order (Zα)6m

The nuclear polarizability contribution of order (Zα)6m was considered abovein Subsect. 6.3.1, and we may directly use the expression for this energy shift in(6.30) for muonic hydrogen. In electronic hydrogen the nuclear size correctionof order (Zα)6m is larger than the nuclear size and structure corrections oforder (Zα)5m. This enhancement is due to the smallness of the electron mass(see discussion in Subsect. 6.3.2). The muon mass is much larger than theelectron mass. As a result this hierarchy of the corrections does not survivein muonic hydrogen, and corrections of order (Zα)6m are smaller than thecorrections of the previous order in Zα. Numerically, the nuclear polarizabilitycontribution of order (Zα)6m to the 2P − 2S Lamb shift in muonic hydrogenis about 5 × 10−6 meV, and is negligible.

Nuclear size corrections of order (Zα)6m to the S levels were calculatedin [59, 53] and were discussed above in Subsect. 6.3.2 for electronic hydrogen.Respective formulae may be directly used in the case of muonic hydrogen. Dueto the smallness of this correction it is sufficient to consider only the leadinglogarithmically enhanced contribution to the energy shift from (6.35) [21]

∆E = −2(Zα)6

3n3m3

r〈r2〉[⟨

lnZα

n

⟩

− 23m2

r〈r2〉]

. (7.69)

We have restored in this equation a small second term from [53] which,due to the smallness of the electron mass was omitted in the case of electronichydrogen in (6.35).

Numerically the respective contribution to the 2P −2S energy shift is [21]

∆E(2P − 2S) = −0.0009 (3)meV . (7.70)

The error of this contribution may easily be reduced if we would use the totalexpressions in (6.35) and (6.36) for its calculation.

The nuclear size correction of order (Zα)6m to P levels from (6.39) givesan additional contribution 4 × 10−5 meV to the 2P 1

2− 2S 1

2energy splitting

and may safely be neglected.

7.6.3 Radiative Corrections to the Nuclear Finite Size Effect

Radiative corrections to the leading nuclear finite size contribution were con-sidered in Subsect. 6.4.1. Respective results may be directly used for muonichydrogen, and numerically we obtain

∆E(2P − 2S) = 0.0006〈r2〉 = 0.0005 meV . (7.71)

This contribution is dominated by the diagrams with radiative photon inser-tions in the muon line. As usual in muonic hydrogen a much larger contri-bution is generated by the electron loop insertions in the external Coulomb


Fig. 7.15. Electron polarization corrections to the leading nuclear size effect

photons. In muonic hydrogen, even after insertion of the electron loop in theexternal photon, the effective integration momenta are still of the atomic scalek ∼ mZα ∼ me, and the respective contribution to the energy shift is of or-der α(Zα)4m3

r〈r2〉, unlike the case with the muonic loop insertions, where therespective contribution is of higher order α(Zα)5m3

r〈r2〉 (compare discussionin Subsect. 6.4.1).

Electron-loop radiative corrections to the leading nuclear finite size con-tribution in light muonic atoms were considered in [60, 20]. Two diagrams inFig. 7.15 give contributions of order α(Zα)4m3

r〈r2〉. The analytic expressionfor the first diagram up to a numerical factor coincides with the expressionfor the mixed electron and muon loops in (7.48), and we obtain

∆Enl =16α(Zα)2

9πn3m2

e〈r2〉Q(3)nl (β)mr , (7.72)

where Q(3)nl (β) is defined in (7.49).

The contribution of the second diagram in Fig. 7.15 was obtained in [20]in the form

∆E =4πZα〈r2〉

3

∫

d3rφ(r)VV P G′(r, 0)φ(0) . (7.73)

Total radiative correction to the nuclear finite size effect has the form [20, 21]

∆E(2P − 2S) = −0.0275〈r2〉 ≈ −0.0220 (9) meV . (7.74)

Collecting all contributions to the Lamb shift proportional to r2p in (7.58)

and (7.74) we obtain the total dependence of the Lamb shift on r2p in the form

∆E(2P − 2S) = −5.225 r2p meV , (7.75)

where rp is measured in fermi. This expression may be used for determinationof the proton rms radius from the pending experimental data on the 2P 1

2−2S 1

2Lamb shift in muonic hydrogen.

Using the dipole form factor one can connect the third Zemach momentwith the proton rms radius, and include the nuclear size correction of order(Zα)5m in (7.62) on par with other contributions in (7.58) and (7.74) depend-ing on the proton radius. Then the total dependence of the Lamb shift on rp

acquires the form [3, 25]

∆E(2P − 2S) = −5.225 r2p + 0.0347 r3

p meV . (7.76)


This expression also may be used for determination of the proton rms ra-dius from the experimental data. Numerically it makes almost no differencebecause contributions in (7.63) and (7.64) with high accuracy coincide. How-ever, the coefficient before r3

p in (7.76) is model dependent, so it is conceptuallyadvantageous to use the experimental value of the third Zemach moment ob-tained in [56] for calculation of the nuclear size correction of order (Zα)5m,and use the expression in (7.75) for determination of the proton radius fromexperimental data.

7.6.4 Radiative Corrections to Nuclear Polarizability Contribution

Radiative corrections to the nuclear polarizability α(Zα)5m to S-levels aredescribed by the diagrams in Fig. 7.16 and in Fig. 7.17 (compare with thediagrams in Fig. 6.4). As usual for muonic hydrogen the dominant polarizationoperator contribution is connected with the electron loops, while heavier loopsare additionally suppressed. The contribution of the diagrams in Fig. 7.16 wascalculated in [52] on the basis of the experimental data on the proton structurefunctions

∆E = −0.00152n3

δl0 meV . (7.77)

The respective correction generated by the graphs with the radiative pho-ton insertions in Fig. 7.17 is even smaller [52]

∆E =0.000092

n3δl0 meV . (7.78)

Fig. 7.16. Electron vacuum polarization correction to nuclear polarizability contri-bution

Fig. 7.17. Electron line radiative correction to nuclear polarizability contribution


Table 7.1. Lamb Shift in Muonic Hydrogen

∆E(nl) ∆E(2P − 2S) meV

One-loop electron polarization

Galanin, Pomeranchuk (1952) [4] − 8α(Zα)2

3πn3 Q(1)nl (β)mr 205.0074

Two-loop electron polarization

Di Giacomo (1969) [10] 4α2(Zα)2

π2n3 Q(2)nl (β)mr 1.5079

Three-loop electron polarizationcontribution, order α3(Zα)2

Kinoshita, Nio (1999) [18] 0.0053

Polarization insertions in twoCoulomb lines, order α2(Zα)2

Pachucki (1996) [20, 21] 0.1509

Polarization insertions intwo and threeCoulomb lines, order α3(Zα)2

Kinoshita, Nio (1999) [18] 0.0023

Relativistic corrections oforder α(Zα)4

Pachucki (1996) [20, 21]Veitia, Pachucki (2004) [25]Borie (2005) [3] 〈V Br

V P 〉 + 2〈VBrG′EV C

V P 〉 0.0169

Wichmann-Kroll, order α(Zα)4

Rinker (1976) [31]Borie, Rinker (1978) [32]

Borie (2005) [3] 4α(Zα)4

πn3 mrQWKnl (β) −0.0010

Radiative photon and electronpolarization in the Coulombline, order α2(Zα)4

Pachucki (1996) [20, 21] 〈∆V V PC 〉 + 2〈V V P

C G′E∆VC〉 −0.005 (1)

Electron loop in the radiativephoton, order α2(Zα)4

Barbieri, et al. (1973) [40] [1.094 26(− 13

mmr

− 1)

Suura, Wichmann (1957) [41] −4 × 2.216 56]

Peterman (1957) [42] ×α2(Zα)4

π2n3 (mrm

)3m −0.0015


Table 7.1. (continued)

Mixed electron and muon loopsorder α2(Zα)2(me

m)2m

Borie (1975) [45] 64α2(Zα)2

45π2n3 (mem

)2Q(3)nl (β)mr 0.00007

Hadronic polarization, order α(Zα)4mFolomeshkin (1974) [46]Friar, Martorell, Sprung (1999) [50]

Faustov, Martynenko (1999) [51, 52] −0.638 (22) 4α(Zα)4

15πn3 (mrm

)3mδl0 0.0108 (4)

Hadronic polarization, order α(Zα)5mFaustov, Martynenko (1999) [51, 52] 0.000047

Hadronic polarization in the radiativephoton, order α2(Zα)4mFaustov, Martynenko (1999) [51, 52] −0.000015

Recoil contribution oforder (Zα)4(m/M)2m

Barker-Glover (1955) [61](Zα)4m3

r2n3M2

(1

j+ 12− 1

l+ 12

)

(1 − δl0) 0.0575

Radiative corrections of

order αn(Zα)km Tables 3.2, 3.3, 3.7, 3.8, 3.9 −0.6677

Recoil corrections of order (Zα)n mM

m Table 4.1 −0.0447

Radiative-recoil correctionsof order α(Zα)n m

Mm Table 5.1 −0.0096

Leading nuclear sizecontribution 2

3n3 (Zα)4m3r〈r2〉δl0−4.163 (188)

Nuclear size correctionof order (Zα)5

Pachucki (1996) [20, 21] − (Zα)5

3n3 m4r〈r3〉(2)δl0 0.0247 (12)

Faustov, Martynenko (1999) [54]Borie (2005) [3]Friar, Sick (2005) [56]

Nuclear structure correctionof order (Zα)5

Startsev, Petrun’kin,Khomkin (1976) [57]Rosenfelder (1999) [58]Faustov, Martynenko (1999) [54]

Pachucki (1999) [21] − 0.095 (18)

n3 δl0 meV 0.012 (2)



Nuclear size correction of order (Zα)6

Borisoglebsky, Trofimenko (1979) [59]

Friar (1979) [53] −[〈ln Zα

n〉 − 2

3m2

r〈r2〉]

Pachucki (1999) [21] × 2(Zα)6

3n3 m3r〈r2〉 −0.0009 (3)

Radiative corrections to nuclearfinite size effect, order α(Zα)4m3

r〈r2〉Friar (1979) [60]Pachucki (1996) [20] −0.0275〈r2〉 meV −0.0220 (9)

Polarization operator induced correctionto nuclear polarizability, order α(Zα)5mFaustov, Martynenko (2001) [52] − 0.00152

n3 δl0 meV 0.000 19

Radiative photon induced correctionto nuclear polarizability, order α(Zα)5mFaustov, Martynenko (2001) [52] 0.000092

n3 δl0 meV −0.000 0115

References

1. J. A. Wheeler, Rev. Mod. Phys. 21, 133 (1949).2. F. Kottmann et al, Hyperfine Interactions 138, 55 (2001).3. E. Borie, Phys. Rev. A 71, 032508 (2005).4. A. D. Galanin and I. Ia. Pomeranchuk, Dokl. Akad. Nauk SSSR 86, 251 (1952).5. V. B. Berestetskii, E. M. Lifshitz, and L. P. Pitaevskii, Quantum electrodynam-

ics, 2nd Edition, Pergamon Press, Oxford, 1982.6. L. D. Landau and E. M. Lifshitz, “Quantum Mechanics”, 3d Edition,

Butterworth-Heinemann, 1997.7. L. Schiff, Quantum Mechanics, 3d ed., McGraw-Hill, New York, 1968.8. A. B. Mickelwait and H. C. Corben, Phys. Rev. 96, 1145 (1954).9. G. E. Pustovalov, Zh. Eksp. Teor. Fiz. 32, 1519 (1957) [Sov. Phys.-JETP 5,

1234 (1957)].10. A. Di Giacomo, Nucl. Phys. B 11, 411 (1969).11. R. Glauber, W. Rarita, and P. Schwed, Phys. Rev. 120, 609 (1960).12. G. Kallen and A. Sabry, Kgl. Dan. Vidensk. Selsk. Mat.-Fis. Medd. 29 (1955)

No.17.13. J. Blomkwist, Nucl. Phys. B 48, 95 (1972).14. K.-N. Huang, Phys. Rev. A 14, 1311 (1976).15. T. Kinoshita and W. B. Lindquist, Phys. Rev. D27, 853 (1983).16. T. Kinoshita and W. B. Lindquist, Phys. Rev. D27, 867 (1983).17. P. A. Baikov and D. J. Broadhurst, preprint OUT-4102-54, hep-ph 9504398,

April 1995, published in the proceedings New Computing Technique in PhysicsResearch IV, ed. B. Denby and D. Perret-Gallix, World Scientific, 1995.

18. T. Kinoshita and M. Nio, Phys. Rev. Lett. 82, 3240 (1999).19. B. J. Laurenzi and A. Flamberg, Int. J. of Quantum Chemistry 11, 869 (1977).20. K. Pachucki, Phys. Rev. A 53, 2092 (1996).21. K. Pachucki, Phys. Rev. A 60, 3593 (1999).22. M. K. Sundaresan and P. J. S. Watson, Phys. Rev. Lett. 29, 15 (1972).

References 159

23. E. Borie and G. A. Rinker, Rev. Mod. Phys. 54, 67 (1982).24. S. G. Karshenboim, Can. J. Phys. 76, 169 (1998); Zh. Eksp. Teor. Fiz. 116,

1575 (1999) [JETP 89, 850 (1999)].25. A. Veitia and K. Pachucki, Phys. Rev. A 69, 042501 (2004).26. E. H. Wichmann and N. M. Kroll, Phys. Rev. 101, 843 (1956) .27. B. Fricke, Z. Phys. 218, 495 (1969).28. P. Vogel, At. Data Nucl. Data Tables 14, 599 (1974).29. M. I. Eides, H. Grotch, and V. A. Shelyuto, Phys. Rep. C 342, 63 (2001).30. G. A. Rinker, Comput. Phys. Commun. 16, 221 (1979).31. G. A. Rinker, Phys. Rev. A 14, 18 (1976).32. E. Borie and G. A. Rinker, Phys. Rev. A 18, 324 (1978).33. M.-Y. Chen, Phys. Rev. Lett. 34, 341 (1975).34. L. Wilets and G. A. Rinker, Jr., Phys. Rev. Lett. 34, 339 (1975).35. D. H. Fujimoto, Phys. Rev. Lett. 35, 341 (1975).36. E. Borie, Nucl. Phys. A 267, 485 (1976).37. J. Calmet and D. A. Owen, J. Phys. B: At. Mol. Opt. Phys. 12, 169 (1979).38. H. A. Bethe, Phys. Rev. 72, 339 (1947).39. J. D. Bjorken and S. D. Drell, Relativistic Quantum Mechanics, McGraw-Hill

Book Co., NY, 1964.40. R. Barbieri, M. Caffo, and E. Remiddi, Lett. Nuovo Cimento 7, 60 (1973).41. H. Suura and E. Wichmann, Phys. Rev. 105, 1930 (1957).42. A. Peterman, Phys. Rev. 105, 1931 (1957).43. H. H. Elend, Phys. Lett. 20, 682 (1966); Errata 21, 720 (1966).44. G. Erickson and H. H. Liu, preprint UCD-CNL-81, 1968.45. E. Borie, Helv. Physica Acta 48, 671 (1975).46. V. N. Folomeshkin, Yad. Fiz. 19, 1157 (1974) [Sov. J. Nucl. Phys. 19, 592

(1974)].47. M. K. Sundaresan and P. J. S. Watson, Phys. Rev. D 11, 230 (1975).48. V. P. Gerdt, A. Karimkhodzhaev, and R. N. Faustov, Proc. of the Int. Workshop

on High Energy Phys. and Quantum Filed Theory, 1978, p.289.49. E. Borie, Z. Phys. A 302, 187 (1981).50. J. L. Friar, J. Martorell, and D. W. L. Sprung, Phys. Rev. A 59, 4061 (1999).51. R. N. Faustov and A. P. Martynenko, Eur. Phys. Jdirect C 6, 1 (1999); Samara

State University preprint SSU-HEP-99/07, hep-ph/9906315, January 2000.52. R. N. Faustov and A. P. Martynenko, Yad. Phys. 64, 1358 (2001) [Phys. Atom.

Nuclei 64, 1282 (2001)].53. J. L. Friar, Ann. Phys. (NY) 122, 151 (1979).54. R. N. Faustov and A. P. Martynenko, Yad. Phys. 63, 915 (2000) [Phys. Atom.

Nuclei 63, 845 (2000)].55. J. L. Friar and G. L. Payne, Phys. Rev. A 56, 5173 (1997).56. J. L. Friar and I. Sick, Phys. Rev. A 72, 040503(R) (2005).57. S. A. Startsev, V. A. Petrun’kin, and A. L. Khomkin, Yad. Fiz. 23, 1233 (1976)

[Sov. J. Nucl. Phys. 23, 656 (1976)].58. R. Rosenfelder, Phys. Lett. B 479, 381 (2000).59. L. A. Borisoglebsky and E. E. Trofimenko, Phys. Lett. B 81, 175 (1979).60. J. L. Friar, Zeit. f. Physik A 292, 1 (1979); ibid. 303, 84 (1981)61. W. A. Barker and F. N. Glover, Phys. Rev. 99, 317 (1955).

8

Physical Origin of the Hyperfine Splittingand the Main Nonrelativistic Contribution

The theory of the atomic energy levels developed in the previous chapter isincomplete, since we systematically ignored the nuclear spin which leads toan additional splitting of the energy levels. This effect will be the subject ofour discussion below.

Fig. 8.1. Scheme of hyperfine energy levels in the ground state

Unlike the Lamb shift, the hyperfine splitting (see Fig. 8.1) can be readilyunderstood in the framework of nonrelativistic quantum mechanics. It origi-nates from the interaction of the magnetic moments of the electron and thenucleus. The classical interaction energy between two magnetic dipoles is givenby the expression (see, e.g., [1, 4])

H = −23µ1 · µ2δ(r) . (8.1)

This effective Hamiltonian for the interaction of two magnetic momentsmay also easily be derived from the one photon exchange diagram in Fig. 8.2.


162 8 Physical Origin of the Hyperfine Splitting

In the leading nonrelativistic approximation the denominator of the photonpropagator cancels the exchanged momentum squared in the numerator, andwe immediately obtain the Hamiltonian for the interaction of two magneticmoments, reproducing the above result of classical electrodynamics.

Fig. 8.2. Leading order contribution to hyperfine splitting

The simple calculation of the matrix element of this Hamiltonian betweenthe nonrelativistic Schrodinger-Coulomb wave functions gives the Fermi result[2] for the splitting between the 13S 1

2and 11S 1

2states1

EF =83(Zα)4(1 + aµ)

m

M

(mr

m

)3

mc2

=163

Z4α2(1 + aµ)m

M

(mr

m

)3

ch R∞ , (8.2)

where m and M are the electron and muon masses respectively,2 Z is thecharge of the muon in terms of the proton charge,3 c is the velocity of light,aµ is the muon anomalous magnetic moment, R∞ is the Rydberg constantand h is the Planck constant.

The sign of this contribution may easily be understood from purely clas-sical considerations, if one thinks about the magnetic dipoles in the contextof the Ampere hypothesis about small loops of current. According to classi-cal electrodynamics parallel currents attract each other and antiparallel onesrepel. Hence, it is clear that the state with antiparallel magnetic moments(parallel spins) should have a higher energy than the state with antiparallelspins and parallel magnetic moments.

As in the case of the Lamb shift, QED provides the framework for system-atic calculation of numerous corrections to the Fermi formula for hyperfinesplitting (see the scheme of muonium energy levels in Fig. 8.3). We again1 In comparison with the Fermi result, we have restored here the proper dependence

of the hyperfine splitting on the reduced mass.2 Here we call the heavy particle the muon, having in mind that the precise theory

of hyperfine splitting finds its main application in comparison with the highlyprecise experimental data on muonium hyperfine splitting. However, the theoryof nonrecoil corrections is valid for any hydrogenlike atom.

3 Of course, Z = 1 for muon, but the Fermi formula is valid for any heavy nucleuswith arbitrary Z. As in the case of the Lamb shift, it is useful to preserve Z as aparameter in all formulae for the different contributions to HFS, since it helps toclarify the origin of different corrections.

8 Physical Origin of the Hyperfine Splitting 163

Fig. 8.3. Muonium energy levels

have the three small parameters, namely, the fine structure constant α, Zαand the small electron-muon mass ratio m/M . Expansion in these parametersgenerates relativistic (binding), radiative, recoil, and radiative-recoil correc-tions. At a certain level of accuracy the weak interactions and, for the caseof hadronic atoms, the nuclear size and structure effects also become impor-tant. Below we will first discuss corrections to hyperfine splitting in the caseof a structureless nucleus, having in mind the special case of muonium wherethe most precise comparison between theory and experiment is possible. In aseparate Chapter we will also consider the nuclear size and structure effectswhich should be taken into account in the case of hyperfine splitting in hydro-gen. We postpone more detailed discussion of the phenomenological situationto Sect. 12.2. Even in the case of muonium, strong interaction contributionsgenerated by the hadron polarization insertions in the exchanged photons andthe weak interaction contribution induced by the Z-boson exchange should betaken into account at the current level of accuracy. The experimental value ofthe hyperfine splitting in muonium is measured with uncertainty ±53 Hz [3](relative accuracy 1.2 × 10−8), and the next task of the theory is to obtainall corrections which could be as large as 10 Hz. This task is made even morechallenging by the fact that only a few years ago reduction of the theoreticalerror below 1 kHz was considered as a great success.

164 8 Physical Origin of the Hyperfine Splitting

References

1. J. D. Bjorken and S. D. Drell, Relativistic Quantum Mechanics, McGraw-HillBook Co., NY, 1964.

2. E. Fermi, Z. Phys. 60, 320 (1930).3. W. Liu, M. G. Boshier, S. Dhawan et al, Phys. Rev. Lett. 82, 711 (1999).4. J. Schwinger, Particles, Sources and Fields, Vol. 2 (Addison-Wesley, Reading,

MA, 1973).

9

Nonrecoil Corrections to HFS

9.1 Relativistic (Binding) Corrections to HFS

Relativistic and radiative corrections depend on the electron and muon massesonly via the explicit mass factors in the electron and muon magnetic moments,and via the reduced mass factor in the Schrodinger wave function. All suchcorrections may be calculated in the framework of the external field approxi-mation.

In the external field approximation the heavy particle magnetic momentfactorizes and the relativistic and radiative corrections have the form

∆EHFS = EF (1 + corrections) . (9.1)

This factorization of the total muon magnetic moment occurs because thevirtual momenta involved in calculation of the relativistic and radiative cor-rections are small in comparison with the muon mass, which sets the naturalmomentum for corrections to the muon magnetic moment.

Purely relativistic corrections are by far the simplest corrections to hyper-fine splitting. As in the case of the Lamb shift, they essentially correspondto the nonrelativistic expansion of the relativistic square root expression forthe energy of the light particle in (1.3), and have the form of a series over(Zα)2 ∼ p2/m2. Calculation of these corrections should be carried out in theframework of the spinor Dirac equation, since clearly there would not be anyhyperfine splitting for a scalar particle.

The binding corrections to hyperfine splitting as well as the main Fermicontribution are contained in the matrix element of the interaction Hamil-tonian of the electron with the external vector potential created by the muonmagnetic moment (A = ∇ × µ/(4πr)). This matrix element should be cal-culated between the Dirac-Coulomb wave functions with the proper reducedmass dependence (these wave functions are discussed at the end of Sect. 1.3).Thus we see that the proper approach to calculation of these corrections is tostart with the EDE (see discussion in Sect. 1.3), solve it with the convenient


166 9 Nonrecoil Corrections to HFS

zero-order potential and obtain the respective Dirac-Coulomb wave functions.Then all binding corrections are given by the matrix element

∆EBr = 〈n|γ0γ · A|n〉 . (9.2)

As discovered by Breit [1] an exact calculation of this matrix element isreally no more difficult than calculation of the leading binding correction ofrelative order (Zα)2. After straightforward calculation one obtains a closedexpression for the hyperfine splitting of an energy level with an arbitraryprincipal quantum number n [1, 2]1 (see also [3])

∆EBr(nS) =1 + 2

√

1 − (Zα)2

N2

N3γ(4γ2 − 1)EF , (9.3)

where N =√

n2 − 2(Zα)2(n − 1)/(1 + γ), γ =√

1 − (Zα)2.Let us emphasize once more that the expression in (9.3) contains all bind-

ing corrections. Expansion of this expression in Zα gives explicitly

∆EBr(nS) =[

1 +11n2 + 9n − 11

6n2(Zα)2

+203n4 + 225n3 − 134n2 − 330n + 189

72n4(Zα)4 + · · ·

]EF

n3,

(9.4)

or

∆EBr(1S) =EF

√1 − (Zα)2

(

2√

1 − (Zα)2 − 1)

=[

1 +32(Zα)2 +

178

(Zα)4 + · · ·]

EF , (9.5)

and

∆EBr(2S) =[

1 +178

(Zα)2 +449128

(Zα)4 + · · ·]

EF

8. (9.6)

Only the first two terms in the series give contributions larger than 1 Hzto the ground state splitting in muonium. As usual, in the Coulomb problem,expansion in the series for the binding corrections goes over the parameter

1 The closed expression for an arbitrary n is calculated in formula (6.14a) in [2],p. 471. While this closed expression is correct, its expansion over Zα printed in[2] after an equality sign, contains two misprints. Namely the sign before (Zα)2 inthe square brackets should be changed to the opposite, and the numerical factorinside these brackets should be −2 instead of −1. After these corrections theexpansion in formula (6.14a) in [2] does not contradict the exact expression inthe same formula, and also coincides with the result in [1].

9.2 Electron Anomalous Magnetic Moment Contributions 167

(Zα)2 without any factors of π in the denominator. This is characteristic forthe Coulomb problem and emphasizes the nonradiative nature of the relativis-tic corrections.

The sum of the main Fermi contribution and the Breit correction is givenin the last line of Table 9.1. The uncertainty of the main Fermi contribu-tion determines the uncertainty of the theoretical prediction of HFS in theground state in muonium, and is in its turn determined by the experimentaluncertainty of the electron-muon mass ratio.

Table 9.1. Relativistic (Binding) Corrections

EF kHz

Fermi (1930) [4] 1 4 459 031.936 (518)

Breit (1930) [1] 1√1−(Zα)2

(2√

1−(Zα)2−1) − 1 356.201

Total Fermi and Breitcontributions 1√

1−(Zα)2(2√

1−(Zα)2−1) 4 459 388.138 (518)

9.2 Electron Anomalous Magnetic MomentContributions (Corrections of Order αnEF)

Leading radiative corrections to HFS are generated either by the electron ormuon anomalous magnetic moments. The muon anomalous magnetic momentcontribution is already taken into account in the expression for the Fermi en-ergy in (8.2) and we will not discuss it here. All corrections of order αnEF aregenerated by the electron anomalous magnetic moment insertion in the elec-tron photon vertex in Fig. 9.1. These are the simplest of the purely radiativecorrections since they are independent of the binding parameter Zα. The valueof the electron anomalous magnetic moment entering in the expression for HFScoincides with the one for the free electron. In this situation the contributionto HFS is given by the matrix element of the electron Pauli form factor be-tween the wave functions which are the products of the Schrodinger-Coulombwave functions and the free electron spinors. Relativistic Breit corrections mayalso be trivially included in this calculation by calculating the matrix elementbetween the Dirac-Coulomb wave functions. However, we will omit here theBreit correction of order α(Zα)2EF to the anomalous moment contribution toHFS, since we will take it into account below, together with other correctionsof order α(Zα)2EF . Then the anomalous moment contribution to HFS hasthe form

∆EF = aeEF , (9.7)

where [5, 6, 7, 8, 9, 10] (see analytic expressions above in (3.11), (3.15), and(3.22))


Fig. 9.1. Electron anomalous magnetic moment contribution to HFS. Bold dotcorresponds to the Pauli form factor

ae = F2(0) =α

2π− 0.328 478 965 . . .

(α

π

)2

+ 1.181 241 456 . . .(α

π

)3

. (9.8)

We have omitted here higher order electron-loop contributions as well as theheavy particle loop contributions to the electron anomalous magnetic moment(see, e.g., [11]) because respective corrections to HFS are smaller than 0.001kHz. Let us note that the electron anomalous magnetic moment contribu-tions to HFS do not introduce any additional uncertainty in the theoreticalexpression for HFS (see also Table 9.2).

In analogy with the case of the Lamb shift discussed in Sect. 2.2 one couldexpect that the polarization insertion in the one-photon exchange would alsogenerate corrections of order αEF . However, due to the short-distance natureof the main contribution to HFS, the leading small momentum (large distance)term in the polarization operator expansion does not produce any contributionto HFS. Only the higher momentum (smaller distance) part of the polarizationoperator generates a contribution to HFS and such a contribution inevitablycontains, besides the factor α, an extra binding factor of Zα. This contributionwill be discussed in the next section.

Table 9.2. Electron AMM Contributions

EF kHz

Schwinger (1948) [5] α2π

5 178.763

Sommerfield (1957) [8][

34ζ(3) − π2

2ln 2 + π2

12+ 197

144

]

(απ)2

Peterman (1957) [7] ≈ −0.328 478 965 . . . (απ)2 −7.903

Kinoshita (1990) [9] { 8372

π2ζ(3) − 21524

ζ(5) + 1003

[(a4 + 124

ln4 2)Laporta, Remiddi (1996) [10] − 1

24π2 ln2 2] − 239

2 160π4 + 139

18ζ(3)

− 2989

π2 ln 2 + 17 101810

π2 + 28 2595 184

}(απ)3

≈ 1.181 241 456 . . . (απ)3 0.066

Total electron AMMcontribution 5 170.926

9.3 Radiative Corrections of Order αn(Zα)EF 169

9.3 Radiative Corrections of Order αn(Zα)EF

9.3.1 Corrections of Order α(Zα)EF

Nontrivial interplay between radiative corrections and binding effects firstarises in calculation of the combined expansion over α and Zα. The simplestcontribution of this type is of order α(Zα)EF and was calculated a long timeago in classical papers [12, 13, 14].

Fig. 9.2. Skeleton two-photon diagram for HFS in the external field approximation

The crucial observation, which greatly facilitates the calculations, is thatthe scattering approximation (skeleton integral approach) is adequate for cal-culation of these corrections (see, e.g., a detailed proof in [15]). As in the caseof the radiative corrections to the Lamb shift discussed in Subsect. 3.3.1, ra-diative corrections to HFS of order α(Zα)EF are given by the matrix elementsof the diagrams with all external electron lines on the mass shell calculatedbetween free electron spinors. The external spinors should be projected onthe respective spin states and multiplied by the square of the Schrodinger-Coulomb wave function at the origin. One may easily understand the physicalreasons beyond this recipe. Radiative insertions in the skeleton two-photon di-agrams in Fig. 9.2 suppress low integration momenta (of atomic order mZα)in the exchange loops and the effective loop integration momenta are of orderm. Account of off mass shell external lines would produce an additional factorZα and thus generate a higher order correction. Let us note that suppressionof low intermediate momenta in the loops takes place only for gauge invariantsets of radiative insertions, and does not happen for all individual diagramsin an arbitrary gauge. Only in the Yennie gauge is the infrared behavior ofindividual diagrams not worse than the infrared behavior of their gauge invari-ant sums. Hence, use of the Yennie gauge greatly facilitates the proof of thevalidity of the skeleton diagram approach [15]. In other gauges the individualdiagrams with on mass shell external lines often contain apparent infrareddivergences, and an intermediate infrared regularization (e.g., with the helpof the infrared photon mass of the radiative photons) is necessary. Due to theabove mentioned theorem about the infrared behavior of the complete gaugeinvariant set of diagrams, the auxiliary infrared regularization may safely belifted after calculation of the sum of all contributions. Cancellation of the in-frared divergent terms may be used as an additional test of the correctness ofcalculations.


The contribution to HFS induced by the skeleton diagram with two exter-nal photons in Fig. 9.2 is given by the infrared divergent integral

8Zα

πn3EF

∫ ∞

0

dk

k2. (9.9)

Insertion in the integrand of the factor F (k) which describes radiative cor-rections, turns the infrared divergent skeleton integral into a convergent one.Hence, the problem of calculating contributions of order α(Zα)EF to HFSturns into the problem of calculating the electron factor describing radiativeinsertions in the electron line. Calculation of the radiative corrections inducedby the polarization insertions in the external photon is straightforward sincethe explicit expression for the polarization operator is well known.

9.3.1.1 Correction Induced by the Radiative Insertionsin the Electron Line

For calculation of the contribution to HFS of order α(Zα)EF induced bythe one-loop radiative insertions in the electron line in Fig. 9.3 we have tosubstitute in the integrand in (9.9) the gauge invariant electron factor F (k).This electron factor is equal to the one loop correction to the amplitude of theforward Compton scattering in Fig. 9.4. Due to absence of bremsstrahlung inthe forward scattering the electron factor is infrared finite.

Fig. 9.3. Diagrams with radiative insertions in the electron line

Convergence of the integral for the contribution to HFS is determined bythe asymptotic behavior of the electron factor at small and large momenta.The ultraviolet (with respect of the large momenta of the external photons)asymptotics of the electron factor is proportional to the ultraviolet asymp-totics of the skeleton graph for the Compton amplitude. This may easily beunderstood in the Landau gauge when all individual radiative insertions inthe electron line do not contain logarithmic enhancements [16]. One may alsoprove the absence of logarithmic enhancement with the help of the Wardidentities. This means that insertion of the electron factor in (9.9) does notspoil the ultraviolet convergence of the integral. More interesting is the lowmomentum behavior of the electron factor. Due to the generalized low energytheorem for the Compton scattering (see, e.g., [15]), the electron factor has apole at small momenta and the residue at this pole is completely determinedby the one loop anomalous magnetic moment. Hence, naive substitution of


Fig. 9.4. One-loop electron factor

the electron factor in (9.9) (see (9.10) below) would produce a linearly in-frared divergent contribution to HFS. This would be infrared divergence of theanomalous magnetic moment contribution should be expected. As discussedin the previous section, the contribution connected with the anomalous mag-netic moment does not contain an extra factor Zα present in our skeletondiagram. The contribution is generated by the region of small (atomic scale)intermediate momenta, and the linear divergence would be cutoff by the wavefunction at the scale k ∼ mZα and will produce the correction of the pre-vious order in Zα induced by the electron anomalous magnetic moment andalready considered above. Hence, to obtain corrections of order α(Zα)EF wesimply have to subtract from the electron factor its part generated by theanomalous magnetic moment. This subtraction reduces to subtraction of theleading pole term in the infrared asymptotics of the electron factor. A closedanalytic expression for this subtracted electron factor as a function of momen-tum k was obtained in [17]. This electron factor was normalized according tothe relationship

1k2

→ α

2πF (k) . (9.10)

The subtracted electron factor generates a finite radiative correction aftersubstitution in the integral in (9.9). The contribution to HFS is equal to

∆Eelf =4Zα

π2n3EF

∫ ∞

0

dkF (k) =(

ln 2 − 134

)

α(Zα)EF , (9.11)

and was first obtained in a different way in [12, 13, 14].This expression should be compared with the correction of order α(Zα)5

to the Lamb shift in (3.34). Both expressions have the same physical origin,they correspond to the radiative insertions in the diagrams with two externalphotons, may be calculated in the skeleton diagram approach, and do notcontain a factor π in the denominators. The reasons for its absence werediscussed in the end of Subsect. 3.3.1.

9.3.1.2 Correction Induced by the Polarization Insertionsin the External Photons

Explicit expression for the electron loop polarization contribution to HFSin Fig. 9.5 is obtained from the skeleton integral in (9.9) by the standardsubstitution in (3.35). One also has to take into account an additional factor2 which corresponds to two possible insertions of the polarization operator in


Fig. 9.5. Diagrams with electron-loop polarization insertions in the external photonlines

either of the external photon lines. The final integral may easily be calculatedand the polarization operator insertion leads to the correction [12, 13, 14]

∆Epol =16α(Zα)

π2n3EF

∫ ∞

0

dkI1(k2) =34α(Zα)EF . (9.12)

There is one subtlety in this result, which should be addressed here. Theskeleton integral in (9.9) may be understood as the heavy muon pole con-tribution in the diagrams with two exchanged photons in Fig. 9.6. In such acase an exact calculation will produce an extra factor 1/(1 + m/M) beforethe skeleton integral in (9.9). We have considered above only the nonrecoilcontributions, and so we have ignored an extra factor of order m/M , keepingin mind that it would be considered together with other recoil corrections oforder α(Zα)EF . This strategy is well suited for consideration of the recoil andnonrecoil corrections generated by the electron factor, but it is less convenientin the case of the polarization insertion.

Fig. 9.6. Skeleton two-photon diagrams for HFS

In the case of the polarization insertions the calculations may be simpli-fied by simultaneous consideration of the insertions of both the electron andmuon polarization loops [18, 19]. In such an approach one explicitly takes intoaccount internal symmetry of the problem at hand with respect to both par-ticles. So, let us preserve the factor 1/(1 + m/M) in (9.9), even in calculationof the nonrecoil polarization operator contribution. Then we will obtain anextra factor mr/m on the right hand side in (9.12). To facilitate further recoilcalculations we could simply declare that the polarization operator contribu-tion with this extra factor mr/m is the result of the nonrecoil calculation butthere exists a better choice. Insertion in the external photon lines of the po-larization loop of a heavy particle with mass M generates correction to HFSsuppressed by an extra recoil factor m/M in comparison with the electronloop contribution. Corrections induced by such heavy particles polarizationloop insertions clearly should be discussed together with other radiative-recoil


corrections. However, as was first observed in [18, 19], the muon loop playsa special role. Its contribution to HFS differs from the result in (9.12) by anextra recoil factor mr/M , and, hence, the sum of the electron loop contribu-tion in Fig. 9.5 (with the extra factor mr/m taken into account) and of themuon loop contribution in Fig. 9.7 is exactly equal to the result in (9.12),which we will call below the nonrecoil polarization operator contribution. Wehave considered here this cancellation of part of the radiative-recoil correctionin order to facilitate consideration of the total radiative-recoil correction gen-erated by the polarization operator insertions below. Let us emphasize thatthere was no need to restore the factor 1/(1 + m/M) in the consideration ofthe electron line radiative corrections, since in the analytic calculation of therespective radiative-recoil corrections to be discussed below we do not use anysubtractions and recalculate the nonrecoil part of these corrections explicitly.

Fig. 9.7. Diagrams with muon-loop polarization insertions in the external photonlines

Recoil corrections induced by the polarization loops containing other heavyparticles will be considered below in Sect. 10.2 together with other radiative-recoil corrections.

9.3.2 Corrections of Order α2(Zα)EF

Calculation of the corrections of order α2(Zα)EF goes in principle along thesame lines as the calculation of the corrections of the previous order in α inthe preceding section. Once again the scattering approximation is adequatefor calculation of these corrections. There exist six gauge invariant sets ofgraphs in Fig. 9.8 which produce corrections of order α2(Zα)/πEF to HFS[17]. Respective contributions once again may be calculated with the help ofthe skeleton integral in (9.9) [17, 20, 21].

Some of the diagrams in Fig. 9.8 also generate corrections of the previousorder in Zα, which would naively induce infrared divergent contributions aftersubstitution in the skeleton integral in (9.9).

The physical nature of these contributions is quite transparent. They cor-respond to the anomalous magnetic moment which is hidden in the two-loopelectron factor. The true order in Zα of these anomalous magnetic momentcontributions is lower than their apparent order and they should be subtractedfrom the electron factor prior to calculation of the contributions to HFS. We


Fig. 9.8. Six gauge invariant sets of diagrams for corrections of order α2(Zα)EF

have already encountered a similar situation above in the case of the cor-rection of order α(Zα)EF induced by the electron factor, and the remedy isthe same. Let us mention that the analogous problem was also discussed inconnection with the Lamb shift calculations in Subsect. 3.3.1.

Technically the lower order contributions to HFS are produced by the con-stant terms in the low-frequency asymptotic expansion of the electron factor.These lower order contributions are connected with integration over externalphoton momenta of the characteristic atomic scale mZα and the approxima-tion based on the skeleton integrals in (9.9) is inadequate for their calculation.In the skeleton integral approach these previous order contributions arise asthe infrared divergences induced by the low-frequency terms in the electronfactors. We subtract leading low-frequency terms in the low-frequency asymp-totic expansions of the electron factors, when necessary, and thus get rid ofthe previous order contributions.

Let us discuss in more detail calculation of different contributions of orderα2(Zα)EF . The reader could notice that the discussion below is quite similar


to the discussion of calculation of the corrections of order α2(Zα)5m to theLamb shift in Subsect. 3.3.3.

9.3.2.1 One-Loop Polarization Insertions in the External Photons

The simplest correction is induced by the diagrams in Fig. 9.8(a) with twoinsertions of the one-loop vacuum polarization in the external photon lines.The respective contribution to HFS is obtained from the skeleton integral in(9.9) by the substitution of the polarization operator squared

1k2

→(α

π

)2

k2I1(k) . (9.13)

Taking into account the multiplicity factor 3 one easily obtains [17]

∆E =24Zα

πn3

(α

π

)2

EF

∫ ∞

0

dkk2I21 (k) =

3635

α2(Zα)πn3

EF . (9.14)

9.3.2.2 Insertions of the Irreducible Two-Loop Polarizationin the External Photons

Expression for the two-loop vacuum polarization contribution to HFS inFig. 9.8(b) is obtained from the skeleton integral in (9.9) by the substitu-tion

1k2

→(α

π

)2

I2(k) . (9.15)

With account of the multiplicity factor 2 one obtains [17]

∆E =16Zα

πn3

(α

π

)2

EF

∫ ∞

0

dkI2(k) =(

22415

ln 2 − 3815

π − 118225

)α2(Zα)

πn3EF

≈ 1.87 . . .α2(Zα)

πn3EF . (9.16)

9.3.2.3 Insertion of One-Loop Electron Factor in the Electron Lineand of the One-Loop Polarization in the External Photons

The next correction of order α2(Zα)EF is generated by the gauge invariantset of diagrams in Fig. 9.8(c). The respective analytic expression is obtainedfrom the skeleton integral by simultaneous insertion in the integrand of theone-loop polarization function I1(k) and of the electron factor F (k).


Then taking into account the multiplicity factor 2 corresponding to twopossible insertions of the one-loop polarization, one obtains

∆E =8Zα

πn3

(α

π

)2

EF

∫ ∞

0

dkk2F (k)I1(k)

=

(

−43

ln2 1 +√

52

− 209

√5 ln

1 +√

52

− 6445

ln 2 +π2

9+

1 043675

)

× α2(Zα)πn3

EF . (9.17)

We used in (9.17) the subtracted electron factor. However, it is easy to seethat the one-loop anomalous magnetic moment term in the electron factorgenerates a correction of order α2(Zα)EF in the diagrams in Fig, and alsoshould be taken into account. An easy direct calculation of the anomalousmagnetic moment contribution leads to the correction

∆E =38

α2(Zα)πn3

EF , (9.18)

which may also be obtained multiplying the result in (9.12) by the one-loopanomalous magnetic moment α/(2π).

Hence, the total correction of order α2(Zα)EF generated by the diagramsin Fig. 9.8 (c) is equal to

∆E =8Zα

πn3

(α

π

)2

EF

∫ ∞

0

dkk2F (k)I1(k)

=

(

−43

ln2 1 +√

52

− 209

√5 ln

1 +√

52

− 6445

ln 2 +π2

9+

38

+1 043675

)

×α2(Zα)πn3

EF ≈ 2.23 . . .α2(Zα)

πn3EF . (9.19)

9.3.2.4 One-Loop Polarization Insertionsin the Radiative Electron Factor

This correction is induced by the gauge invariant set of diagrams in Fig. 9.8(d)with the polarization operator insertions in the radiative photon. The two-loopanomalous magnetic moment generates correction of order α2EF to HFS andthe respective leading pole term in the infrared asymptotics of the electronfactor should be subtracted to avoid infrared divergence and double counting.

The subtracted radiatively corrected electron factor may be obtained fromthe subtracted one-loop electron factor in (9.10). To this end, one shouldrestore the radiative photon mass in the one-loop electron factor, and then thepolarization operator insertion in the photon line is taken into account withthe help of the dispersion integral like one in (3.44) for the spin-independent


electron factor. In terms of the electron factor F (k, λ) with a massive radiativephoton with mass λ = 2/

√1 − v2 the contribution to HFS has the form [20]

∆E =4α2(Zα)

π2n3EF

∫ ∞

0

dk

∫ 1

0

dvv2(1 − v2

3 )1 − v2

L(k, λ) . (9.20)

This integral was analytically simplified to a one-dimensional integral of acomplete elliptic integral, which admits numerical evaluation with an arbitraryprecision [20]

∆E = −0.310 742 . . .α2(Zα)

πn3EF . (9.21)

9.3.2.5 Light by Light Scattering Insertionsin the External Photons

The diagrams in Fig. 9.8(e) with the light by light scattering insertions inthe external photons do not generate corrections of the previous order inZα. As is well known, the light by light scattering diagrams are apparentlylogarithmically ultraviolet divergent, but due to gauge invariance the diagramsare really ultraviolet convergent. Also, as a result of gauge invariance thelight by light scattering tensor is strongly suppressed at small momenta of theexternal photons. The contribution to HFS can easily be expressed in termsof a weighted integral of the light by light scattering tensor [21], and furthercalculations are in principle quite straightforward though technically involved.This integral was analytically simplified to a three-dimensional integral whichmay be calculated with high accuracy [21]

∆E = −0.472 514(1)α2(Zα)

πn3EF . (9.22)

The original result in [21] differed from the one in (9.22) by two percent.A later purely numerical calculation of the light by light contribution in [22,23] produced a less precise result which, however, differed from the originalresult in [21] by two percent. After a thorough check of the calculations in[21] a minor arithmetic mistake in one of the intermediate expressions inthe original version of [21] was discovered. After correction of this mistake,the semianalytic calculations in [21] lead to the result in (9.22) in excellentagreement with the somewhat less precise purely numerical result in [22, 23].

9.3.2.6 Diagrams with Insertions of Two Radiative Photonsin the Electron Line

By far the most difficult task in calculations of corrections of order α2(Zα)EF

to HFS is connected with the last gauge invariant set of diagrams in Fig. 9.8(f),which consists of nineteen topologically different diagrams [17] presented inFig. 9.9 (compare a similar set of diagrams in Fig. 3.12 in the case of the


Lamb shift). These nineteen graphs may be obtained from the three graphsfor the two-loop electron self-energy by insertion of two external photons inall possible ways. The graphs in Fig. 9.9(a–c) are obtained from the two-loopreducible electron self-energy diagram, graphs in Fig. 9.9(d–k) are the resultof all possible insertions of two external photons in the rainbow self-energydiagram, and diagrams in Fig. 9.9(l–s) are connected with the overlappingtwo-loop self-energy graph.

Calculation of the respective contribution to HFS in the skeleton inte-gral approach was initiated in [24, 25], where contributions induced by thediagrams in Fig. 9.9(a–h) and Fig. 9.9(l) were obtained. In order to avoidspurious infrared divergences in the individual diagrams the semianalytic cal-culations in [24, 25] were performed in the Yennie gauge. The diagrams underconsideration contain anomalous magnetic moment contributions which weresubtracted before taking the scattering approximation integrals.

The total contribution of all nineteen diagrams to HFS was first calculatedpurely numerically in the Feynman gauge in the NRQED framework in [22,23]. The semianalytic skeleton integral calculation in the Yennie gauge wascompleted a bit later in [26, 27]

∆E = −0.672 6(4) . . .α2 (Zα)

πn3EF . (9.23)

This semianalytic result is consistent with the purely numerical result in [22,23] but more than an order of magnitude more precise. It is remarkable thatthe results of two complicated calculations performed in completely differentapproaches turned out to be in excellent agreement.

9.3.2.7 Total Correction of Order α2(Zα)EF

The total contribution of order α2(Zα)EF is given by the sum of contributionsin (9.14), (9.16), (9.19), (9.21), (9.22), and (9.23)

∆E =[

− 43

ln2 1 +√

52

− 209

√5 ln

1 +√

52

+60845

ln 2 +π2

9− 38

15π

+91 63937 800

− 0.310 742 − 0.472 514(1) − 0.672 6(4)]α2(Zα)

πEF

≈ 0.771 7(4)α2(Zα)

πEF , (9.24)

or numerically∆E = 0.425 6(2) kHz . (9.25)

As we have already mentioned consistent results for this correction wereobtained independently in different approaches by two groups in [17, 20, 21,24, 25, 26, 27] and [22, 23].


Fig. 9.9. Nineteen topologically different diagrams with two radiative photons in-sertions in the electron line

9.3.3 Corrections of Order α3(Zα)EF

Corrections of order α3(Zα)EF are similar to the corrections of order α2(Zα)EF , and can be calculated in the same way. These corrections are generatedby three-loop radiative insertions in the skeleton diagram in Fig. 9.2. Theirnatural scale is determined by the factor α3(Zα)/π2EF , that is about 1 kHz.

All corrections of order α3(Zα)EF connected with the diagrams containingat least one one-loop or two-loop polarization insertion were obtained in [28]


Fig. 9.10. Reducible three-loop diagrams

∆E = −1.358 (1)α3(Zα)

π2EF . (9.26)

Calculation of corrections generated by the gauge invariant set of diagramswith insertions of three radiative photons in the electron line in the skeletondiagrams in Fig. 9.2 is more complicated. Only insertions of the three-loopone-particle reducible diagrams with radiative photons in the electron linein Fig. 9.10 were considered thus far. Contribution of these diagrams in theYennie gauge is equal to [29]

∆E = 0.104 23 (1)α3(Zα)

π2EF . (9.27)

Work on calculation of the remaining contributions of order α3(Zα)5 is inprogress now.

Note that the uncertainty of the total contribution in the last line inTable 9.3 is determined not by the uncertainties of any of the entries in theupper lines of this table, that are too small, but just by the uncalculatedcontribution of order α3(Zα)EF .

9.4 Radiative Corrections of Order αn(Zα)2EF

9.4.1 Corrections of Order α(Zα)2EF

9.4.1.1 Electron-Line Logarithmic Contributions

Binding effects are crucial in calculation of the corrections of order α(Zα)2EF .Unlike the corrections of the first order in the binding parameter Zα, in thiscase the exchanged photon loops with low (of order ∼mZα) exchanged mo-menta give a significant contribution and the external wave functions at theorigin do not factorize as in the scattering approximation. The anticipatedlow momentum logarithmic divergence in the loop integration is cut off bythe wave functions at the atomic scale and, hence, the contribution of orderα(Zα)2EF is enhanced by the low-frequency logarithmic terms ln2 Zα andln Zα. The situation for this calculation resembles the case of the main con-tribution to the Lamb shift of order α(Zα)4 and also corrections to the Lambshift of order α(Zα)6. Once again, the factors before logarithmic terms origi-nate from the electron form factor and from the logarithmic integration over

9.4 Radiative Corrections of Order αn(Zα)2EF 181

Table 9.3. Radiative Corrections of Order αn(Zα)EF

α(Zα)EF kHz

Electron-Line InsertionsKroll, Pollock (1951) [12]Karplus, Klein, (ln 2 − 13

4) −607.123

Schwinger (1951) [13, 14]

Polarization InsertionKroll, Pollock (1951) [12]Karplus, Klein, 3

4178.087

Schwinger (1951) [13, 14]

One-Loop PolarizationEides, Karshenboim,Shelyuto (1989) [17] 36

35απ

0.567

Two-Loop PolarizationEides, Karshenboim,Shelyuto (1989) [17] ( 224

15ln 2 − 38π

15− 118

225)α

π1.030

One-Loop Polarizationand Electron Factor

Eides,Karshenboim,{− 4

3ln2 1+

√5

2− 20

9

√5 ln 1+

√5

2

Shelyuto (1989) [17] − 6445

ln 2 + π2

9+ 3

8+ 1 043

675

}απ

−0.369

Polarization insertionin the Electron FactorEides, Karshenboim,Shelyuto (1990) [20] −0.310 742 . . . α

π−0.171

Light by Light ScatteringEides, Karshenboim,Shelyuto (1991, 1993) [21] −0.472 514(1)α

π−0.261

Kinoshita, Nio (1994, 1996) [22, 23]

Insertions of Two RadiativePhotons in the Electron LineKinoshita, Nio (1994, 1996) [22, 23]Eides, Shelyuto (1995) [26, 27] −0.672 6(4)α

π−0.371

Polarization Insertions

Eides, Shelyuto (2003) [28] −1.358(1)(

απ

)2 −0.002

Total correction oforder αn(Zα)EF −428.612 (1)


Fig. 9.11. Leading logarithm squared contribution of order α(Zα)2EF to HFS

the loop momenta in the diagrams with three exchanged photons. The leadinglogarithm squared term is state independent and can easily be calculated bythe same methods as the leading logarithm cube contribution of order α2(Zα)6

(see Subsubsect. 3.4.2.1). The only difference is that this time it is necessaryto take as one of the perturbation potentials the potential responsible for themain Fermi contribution to HFS

VF =83

πZα

mM(1 + aµ) . (9.28)

This calculation of the leading logarithm squared term [30] (see Fig. 9.11)also produces a recoil correction to the nonrecoil logarithm squared contri-bution. We will discuss this radiative-recoil correction below in the Subsect.10.2.11 dealing with other radiative-recoil corrections, and we will consider inthis section only the nonrecoil part of the logarithm squared term.

All logarithmic terms for the 1S state were originally calculated in [31, 32].The author of [31] also calculated the logarithmic contribution to the 2Shyperfine splitting

∆E(1S) =[

−23

ln2(Zα)−2 −(

83

ln 2 − 3772

)

ln(Zα)−2

]α(Zα)2

πEF , (9.29)

∆E(2S) =[

−23

ln2(Zα)−2 −(

163

ln 2 − 4 − 172

)

ln(Zα)−2

]α(Zα)2

8πEF .

(9.30)

9.4.1.2 Nonlogarithmic Electron-Line Corrections

Calculation of the nonlogarithmic part of the contribution of order α(Zα)2EF

is a more complicated task than obtaining the leading logarithmic terms. Theshort distance leading logarithm squared contributions cancel in the difference∆E(1S)− 8∆E(2S) and it is this difference, containing both logarithmic andnonlogarithmic corrections, which was calculated first in [33]. An estimate ofthe nonlogarithmic terms for n = 1 and n = 2 in accordance with [33] wasobtained in [34]. This work also confirmed the results of [31, 32] for the log-arithmic terms. The first complete calculation of the nonlogarithmic termswas done in a purely numerical approach in [35]. The idea of this calculationis similar to the one used for calculation of the α(Zα)6 contribution to theLamb shift in [36]. In this calculation the electron-line radiative corrections

9.4 Radiative Corrections of Order αn(Zα)2EF 183

were written in the form of the diagrams with the electron propagator in theexternal field, and then an approximation scheme for the relativistic electronpropagator in the Coulomb field was set with the help of the well known rep-resentation [37, 38] for the nonrelativistic propagator in the Coulomb field. Inview of the significant theoretical progress achieved in calculation of contri-butions to HFS of order α2(Zα)EF , insufficient accuracy in the calculationin [35] (about 0.2 kHz) became for some time the main source of uncertaintyin the theoretical expression for muonium HFS. The accuracy of this correc-tion was significantly improved in two independent calculations [39, 40]. Theauthor of [39] used the approach he developed for calculation of correctionsof order α(Zα)6 to the Lamb shift. The main ideas of this approach were dis-cussed above in Subsubsect. 3.4.1.2. Calculation in [40] was performed in thecompletely different framework of nonrelativistic QED (see, e.g., [41, 22, 23]).Results of both calculations are in remarkable agreement, the result of [39]being equal to

∆E = 17.122α(Zα)2

πEF , (9.31)

and the result of [40] is

∆E = 17.122 7(11)α(Zα)2

πEF . (9.32)

9.4.1.3 Logarithmic Contribution Inducedby the Polarization Operator

Calculation of the leading logarithmic contribution induced by the polar-ization operator insertion (see Fig. 9.12) proceeds along the same lines asthe calculation of the logarithmic polarization operator contribution of or-der α(Zα)6 to the Lamb shift (compare Subsubsect. 3.4.1.3). Again, only theleading term in the small momentum expansion of the polarization operator(see (2.6), (3.63)) produces a contribution of the order under considerationand only the state-independent logarithmic corrections to the Schrodinger-Coulomb function are relevant. The only difference is that the correction tothe wave function is produced, not by the Darwin term as in the case of theLamb shift (compare (3.65)) but, by the external magnetic moment perturba-tion in (9.28). This correction to the wave function was calculated in [33, 42],and with its help one immediately obtains the state-independent logarithmic

Fig. 9.12. Leading logarithm polarization contribution of order α(Zα)2EF to HFS


contribution α(Zα)2EF to HFS induced by the polarization operator insertion[31, 32]

∆Epol =415

ln(Zα)−2 α(Zα)2

πn3EF . (9.33)

9.4.1.4 Nonlogarithmic Corrections Inducedby the Polarization Operator

Calculation of the nonlogarithmic polarization operator contribution is quitestraightforward. One simply has to calculate two terms given by ordinaryperturbation theory, one is the matrix element of the radiatively correctedexternal magnetic field, and another is the matrix element of the radiativelycorrected external Coulomb field between wave functions corrected by theexternal magnetic field (see Fig. 9.13). The first calculation of the respectivematrix elements was performed in [34]. Later a number of inaccuracies in[34] were uncovered [22, 23, 40, 43, 44, 45] and the correct result for thenonlogarithmic contribution of order α(Zα)2EF to HFS is given by

∆E =(

− 815

ln 2 +34225

)α(Zα)2

πEF . (9.34)

Fig. 9.13. Corrections of order α(Zα)2EF induced by the polarization operator

9.4.2 Corrections of Order α2(Zα)2EF

9.4.2.1 Leading Double Logarithm Corrections

Corrections of order α2(Zα)2EF are again enhanced by a logarithm squaredterm and one should expect that they are smaller by the factor α/π thanthe corrections of order α(Zα)2EF considered above. Calculation of the lead-ing logarithm squared contribution to HFS may be performed in exactly thesame way as the calculation of the leading logarithm cube contribution of or-der α2(Zα)6 to the Lamb shift considered above in Subsubsect. 3.4.2.1. Bothresults were originally obtained in one and the same work [30]. The logarithmcube term is missing in the case of HFS, since now at least one of the per-turbation operators in Fig. 9.14 should contain a magnetic exchange photonand the respective anomalous magnetic moment is infrared finite. It is easy

9.5 Radiative Corrections of Order α(Zα)3EF and of Higher Orders 185

Fig. 9.14. Leading double logarithm contribution of order α2(Zα)2EF

to realize that as the result of this calculation one obtains simply the lead-ing logarithmic contribution of order α(Zα)2EF in (9.29), multiplied by theelectron anomalous magnetic moment α/(2π) [30]

∆E = −13

ln2(Zα)−2 α2(Zα)2

π2EF . (9.35)

Numerically this correction is about −0.04 kHz, and this contribution is largeenough to justify calculation of the single-logarithmic contributions.

9.4.2.2 Single-Logarithmic and Nonlogarithmic Contributions

Terms linear in the large logarithm were calculated in the NRQED framework[46]

∆E =[(

94ζ(3) − 3

2π2 ln 2 +

1027

π2 +4 3581 296

)

−34

(34ζ(3) − π2

2ln 2 +

π2

12+

197144

)

−43

(

ln 2 − 34

)

+14

(

−119

− 1 +815

)]

ln(Zα)−2 α2(Zα)2

π2EF

≈ −0.639 000 544 . . .α2(Zα)2

π2EF . (9.36)

The nonlogarithmic contributions of order α2(Zα)2EF are also estimatedin [46]

∆E = (10 ± 2.5)α2(Zα)2

π2EF . (9.37)

The numerical uncertainty of the last contribution is about 0.003 kHz.Corrections of order α3(Zα)2EF are suppressed by an extra factor α/π in

comparison with the leading contributions of order α2(Zα)2EF and are toosmall to be of any phenomenological interest now. All corrections of orderαn(Zα)2EF are collected in Table 9.4, and their total uncertainty is deter-mined by the error of the nonlogarithmic contribution of order α2(Zα)2EF .

9.5 Radiative Corrections of Order α(Zα)3EF

and of Higher Orders

As we have repeatedly observed corrections to the energy levels suppressedby an additional power of the binding parameter Zα are usually numerically


Table 9.4. Radiative Corrections of Order αn(Zα)2EF

α(Zα)2

πEF kHz

Logarithmic Electron-LineContributionZwanziger (1964) [32]Layzer (1964) [31] − 2

3ln2(Zα)−2 − ( 8

3ln 2 − 37

72)

× ln(Zα)−2 −42.850

Nonlogarithmic Electron-LineContributionPachucki (1996) [39]Kinoshita, Nio (1997) [40] 17.122 7(11) 9.444

Logarithmic PolarizationOperator ContributionZwanziger (1964) [32]Layzer (1964) [31] 4

15ln(Zα)−2 1.447

Nonlogarithmic PolarizationOperator ContributionKinoshita, Nio (1996) [22, 23, 40]Sapirstein (1996) [43],Brodsky (1996) [44] (− 8

15ln 2 + 34

225) −0.121

Schneider, Greiner, Soff (1994) [45]

Leading LogarithmicContribution of order α2(Zα)2EF

Karshenboim (1993) [30] − 13

ln2(Zα)−2 απ

−0.041

Single-logarithmicContributions of order α2(Zα)2EF

Kinoshita, Nio (1998) [46] [( 94ζ(3) − 3

2π2 ln 2 + 10

27π2 + 4358

1296)

− 34( 34ζ(3) − π2

2ln 2 + π2

12+ 197

144)

− 43(ln 2 − 3

4) + 1

4(− 11

9− 1 + 8

15)]

× ln(Zα)−2 απ

−0.008

Nonlogarithmic Contributionsof order α2(Zα)2EF

Kinoshita, Nio (1998) (10 ± 2.5)απ

0.013(3)

Total correction oforder αn(Zα)2EF −32.115 (3)


larger than the corrections suppressed by an additional power of α/π, inducedby radiative insertions. With this perspective one could expect that the cor-rections of order α(Zα)3EF would be numerically larger than the correctionsof order α2(Zα)2EF considered above.

9.5.1 Corrections of Order α(Zα)3EF

9.5.1.1 Leading Logarithmic Contributions Inducedby the Radiative Insertions in the Electron Line

Calculation of the leading logarithmic corrections of order α(Zα)3EF to HFSparallels the calculation of the leading logarithmic corrections of order α(Zα)7

to the Lamb shift, described above in Subsect. 3.5.1. Again all leading loga-rithmic contributions may be calculated with the help of second order pertur-bation theory (see (3.71)).

Fig. 9.15. Leading logarithmic electron-line contributions of order α(Zα)3EF

It is easy to check that the leading contribution is linear in the large loga-rithm. Due to the presence of the local potential in (9.28) which correspondsto the main contribution to HFS, and of the potential

VKP =83

(

ln 2 − 134

)

α(Zα)πα(Zα)2

mM(1 + aµ) , (9.38)

which corresponds to the contribution of order α(Zα)EF , we now have twocombinations of local potentials V1 ((3.95)), VF ((9.28)), and V2 ((3.96)), VKP

((9.38)) in Fig. 9.15, which generate logarithmic contributions. This differsfrom the case of the Lamb shift when only one combination of local operatorswas relevant for calculation of the leading logarithmic contribution. An easycalculation produces [47, 48]

∆E = 4(

12

ln 2 − 1 − 11128

)


n3

(mr

m

)2

EF , (9.39)

and [48]

∆E =12

(

ln 2 − 134

)


n3EF , (9.40)

for the first and second combinations of the perturbation potentials, respec-tively.


Fig. 9.16. Leading logarithmic photon-line contributions of order α(Zα)3EF

9.5.1.2 Leading Logarithmic Contributions Inducedby the Polarization Insertions in the External Photon Lines

The leading logarithmic correction induced by the radiative insertions in theexternal photon in Fig. 9.16 is calculated in exactly the same way as wasdone above in the case of radiative insertions in the electron line. The onlydifference is that instead of the potential V1 in (3.95) we have to use therespective potential in (3.99), and instead of the potential VKP in (9.38) wehave to use the potential

VKP = 2α(Zα)πα(Zα)2

mM(1 + aµ) , (9.41)

connected with the external photons. Then we immediately obtain [47, 48]

∆E = − 548


n3

(mr

m

)2

EF , (9.42)

and [48]

∆E =38


n3EF , (9.43)

for the first and second combinations of the perturbation potentials.

9.5.1.3 Nonlogarithmic Contributions of Order α(Zα)3EF

and of Higher Orders in Zα

The large magnitude of the leading logarithmic contributions of order α(Zα)3

EF generated by the radiative photon insertions in the electron line warrantsconsideration of the respective nonlogarithmic corrections. Results of NRQEDcalculation of radiative corrections to HFS of order α(Zα)3EF generated byinsertions of one radiative photon in the electron line were reported in [49].The infrared divergences in this calculation were regulated by a finite photonmass, and purely numerical calculations lead to the contribution [49]

∆E = −5.07 (50) α(Zα)3EF . (9.44)


The uncertainty of this result arises from extrapolation to the zero photonmass limit. The magnitude of the nonlogarithmic coefficient in (9.44) seemsto be quite reasonable qualitatively. Numerically the contribution to HFS in(9.44) is about 0.12 of the leading logarithmic contribution.

Numerical calculations of radiative corrections to HFS generated by inser-tion of one radiative photon in the electron line without expansion over Zα,which are routinely used for the high Z atoms, were also extended for Z = 1.Initial disagreements between the results of [50] and [51] were resolved in [52],which confirmed (but with an order of magnitude worse accuracy) the resultof [51]

∆E = −3.82 (63) α(Zα)3EF . (9.45)

Another numerical calculation without expansion in Zα was performed forZ ≥ 4 in [53, 54]. After extrapolation of these results to Z = 1 the authorsobtain [54]

∆E = −4.393 (95) α(Zα)3EF . (9.46)

The results in (9.45) and (9.46) include not only nonlogarithmic correctionsof order α(Zα)3EF but also contributions of all terms of the form α(Zα)nEF

with n ≥ 4. Comparing contributions in (9.44) and in (9.45) with the contri-bution of order α(Zα)3EF in (9.44) we could come to the conclusion that thecontribution of the corrections of order α(Zα)nEF with n ≥ 4 is relativelylarge, about 16 Hz. On the other hand, even allowing for the presence of higherorder terms in (9.45) and (9.46), all three results [49, 51, 54] are compatiblewithin the error bars. Clearly further work on nonlogarithmic corrections oforder α(Zα)3EF is warranted. We use the result [54] for numerical calculationof the magnitude of HFS in muonium.

Calculation of the nonlogarithmic polarization operator contributions oforder α(Zα)3EF goes exactly like calculation of the respective correctionsof order α(Zα)2EF and is connected with the same diagrams. In fact bothlogarithmic and nonlogarithmic polarization operator corrections of ordersα(Zα)2EF and α(Zα)3EF were obtained in one and the same calculation in[55]. Nonlogarithmic corrections of order α(Zα)3EF have the form

∆E =(

1324

ln 2 +539288

)

α(Zα)3EF . (9.47)

Numerical results obtained in [52] without expansion in Zα are consistentwith the result in (9.47).

Thus all corrections of order α(Zα)nEF collected in Table 9.5 are nowknown with an uncertainty of about 0.008 kHz. Scattering of the results in[49, 51, 53, 54] for the nonlogarithmic contribution of order α(Zα)3EF showsthat due to complexity of the numerical calculations a new consideration ofthis correction would be helpful.


Table 9.5. Radiative Corrections of Order α(Zα)3EF

α(Zα)3EF kHz

Logarithmic Electron-LineContributionLepage (1994) [47], Karshenboim (1996) [48] ( 5

2ln 2 − 191

32) −0.527

× ln(Zα)−2

Logarithmic PolarizationOperator ContributionLepage (1994) [47], Karshenboim (1996) [48] 13

48ln(Zα)−2 0.034

Nonlogarithmic Electron-LineContribution (including terms α(Zα)nEF , n ≥ 4)Yerokhin, Artemyev,Shabaev, Plunien (2001) [54] −4.393 (95) −0.056 (1)

Nonlogarithmic PolarizationOperator ContributionKarshenboim, Ivanov, Shabaev (1999) [55] 13

24ln 2 + 539

2880.028

Total correction oforder α(Zα)3EF −0.520 (1)

9.5.2 Corrections of Order α2(Zα)3EF and of Higher Orders in α

One should expect that corrections of order α2(Zα)3EF are suppressed rela-tive to the contributions of order α(Zα)3EF by the factor α/π. This meansthat at the present level of experimental accuracy one may safely neglect thesecorrections, as well as corrections of even higher orders in α.

References

1. G. Breit, Phys. Rev. 35, 1477 (1930).2. G. W. Erickson and D. R. Yennie, Ann. Phys. (NY) 35, 447 (1965).3. J. Schwinger, Particles, Sources and Fields, Vol.2 (Addison-Wesley, Reading,

MA, 1973).4. E. Fermi, Z. Phys. 60, 320 (1930).5. J. Schwinger, Phys. Rev. 73, 416 (1948).6. R. Karplus and N. M. Kroll, Phys. Rev. 77, 536 (1950).7. A. Peterman, Helv. Phys. Acta 30, 407 (1957); Nucl. Phys. 3, 689 (1957).8. C. M. Sommerfield, Phys. Rev. 107, 328 (1957); Ann. Phys. (NY) 5, 26 (1958).9. T. Kinoshita, in Quantum Electrodynamics, ed. T. Kinoshita (World Scientific,

Singapore, 1990), p.218.10. S. Laporta and E. Remiddi, Phys. Lett. B 379, 283 (1996).11. T. Kinoshita, Rep. Prog. Phys. 59, 3803 (1996).12. N. Kroll and F. Pollock, Phys. Rev. 84, 594 (1951); ibid. 86, 876 (1952).13. R. Karplus, A. Klein, and J. Schwinger, Phys. Rev. 84, 597 (1951).

References 191

14. R. Karplus and A. Klein, Phys. Rev. 85, 972 (1952).15. M. I. Eides, S. G. Karshenboim, and V. A. Shelyuto, Ann. Phys. (NY) 205,

231, 291 (1991).16. V. B. Berestetskii, E. M. Lifshitz, and L. P. Pitaevskii, Quantum electrodynam-

ics, 2nd Edition, Pergamon Press, Oxford, 1982.17. M. I. Eides, S. G. Karshenboim, and V. A. Shelyuto, Phys. Lett. B 229, 285

(1989); Pis’ma Zh. Eksp. Teor. Fiz. 50, 3 (1989) [JETP Lett. 50, 1 (1989)]; Yad.Fiz. 50, 1636 (1989) [Sov. J. Nucl. Phys.50, 1015 (1989)].

18. E. A. Terray and D. R. Yennie, Phys. Rev. Lett. 48, 1803 (1982).19. J. R. Sapirstein, E. A. Terray, and D. R. Yennie, Phys. Rev. D29, 2290 (1984).20. M. I. Eides, S. G. Karshenboim, and V. A. Shelyuto, Phys. Lett. B 249, 519

(1990); Pis’ma Zh. Eksp. Teor. Fiz. 52, 937 (1990) [JETP Lett. 52, 317 (1990)].21. M. I. Eides, S. G. Karshenboim, and V. A. Shelyuto, Phys. Lett. B 268, 433

(1991); 316, 631 (E) (1993); 319, 545 (E) (1993); Yad. Fiz. 55, 466 (1992); 57,1343 (E) (1994) [Sov. J. Nucl. Phys. 55, 257 (1992); 57, 1275 (E) (1994)].

22. T. Kinoshita and M. Nio, Phys. Rev. Lett. 72, 3803 (1994).23. T. Kinoshita and M. Nio, Phys. Rev. D 53, 4909 (1996).24. M. I. Eides, S. G. Karshenboim, and V. A. Shelyuto, Phys. Lett. B 312, 358

(1993); Yad. Phys. 57, 1309 (1994) [Phys. Atom. Nuclei 57, 1240 (1994)].25. M. I. Eides, S. G. Karshenboim, and V. A. Shelyuto, Yad. Phys. 57, 2246 (1994)

[Phys. Atom. Nuclei 57, 2158 (1994)].26. M. I. Eides and V. A. Shelyuto, Pis’ma Zh. Eksp. Teor. Fiz. 61, 465 (1995)

[JETP Letters 61, 478 (1995)].27. M. I. Eides and V. A. Shelyuto, Phys. Rev. A 52, 954 (1995).28. M. I. Eides and V. A. Shelyuto, Phys. Rev. A 68, 042106 (2003).29. M. I. Eides and V. A. Shelyuto, Phys. Rev. A 70, 022506 (2004).30. S. G. Karshenboim, Zh. Eksp. Teor. Fiz. 103, 1105 (1993) [JETP 76, 541

(1993)].31. A. Layzer, Nuovo Cim. 33, 1538 (1964).32. D. Zwanziger, Nuovo Cim. 34, 77 (1964).33. D. E. Zwanziger, Phys. Rev. 121, 1128 (1961).34. S. J. Brodsky and G. W. Erickson, Phys. Rev. 148, 148 (1966).35. J. R. Sapirstein, Phys. Rev. Lett. 51, 985 (1983).36. J. R. Sapirstein, Phys. Rev. Lett. 47, 1723 (1981).37. L. Hostler, J. Math. Phys. 5, 1235 (1964).38. J. Schwinger, J. Math. Phys. 5, 1606 (1964).39. K. Pachucki, Phys. Rev. A 54, 1994 (1996).40. T. Kinoshita and M. Nio, Phys. Rev. D 55, 7267 (1997).41. W. E. Caswell and G. P. Lepage, Phys. Lett. B 167, 437 (1986).42. S. J. Brodsky and G. W. Erickson, Phys. Rev. 148, 26 (1966).43. J. R. Sapirstein, unpublished, as cited in [22] .44. S. J. Brodsky, unpublished, as cited in [40] .45. S. M. Schneider, W. Greiner, and G. Soff, Phys. Rev. A A50, 118 (1994).46. T. Kinoshita, hep-ph/9808351, Cornell preprint, 1998.47. P. Lepage, unpublished, as cited in [22] .48. S. G. Karshenboim, Z. Phys. D 36, 11 (1996).49. M. Nio, in Quantum Electrodynamics and Physics of the Vacuum: QED 2000.

AIP CP 564, ed. G. Cantatore (AIP, 2001), p.178.50. H. Persson, S. M. Schneider, W. Greiner et al, Phys. Rev. Lett. 76, 1433 (1996).


51. S. A. Blundell, K. T. Cheng, and J. Sapirstein, Phys. Rev. Lett. 78, 4914 (1997).52. P. Sinnergen, H. Persson, S. Salomoson et al, Phys. Rev. A 58, 1055 (1998).53. V. A. Yerokhin and V. M. Shabaev, Phys. Rev. A 64, 012506 (2001).54. V. A. Yerokhin, A. N. Artemyev, V. M. Shabaev, and G. Plunien, Phys. Rev.

A 72, 052510 (2005).55. S. G. Karshenboim, V. G. Ivanov, and V. M. Shabaev, Physica Scripta T 80,

491 (1999); Zh. Eksp. Teor. Fiz. 117, 67 (2000) [JETP 90, 59 (2000)].

10

Essentially Two-Body Corrections to HFS

10.1 Recoil Corrections to HFS

The very presence of the recoil factor m/M emphasizes that the external fieldapproach is inadequate for calculation of recoil corrections and, in principle,one needs the complete machinery of the two-particle equation in this case.However, many results may be understood without a cumbersome formalism.

Technically, the recoil factor m/M arises because the integration over theexchanged momenta in the diagrams which generate the recoil corrections goesover a large interval up to the muon mass, and not just up to the electron mass,as was the case of the nonrecoil radiative corrections. Due to large intermediatemomenta in the general expression for the recoil corrections only the Diracmagnetic moment of the muon factorizes naturally in the general expressionfor the recoil corrections

∆E = EF (1 + corrections) , (10.1)

whereEF =

163

Z4α2 m

M

(mr

m

)3

ch R∞ . (10.2)

Here EF does not include, unlike (8.2), the muon anomalous magnetic momentaµ which should now be considered on the same grounds as other correctionsto hyperfine splitting. Nonfactorization of the muon anomalous magnetic mo-ment is a natural consequence of the presence of the large integration regionmentioned above. It is worth mentioning that the expression for the Fermienergy EF is symmetric with respect to the light and heavy particles, anddoes not change under exchange of the particles m ↔ M .

10.1.1 Leading Recoil Correction

The leading recoil correction of order Zα(m/M)EF is generated by the graphswith two exchanged photons in Fig. 10.1, similar to the case of the recoil


194 10 Essentially Two-Body Corrections to HFS

Fig. 10.1. Diagrams with two-photon exchanges

correction to the Lamb shift of order (Zα)5(m/M)m considered in Sect. 4.1.However, calculations in the case of hyperfine splitting are much simpler incomparison with the Lamb shift, since the region of extreme nonrelativisticexchange momenta m(Zα)2 < k < m(Zα) does not generate any correction oforder Zα(m/M)EF . This is almost obvious in the nonrelativistic perturbationtheory framework, which is quite sufficient for calculation of all correctionsgenerated at such small momenta. Unlike the case of the Lamb shift theleading contribution which is due to the one-transverse quanta exchange inthe nonrelativistic dipole approximation is given by the Breit potential. Thiscontribution is simply the Fermi energy, and all nonrelativistic corrections tothe Fermi energy are suppressed at least by the additional factor (Zα)2. Thenthe leading recoil correction to hyperfine splitting may reliably be calculatedin the scattering approximation, ignoring even the wave function momentaof order mZα. The formal proof has been given, e.g., in [1], but this mayeasily be understood at the qualitative level. The skeleton integral is linearlyinfrared divergent and this divergence has a clear origin since it correspondsto the classical Fermi contribution to HFS. This divergence is produced by theheavy particle pole contribution and after subtraction (note that we effectivelysubtract the skeleton integral in (9.9) with restored factor 1/(1 + m/M), seediscussion in Subsect. 9.3.1.2) we obtain the convergent skeleton integral

∆E = EFmM

M2 − m2

Zα

2π

∫ ∞

0

dk

k[f(k) − f(µk)] , (10.3)

where

f(k) =k4 − 4k2 − 32

k√

4 + k2− k2 +

16k

, (10.4)

and µ = m/M . One may easily perform the momentum integration in thisinfrared finite integral and obtain [2, 3, 4]

∆Erec = − 3mM

M2 − m2

Zα

πln

M

mEF . (10.5)

The subtracted heavy pole (Fermi) contribution is generated by the ex-change of a photon with a small (atomic scale ∼mZα) momentum and aftersubtraction of this contribution only high loop momenta k (m < k < M)contribute to the integral for the recoil correction. Then the exchange loopmomenta are comparable to the virtual momenta determining the anomalousmagnetic moment of the muon and there are no reasons to expect that the

10.1 Recoil Corrections to HFS 195

anomalous magnetic moment will enter as a factor in the formula for the recoilcorrections. It is clear that the contribution of the muon anomalous magneticmoment in this case cannot be separated from contributions of other radiative-recoil corrections.

Let us emphasize that, unlike the other cases where we encountered thelogarithmic contributions, the result in (10.5) is exact in the sense that this isa complete contribution of order Zα(m/M)EF . There are no nonlogarithmiccontributions of this order.

Despite its nonsymmetric appearance the recoil correction in (10.5) is sym-metric with respect to the electron and muon masses. As in the case of the lead-ing recoil corrections to the Lamb shift coming from one-photon exchanges,this formula generated by the two-photon exchange is exact in the electron-muon mass ratio. This is crucial from the phenomenological point of view,taking into account the large value of the correction under consideration andthe high precision of the current experimental results.

10.1.2 Recoil Correction of Relative Order (Zα)2(m/M)

Recoil corrections of relative order (Zα)2(m/M) are generated by the kernelswith three exchanged photons (see Fig. 10.2). One might expect, similar tothe case of the leading recoil correction, emergence of a recoil contributionof order (Zα)2(m/M)EF logarithmic in the mass ratio. The logarithm of themass ratio could originate only from the integration region m � k � M ,where one can safely omit electron masses in the integrand. The integrandsimplifies, and it turns out that despite the fact that the individual diagramsproduce logarithmic contributions, these contributions sum to zero [5, 6]. It isnot difficult to understand the technical reason for this effect which is calledthe Caswell-Lepage cancellation. For each exchanged diagram there exists apair diagram where the exchanged photons are attached to the electron linein an opposite order. Respective electron line contributions to the logarithmicintegrands generated by these diagrams differ only by sign [6] and, hence, thetotal contribution logarithmic in the mass ratio vanishes.

This means that all pure recoil corrections of relative order (Zα)2(m/M)originate from the exchanged momenta of order of the electron mass andsmaller. One might think that as a result the muon anomalous magnetic mo-ment would enter the expression for these corrections in a factorized form,and the respective corrections should be written in terms of EF and not ofEF , as was in the case of the leading recoil correction. However, if we includethe muon anomalous magnetic moment in the kernels with three exchangedphotons, they would generate the contributions proportional not only to themuon anomalous magnetic moment but to the anomalous magnetic momentsquared. This makes any attempt to write the recoil correction of relative or-der (Zα)2(m/M) in terms of EF unnatural, and it is usually written in termsof EF (see however discussion of this correction in the case of hydrogen inSubsect. 11.1.2). The corrections to this result due to the muon anomalous


magnetic moment should be considered separately as the radiative-recoil cor-rections of order (Z2α)α(Zα)2(m/M)EF . A naive attempt to write the recoilcorrection of relative order (Zα)2(m/M) in terms of EF would shift the mag-nitude of this correction by about 10 Hz. This shift could be understoodas an indication that calculation of the radiative-recoil corrections of order(Z2α)α(Zα)2(m/M)EF could be of some phenomenological interest at thepresent level of experimental accuracy.

Fig. 10.2. Recoil corrections of order (Zα)2(m/M)EF

Leading terms logarithmic in Zα were first considered in [7], and the com-plete logarithmic contribution was obtained in [8, 9]

∆E = 2 ln(Zα)−1(Zα)2m2

r

mMEF . (10.6)

As usual this logarithmic contribution is state-independent. Calculation of thenonlogarithmic contribution turned out to be a much more complicated taskand the whole machinery of the relativistic two-particle equations was usedin this work. First, the difference ∆E(1S) − 8∆E(2S) was calculated [10],then some nonlogarithmic contributions of this order for the 1S level wereobtained [11], and only later the total nonlogarithmic contribution [12, 13]was obtained

∆E =(

−8 ln 2 +6518

)

(Zα)2m2

r

mMEF . (10.7)

This result was later confirmed in a purely numerical calculation [14] in theframework of NRQED.

Recoil contributions in (10.6), and (10.7) are symmetric with respect tomasses of the light and heavy particles. As in the case of the leading recoilcorrection, they were obtained without expansion in the mass ratio, and hencean exact dependence on the mass ratio is known (not just the first termin the expansion over m/M). Let us mention that while for the nonrecoilnonlogarithmic contributions of order (Zα)6, both to HFS and the Lambshift, only numerical results were obtained, the respective recoil contributionsare known analytically in both cases (compare discussion of the Lamb shiftcontributions in Subsect. 4.2.3).

10.1 Recoil Corrections to HFS 197

10.1.3 Higher Order in Mass Ratio Recoil Correctionof Relative Order (Zα)2

The diagrams with three exchanged photons in Fig. 10.2 also generate cor-rections of higher order in the mass ratio. Originally these corrections werecalculated numerically with the help of an effective Hamiltonian in [15]. Laterthe NRQED methods for calculation of recoil corrections [16] developed orig-inally for the Lamb shift were applied to hyperfine splitting [17]. In this workthe authors derived a regular expansion over the small mass parameter m/M ,and obtained recoil corrections of relative order (m/M)2

∆E =[92

ln2 M

m+

(272

− π2

)

lnM

m+ 33ζ(3) − 12π2 ln 2 − 13π2

12+

934

]

× (Zα)2

π2

m

M

m2r

mMEF , (10.8)

and (m/M)3

∆E =[

9 ln2 M

m+

(27 − 2π2

)ln

M

m+ 66ζ(3) − 13π2 ln 2 − 4π2

3+

932

]

× (Zα)2

π2

( m

M

)2 m2r

mMEF . (10.9)

10.1.4 Recoil Corrections of Order (Zα)3(m/M)EF

There are two different double logarithm contributions of order (Zα)3

(m/M)EF , one contains a regular low-frequency logarithm squared, and thesecond depends on the product of the low-frequency logarithm and the loga-rithm of the mass ratio. Calculation of these contributions is quite straightfor-ward and goes along the same lines as calculation of the leading logarithmiccontribution of order α2(Zα)6 to the Lamb shift (see Subsubsect. 3.4.2.1).Taking as one of the perturbation potentials the potential corresponding tothe logarithmic recoil contribution of order (Zα)5 to the Lamb shift in (4.12)and as the other the potential responsible for the main Fermi contributionto HFS in (9.28) (see Fig. 10.3), one obtains [18] a small logarithm squaredcontribution

∆E = −23

(Zα)3

π

m2r

mMln2(Zα)−1EF . (10.10)

A significantly larger double mixed logarithm correction is generated bythe potential corresponding to the leading recoil correction to hyperfine split-ting in (10.5) and the leading logarithmic Dirac correction to the Coulomb-Schrodinger wave function [19, 20, 21] (see Fig. 10.4)

∆E = −3(Zα)3

π

m

Mln(Zα)−1 ln

M

mEF . (10.11)


Fig. 10.3. Leading logarithm squared contribution of order (Zα)3(m/M)EF

Fig. 10.4. Leading mixed double logarithm contribution of order (Zα)3(m/M)EF

The single-logarithmic recoil correction was first estimated in [22], andcalculated in [23, 24]

∆Elog =(

−20 ln 2 +1019

)(Zα)3

π

m

Mln(Zα)−1EF . (10.12)

An estimate of nonlogarithmic contributions of relative order (Zα)3 was ob-tained in [22]

∆Enonlog = (40 ± 20)(Zα)3

π

m

MEF . (10.13)

The error bars of this last estimate are still rather large, and more accuratecalculation would be necessary in pursuit of all corrections of the scale of 10Hz. The relatively large magnitude of the mixed logarithm contribution in(10.11) warrants calculation of the contribution which is linear in the loga-rithm of the mass ratio and nonlogarithmic in Zα. All recoil corrections arecollected in Table 10.1. The uncertainty of the total recoil correction in the lastline of this Table includes an estimate of the magnitude of the yet unknownrecoil contributions.

10.2 Radiative-Recoil Corrections to HFS

10.2.1 Corrections of Order α(Zα)(m/M)EF

and (Z2α)(Zα)(m/M)EF

As in the case of the purely radiative corrections of order α(Zα)EF , all di-agrams relevant for calculation of radiative-recoil corrections of order α(Zα)(m/M)EF may be obtained by radiative insertions in the skeleton diagrams.The only difference is that now the heavy particle line is also dynamical. Theskeleton diagrams for this case coincide with the diagrams for the leading re-coil corrections in Fig. 10.1. Note that even the leading recoil correction toHFS may be calculated in the scattering approximation. Insertion of radia-tive corrections in the skeleton diagrams emphasizes the high momenta region

10.2 Radiative-Recoil Corrections to HFS 199

Table 10.1. Recoil Corrections

EF kHz

Leading recoil correctionArnowitt (1953) [2]

Fulton, Martin (1954) [3] − 3(Zα)π

mMM2−m2 ln M

m−800.304

Newcomb, Salpeter (1955) [4]

Leading logarithmic recoilcorrection, relativeorder (Zα)2

Lepage (1977) [8]

Bodwin, Yennie (1978) [9] 2(Zα)2m2

rmM

ln(Zα)−1 11.179

Nonlogarithmic recoil correction,relative order (Zα)2

Bodwin, Yennie,

Gregorio (1982) [12, 13] (Zα)2m2

rmM

(−8 ln 2 + 6518

) −2.197

Second order in mass ratio,relative order (Zα)2

Blokland, Czarnecki,[

92

ln2 Mm

+(

272− π2

)ln M

m

Melnikov (2002) [17] +33ζ(3) − 12π2 ln 2 − 13π2

12+ 93

4

]

× (Zα)2

π2mM

m2r

mM0.065

Third order in mass ratio,relative order (Zα)2

Blokland, Czarnecki,[9 ln2 M

m+

(27 − 2π2

)ln M

m

Melnikov (2002) [17] +66ζ(3) − 13π2 ln 2 − 4π2

3+ 93

2

]

× (Zα)2

π2

(mM

)2 m2r

mM0.001

Logarithm squared correction,relative order (Zα)3

Karshenboim (1993) [18] − 23

(Zα)3

π

m2r

mMln2(Zα)−1(1 + aµ) −0.043

Mixed logarithm correction,relative order (Zα)3

Kinoshita, Nio (1994) [19, 20]

Karshenboim (1996) [21] −3 (Zα)3

πmM

ln(Zα)−1 ln Mm

−0.210

Single-logarithmic correction,relative order (Zα)3


Hill (2001) [24](−20 ln 2 + 101

9

)(Zα)3

πmM

ln(Zα)−1 −0.035

Nonlogarithmic correction,relative order (Zα)3

Kinoshita (1998) [22] (40 ± 20) (Zα)3

πmM

0.107 (59)

Total recoil correction −791.436 (80)


even more and, hence, the radiative-recoil correction to HFS splitting may becalculated in the scattering approximation. The diagrams for contributions oforder α(Zα)(m/M)EF are presented in Fig. 10.5, in Fig. 10.6 and in Fig. 10.7,and they coincide topologically with the set of diagrams used for calculationof the radiative-recoil corrections to the Lamb shift (formal selection of rele-vant diagrams and proof of validity of the scattering approximation based onthe relativistic two-particle equation, see, e.g., in [1, 25, 26]). We will considerbelow separately the corrections generated by the three types of diagrams:polarization insertions in the exchanged photons, radiative insertions in theelectron line, and radiative insertions in the muon line.

Fig. 10.5. Electron-line radiative-recoil corrections

10.2.2 Electron-Line Logarithmic Contributionsof Order α(Zα)(m/M)EF

The leading recoil correction to hyperfine splitting is generated in the broadintegration region over exchanged momenta m � k � M , and one might ex-pect that insertion of radiative corrections in the skeleton diagrams in Fig. 10.1would produce double logarithmic contributions since the radiative insertionsare themselves logarithmic when the characteristic momentum is larger thanthe electron mass. However, this is only partially true, since the sum of theradiative insertions in the electron line does not have a logarithmic asymp-totic behavior. The simplest way to see this is to work in the Landau gaugewhere respective radiative insertions are nonlogarithmic [27]. In other gauges,individual radiative insertions have leading logarithmic terms, but it is easyto see that due to the Ward identities these logarithmic terms cancel. Thefirst time this cancellation was observed in the direct calculation in [6]. Inthe absence of the leading logarithmic contribution to the electron factor thelogarithmic contribution to HFS is equal to the product of the leading con-stant term −5α/4π in the electron factor [28] and the leading recoil correction(10.5), as calculated in [29]

∆E =154

α(Zα)π2

m

Mln

M

mEF . (10.14)


10.2.3 Electron-Line Nonlogarithmic Contributionsof Order α(Zα)(m/M)EF

The validity of the scattering approximation for calculation of all radiative andradiative-recoil corrections of order α(Zα)EF greatly facilitates the calcula-tions. One may obtain a compact general expression for all such corrections(both logarithmic and nonlogarithmic) induced by the radiative insertions inthe electron line in Fig. 10.5 (see, e.g., [30])

∆Ee-line =Zα

πEF

(

− 316µ

)∫d4k

iπ2

1(k2 + i0)2

×(

1k2 + µ−1k0 + i0

+1

k2 − µ−1k0 + i0

)

〈γµkγν〉(µ) Lµν ,

(10.15)

where the electron factor Lµν describes all radiative corrections in Fig. 9.4,〈γµkγν〉(µ) is the projection of the muon-line numerator on the spinor struc-ture relevant to HFS, and µ = m/(2M).

The integral in (10.15) contains nonrecoil radiative corrections of orderα(Zα)EF , as well as radiative-recoil corrections of all orders in the electron-muon mass ratio generated by the radiative insertions in the electron line.This integral admits in principle a direct brute force numerical calculation.The complicated structure of the integrand makes analytic extraction of thecorrections of definite order in the mass ratio more involved. Direct applica-tion of the standard Feynman parameter methods leads to integrals for theradiative-recoil corrections which do not admit expansion of the integrandover the small mass ratio prior to integration, thus making the analytic calcu-lation virtually impossible. Analytic results may be obtained with the help ofthe approach developed in [31] (see also review in [1]). The idea is to performintegration over the exchanged momentum directly in spherical coordinates.Following this route, we come to the expression of the type

∆E = α(Zα)EF

∫ 1

0

dx

∫ x

0

dy

∫ ∞

0

dk2 Φ(k, x, y)(k2 + a2)2 − 4µ2b2k4

, (10.16)

where a(x, y), b(x, y), and Φ(k) are explicitly known functions.The crucial property of the integrand in Eq. (10.16), which facilitates

calculation, is that the denominator admits expansion in the small para-meter µ prior to momentum integration. This is true due to the inequality4µ2b2k4/(k2 + a2)2 ≤ 4µ2, which is valid according to the definitions of thefunctions a and b. In this way, we may easily reproduce the nonrecoil skeletonintegral in (9.9), and obtain once again the nonrecoil corrections induced bythe radiative insertions in the electron line [32, 33, 34]. This approach admitsalso an analytic calculation of the radiative-recoil corrections of the first orderin the mass ratio.


Nonlogarithmic radiative-recoil corrections to HFS were first calculatednumerically in the Yennie gauge [35, 25] and then analytically in the Feynmangauge [31]

∆E =(

6ζ(3) + 3π2 ln 2 +π2

2+

178

)α(Zα)

π2

m

MEF , (10.17)

where ζ(3) is the Riemann ζ-function.This expression contains all characteristic structures (ζ-function, π2 ln 2,

π2 and a rational number) which one usually encounters in the results of theloop calculations. Let us emphasize that the relative scale of these subleadingterms is rather large, of order π2, which is just what one should expect forthe constants accompanying the large logarithm.

Originally there was a discrepancy between the analytic result in the Feyn-man gauge [31] (6ζ(3) + 3π2 ln 2 + π2/2 + 17/8)/π2 = 3.526 and the numer-ical result in the Yennie gauge [25] 3.335 ± 0.058. When both works werecompleted this discrepancy which is as large as three standard deviations ofthe accuracy of the numerical integration in [25] was purely academic. Butnowadays, with the increase of the accuracy of the experimental data the dis-crepancy of about 0.22 kHz has a phenomenological significance. In order toresolve this discrepancy, an independent analytic calculation of the electron-line contribution in the Yennie gauge was undertaken [30]. This new work waslogically independent of [31], the new calculation was performed in the Yenniegauge, and the expression for the electron factor from [25, 35] was used as theinitial point of the calculation. The result of [30] confirmed the earlier analyticresult in [31], later it was also confirmed in the NRQED framework in [16, 17],and the result in (10.17) is now firmly established1.

10.2.4 Muon-Line Contribution of Order (Z2α)(Zα)(m/M)EF

The radiative-recoil correction to HFS generated by the diagrams in Fig. 10.6with insertions of the radiative photons in the muon line is given by an ex-pression similar to the one in (10.15) [1, 36], the only difference is that nowwe have to insert the muon factor instead of the heavy line numerator and topreserve the skeleton electron-factor numerator. Unlike the case of the elec-tron line, radiative insertions in the heavy muon line do not generate nonrecoilcorrections. This is easy to realize if one recalls that the nonrecoil electronfactor contribution is generated by the muon pole, which is absent in the di-agrams with two exchanged photons and the muon-line fermion factor (muonanomalous magnetic moment is subtracted from the muon factor, since in thesame way as in the case of the electron factor it generates corrections of lowerorder in α). Radiative insertions in the heavy fermion line do not generatelogarithmic terms [29] either. This can be understood with the help of the low1 See one more comment on this discrepancy below in Subsect. 10.2.10 where the

radiative-recoil correction of order α(Zα)(m/M)2EF is discussed.


Fig. 10.6. Muon-line radiative-recoil corrections

energy theorem for the Compton scattering. The effective momenta in theintegral for the radiative-recoil corrections are smaller than the muon massand, hence, the muon-line factor enters the integral in the low momenta limit.The classical low energy theorem for Compton scattering cannot be used di-rectly in this case since the exchanged photons are virtual, but neverthelessit is not difficult to prove the validity of a generalized low energy theorem inthis case [1, 18]. Then we see that the logarithmic skeleton integrand gets anextra factor k2 after insertion of the muon factor in the integrand, and thisextra factor changes the logarithmic nature of the integral.

Analytic calculations of the muon-line radiative-recoil correction are car-ried out in the same way as in the electron-line case and the purely numerical[25, 35] and analytic [36, 1] results for this contribution are in excellent agree-ment

∆E =(

92ζ(3) − 3π2 ln 2 +

398

)(Z2α)(Zα)

π2

m

MEF . (10.18)

10.2.5 Leading Photon-Line Double Logarithmic Contributionof Order α(Zα)(m/M)EF

The only double-logarithmic radiative-recoil contribution of order α(Zα)(m/M)EF is generated by the leading logarithmic term in the polarization op-erator. Substitution of this leading logarithmic term in the logarithmic skele-ton integral in (10.3) immediately leads to the double logarithmic contribution[6]

∆E = −2α(Zα)

π2

m

Mln2 M

mEF . (10.19)

As was noted in [28] this contribution may be obtained without any cal-culations at all. It is sufficient to realize that with logarithmic accuracy thecharacteristic momenta in the leading recoil correction in (10.3) are of orderM and, in order to account for the leading logarithmic contribution gener-ated by the polarization insertions, it is sufficient to substitute in (10.5) therunning value of α at the muon mass instead of the fine structure α. Thisalgebraic operation immediately reproduces the result above.


Fig. 10.7. Photon-line radiative-recoil corrections

10.2.6 Photon-Line Single-Logarithmic and NonlogarithmicContributions of Order α(Zα)(m/M)EF

Calculation of the nonrecoil radiative correction of order α(Zα)EF was fa-cilitated by simultaneous consideration of the electron and muon loop po-larization insertions in the exchanged photons. Similarly calculation of theradiative-recoil corrections generated by the diagrams in Fig. 10.7 with in-sertions of the vacuum polarizations, is technically simpler if one considerssimultaneously both electron and muon vacuum polarizations. All correctionsmay be obtained by substituting the explicit expression for the sum of vacuumpolarizations in the skeleton integral (10.3). In this skeleton integral, part ofthe recoil correction corresponding to the factor 1/(1 + m/M) is subtractedand this explains why we have restored this factor in consideration of thenonrecoil part of the vacuum polarization. Technically, consideration of thesum of the electron and muon vacuum polarizations leads to simplification ofthe integrand for the radiative-recoil corrections, and after an easy calcula-tion one obtains the single-logarithmic and nonlogarithmic contributions tothe total radiative-recoil correction induced by the sum of the electron andmuon vacuum polarizations [18, 25]

∆E =(

−83

lnM

m− 28

9− π2

3

)α(Zα)

π2

M

mEF . (10.20)

Note that in the parenthesis we have parted with our usual practice ofconsidering the muon as a particle with charge Ze, and assumed Z = 1.Technically this is inspired by the cancellation of certain contributions betweenthe electron and muon polarization loops mentioned above, and from thephysical point of view it is not necessary to preserve a nontrivial factor Z here,since we need it only as a reference to an interaction with the “constituent”muon and not with the one emerging in the polarization loops.

As was explained above, the coefficient before the leading logarithmsquared term in (10.19) may easily be obtained almost without any calcu-lations. Simultaneous account for the electron and muon loops does not affectthis contribution, since all logarithmic contributions are generated only bythe insertion of the electron loop. It may be shown also that the coefficientbefore the subleading logarithm originates from the first two terms in the as-ymptotic expansion of the polarization operator 2α/(3π) ln(k/m) − 5/9 [28].Substituting the polarization operator asymptotics in the skeleton integral in(10.3) and, multiplying the result by the factor 2 in order to take into accountall possible insertion of the polarization loop in the exchanged photons, one


Fig. 10.8. Heavy-particle polarization contribution to HFS

obtains −8/3 for the coefficient before the single-logarithmic term, in accor-dance with the result above. The factor −6 comes from the leading logarithmicterm in the polarization operator expansion, and the factor 10/3 is generatedby the subleading constant, their sum being equal to −8/3.

10.2.7 Heavy Particle Polarization Contributionsof Order α(Zα)EF

The contribution of the muon polarization operator was already consideredabove. One might expect that contributions of the diagrams in Fig. 10.8 withthe heavy particle polarization loops are of the same order of magnitude as thecontribution of the muon loop, so it is natural to consider this contributionhere. Respective corrections could easily be calculated by substituting theexpressions for the heavy particle polarizations in the unsubtracted skeletonintegral in (10.3). The contribution of the heavy lepton τ polarization operatorwas obtained in [37, 38] both numerically and analytically

∆E = 0.019 190 6α(Zα)

π2

m

MEF =

(65

lnmτ

mµ+

5125

)α(Zα)

π2

memµ

m2τ

EF .

(10.21)Polarization contributions due to the loops of pions and other hadrons

cannot be calculated with the help of the QCD perturbation theory. It is easyto estimate hadron vacuum polarization using some low energy model likevector dominance [25]. In a more accurate approach one uses experimentaldata for e+e− annihilation into hadrons to find the polarization operator atlow energy, and approximates the hadron vacuum polarization by the pertur-bative formula at high energy. The accuracy of such calculations is limitedby the accuracy of experimental results for e+e− annihilation, and by theproper choice of the threshold above which to use the perturbative expres-sions. Many independent calculations along these lines were performed in theliterature [37, 39, 40, 41] which all produced compatible results. Currentlythe most accurate result for the hadron polarization operator contribution toHFS is [37]

∆E = 0.233 (3) kHz . (10.22)

There exist also higher order in α hadronic contributions, e.g., insertionsof hadronic light by light scattering in the two-photon exchange graphs, inser-tions of two hadron vacuum polarizations in the two-photon exchange graphs,etc. These contributions are suppressed by at least one additional power of α


in comparison with contributions from the diagram in Fig. 10.8. They wereconsidered in [42, 43, 44] with the result [44]

∆E = 0.005 (2) kHz . (10.23)

The total hadronic radiative-recoil correction is just the sum of the resultsin (10.22) and (10.23)

∆Ehad = 0.238 (4) kHz . (10.24)

10.2.8 Leading Logarithmic Contributionsof Order α2(Zα)(m/M)EF

The leading contribution of order α2(Zα)(m/M)EF is enhanced by the cubeof the large logarithm of the electron-muon mass ratio. One could expectthat logarithm cube terms would be generated by a few types of radiativeinsertions in the skeleton graphs with two exchanged photons: insertions ofthe first and second order polarization operators in the exchanged photons inFig. 10.9 and in Fig. 10.10, insertions of the light-by-light scattering contri-butions in Fig. 10.11, insertions of two radiative photons in the electron linein Fig. 10.12, and insertions of the polarization operator in the radiative pho-ton in Fig. 10.13. In [45], where the leading logarithm cube contribution wascalculated explicitly, it was shown that only the graphs in Fig. 10.9 with in-sertions of the one-loop polarization operators generate logarithm cube terms.This leading contribution may be obtained without any calculations by sim-ply substituting the effective charge α(M) defined at the characteristic scaleM in the leading recoil correction of order (Zα)(m/M)EF instead of the finestructure constant α and expanding the resulting expression in the power se-ries over α [28] (compare with a similar remark above concerning the leadinglogarithm squared term of order α(Zα)EF ).

Calculation of the logarithm squared term of order α2(Zα)(m/M)EF ismore challenging [28]. All graphs in Figs. 10.9, 10.10, 10.11, 10.12, and 10.13

Fig. 10.9. Graphs with two one-loop polarization insertions

Fig. 10.10. Graphs with two–loop polarization insertions


Fig. 10.11. Graphs with light by light scattering insertions

Fig. 10.12. Graphs with radiative photon insertions

Fig. 10.13. Graphs with polarization insertions in the radiative photon

generate corrections of this order. The contribution induced by the irreducibletwo-loop vacuum polarization in Fig. 10.10 is again given by the effectivecharge expression. Subleading logarithm squared terms generated by the one-loop polarization insertions in Fig. 10.9 may easily be calculated with the helpof the two leading asymptotic terms in the polarization operator expansionand the skeleton integral. An interesting effect takes place in calculation ofthe logarithm squared term generated by the polarization insertions in theradiative photon in Fig. 10.13. One might expect that the high energy as-ymptote of the electron factor with the polarization insertion is given by theproduct of the leading constant term of the electron factor −5α/(4π) and theleading polarization operator term. However, this expectation turns out tobe wrong. One may check explicitly that instead of the naive factor aboveone has to multiply the polarization operator by the factor −3α/(4π). Thereason for this effect may easily be understood. The factor −3α/(4π) is theasymptote of the electron factor in massless QED and it gives a contributionto the logarithmic asymptotics only after the polarization operator insertion.This means that in massive QED the part −2α/(4π) of the constant electronfactor originates from the integration region where the integration momentumis of order of the electron mass. Naturally this integration region does not giveany contribution to the logarithmic asymptotics of the radiatively correctedelectron factor. The least trivial logarithm squared contribution is generatedby the three-loop diagrams in Fig. 10.11 with the insertions of light by lightscattering block. Their contribution was calculated explicitly in [28]. Later it


Fig. 10.14. Renormalization of the fifth current

was realized that these contributions are intimately connected with the wellknown anomalous renormalization of the axial current in QED [46]. Due tothe projection on the HFS spin structure in the logarithmic integration regionthe heavy particle propagator effectively shrinks to an axial current vertex,and in this situation calculation of the respective contribution to HFS re-duces to substitution of the well known two-loop axial renormalization factorin Fig. 10.14 [47] in the recoil skeleton diagram. Of course, this calculationreproduces the same contribution as obtained by direct calculation of the di-agrams with light by light scattering expressions. From the theoretical pointof view it is interesting that having sufficiently accurate experimental dataone can in principle measure anomalous two-loop renormalization of the axialcurrent in the atomic physics experiment.

The sum of all logarithm cubed and logarithm squared contributions oforder α2(Zα)(m/M)EF is given by the expression [45, 28]

∆E =(

−43

ln3 M

m+

43

ln2 M

m

)α2(Zα)

π3

m

MEF . (10.25)

It was also shown in [28] that there are no other contributions with thelarge logarithm of the mass ratio squared accompanied by the factor α3, evenif the factor Z enters in another manner than in the equation above.

Fig. 10.15. Graphs with simultaneous insertions of the electron and muon loops

Unlike the logarithm cube and logarithm squared terms which are gener-ated only by a small number of diagrams discussed above, there are numeroussources of the single-logarithmic terms, e.g., diagrams in Fig. 10.15 and manyothers. The first estimate of such contributions in [48] demonstrated that theirmagnitude could be about a few dozen hertz, and therefore they may be phe-nomenologically relevant at the current level of experimental and theoreticalaccuracy. First results of a systematic calculation of all single-logarithmic andnonlogarithmic contributions are reported in [49, 50, 51]

∆E =[(

−4π2 ln 2 − 2912

)

lnM

m+ 47.7213

]α3

π3

m

MEF . (10.26)


This result is still incomplete, and work on calculation of the remainingsingle-logarithmic and nonlogarithmic contributions is in progress.

10.2.9 Leading Four-Loop Contributionof Order α3(Zα)(m/M)EF

The leading four-loop radiative-recoil correction is generated by the diagramswith three polarization operator insertions in the exchanged photons similarto the diagrams with the two polarization insertions in Fig. 10.10. It containsthe large logarithm to the fourth power, and like the leading logarithm cubedcontribution may be easily obtained either by direct calculation or by substi-tution of the effective charge α(M) in the leading recoil correction of order(Zα)(m/M)EF [49]

∆E = −89

ln4 M

m

α3(Zα)π4

m

MEF . (10.27)

Due to the presence of the large logarithm this correction of order α3(Zα)numerically is of the same order as corrections of order α2(Zα)

∆E = −1.668α2(Zα)

π3

m

MEF . (10.28)

10.2.10 Corrections of Order α(Zα)(m/M)nEF

Higher order in mass ratio radiative-recoil corrections of order α(Zα)(m/M)nEF , n ≥ 2, are generated by the same set of diagrams in Fig. 10.5,in Fig. 10.6 and in Fig. 10.7 with the radiative insertions in the electron andmuon lines, and with the polarization insertions in the photon lines, as therespective corrections of the previous order in the mass ratio. Analytic cal-culation of the correction of order α(Zα)(m/M)2EF in [52] proceeds as inthat case, the only difference is that now one has to preserve all contributionswhich are of second order in the small mass ratio. It turns out that all suchcorrections are generated at the scale of the electron mass, and one obtainsfor the sum of all corrections [52]

∆E =[(

−6 ln 2 − 34

)

α(Zα) − 1712

(Z2α)(Zα)](

m

M

)2

EF . (10.29)

The electron-line contribution from [52] (confirmed later independently in[16, 17])

∆E =(

−6 ln 2 − 32

)

α(Zα)(

m

M

)2

EF ≈ −0.03 kHz , (10.30)

is of special interest in view of the discrepancy between the results for theelectron line contributions of order α(Zα)(m/M)EF in [31, 30] and in [25]


(see discussion in Subsect. 10.2.3). The result for the electron line contributionin [25] was obtained without expansion in m/M , so one could try to ascribethe discrepancy to the contribution of these higher order terms. To check thishypothesis we can use the systematic expansion of the contribution of relativeorder α(Zα) in mass ratio which was obtained in [16, 17] in the NRQEDframework. The next, third in the mass ratio, term in this expansion has theform [16, 17]

∆E =(

6112

ln2 M

m+

103772

lnM

m+

9ζ(3)2

+ 3π2 ln 2 +133π2

72+

5521288

)

×α(Zα)π2

(m

M

)3

EF ≈ 0.0008 kHz . (10.31)

We see that the contributions of higher order in the mass ratio in (10.30)and (10.31) are by far too small to explain the discrepancy between the resultsof [31, 30] and [25].

10.2.11 Corrections of Orders α(Zα)2(m/M)EF

and Z2α(Zα)2(m/M)EF

Radiative-recoil corrections of order α(Zα)2(m/M)EF were never calculatedcompletely. As we have mentioned in Subsect. 9.4.1.1 the leading logarithmsquared contribution of order α(Zα)2EF may easily be calculated if one takesas one of the perturbation potentials the potential corresponding to the elec-tron electric form factor and as the other the potential responsible for themain Fermi contribution to HFS (see Fig. 10.16). Then one obtains the lead-ing logarithm squared contribution in the form [18]

∆E = −23

ln2(Zα)−2 α(Zα)2

π

(mr

m

)2

EF , (10.32)

which differs from the leading logarithm squared term in (9.29) by the recoilfactor (mr/m)2. Preserving only the linear term in the expansion of this resultover the mass ratio one obtains

∆E =43

ln2(Zα)−2 α(Zα)2

π

m

MEF . (10.33)

Numerically this contribution is about 0.3 kHz, and clearly has to be takeninto account in comparison of the theory with the experimental data. In this

Fig. 10.16. Leading logarithm squared contribution of order α(Zα)2EF


situation it is better simply to use in the theoretical formulae the leadinglogarithm squared contribution of order α(Zα)2EF in the form in (10.32)instead of the expression for this logarithmic term in (9.29).

The single-logarithmic correction of order α(Zα)2(m/M)EF was esti-mated in [22], and calculated in [23, 24]

∆E =(

323

ln 2 − 43190

)


π

m

MEF . (10.34)

The single-logarithmic correction of order Z2α(Zα)2(m/M)EF originatingfrom radiative insertion in the muon line was calculated in [23, 24]

∆E = ln(Zα)−1 Z2α(Zα)2

π

m

MEF . (10.35)

An estimate of the nonlogarithmic contribution was obtained in [22]

∆E = (−40 ± 10)α(Zα)2

π

m

MEF . (10.36)

The relatively large magnitude of the correction in (10.33) demonstratesthat a calculation of all radiative-recoil corrections of order α(Zα)2(m/M)EF

is warranted. The error of the total radiative-recoil correction in the last linein Table 10.2 includes, besides the errors of individual contributions in theupper lines of this Table, also an educated guess on the magnitude of yetuncalculated contributions.


The weak interaction contribution to hyperfine splitting is due to Z-bosonexchange between the electron and muon in Fig. 6.7. Due to the large massof the Z-boson this exchange is effectively described by the local four-fermioninteraction Hamiltonian

Hint = − 12M2

Z

∫

d3x (j · j) , (10.37)

where j is the spatial part of the weak current (j0, j), weak charge is includedin the definition of the weak current, and MZ is the Z-boson mass.

The weak interaction contribution to HFS was calculated many years ago[53, 54], and even radiative corrections to the leading term were discussed inthe literature [55, 56, 57]. However, this weak contribution to HFS was cited inthe later literature with different signs [9, 19]. This happened probably becausethe weak correction was of purely academic interest for early researchers. Thediscrepancy in sign was subjected to scrutiny in a number of works [20, 58, 59]which all produced the result in agreement with [54]


Table 10.2. Radiative-Recoil Corrections

EF kHz

Leading logarithmicelectron-line correction

Terray, Yennie (1982) [29] 154

α(Zα)

π2mM

ln Mm

2.324

Nonlogarithmicelectron-line correction

Eides, Karshenboim, Shelyuto (1986) [31](6ζ(3) + 3π2 ln 2 + π2

2+ 17

8

)

×α(Zα)

π2mM

4.044

Muon-line contribution

Eides, Karshenboim, Shelyuto (1988) [36](

92ζ(3) − 3π2 ln 2 + 39

8

)

× (Z2α)(Zα)

π2mM

−1.190

Logarithm squaredpolarization contribution

Caswell, Lepage (1978) [6] −2 ln2 Mm

α(Zα)

π2mM

−6.607

Single-logarithmic and nonlogarithmicpolarization contributions

Terray, Yennie (1982) [29](− 8

3ln M

m− π2

3− 28

9

)α(Zα)

π2mM

−2.396

τ lepton polarization contributionCzarnecki, Eidelman,Karshenboim (2002) [37]

Eides, Shelyuto (2003) [38] 0.019 190 6α(Zα)

π2mM

0.002

Hadron polarization contributionCzarnecki, Eidelman,Karshenboim (2002) [37]Eidelman, Karshenboim,Shelyuto (2002) [44] 0.238(4)

Leading log cube correction

Eides, Shelyuto (1984) [45] − 43

α2(Zα)

π3mM

ln3 Mm

−0.055

Log square correction

Eides, Karshenboim, Shelyuto (1989) [28] 43

α2(Zα)

π3mM

ln2 Mm

0.010

Single-logarithmic andnonlogarithmic corrections

Eides, Grotch, Shelyuto (2005) [51][(

− 4π2 ln 2 − 2912

)

× ln Mm

+ 47.7213]

α3

π3mM

−0.0300

Leading quartic log correction

Eides, Grotch, Shelyuto (2001) [49] − 89

ln4 Mm

α3(Zα)

π4mM

−0.0004

References 213


EF kHz

Second order in mass ratiocontribution

Eides, Grotch, Shelyuto (1998) [52][(

− 6 ln 2 − 34

)α(Zα)

− 1712

(Z2α)(Zα))]

( mM

)2 −0.035

Electron line third order

in mass ratio contribution(

6112

ln2 Mm

+ 103772

ln Mm

+ 9ζ(3)2

Czarnecki, Melnikov (2001) [16] +3π2 ln 2 + 133π2

72+ 5521

288

)

×α(Zα)

π2

(mM

)30.0008

Leading logarithmicα(Zα)2 correction

Karshenboim (1993) [18] 43

α(Zα)2

πmM

ln2(Zα)−2 0.344

Single-logarithmicα(Zα)2 correctionMelnikov, Yelkhovsky (2001) [23]

Hill (2001) [24](

323

ln 2 − 43190

)ln(Zα)−1 α(Zα)2

πmM

0.034

Single-logarithmicZ2α(Zα)2 correctionMelnikov, Yelkhovsky (2001) [23]

Hill (2001) [24] ln(Zα)−1 Z2α(Zα)2

πmM

0.013

Nonlogarithmic α(Zα)2

radiative-recoil correction

Kinoshita (1998) [22] (−40 ± 10)α(Zα)2

πmM

−0.107(27)

Total radiative-recoil correction −3.410 (70)

∆E = −GF√2

3mM

4πZαEF ≈ −0.065 kHz . (10.38)

It is easy to see from [59] that for nuclei with more than one nucleon theexpression in (10.38) would contain an extra factor equal to the eigenvalue ofthe doubled third component 2T3 of the weak isospin operator.

References

1. M. I. Eides, S. G. Karshenboim, and V. A. Shelyuto, Ann. Phys. (NY) 205,231, 291 (1991).

2. R. Arnowitt, Phys. Rev. 92, 1002 (1953).3. T. Fulton and P. C. Martin, Phys. Rev. 95, 811 (1954).4. W. A. Newcomb and E. E. Salpeter, Phys. Rev. 97, 1146 (1955).


5. G. T. Bodwin, D. R. Yennie, and M. A. Gregorio, Phys. Rev. Lett. 41, 1088(1978).

6. W. E. Caswell and G. P. Lepage, Phys. Rev. Lett. 41, 1092 (1978).7. T. Fulton, D. A. Owen, and W. W. Repko, Phys. Rev. Lett. 26, 61 (1971).8. P. Lepage, Phys. Rev. A 16, 863 (1977).9. G. T. Bodwin and D. R. Yennie, Phys. Rep. C 43, 267 (1978).

10. M. M. Sternheim, Phys. Rev. 130, 211 (1963).11. W. E. Caswell and G. P. Lepage, Phys. Rev. A 18, 810 (1978).12. G. T. Bodwin, D. R. Yennie, and M. A. Gregorio, Phys. Rev. Lett. 48, 1799

(1982).13. G. T. Bodwin, D. R. Yennie, and M. A. Gregorio, Rev. Mod Phys. 57, 723

(1985).14. W. E. Caswell and G. P. Lepage, Phys. Lett. B 167, 437 (1986).15. K. Pachucki, Phys. Rev. A 56, 297 (1997).16. A. Czarnecki and K. Melnikov, Phys. Rev. Lett. 87, 013001 (2001).17. I. Blokland, A. Czarnecki, and K. Melnikov, Phys. Rev. D 65, 073015 (2002).18. S. G. Karshenboim, Zh. Eksp. Teor. Fiz. 103, 1105 (1993) [JETP 76, 541

(1993)].19. T. Kinoshita and M. Nio, Phys. Rev. Lett. 72, 3803 (1994).20. T. Kinoshita and M. Nio, Phys. Rev. D 53, 4909 (1996).21. S. G. Karshenboim, Z. Phys. D 36, 11 (1996).22. T. Kinoshita, hep-ph/9808351, Cornell preprint, 1998.23. K. Melnikov and A. S. Yelkhovsky, Phys. Rev. Lett. 86, 1498 (2001).24. R. J. Hill, Phys. Rev. Lett. 86, 3280 (2001).25. J. R. Sapirstein, E. A. Terray, and D. R. Yennie, Phys. Rev. D29, 2290 (1984).26. S. G. Karshenboim, M. I. Eides, and V. A. Shelyuto, Yad. Fiz. 47, 454 (1988)

[Sov. J. Nucl. Phys. 47, 287 (1988)]; Yad. Fiz. 48, 769 (1988) [Sov. J. Nucl.Phys. 48, 490 (1988)].

27. V. B. Berestetskii, E. M. Lifshitz, and L. P. Pitaevskii, Quantum electrodynam-ics, 2nd Edition, Pergamon Press, Oxford, 1982.

28. M. I. Eides, S. G. Karshenboim, and V. A. Shelyuto, Phys. Lett. B 216, 405(1989); Yad. Fiz. 49, 493 (1989) [Sov. J. Nucl. Phys. 49, 309 (1989)].

29. E. A. Terray and D. R. Yennie, Phys. Rev. Lett. 48, 1803 (1982).30. V. Yu. Brook, M. I. Eides, S. G. Karshenboim, and V. A. Shelyuto, Phys. Lett.

B 216, 401 (1989).31. M. I. Eides, S. G. Karshenboim, and V. A. Shelyuto, Phys. Lett. B 177, 425

(1986); Yad. Fiz. 44, 1118 (1986) [Sov. J. Nucl. Phys. 44, 723 (1986)]; Zh. Eksp.Teor. Fiz. 92, 1188 (1987) [Sov. Phys.-JETP 65, 664 (1987)]; Yad. Fiz. 48, 1039(1988) [Sov. J. Nucl. Phys. 48, 661 (1988)].

32. N. Kroll and F. Pollock, Phys. Rev. 84, 594 (1951); ibid. 86, 876 (1952).33. R. Karplus, A. Klein, and J. Schwinger, Phys. Rev. 84, 597 (1951).34. R. Karplus and A. Klein, Phys. Rev. 85, 972 (1952).35. J. R. Sapirstein, E. A. Terray, and D. R. Yennie, Phys. Rev. Lett. 51, 982 (1983).36. M. I. Eides, S. G. Karshenboim, and V. A. Shelyuto, Phys. Lett. B 202, 572

(1988); Zh. Eksp. Teor. Fiz. 94, 42 (1988) [Sov.Phys.-JETP 67, 671 (1988)].37. A. Czarnecki, S. I. Eidelman, and S. G. Karshenboim, Phys. Rev. D 65, 053004

(2002).38. M. I. Eides and V. A. Shelyuto, Phys. Rev. A 68, 042106 (2003).39. A. Karimkhodzhaev and R. N. Faustov, Yad. Phys. 53, 1012 (1991) [Sov. J.

Nucl. Phys. 53, 626 (1991)].

References 215

40. R. N. Faustov, A. Karimkhodzhaev, and A. P. Martynenko, Phys. Rev. A 59,2498 (1999); Yad. Phys. 62, 2284 (1999) [Phys. Atom. Nuclei 62, 2103 (1999)].

41. S. Narison, preprint hep-ph/0203053, 2002.42. S. G. Karshenboim, and V. A. Shelyuto, Phys. Lett. B 517, 32 (2001).43. R. N. Faustov and A. P. Martynenko, Phys. Lett. B 541, 135 (2002).44. S. I. Eidelman, S. G. Karshenboim, and V. A. Shelyuto, Can. J. Phys. 80, 1297

(2002).45. M. I. Eides and V. A. Shelyuto, Phys. Lett. B 146, 241 (1984).46. S. G. Karshenboim, M. I. Eides, and V. A. Shelyuto, Yad. Fiz. 52, 1066 (1990)

[Sov. J. Nucl. Phys. 52, 679 (1990)].47. S. L. Adler, Phys. Rev. 177, 2426 (1969).48. G. Li, M. A. Samuel, and M. I. Eides, Phys. Rev. A 47, 876 (1993).49. M. I. Eides, H. Grotch, and V. A. Shelyuto, Phys. Rev. D 65, 013003 (2001).50. M. I. Eides, H. Grotch, and V. A. Shelyuto, Phys. Rev. D 67, 113003 (2003).51. M. I. Eides, H. Grotch, and V. A. Shelyuto, Can. J. Phys. 83, 363 (2005).52. M. I. Eides, H. Grotch, and V. A. Shelyuto, Phys. Rev. D 58, 013008 (1998).53. J. Barclay Adams, Phys. Rev. 139, B1050 (1965).54. M. A. Beg and G. Feinberg, Phys. Rev. Lett. 33, 606 (1974); 35, 130(E) (1975).55. W. W. Repko, Phys. Rev. D 7, 279 (1973).56. H. Grotch, Phys. Rev. D 9, 311 (1974).57. R. Alcotra and J. A. Grifols, Ann. Phys.(NY) 229, 109 (1993).58. J. R. Sapirstein, unpublished, as cited in [19] .59. M. I. Eides, Phys. Rev. A 53, 2953 (1996).

11

Hyperfine Splitting in Hydrogen

The hyperfine splitting in the ground state of hydrogen is one of the mostprecisely measured quantities in modern physics [1, 2] (see for more detailsSubsect. 12.2.1 below), and to describe it theoretically we need to consideradditional contributions to HFS connected with the bound state nature of theproton.

Dominant nonrecoil contributions to the hydrogen hyperfine splitting areessentially the same as in the case of muonium. The only differences are thatnow in all formulae the proton mass replaces the muon mass, and we have tosubstitute the proton anomalous magnetic moment κ = 1.792 847 351 (28)[3], measured in nuclear magnetons, instead of the muon anomalous magneticmoment aµ, measured in the Bohr magnetons, in the expression for the hy-drogen Fermi energy in (8.2). After this substitution one can use the nonrecoilcorrections collected in Tables 9.1–9.5 for the case of hydrogen.

As in the case of the Lamb shift the composite nature of the nucleus re-veals itself first of all via a relatively large finite size correction. It is alsonecessary to reconsider all recoil and radiative-recoil corrections. Due to exis-tence of the proton anomalous magnetic moment and nontrivial proton formfactors, simple minded insertions of the hydrogen Fermi energy instead of themuonium Fermi energy in the muonium expressions for these corrections leadsto the wrong results. As we have seen in Sects. 10.1–10.2 leading recoil andradiative-recoil corrections originate from distances small on the atomic scale,between the characteristic Compton lengths of the heavy nucleus 1/M and thelight electron 1/m. Proton contributions to hyperfine splitting coming fromthese distances cannot be satisfactorily described only in terms of such globalcharacteristics as its electric and magnetic form factors. Notice that the lead-ing recoil and radiative-recoil contributions to the Lamb shift are softer, theycome from larger distances (see Sects. 4.1–5.2), and respective formulae arevalid both for elementary and composite nuclei. Theoretical distinctions be-tween the case of elementary and composite nuclei are much more importantfor HFS than for the Lamb shift.


218 11 Hyperfine Splitting in Hydrogen

Despite the difference between the two cases, discussion of the protonsize and structure corrections to HFS in hydrogen below is in many respectsparallel to the discussion of the respective corrections to the Lamb shift inChap. 6.

11.1 Nuclear Size, Recoil and Structure Correctionsof Orders (Zα)EF and (Zα)2EF

Nontrivial nuclear structure beyond the nuclear anomalous magnetic momentfirst becomes important for corrections to hyperfine splitting of order (Zα)5

(compare respective corrections to the Lamb shift in Subsubsect. 6.2). Cor-rections of order (Zα)EF connected with the nonelementarity of the nucleusare generated by the diagrams with two-photon exchanges. Insertion of theperturbation corresponding to the magnetic or electric form factors in oneof the legs of the skeleton diagram in Fig. 9.2 described by the infrared di-vergent integral in (9.9) makes the integral infrared convergent and pushescharacteristic integration momenta to the high scale determined by the char-acteristic scale of the hadron form factor. Due to the composite nature of thenucleus, besides intermediate elastic nuclear states, we also have to considerthe contribution of the diagrams with inelastic intermediate states.

We will first consider the contributions generated only by the elastic in-termediate nuclear states. This means that calculating this correction we willtreat the nucleus as a particle which interacts with the photons via its non-trivial Sachs electric and magnetic form factors in (6.8).

11.1.1 Corrections of Order (Zα)EF

11.1.1.1 Correction of Order (Zα)(m/Λ)EF (Zemach Correction)

As usual we start consideration of the contributions of order (Zα)EF withthe infrared divergent integral (9.9) corresponding to the two-photon skeletondiagram in Fig. 9.2. Insertion of factors GE(−k2)−1 or GM (−k2)/(1+κ)−1in one of the external proton legs corresponds to the presence of a nontrivialproton form factor.1

We need to consider diagrams in Fig. 11.1 with insertion of one factorGE(M)(−k2) − 1 in the proton vertex2

1 Subtraction is necessary in order to avoid double counting since the diagramswith the subtracted term correspond to the pointlike proton contribution, alreadytaken into account in the expression for the Fermi energy in (8.2).

2 Dimensionless integration momentum in (9.9) is measured in electron mass. Wereturn here to dimensionful integration momenta, which results in an extra factorm in the numerators in (11.1) and (11.2). Notice also the minus sign before themomentum in the arguments of form factors, it arises because in the equationsbelow k = |k|.

11.1 Nuclear Size, Recoil and Structure Corrections 219

Fig. 11.1. Elastic nuclear size corrections of order (Zα)EF with one form factorinsertion. Empty dot corresponds either to GE(−k2) − 1 or GM (−k2)/(1 + κ) − 1

∆E =8(Zα)m

πn3EF

∫ ∞

0

dk

k2

{[GE(−k2) − 1

]+

[GM (−k2)

1 + κ− 1

]}

, (11.1)

and the diagram in Fig. 11.2 with simultaneous insertion of both factorsGE(−k2) − 1 and GM (−k2) − 1 in two proton vertices

∆E =8(Zα)m

πn3EF

∫ ∞

0

dk

k2

[GE(−k2) − 1

][GM (−k2)

1 + κ− 1

]

. (11.2)

Fig. 11.2. Elastic nuclear size correction of order (Zα)EF with two form factorinsertions. Empty dot corresponds either to GE(−k2) − 1 or GM (−k2)/(1 + κ) − 1

Effectively the integration in (11.2), and (11.1) goes up to the form factorscale. This scale is much higher than the electron mass and high momenta, inthis section, means momenta much higher than the electron mass. In earliersections high momenta often meant momenta of the scale of the electron mass.

The total proton size dependent contribution of order (Zα)EF , which isoften called the Zemach correction, has the form

∆E =8(Zα)m

πn3EF

∫ ∞

0

dk

k2

[GE(−k2)GM (−k2)

1 + κ− 1

]

, (11.3)

or in the coordinate space [4]

∆E = −2(Zα)m〈r〉(2)EF , (11.4)

where 〈r〉(2) is the first Zemach moment (or the Zemach radius) [4], definedvia the weighted convolution of the electric and magnetic densities ρE(M)(r)corresponding to the respective form factors (compare with the third Zemachmoment in (6.15))

〈r〉(2) ≡∫

d3rd3rρE(r)ρM (r)|r − r| . (11.5)


Parametrically the result in (11.4) is of order (Zα)(m/Λ)EF , where Λ isthe form factor scale. This means that this correction should be consideredtogether with other recoil corrections, even though it was obtained from anonrecoil skeleton integral.

The simple coordinate form of the result in (11.4) suggests that it mighthave an intuitively clear interpretation. This is the case and the expression forthe Zemach correction was originally derived from simple nonrelativistic quan-tum mechanics without any field theory [4]. Let us describe the main steps ofthis quite transparent derivation. Recall first that the main Fermi contributionto hyperfine splitting in (8.2) is simply a matrix element of the one-photonexchange which, due to the local nature of the magnetic interaction, is simplyproportional to the value of the Schrodinger wave function squared at theorigin |ψ(0)|2. However, the nuclear magnetic moment is not pointlike, butdistributed over a finite region described by the magnetic moment densityρM (r). This effect can be taken into account in the matrix element for theleading contribution to HFS with the help of the obvious substitution

|ψ(0)|2 →∫

d3rρM (r)|ψ(r)|2 . (11.6)

Hence, the correction to the leading contribution to HFS depends on thebehavior of the bound-state wave function near the origin. The ordinarySchrodinger-Coulomb wave function for the ground state behaves near theorigin as exp(−mrZαr) ≈ 1−mrZαr. For a very short-range nonlocal sourceof the electric field, the wave function behaves as

ψ(r) ≈ 1 − mrZα

∫

d3r|r − r|ρE(r) , (11.7)

as may easily be checked using the Green function of the Laplacian operator[4]. Substituting (11.7) in (11.6) we again come to the Zemach correctionbut with the reduced mass factor instead of the electron mass in (11.4). Thedifference between these two results is of order m/M , and might becomeimportant in a systematic treatment of the corrections of second order in theelectron-proton mass ratio.

The quantum mechanical derivation also explains the sign of the Zemachcorrection. The spreading of the magnetic and electric charge densities weak-ens the interaction and consequently diminishes hyperfine splitting in accor-dance with the analytic result in (11.4).

The Zemach correction is essentially a nontrivial weighted integral of theproduct of electric and magnetic densities, normalized to unity. It cannot bemeasured directly, like the rms proton charge radius which determines themain proton size correction to the Lamb shift (compare the case of the protonsize correction to the Lamb shift of order (Zα)5 in (6.13) which depends onthe third Zemach moment). This means that the correction in (11.4) may onlyconditionally be called the proton size contribution.


The Zemach correction was calculated numerically in a number of papers,see, e.g., [5, 6, 7]. The most straightforward approach is to use the phenom-enological dipole fit for the Sachs form factors of the proton

GE(k2) =GM (k2)1 + κ

=1

(1 − k2

Λ2

)2 , (11.8)

with the parameter Λ = 0.898 (13)M , where M is the proton mass.3 Sub-stituting this parametrization in the integral in (11.3) one obtains ∆E =−38.72 (56) × 10−6EF [6] for the Zemach correction. The uncertainty in thebrackets accounts only for the uncertainty in the value of the parameter Λand the uncertainty introduced by the approximate nature of the dipole fitis completely ignored. This last uncertainty could be significantly underesti-mated in such approach. As was emphasized in [7] the integration momentawhich are small in comparison with the proton mass play an important rolein the integral in (11.3). About fifty percent of the integral comes from theintegration region where k < Λ/2. But the dipole form factor parameter Λis simply related to the rms proton radius r2

p = 12/Λ2, and one can try touse the empirical value of the proton radius as an input for calculation ofthe low momentum contribution to the Zemach correction. Such an approach,which assumes validity of the dipole parametrization for both form factors atsmall momenta transfers, but with the parameter Λ determined by the protonradius leads to the Zemach correction ∆E = −41.07 (75) × 10−6EF [7] forrp = 0.862 fm (Λ = 0.845 (12)M). For an even higher value of the protoncharge radius rp = 0.891 (18) fm the respective Zemach correction turns outto be ∆E = −42.4 (1.1) × 10−6EF .

Model dependence of the Zemach correction, as well as its dependence onthe proton radius is theoretically unsatisfactory. A much better approach issuggested in [8], where the values of the proton and deuteron first Zemachmoments were determined in a model independent way from the analysis ofthe world data on the elastic electron-proton and electron-deuteron scattering.The respective moments turned out to be [8]

〈r〉p(2) = 1.086 (12) fm , (11.9)

〈r〉d(2) = 2.593 (16) fm . (11.10)

The Zemach correction to HFS in hydrogen is then

∆E = −2(Zα)m〈r〉p(2)EF = −41.0 (5) × 10−6EF , (11.11)

and we will use this value of the Zemach correction for further numericalestimates.3 We used the same value of Λ in Subsect. 5.1.3 for calculation of the correction to

the Lamb shift in hydrogen generated by the radiative insertions in the protonline. Due to the logarithmic dependence of this correction on Λ small changes ofits value do not affect the result for the proton line contribution to the Lambshift.


11.1.1.2 Recoil Correction of Order (Zα)(m/M)EF

For muonium the skeleton two-photon exchange diagrams in Fig. 10.1 gen-erated, after subtraction of the heavy pole contribution, a recoil correctionof order (Zα)(m/M)EF to hyperfine splitting. Calculation of the respectiverecoil corrections for hydrogen does not reduce to substitution of the protonmass instead of the muon mass in (10.5), but requires an account of the protonanomalous magnetic moment and the proton form factors. As we have seen,insertions of the nontrivial proton form factors in the external field skeletondiagram pushes the characteristic integration momenta into the region de-termined by the proton size and, as a result respective contribution to HFSin (11.4) contains the small proton size factor m/Λ. The scale of the protonform factor is of order of the proton mass and thus the Zemach correctionin Fig. 11.1 and Fig. 11.2 is of the same order as the recoil contributions inFig. 11.3 and Fig. 11.4 generated by the form factor insertions in the skele-ton diagrams in Fig. 10.1. It is natural to consider all contributions of order(Zα)FF together and to call the sum of these corrections the total proton sizecorrection. However, we have two different parameters m/Λ and m/M , andthe Zemach and the recoil corrections admit separate consideration.

Fig. 11.3. Diagrams with one form factor insertion for total elastic nuclear sizecorrections of order (Zα)EF

Fig. 11.4. Diagrams with two form factor insertions for total elastic nuclear sizecorrections of order (Zα)EF

The recoil part of the proton size correction of order (Zα)EF was firstconsidered in [9, 10]. In these works existence of the nontrivial nuclear formfactors was ignored and the proton was considered as a heavy particle withoutnontrivial momentum dependent form factors but with an anomalous mag-netic moment. The result of such a calculation is most conveniently writtenin terms of the “elementary” proton Fermi energy EF which does not includethe contribution of the proton anomalous magnetic moment (compare (10.2)in the muonium case). Calculation of this correction coincides almost exactly


with the one in the case of the leading muonium recoil correction in (10.5)4

and generates an ultraviolet divergent result [9, 10, 11]

∆E = − 3mM

M2 − m2

Zα

π

{(

1 − κ2

4

)

lnM

m+

κ2

4

(

−16

+ 3 lnMM

)}

EF ,

(11.12)where M is an arbitrary ultraviolet cutoff.

The ultraviolet divergence is generated by the diagrams with insertions oftwo anomalous magnetic moments in the heavy particle line. This should beexpected since quantum electrodynamics of elementary particles with nonva-nishing anomalous magnetic moments is nonrenormalizable.

For the real proton we have to include in the vertices the proton formfactor, which decays fast enough at large momenta transfer and neutralizesthe divergence. Insertion of the form factors will effectively cut off momen-tum integration at the form factor scale Λ, which is slightly smaller than theproton mass. Precise calculation needs an accurate rederivation of the recoilcorrection using the form factors, but one important feature of the expectedresult is obvious immediately. The factor in the braces in (11.12) is numericallysmall just for the physical value of the proton anomalous magnetic momentκ = 1.792 847 351 (28) measured in nuclear magnetons and the ultravioletcutoff of the order of the proton mass [6]. We should expect that this acciden-tal suppression of the recoil correction would survive on account of the formfactors, and this correction will at the end of the day be numerically muchsmaller than the Zemach correction, even though these two corrections areparametrically of the same order.

Since the Zemach and recoil corrections are parametrically of the sameorder of magnitude only their sum was often considered in the literature. Thefirst calculation of the total proton size correction of order (Zα)EF with formfactors was done in [12], followed by the calculations in [13, 11]. Separatelythe Zemach and recoil corrections were calculated in [5, 6]. Results of all theseworks essentially coincide, but some minor differences are due to the differ-ences in the parameters of the dipole nucleon form factors used for numericalcalculations.

We will present here results in the form obtained in [13, 11]. They areindependent of the specific parametrization of the form factors and especiallyconvenient for further consideration of the inelastic polarizability contribu-tions. The total proton size correction of order (Zα)EF generated by thediagrams with the insertions of the total proton form factors in Fig. 11.3and in Fig. 11.4 may easily be calculated. The resulting integral contains twocontributions of the pointlike proton with the anomalous magnetic momentwhich were already taken into account. One is the infrared divergent nonrecoilcontribution corresponding to the external field skeleton integral in Fig. 9.24 Really the original works [9, 10] contain just the elementary proton ultraviolet

divergent result in (11.12) which turns into the ultraviolet finite muonium resultin (10.5) if the anomalous magnetic moment κ is equal zero.


with insertion of the anomalous magnetic moment, the other is the pointlikeproton recoil correction in (11.12). After subtraction of these contributionswe obtain an expression for the remaining proton size correction in the formof the Euclidean four-dimensional integral [13, 11]

∆E =∫

d4k

k6

{

2k2(2k2 + k20) [F1(F1 + F2) − (1 + κ)]

4k20 + k4

M2

+6k2k2

0 [F2(F1 + F2) − κ(1 + κ)]4k2

0 + k4

M2

− (2k2 + k20)(F

22 − κ2)

2

}

× Zα

π

m

MEF . (11.13)

The last term in the braces is ultraviolet divergent, but it exactly cancelsin the sum with the point proton contribution in (11.12). The sum of con-tributions in (11.12) and (11.13) is the total proton size correction, includingthe Zemach correction. According to the numerical calculation in [6] this isequal to ∆E = −33.50 (55) × 10−6EF . As was discussed above, the Zemachcorrection included in this result strongly depends on the precise value ofthe proton radius, while numerically the much smaller recoil correction is lesssensitive to the small momenta behavior of the proton form factor and hassmaller uncertainty. For further numerical estimates we will use the estimate∆E = 5.22 (1) × 10−6EF of the recoil correction obtained in [6].

11.1.1.3 Nuclear Polarizability Contribution of Order (Zα)EF

Up to now we considered only the contributions of order (Zα)EF to hyperfinesplitting in hydrogen generated by the elastic intermediate nuclear states. Aswas first realized by Iddings [13] inelastic contributions in Fig. 11.5 admit anice representation in terms of spin-dependent proton structure functions G1

and G2 [13, 11]

∆E = (∆1 + ∆2)Zα

2π

m

MEF ≡ δpolEF , (11.14)

where

∆1 =∫ ∞

0

dQ2

Q2

{

94F 2

2 (Q2) + 4M2

∫ ∞

ν0(Q2)

dν

ν2β1(

ν2

Q2)G1(Q2, ν)

}

, (11.15)

∆2 = 12M2

∫ ∞

0

dQ2

Q2

∫ ∞

ν0(Q2)

dν

ν2β2(

ν2

Q2)G2(Q2, ν) . (11.16)

The inelastic pion-nucleon threshold ν0 may be written as

ν0(Q2) = mπ +m2

π + Q2

2M, (11.17)


Fig. 11.5. Diagrams for nuclear polarizability correction of order (Zα)EF

and the auxiliary functions βi have the form

β1(x) = −3x + 2x2 + 2(2 − x)√

x(1 + x) , (11.18)

β2(x) = 1 + 2x − 2√

x(1 + x) . (11.19)

The structure functions G1 and G2 may be measured in inelastic scatteringof polarized electrons on polarized protons. The difference between the spinantiparallel and spin parallel cross sections has the form

d2σ↑↓

dq2dE′ −d2σ↑↑

dq2dE′ =4πα2

E2Q2

[E + E′ cos θ

νG1(Q2, ν) + G2(Q2, ν)

]

, (11.20)

where E,E′ are the initial and final electron energies, ν = E − E′, and θ isthe electron scattering angle in the laboratory frame.

About thirty years ago the general properties of the structure functionsand the known experimental data were used to set rigorous bounds on thepolarizability contribution in (11.14) [14, 15] (see also reviews in [16, 6])

|δpol| ≤ 4 × 10−6 . (11.21)

Experimental data on the proton spin-dependent structure functions shouldbe used to calculate the integrals in (11.14) directly. First estimates of the po-larizability correction were obtained a long time ago [11, 13, 17, 18, 19, 20].One popular approach was to consider the polarizability correction directlyas a sum of contributions of all intermediate nuclear states. Then the leadingcontributions to this correction are generated by the low lying states, and thecontribution of the ∆ isobar was estimated many times [11, 13, 17, 18, 19, 21].The total polarizability contribution in this approach may be obtained aftersummation over contributions of all relevant intermediate states. The firstreal calculation of the polarizability correction was done in [22]. In this paperthe authors accounted for existing experimental data on the spin-dependentstructure functions from SLAC, CERN, and DESY, as well as for five low-energy resonances. The polarizability contribution was obtained in the form[22]

δpol = 1.4 (6) × 10−6 . (11.22)

In a later paper [23], new lower momentum transfer data from JLab was alsotaken into account, which resulted in the polarizability contribution

δpol = 1.3 (3) × 10−6 . (11.23)


11.1.2 Recoil Corrections of Order (Zα)2(m/M)EF

Recoil corrections of relative order (Zα)2(m/M) are connected with the di-agrams with three exchanged photons (see Fig. 11.6). Due to the Caswell-Lepage cancellation [24, 25] recoil corrections of order (Zα)2(m/M)EF inmuonium (see discussion in Subsect. 10.1.2) originate from the exchangedmomenta of order of the electron mass and smaller. The same small ex-changed momenta are also relevant in the case of hydrogen. This means thatunlike the case of the recoil correction of order (Zα)(m/M)EF consideredabove, the proton structure is irrelevant in calculation of corrections of order(Zα)2(m/M)EF . However, we cannot simply use the muonium formulae forhydrogen because the muonium calculations ignored the anomalous magneticmoment of the heavy particle. A new consideration [6] of the recoil correctionsof order (Zα)2(m/M)EF in the case of a heavy particle with an anomalousmagnetic moment resulted in the correction

∆E ={[

2(1 + κ) +7κ2

4

]

ln(Zα)−1 −[

8(1 + κ) − κ(12 − 11κ)4

]

ln 2

+6518

+κ(11 + 31κ)

36

}

(Zα)2m2

r

mMEF . (11.24)

For a vanishing anomalous magnetic moment of the heavy particle (κ = 0)this correction turns into the muonium result in (10.6) and (10.7).

Fig. 11.6. Diagrams for recoil corrections of order (Zα)2EF

Calculation of the respective radiative-recoil correction of order α(Zα)2

(m/M)EF in the skeleton integral approach is quite straightforward and mayreadily be done. However, numerically the correction in (11.24) is smaller thanthe uncertainty of the Zemach correction, and calculation of corrections to thisresult does not seem to be an urgent task.

11.1.3 Correction of Order (Zα)2m2r2EF

The leading nuclear size correction of order (Zα)2m2〈r2〉EF may easily becalculated in the framework of nonrelativistic perturbation theory if one takesas one of the perturbation potentials the potential corresponding to the mainproton size contribution to the Lamb shift in (6.3). The other perturbationpotential is the potential in (9.28) responsible for the main Fermi contribution

11.2 Radiative Corrections to Nuclear Size and Recoil Effects 227

to HFS (compare calculation of the leading logarithmic contribution of orderα(Zα)2(m/M)EF in Subsect. 10.2.11). The result is [7]

∆E = −23(Zα)2 ln(Zα)−2m2〈r2〉EF = −1.7 × 10−9EF . (11.25)

This tiny correction is too small to be of any phenomenological interest forhydrogen.

11.1.4 Correction of Order (Zα)3(m/Λ)EF

The logarithmic nuclear size correction of order (Zα)3EF may simply be ob-tained from the Zemach correction if one takes into account the Dirac correc-tion to the Schrodinger-Coulomb wave function in (3.65) [7]

∆E = −(Zα)3 ln(Zα)−2m〈r〉(2)EF . (11.26)

The corrections in (11.25) and (11.26) are negligible for ground state hy-perfine splitting in hydrogen. However, it is easy to see that these correctionsare state dependent and give contributions to the difference of hyperfine split-tings in the 2S and 1S states 8∆E(2S) −∆E(1S). Respective formulae wereobtained in [7, 26] and are of phenomenological interest in the case of HFSsplitting in the 2S state in hydrogen [27, 28], in deuterium [29], and in the3He+ ion [30], and also for HFS in the 2P state [31] (see also review in [32]).

11.2 Radiative Corrections to Nuclear Sizeand Recoil Effects

11.2.1 Radiative-Recoil Corrections of Order α(Zα)(m/Λ)EF

Diagrams for the radiative corrections to the Zemach contribution in Fig. 11.7and in Fig. 11.8 are obtained from the diagrams in Fig. 11.1 and in Fig. 11.2by insertions of the radiative photons in the electron line or of the polarizationoperator in the external photon legs. Analytic expressions for the nuclear sizecorrections of order α(Zα)EF are obtained from the integral for the Zemachcorrection in (11.3) by insertions of the electron factor or the one-loop polar-ization operator in the integrand in (11.3). Effective integration momenta in(11.3) are determined by the scale of the proton form factor, and so we needonly the leading terms in the high-momentum expansion of the polarizationoperator and the electron factor for calculation of the radiative correctionsto the Zemach correction. The leading term in the high-momentum asymp-totic expansion of the electron factor is simply a constant (see the text above(10.14)) and the correction to hyperfine splitting is the product of this con-stant and the Zemach correction [7]


Fig. 11.7. Electron-line radiative correction to the Zemach contribution

Fig. 11.8. Photon-line radiative correction to the Zemach contribution

∆E =52

α(Zα)π

m〈r〉(2)EF . (11.27)

The contribution of the polarization operator is logarithmically enhanceddue to the logarithmic asymptotics of the polarization operator. This loga-rithmically enhanced contribution of the polarization operator is equal to thedoubled product of the Zemach correction and the leading term in the po-larization operator expansion (an extra factor two is necessary to take intoaccount two ways to insert the polarization operator in the external photonlegs in Fig. 11.1 and in Fig. 11.2)

∆E = −43

(

lnΛ2

m2

)α(Zα)

πm〈r〉(2)EF . (11.28)

Calculation of the nonlogarithmic part of the polarization operator inser-tion requires more detailed information on the proton form factors, and usingthe dipole parametrization one obtains [7]

∆E = −43

(

lnΛ2

m2− 317

105

)α(Zα)

πm〈r〉(2)EF . (11.29)

11.2.2 Radiative-Recoil Corrections of Order α(Zα)(m/M)EF

Radiative-recoil corrections of order α(Zα)(m/M)EF are similar to the radia-tive corrections to the Zemach contribution, and in principle admit a straight-forward calculation in the framework of the skeleton integral approach. Lead-ing logarithmic contributions of this order were considered in [6, 7]. The log-arithmic estimate in [7] gives

∆E = 0.11 (2) × 10−6EF , (11.30)

for the contribution of the electron-line radiative insertions, and

∆E = −0.02 × 10−6EF , (11.31)


for the contribution of the vacuum polarization insertions in the exchangedphotons.

Numerically these contributions are much smaller than the uncertainty ofthe Zemach correction.

11.2.3 Heavy Particle Polarization Contributions

Muon and heavy particle polarization contributions to hyperfine splitting inmuonium were considered in Subsubsect. 9.3.1.2 and Subsect. 10.2.7.

In the external field approximation the skeleton integral with the muon po-larization insertion coincides with the respective integral for muonium (com-pare (9.12) and the discussion after this equation) and one easily obtains [33]

∆E =34α(Zα)

m

mµEF . (11.32)

This result gives a good idea of the magnitude of the muon polarizationcontribution since the muon is relatively light in comparison to the scale ofthe proton form factor which was ignored in this calculation.

The total muon polarization contribution may be calculated without greatefforts but due to its small magnitude such a calculation is of minor phenom-enological significance and was never done. Only an estimate of the total muonpolarization contribution exists in the literature [7]

∆E = 0.07 (2) × 10−6EF . (11.33)

Hadronic vacuum polarization in the external field approximation for thepointlike proton also was calculated in [33]. Such a calculation may serve onlyas an order of magnitude estimate since both the external field approximationand the neglect of the proton form factor are not justified in this case, becausethe scale of the hadron polarization contribution is determined by the sameρ-meson mass which determines the scale of the proton form factor. Againa more accurate calculation is feasible but does not seem to be warranted,and only an estimate of the hadronic polarization contribution appears in theliterature [7]

∆E = 0.03 (1) × 10−6EF . (11.34)


The weak interaction contribution to hyperfine splitting in hydrogen is easilyobtained by generalization of the muonium result in (10.38)

∆E =gA

1 + κ

GF√2

3mM

4πZαEF ≈ 5.8 × 10−8 kHz . (11.35)


Table 11.1. Hyperfine Splitting in Hydrogen

EF = 1 418 840.10 (2) kHz kHz

Total nonrecoil contributionTables 9.1, 9.2, 9.3, 9.4, 9.5 1.001 136 093 9 (3) 1 420 452.03 (2)

Proton size correction,relative order (Zα)(m/Λ)Zemach (1956) [4] −2(Zα)m〈r〉(2)

= −41.0 (5) × 10−6 −58.2 (6)

Recoil correction,relative order (Zα)(m/M)Arnowitt (1953) [9]Newcomb, Salpeter (1955) [10]Iddings, Platzman (1959) [12] 5.22 (1) × 10−6 7.41 (1)

Nuclear polarizability,relative order (Zα)Iddings (1965) [13]Faustov, Martynenko (2002) [22]Nazaryan, Carlson,Griffion (2006) [23] 1.3 (3) × 10−6 1.8 (4)

Recoil correction,{[

2(1 + κ) + 7κ2

4

]ln(Zα)−1

relative order (Zα)2(m/M) −[8(1 + κ) − κ(12−11κ)

4

]ln 2

Bodwin, Yennie (1988) [6] + 6518

+ κ(11+31κ)36

}(Zα)2

1+κ

m2r

mM

= 0.4585 × 10−6 0.65

Leading logarithmic correction,relative order (Zα)2m2r2

p

Karshenboim (1997) [7] − 23(Zα)2 ln(Zα)−2m2〈r2〉

= −0.002 × 10−6 -0.003

Leading logarithmic correction,relative order (Zα)3(m/Λ)Karshenboim (1997) [7] −(Zα)3 ln(Zα)−2m〈r〉(2)

= −0.01 × 10−6 −0.015

Electron-line correction,relative order α(Zα)(m/Λ)

Karshenboim (1997) [7] 52

α(Zα)π

m〈r〉(2) = 0.12 × 10−6 0.17

Photon-line correction,relative order α(Zα)(m/Λ)

Karshenboim (1997) [7] − 43

(ln Λ2

m2 − 317105

)α(Zα)

πm〈r〉(2)

= −0.75 × 10−6 −1.06

Leading electron-line correction,relative order α(Zα)(m/M)Karshenboim (1997) [7] 0.11 (2) × 10−6 0.16

References 231


Leading photon-line correction,relative order α(Zα)(m/M)Karshenboim (1997) [7] −0.02 × 10−6 −0.03

Muon vacuum polarization,relative order α(Zα)(m/mµ)Karshenboim (1997) [7, 37] 0.07 (2) × 10−6 0.10 (3)

Hadron vacuum polarization,Karshenboim (1997) [7, 37] 0.03 (1) × 10−6 0.04 (1)

Weak interaction contribution,

Beg, Feinberg (1975) [35] gA1+κ

GF√2

3mM4πZα

= 0.06 × 10−6 0.08

Total theoretical HFS 1 420 403.1 (8)

Two features of this result deserve some comment. First, the axial couplingconstant for the composite proton is renormalized by the strong interactionsand its experimental value is gA = 1.267, unlike the case of the elementarymuon when it was equal unity. Second the signs of the weak interaction cor-rection are different in the case of muonium and hydrogen [34].

References

1. H. Hellwig, R. F. C. Vessot, M. W. Levine et al, IEEE Trans. IM-19, 200 (1970).2. L. Essen, R. W. Donaldson, M. J. Bangham et al, Nature 229, 110 (1971).3. P. J. Mohr and B. N. Taylor, Rev. Mod. Phys. 77, 1 (2005).4. C. Zemach, Phys. Rev. 104, 1771 (1956).5. H. Grotch and D. R. Yennie, Rev. Mod. Phys. 41, 350 (1969).6. G. T. Bodwin and D. R. Yennie, Phys. Rev. D 37, 498 (1988).7. S. G. Karshenboim, Phys. Lett. A 225, 97 (1997).8. J. L. Friar and I. Sick, Phys. Lett. B 579, 285 (2004).9. R. Arnowitt, Phys. Rev. 92, 1002 (1953).

10. W. A. Newcomb and E. E. Salpeter, Phys. Rev. 97, 1146 (1955).11. S. D. Drell and J. D. Sullivan, Phys. Rev. 154, 1477 (1967).12. C. K. Iddings and P. M. Platzman, Phys. Rev. 113, 192 (1959).13. C. K. Iddings, Phys. Rev. 138, B446 (1965).14. E. de Rafael, Phys. Lett. B 37, 201 (1971).15. P. Gnadig and J. Kuti, Phys. Lett. B 42, 241 (1972).16. V. W. Hughes and J. Kuti, Ann. Rev. Nucl. Part. Sci., 33, 611 (1983).17. C. K. Iddings and P. M. Platzman, Phys. Rev. 115, 919 (1959).18. A. Verganalakis and D. Zwanziger, Nuovo Cim. 39, 613 (1965).19. F. Guerin, Nuovo Cim. A 50, 1 (1967).20. G. M. Zinov’ev, B. V. Struminskii, R. N. Faustov et al, Yad. Fiz. 11, 1284 (1970)

[Sov. J. Nucl. Phys. 11, 715 (1970)].21. R. N. Faustov, A. P. Martynenko, and V. A. Saleev, Yad. Phys. 62, 2280 (1999)

[Phys. Atom. Nuclei 62, 2099 (1999)].


22. R. N. Faustov and A. P. Martynenko, Eur. Phys. J. C 24, 281 (2002); Yad.Phys. 65, 291 (2002) [Phys. Atom. Nuclei 65, 265 (2002)].

23. V. Nazaryan, C. E. Carlson, and K. A. Griffion, Phys. Rev. Lett 96, 163001(2006).

24. G. T. Bodwin, D. R. Yennie, and M. A. Gregorio, Phys. Rev. Lett. 41, 1088(1978).

25. W. E. Caswell and G. P. Lepage, Phys. Rev. Lett. 41, 1092 (1978).26. E. E. Trofimenko, Phys. Lett. A 73, 383 (1979).27. J. W. Heberle, H. A. Reich, and P. Kusch, Phys. Rev. 101, 612 (1956).28. N. E. Rothery and E. A. Hessels, Phys. Rev. A 61, 044501 (2000).29. J. W. Heberle, H. A. Reich, and P. Kusch, Phys. Rev. 104, 1585 (1956).30. M. H. Prior and E. C. Wang, Phys. Rev. A 16, 6 (1977).31. S. R. Lundeen, P. E. Jessop, and F. M. Pipkin, Phys. Rev. Lett. 34, 377 (1975)32. N. F. Ramsey, in Quantum Electrodynamics, ed. T. Kinoshita (World Scientific,

Singapore, 1990), p.673.33. S. G. Karshenboim, J. Phys. B: At. Mol. Opt. Phys. 28, L77 (1995).34. M. I. Eides, Phys. Rev. A 53, 2953 (1996).35. M. A. Beg and G. Feinberg, Phys. Rev. Lett. 33, 606 (1974); 35, 130(E) (1975).

12

Notes on Phenomenology

12.1 Lamb Shifts of the Energy Levels

Theoretical results described above find applications in numerous high pre-cision experiments with hydrogen, deuterium, helium, muonium, muonic hy-drogen, etc. Detailed discussion of all experimental results in comparison withtheory would require as much space as the purely theoretical discussion above.We will consider below only some applications of the theory, intended to serveas illustrations, their choice being necessarily somewhat subjective and in-complete (see also detailed discussion of phenomenology in the recent reviews[1, 2]).

12.1.1 Values of Some Physical Constants

In numerical calculations below we will use the most precise modern valuesof the fundamental physical constants as obtained in [1]. The value of theRydberg constant is

R∞ = 10 973 731.568 525 (73) m−1 δ = 6.6 × 10−12 , (12.1)

the fine structure constant is equal to

α−1 = 137.035 999 11 (46) δ = 3.3 × 10−9 , (12.2)

the proton-electron mass ratio is equal to

M

m= 1 836.152 672 61 (85) δ = 4.6 × 10−10 , (12.3)

the muon-electron mass ratio is equal to

M

m= 206.768 283 8 (54) δ = 2.6 × 10−8 , (12.4)


234 12 Notes on Phenomenology

the proton-muon mass ratio is equal to

M

m= 8.880 243 33 (23) δ = 2.6 × 10−8 , (12.5)

and the deuteron-proton mass ratio is equal to

M

m= 1.999 007 500 82 (41) δ = 2.0 × 10−10 . (12.6)

12.1.2 Theoretical Accuracy of S-State Lamb Shifts

From the theoretical point of view the accuracy of calculations is limitedby the magnitude of the yet uncalculated contributions to the Lamb shift.Corrections to the P levels are known now with a higher accuracy than thecorrections to the S levels, and do not limit the results of the comparisonbetween theory and experiment.

The largest source of theoretical uncertainty in the value of the S-stateLamb shift is connected with the corrections of order α2(Zα)nm, n ≥ 6.Corrections of order α2(Zα)6 are a polynomial in ln(Zα)−2, starting with thelogarithm cubed term. The logarithm cubed term, logarithm squared, and thesingle-logarthmic terms are known (for more details on these corrections seediscussion in Subsect. 3.4.2). However, the nonlogarithmic term is known onlywith a 15% accuracy [3, 110] which translates into about 1 kHz for the 1S-stateand about 0.1 kHz for the 2S-state. Corrections of order α2(Zα)7m were nevercalculated perturbatively. They should be suppressed in comparison with thecorrections of order α(Zα)7 by at least the factor α/π. Taking into accountpossible logarithmic enhancements, one could expect that these corrections areas large as 1 kHz for the 1S-state and 0.1 kHz for 2S-state in hydrogen. Thisestimate means that these corrections deserve calculation. Purely numericalcalculation of all corrections of order α2(Zα)nm, n ≥ 6, without expansion inZα was performed in [5]. It overestimates the known coefficient B60 [3, 4] (see(3.76)) by about a factor of two. The result in [5] shifts the value of the 1SLamb shift in hydrogen by about 7 kHz. In this situation we assume that thetheoretical uncertainty connected with contributions of order α2(Zα)nm isabout 4 kHz for 1S state and about 0.5 kHz for 2S state in hydrogen. Clearlythe problem of corrections of order α2(Zα)nm, n ≥ 6, requires more work.

Corrections of order α3(Zα)5m are the largest uncalculated contributionsto the energy levels for S-states. These corrections are pure numbers andare estimated by comparing them with corrections of order α2(Zα)5m. Theycould be as large as 1 kHz for the 1S-state and about 0.1 kHz for the 2S-state.

Only the leading logarithm squared contribution to the recoil correctionof order (Zα)7(m/M) is known now [6, 7]. Numerically the contribution in(4.24) is below 1 kHz. Due to linear dependence of the recoil correction onthe electron-nucleus mass ratio, the respective contribution to the hydrogen-deuterium isotope shift (see Subsect. 12.1.7 below) is phenomenologically

12.1 Lamb Shifts of the Energy Levels 235

much more important, it is larger than the experimental uncertainty, andshould be taken into account in comparison between theory and experimentat the current level of experimental uncertainty.

Numerical results for recoil corrections obtained without expansion overZα [8, 9] indicate that the contribution of the single logarithmic and nonlog-arithmic recoil corrections of order (Zα)7(m/M) is about 0.64(1) kHz for the1S level in hydrogen, and exceeds the total logarithm squared contribution.It would be interesting to calculate respective coefficients perturbatively.

Only the leading logarithm squared radiative-recoil contribution of orderα(Zα)6(m/M) is known now [6, 7]. Single logarithmic and nonlogarithmicradiative-recoil contributions of order α(Zα)6(m/M)m may be estimated asone half of the leading logarithm squared contribution. This constitutes about0.8 kHz and 0.1 kHz for the 1S and 2S levels in hydrogen, respectively. How-ever, experience with the large contributions generated by the nonleadingterms of order α2(Zα)6(m/M)m and of order (Zα)7(m/M)m shows thatwe probably underestimate the magnitude of the nonleading terms in thisway. In view of the rapid experimental progress in the Lamb shift and iso-tope shift measurements calculation of these nonleading corrections of orderα(Zα)6(m/M)m deserves further theoretical efforts.

All other unknown theoretical contributions to the Lamb shift are muchsmaller, and 6 kHz for the 1S-state and 0.8 kHz for the 2S-state are reasonableestimates of the total theoretical uncertainty of the expression for the Lambshift. Theoretical uncertainties for the higher S levels may be obtained fromthe 1S-state uncertainty ignoring its state-dependence and scaling it with theprincipal quantum number n.

12.1.3 Theoretical Accuracy of P -State Lamb Shifts

The Lamb shift theory of P -states is in a better shape than the theory ofS-states. Corrections of order α(Zα)n are now known with uncertainty about1 Hz for 2P states [10, 11, 12, 13].

The largest unknown corrections to the P -state energies are the nonlog-arithmic contributions of the form α2(Zα)6, induced by radiative insertionsin the electron and external photon lines. The single logarithmic contributionof this order is about 0.02 kHz for 2P states in hydrogen. To be on the safeside we would assume that the nonlogarithmic term is about 0.05 kHz for2P . This contribution determines the theoretical uncertainty of the 2P Lambshift in hydrogen. Theoretical uncertainties for the higher non-S levels maybe obtained from the 2P -state uncertainty ignoring its state-dependence andscaling it with the principal quantum number n.

12.1.4 Theoretical Accuracy of the Interval ∆n = n3L(nS)−L(1S)

State-independent contributions to the Lamb shift scale as 1/n3 and vanishin the difference ∆n = n3EL(nS) − EL(nS), which may be calculated more


accurately than the positions of the individual energy levels (see discussion inSubsect. 12.1.4). All main sources of theoretical uncertainty of the individualenergy levels, namely, proton charge radius contributions and yet uncalculatedstate-independent corrections to the Lamb shift vanish in this difference. Ear-lier the practical usefulness of the theoretical value of the interval ∆n forextraction of the experimental value of the 1S Lamb shift was impeded bythe insufficient theoretical accuracy of this interval and by the insufficientaccuracy of the frequency measurement. Significant progress was achieved inboth respects. On the theoretical side the uncertainty of the interval ∆n forn = 2 − 21 was reduced recently to about 0.1 kHz [14]. We reproduce thelist of values of the interval ∆n from [14] in Table 12.1. Nowadays the theo-retical value of ∆n plays an important role in extracting the precise value ofthe Rydberg constant and the 1S Lamb shift from modern highly accurateexperimental data (see discussion below in Subsect. 12.1.6).

Table 12.1. Values of the interval ∆n = n3EL(nS) − EL(nS)

n ∆n kHz n ∆n kHz

2 187 225.70 (5) 17 281 845.77 (11)

3 235 070.90 (7) 18 282 049.05 (11)

4 254 419.32 (8) 19 282 221.81 (11)

5 264 154.03 (9) 20 282 369.85 (11)

6 269 738.49 (9) 21 282 497.67 (11)

7 273 237.83 (9) 22 282 608.78 (11)

8 275 574.90 (10) 23 282 705.98 (11)

9 277 212.89 (10) 24 282 791.50 (11)

10 278 405.21 (10) 25 282 867.11 (11)

11 279 300.01 (10) 26 282 934.29 (11)

12 279 988.60 (10) 27 282 994.18 (11)

13 280 529.77 (10) 28 283 048.01 (11)

14 280 962.77 (10) 29 283 096.35 (11)

15 281 314.61 (10) 30 283 140.01 (11)

16 281 604.34 (11) 31 283 179.54 (11)

12.1.5 Classic Lamb Shift 2S 12

− 2P 12

Discovery of the classic Lamb shift, i.e. splitting of the 2S 12

and the 2P 12

energy levels in hydrogen triggered a new stage in the development of modernphysics. In the terminology accepted in this paper the classic Lamb shift isequal to the difference of Lamb shifts in the respective states ∆E(2S 1

2−2P 1

2) =

L(2S 12) − L(2P 1

2). Unlike the much larger Lamb shift in the 1S state, the

classic Lamb shift is directly observable as a small splitting of energy levelswhich should be degenerate according to Dirac theory. This greatly simplifiescomparison between the theory and experiment for the classic Lamb shift,


since the theoretical predictions are practically independent of the exact valueof the Rydberg constant, which can be measured independently.

Many experiments on the precise measurement of the classic Lamb shiftwere performed since its experimental discovery in 1947. We have collectedmodern post 1979 experimental results in Table 12.2. Two entries in this Tableare changed compared to the original published experimental results [16, 15].These alterations reflect recent improvements of the theory used for extractionof the Lamb shift value from the raw experimental data.

The magnitude of the Lamb shift in [16] was derived from the ratio ofthe 2P 1

2decay width and the ∆E(2S 1

2− 2P 1

2) energy splitting which was

directly measured by the atomic-interferometer method. The theoretical ex-pression for the 2P 1

2-state lifetime was used for extraction of the magnitude

of the Lamb shift. An additional leading logarithmic correction to the widthof the 2P 1

2state of relative order α(Zα)2 ln(Zα)−2, not taken into account in

the original analysis of the experiment, was obtained in [17]. This correctionslightly changes the original experimental result [16] ∆E = 1 057 851.4 (1.9)kHz, and we cite this corrected value in Table 12.2. The magnitude of the newcorrection [17] triggered a certain discussion in the literature [18, 19]. Fromthe phenomenological point of view the new correction [17] is so small thatneither of our conclusions below about the result in [16] is affected by thiscorrection.

The Lamb shift value [15] was obtained from the measurement of interval2P 3

2− 2S 1

2, and the value of the classical Lamb shift was extracted by sub-

traction of this energy splitting from the theoretical value of the fine struc-ture interval 2P 3

2− 2P 1

2. As was first noted in [20], recent progress in the

Lamb shift theory for P -states requires reconsideration of the original value∆E = 1 057 839 (12) kHz of the classical Lamb shift obtained in [15]. The the-oretical value of the fine structure interval can now be calculated with higheraccuracy than in [15]. Even nonlogarithmic contribution of order α2(Zα)6

(which is not known for individual P levels) was calculated in [21], so thetotal theoretical uncertainty of the fine structure interval is determined bythe uncertainty of the fine structure constant in [22] (see (12.2)). Calculatingthe theoretical value of the fine structure interval with all corrections takeninto account we obtain ∆E(2P 3

2− 2P 1

2) = 10 969 038.94 (7) kHz, which co-

incides very well with the result used in [15]. As a consequence the originalexperimental value [15] of the classic Lamb shift does not change, and we citethe original result in Table 12.2.

Accuracy of the radiofrequency measurements of the classic 2S−2P Lambshift [15, 16, 23, 24, 25] is limited by the large (about 100 MHz) naturalwidth of the 2P state, and cannot be significantly improved. New perspec-tives in reducing the experimental error bars of the classic 2S − 2P Lambshift were opened with the development of the Doppler-free two-photon laserspectroscopy for measurements of the transitions between the energy levelswith different principal quantum numbers. Narrow linewidth of such transi-


tions allows very precise measurement of the respective transition frequencies,and indirect accurate determination of 2S−2P splitting from this data.1 Theexperimental value [26] in the sixth line of Table 12.2 was obtained by suchmethods.

Both the theoretical and experimental data for the classic 2S1/2 − 2P1/2

Lamb shift are collected in Table 12.2. Theoretical results for the energy shiftsin this Table contain errors in the parenthesis where the first error is deter-mined by the yet uncalculated contributions to the Lamb shift, discussedabove, and the second reflects the experimental uncertainty in the measure-ment of the proton rms charge radius. We see that the uncertainty of theproton rms radius is the largest source of error in the theoretical predic-tion of the classical Lamb shift. An immediate conclusion from the data inTable 12.2 is that the value of the proton radius [27] recently derived form theanalysis of the world data on the electron-proton scattering seems compatiblewith the experimental data on the Lamb shift, while the values of the rmsproton radius popular earlier [28, 29] are clearly too small to accommodatethe experimental data on the Lamb shift. Unfortunately, these experimentalresults are rather widely scattered and have rather large experimental errors.Their internal consistency leaves much to be desired.

We will return to the numbers in the seven last lines in Table 12.2 below.

12.1.6 1S Lamb Shift and the Rydberg Constant

Unlike the case of the classic Lamb shift above, the Lamb shift in the 1S is notamenable to a direct measurement as a splitting between certain energy levels.In principle, it could be extracted from the experimental data on the tran-sition frequencies between the energy levels with different principal quantumnumbers. Such an approach requires very precise measurement of the grossstructure intervals, and became practical only with the recent development ofDoppler-free two-photon laser spectroscopy. These methods allow very precisemeasurements of the gross structure intervals in hydrogen with an accuracywhich is limited in principle only by the small natural linewidths of respectivetransitions. For example, the 2S − 1S transition in hydrogen is banned as asingle photon process in the electric dipole and quadrupole approximations,and also in the nonrelativistic magnetic dipole approximation. As a resultthe natural linewidth of this transition is determined by the process withsimultaneous emission of two electric dipole photons [37, 2], which leads tothe natural linewidth of the 2S − 1S transition in hydrogen about 1.3 Hz.Many recent spectacular experimental successes were achieved in an attemptto achieve an experimental accuracy comparable with this extremely smallnatural linewidth.

The intervals of gross structure are mainly determined by the Rydbergconstant, and the same transition frequencies should be used both for mea-surement of the Rydberg constant and for measurement of the 1S Lamb shift.1 See more discussion of this method below in Subsect. 12.1.6.


Table 12.2. Classic 2S1/2 − 2P1/2 Lamb Shift

∆E (kHz)

Newton, Andrews,Unsworth (1979) [23] 1 057 862 (20) Experiment

Lundeen, Pipkin (1981) [24] 1 057 845 (9) Experiment

Palchikov, Sokolov,Yakovlev (1983) [16] 1 057 857. 6 (2.1) Experiment

Hagley, Pipkin (1994) [15] 1 057 839 (12) Experiment

Wijngaarden, Holuj,Drake (1998) [25] 1 057 852 (15) Experiment

Schwob, Jozefovski,de Beauvoir,et al. (1999) [26] 1 057 845 (3) Exp., [30, 31, 32, 33, 34]

[24, 15, 25]

1 057 821.9 (0.8) (3.5) Theory, rp = 0.805 (11) fm [28]

1 057 840.5 (0.8) (4.0) Theory, rp = 0.862 (12) fm [29]

1 057 851.8 (0.8) (6.3) Theory, rp = 0.895 (18) fm [27]

Weitz, Huber, Schmidt-Kahler et al (1995) [31] 1 057 851 (11) Self-consistent value

Berkeland, Hinds,Boshier (1995) [32] 1 057 839 (11) Self-consistent value

Bourzeix, de Beauvoir,Nez, et al. (1996) [33] 1 057 835 (9) Self-consistent value

Udem, Huber,Gross, et al. (1997) [34] 1 057 850 (4) Self-consistent value

Schwob, Jozefovski,de Beauvoir,et al. (1999) [26] 1 057 839 (3) Self-consistent value

Reichert, Niering,Holzwarth,et al. (2000) [35] 1 057 850 (4) Self-consistent value

Niering, Holzwarth,Reichert,et al. (2000) [36] 1 057 850 (4) Self-consistent value


The first task is to obtain an experimental value of the 1S Lamb shift whichis independent of the precise value of the Rydberg constant. This goal may beachieved by measuring two intervals with different principal quantum num-bers.

For example, measurement of the 1S Lamb shift in [34] is disentangledfrom the measurement of the Rydberg constant by using the experimentaldata on two different intervals of the hydrogen gross structure [34]

f1S−2S = 2 466 061 413 187.34 (84) kHz δ = 3.4 × 10−13 , (12.7)

and [30, 26]2

f2S 12−8D 5

2= 770 649 561 581.1 (5.9) kHz δ = 7.7 × 10−12 . (12.8)

Theoretically these intervals are given by the expression in (3.6)

E1S−2S =[

EDR2S 1

2

− EDR1S 1

2

]

+ L2S 12− L1S 1

2, (12.9)

E2S−8D =[

EDR8D 5

2

− EDR2S 1

2

]

+ L8D 52− L2S 1

2,

where EDRnlj

is the leading Dirac and recoil contribution to the position of therespective energy level (first two terms in (3.6)).

The differences of the leading Dirac and recoil contribution on the righthand sides in equations (12.9) are proportional to the Rydberg constant pluscorrections of order α2R and higher. Then it is easy to construct a linear com-bination of these measured intervals which is proportional α2R plus higherorder terms (as opposed to ∼ R leading contributions to the intervals them-selves)

E1S−2S − 165

E2S−8D =[

EDR2S 1

2

− EDR1S 1

2

]

− 165

[

EDR8D 5

2

− EDR2S 1

2

]

− L1S 12

+215

L2S 12− 16

5L8D 5

2, (12.10)

Due to the suppression factor α2 the difference of the leading Dirac andrecoil contribution of the RHS in (12.10) may be calculated with high accuracy,and practically does not depend on the exact value of the Rydberg constant.Then the precise magnitude of the linear combination of the Lamb shiftson the RHS extracted from the experimentally measured frequencies on theLHS and calculated difference of the leading Dirac and recoil contribution of

2 The original experimental value f2S 12−8D 5

2= 770 649 561 585.0 (4.9) kHz [30]

used in [34] was revised in [26], and we give in (12.8) this later value. The valuesof the Lamb shifts obtained in [34] change respectively and Tables 11.1 and 12.2contain these revised values.


the RHS also practically does not depend on the exact value of the Rydbergconstant.

The linear combination of the 1S, 2S and 8D 52

Lamb shifts obtained inthis manner admits direct comparison with the Lamb shift theory withoutany further complications. However, to make a comparison between the re-sults of different experiments feasible (different intervals of the hydrogen grossstructure are measured in different experiments) the final experimental resultsare usually expressed in terms of the 1S Lamb shift measurement. The bulkcontribution to the Lamb shift scales as 1/n3 which allows one to use thetheoretical value L8D 5

2= 71.51 kHz for the D-state Lamb shift without loss

of accuracy. Then a linear combination of the Lamb shifts in 1S and 2S statesmay be directly expressed in terms of the experimental data. All other recentmeasurements of the 1S Lamb shift [26, 31, 32, 33] also end up with an exper-imental number for a linear combination of the 1S, 2S and higher level Lambshifts. An unbiased extraction of the 1S Lamb shift from the experimentaldata remains a problem even after an experimental decoupling of the Lambshift measurement from the measurement of the Rydberg constant.

Historically the most popular approach to extraction of the value of the1S Lamb shift was to use the experimental value of the classic 2S − 2P Lambshift (see three first lines in Table 12.3). Due to the large natural width of the2P state the experimental values of the classical Lamb shift have relativelylarge experimental errors (see Table 12.2), and unfortunately different resultsare not too consistent. Such a situation clearly warrants another approachto extraction of the 1S Lamb shift, one which should be independent of themagnitude of the classic Lamb shift. A natural way to obtain a self-consistentvalue of the 1S Lamb shift independent of the experimental data on the 2S −2P splitting, is provided by the theoretical relation between the 1S and nSLamb shifts discussed above in Subsect. 12.1.4. An important advantage of theself-consistent method is that it produces an unbiased value of the L(1S) Lambshift independent of the widely scattered experimental data on the 2S − 2Pinterval. The specific difference ∆n = n3L(nS) − L(1S) is now calculatedtheoretically for n = 2 − 21 with uncertainty not exceeding 0.1 kHz [14].Spectacular experimental progress in the frequency measurement combinedwith the high accuracy of the theoretical calculation of the specific difference∆n allows one to obtain self-consistent values of 1S and L(2S) Lamb shiftsfrom the experimental data, with comparable or even better accuracy (seeseven lines in Tables 12.2 and 12.3 below the theoretical values in the middleof the Table3) than in the method based on experimental results of the classicLamb shift in [15, 24].

The original experimental numbers from [26, 34] in the fourth and fifthlines in Table 12.3 are averages of the self-consistent values and the valuesbased on the classic Lamb shift. The result in [34] is based on the f1S−2S

3 To obtain the Lamb shifts in the two last lines we used besides the results in[35, 36] also the 2S − 8D transition frequency from (12.8) measured in [26, 30].


frequency measurement, f2S−8D/S frequency from [26, 30], and the classicLamb shift measurements [15, 24], while the result in [26] is based on thef2S−12D frequency measurement, as well as on the frequencies measured in[26, 30, 31, 32, 33, 34], and the classic Lamb shift measurements [15, 24, 25].The respective value of the classic 2S − 2P Lamb shift is presented in thesixth line of Table 12.2. Unlike other experimental numbers in Table 12.2,this value of the classic Lamb shift depends on other experimental results inthis Table.

The experimental data on the 1S Lamb shift should be compared with thetheoretical prediction

∆EL(1S) = 8 172 902 (6) (50) kHz , (12.11)

calculated for rp = 0.895 (18) fm [27]. The first error in this result is deter-mined by the yet uncalculated contributions to the Lamb shift and the sec-ond reflects the experimental uncertainty in the measurement of the protonrms charge radius. Note that the uncertainty of the experimentally measuredproton charge radius currently determines the uncertainty of the theoreticalprediction for the Lamb shift.

The experimental results in the first five lines in Table 12.3 seem to becompatible with the theoretical value in (12.11). This compatibility cruciallydepends on the value of the proton charge radius [27]. We presented in Table12.3 also theoretical values of the 1S Lamb shift calculated with the oncepopular earlier lower values of the radius, and they are clearly incompatiblewith the experimental data on the 1S Lamb shift. However, it is necessary toremember that the “experimental” results in the first five lines in the Table are“biased”, namely they depend on the experimental value of the 2S1/2 − 2P1/2

Lamb shift [15, 24, 25]. In view of a rather large scattering of the results forthe classic Lamb shift such dependence is unwelcome.

To obtain unbiased results we have calculated self-consistent values of the1S Lamb shift which are collected in last seven lines of Table 12.3. Thesevalues being formally consistent are rather widely scattered. Respective self-consistent values of the classic Lamb shift obtained from the experimentaldata in [31, 32, 33, 34, 35, 36] are presented in Table 12.2. The uncertaintyof the self-consistent Lamb shifts is determined by the uncertainties of exper-imentally measured frequencies used for their determination. Typically thereare two such frequencies. One is usually f1S−2S , and it is now measured witha very high accuracy of 1.8 parts in 1014 [36]. The other frequency is measuredless precisely and its experimental uncertainty determines the uncertainty ofthe self-consistent Lamb shifts. There are good experimental perspectives formeasuring the second frequency with higher accuracy [33, 36].

Since the main contribution to the uncertainty of the theoretical value ofthe 1S Lamb shift comes from the uncertainty of the proton charge radius wecan invert the problem and calculate the proton radius using the average ofthe self-consistent Lamb shifts in Table 12.3 L(1S) = 8 172 847 (14) as input.Then we obtain the “optical” value of the proton charge radius


rp = 0.875 (6) fm . (12.12)

The major contribution to the uncertainty of the proton charge radius in(12.12) is due to the uncertainty of the self-consistent Lamb shift. One shouldtake this value of the proton charge radius with a grain of salt, its accuracycould be overestimated. It is easy to see that this value of the proton radiuswould change significantly had we used for extraction of the self-consistentLamb shifts from [35, 36] experimentally measured interval 2S − 8S insteadof 2S − 8D. True optical determination of the proton charge radius should bedone on the basis of the Lamb shift measurements in muonic hydrogen (seediscussion in Subsect. 12.1.10). Experiments with muonic hydrogen providethe best approach to measurement of the proton charge radius, and wouldallow reduction of error bars in (12.12) by about an order of magnitude.Nevertheless even at the current level of accuracy it is useful to notice thatthe optical proton charge radius in (12.12) is compatible with other highervalues of the proton charge radius popular recently [27, 39, 40].

The same two (or more) frequencies used for determination of the Lambshifts, may be used for determination of the Rydberg constant. The leadingcontribution to the energy levels in hydrogen in (3.6) is determined by theRydberg constant, and, hence, any measurement of the gross structure intervalin hydrogen and deuterium may be used for determination of the value ofthe Rydberg constant, if the magnitudes of the Lamb shifts of respectiveenergy levels are known. All recent values of the Rydberg constant are derivedfrom experimental data on at least two gross structure intervals in hydrogenand/or deuterium. This allows simultaneous experimental determination ofboth the 1S Lamb shift and the Rydberg constant from the experimentaldata, and makes the obtained value of the Rydberg virtually independent ofthe Lamb shift theory and, of the experimental data on the proton chargeradius. Accuracy of the determination of the Rydberg constant is limited bythe accuracy of the gross frequency measurements and by the accuracy ofthe theory used for determination of the Lamb shifts and/or specific interval∆n = n3L(nS) − L(1S) which is now calculated theoretically for n = 2 − 21with uncertainty not exceeding 0.1 kHz [14]. The best measured frequencytoday is the 1S − 2S transition frequency in hydrogen [36]

f1S−2S = 2 466 061 413 187.103 (46) kHz δ = 1.8 × 10−14 . (12.13)

The uncertainty of this result is 46 Hz, which is only about ten timeslarger than the natural linewidth 4 Hz. The experimentalists envisage furtherimprovement of the accuracy of this measurement with the perspective toachieve an accuracy better than 1 part in 1016 [36, 41].

Determination of the most precise value of the Rydberg constant requiresa comprehensive analysis of results of the same experiments used for deter-mination of the 1S Lamb shift [26, 31, 32, 33, 34, 35, 36]. This analysis wasperformed in [1], and resulted in the value in (12.1) which has relative un-certainty δ = 6.6 × 10−12. This uncertainty is limited by the experimental


Table 12.3. 1S Lamb Shift

∆E (kHz)

Weitz, Huber, Schmidt-Kahler, et al. (1995) [31] 8 172 874 (60) Exp., L2S2P [24]

Berkeland, Hinds,Boshier (1995) [32] 8 172 827 (51) Exp., L2S2P [24, 15]

Bourzeix, de Beauvoir,Nez, et al. (1996) [33] 8 172 798 (46) Exp., L2S2P [24, 15]

Udem, Huber,Gross, et al. (1997) [34] 8 172 851 (30) Exp., L2S2P [24, 15]

Schwob, Jozefovski,de Beauvoir, et al. (1999) [26] 8 172 837 (22) Exp., [30, 31, 32, 33, 34]

[24, 15, 25]

8 172 663 (6) (28) Theory, rp = 0.805 (11) fm [28]

8 172 811 (6) (32) Theory, rp = 0.862 (12) fm [29]

8 172 902 (6) (50) Theory, rp = 0.895 (18) fm [27]

Weitz, Huber, Schmidt-Kahler, et al. (1995) [31] 8 172 916 (92) Self-consistent value

Berkeland, Hinds,Boshier (1995) [32] 8 172 800 (89) Self-consistent value

Bourzeix, de Beauvoir,Nez, et al. (1996) [33] 8 172 763 (70) Self-consistent value

Udem, Huber,Gross, et al. (1997) [34] 8 172 885 (33) Self-consistent value

Schwob, Jozefovski,de Beauvoir, et al. (1999) [26] 8 172 796 (24) Self-consistent value

Reichert, Niering,Holzwarth, et al. (2000) [35] 8 172 884 (33) Self-consistent value

Niering, Holzwarth,Reichert, et al. (2000) [36] 8 172 885 (33) Self-consistent value

uncertainties of the intervals 2S − 8D and 2S − 12D. Current state of theLamb shift theory and especially current theoretical accuracy of the specificinterval ∆n allows in principle to obtain a value of the Rydberg constant withaccuracy about a few parts in 1014 [14], if and when the second basic grossstructure frequency, e.g. 1S − 3S [14, 33], will be measured with respectiveprecision.


12.1.7 Isotope Shift

The methods of Doppler-free two-photon laser spectroscopy allow very precisecomparison of the frequencies of the 1S − 2S transitions in hydrogen anddeuterium. The frequency difference

∆E = [E(2S) − E(1S)]D − [E(2S) − E(1S)]H (12.14)

is called the hydrogen-deuterium isotope shift. Experimental accuracy of theisotope shift measurements was improved by three orders of magnitude duringthe period from 1989 to 1998 (see Table 12.4) and the uncertainty of the mostrecent experimental result [42] was reduced to 0.15 kHz.

The main contribution to the hydrogen-deuterium isotope shift is a puremass effect and is determined by the term EDR

nj in (3.6). Other contributionscoincide with the respective contributions to the Lamb shifts in Tables 3.2,3.3, 3.7, 3.9, 4.1, 5.1, and 6.1. Deuteron specific corrections discussed in Sub-subsect. 6 and collected in (6.16), (6.28), (6.29), and (6.37) also should beincluded in the theoretical expression for the isotope shift.

All yet uncalculated nonrecoil corrections to the Lamb shift almost cancelin the formula for the isotope shift, which is thus much more accurate thanthe theoretical expressions for the Lamb shifts. Theoretical uncertainty ofthe isotope shift is mainly determined by the unknown single logarithmicand nonlogarithmic contributions of order (Zα)7(m/M) and α(Zα)6(m/M)(see Sects. 4.3 and 5.2), and also by the uncertainties of the deuteron size andstructure contributions discussed in Chap. 6. Overall theoretical uncertainty ofall contributions to the isotope shift, besides the leading proton and deuteronsize corrections does not exceed 0.8 kHz.

Numerically the sum of all theoretical contributions to the isotope shift,besides the leading nuclear size contributions in (6.3), is equal to

∆E = 670 999 567. 2 (0.3) (0.8) kHz . (12.15)

The uncertainty in the first parenthesis is defined by the error of the electron-proton and proton-deuteron mass ratios, and the uncertainty in the secondparenthesis is the theoretical uncertainty discussed above.

Individual uncertainties of the proton and deuteron charge radii introduceby far the largest contributions in the uncertainty of the theoretical value ofthe isotope shift. Uncertainty of the charge radii are much larger than theexperimental error of the isotope shift measurement or the uncertainties ofother theoretical contributions. It is sufficient to recall that uncertainty of the1S Lamb shift due to the experimental error of the proton charge radius is aslarge as 50 kHz (see (12.11)), even if we ignore all problems connected withthe proton radius contribution (see discussion in Subsects. 12.1.5, 12.1.6).

In such a situation it is natural to invert the problem and to use the highaccuracy of the optical measurements and isotope shift theory for determina-tion of the difference of charge radii squared of the deuteron and proton. Weobtain


Table 12.4. Isotope Shift

∆E (kHz)

Boshier, Baird, Foot, et al. (1989) [48] 670 994 33 (64)

Schmidt-Kaler, Leibfried, Weitz, et al. (1993) [49] 670 994 414 (22)

Huber, Udem, Gross, et al. (1998) [42] 670 994 334. 64 (15)

r2D − r2

p = 3.820 3 (1) (2) (4) fm2 . (12.16)

Here the first contribution to the uncertainty is due to the experimental errorof the isotope shift measurement, the second uncertainty is due to the errorof the electron-proton mass ratio determination, and the third is generatedby the theoretical uncertainty of the isotope shift.

The difference of the deuteron and proton charge radii squared is connectedto the so called deuteron mean square matter radius (see, e.g., [43, 44]), whichmay be extracted on one hand from the experimental data on the low energynucleon-nucleon interaction, and on the other hand from the experiments onlow energy elastic electron-deuteron scattering. These two kinds of experimen-tal data used to generate inconsistent results for the deuteron matter radiusas was first discovered in [43]. The discrepancy was resolved in [45], wherethe Coulomb distortion in the second order Born approximation was takeninto account in the analysis of the electron-deuteron elastic scattering. Thisanalysis was further improved in [46] where also the virtual excitations of thedeuteron in the electron-deuteron scattering were considered. Now the valuesof the deuteron matter radius extracted from the low energy nucleon-nucleoninteraction [44] and from the low energy elastic electron-deuteron scattering[45, 46] are in agreement, and do not contradict the optical data in (12.16) (seealso discussion in [47]). The isotope shift measurements are today the sourceof the most precise experimental data on the charge radii squared difference,and the deuteron matter radius.

12.1.8 Lamb Shift in Helium Ion He+

The theory of high order corrections to the Lamb shift described above for Hand D may also be applied to other light hydrogenlike ions. The simplest suchion is He+. Originally the classic Lamb shift in He+ was measured in [50]by the quenching-anisotropy method with the result L(2S 1

2− 2P 1

2,He+) =

14 042.52 (16) MHz. Later the authors of [50] discovered a previously unsus-pected source of systematic error in their experiment. Their new measurementof the classic Lamb shift in He+ by the anisotropy method resulted in thevalue L(2S 1

2− 2P 1

2,He+) = 14 041.13 (17) MHz [51]. Besides the experimen-

tal data this result depends also on the theoretical value of the fine structureinterval. In [51] the value ∆E(2P 3

2−2P 1

2) = 175 593.50 (2) MHz was used. We

recalculated this interval using the latest theoretical results discussed aboveand obtained ∆E(2P 3

2− 2P 1

2) = 175 593.33 (1) MHz. Then the value of the


Lamb shift obtained in [51] changes to L(2S 12− 2P 1

2,He+) = 14 041.12 (17)

MHz.Theoretical calculation of the He+ Lamb shift is straightforward with all

the formulae given above. It is only necessary to recall that all contributionsscale with the power of Z, and the terms with high power of Z are enhanced incomparison with the hydrogen case. Theoretical uncertainty is estimated byscaling with Z the uncertainty of the hydrogen formulae. After calculation weobtain Lth(2S − 2P,He+) = 14 041.46 (3) MHz. We see that the theoreticaland experimental results for the classic Lamb shift in helium differ by abouttwo standard deviations of the experimental result. Clearly, further reductionof the experimental error is desirable, and the reason for this mild discrepancyshould be clarified.

12.1.9 1S − 2S Transition in Muonium

Starting with the pioneering work [52] Doppler-free two-photon laser spec-troscopy was also applied for measurements of the gross structure interval inmuonium. Experimental results [52, 53, 54, 55] are collected in Table 12.5,where the error in the first brackets is due to statistics and the second erroris due to systematic effects. The highest accuracy was achieved in the latestexperiment [55]

∆E = 2 455 528 941. 0 (9.8) MHz . (12.17)

Theoretically, muonium differs from hydrogen in two main respects. First,the nucleus in the muonium atom is an elementary structureless particle un-like the composite proton which is a quantum chromodynamic bound stateof quarks. Hence nuclear size and structure corrections in Table 6.1 do notcontribute to the muonium energy levels. Second, the muon is about ten timeslighter than the proton, and recoil and radiative-recoil corrections are numer-ically much more important for muonium than for hydrogen. In almost allother respects, muonium looks exactly like hydrogen with a somewhat lighternucleus, and the theoretical expression for the 1S − 2S transition frequencymay easily be obtained from the leading external field contribution in (3.6)and different contributions to the energy levels collected in Tables 3.2, 3.3, 3.7,3.9, 4.1, and 5.1, after a trivial substitution of the muon mass. Unlike the caseof hydrogen, for muonium we cannot ignore corrections in the two last linesof Table 3.2, and we have to substitute the classical elementary particle con-tributions in (5.6) and (5.7) instead of the composite proton contribution inthe fourth line in Table 5.1. After these modifications we obtain a theoreticalprediction for the frequency of the 1S − 2S transition in muonium

∆E = 2 455 528 935. 7 (0.3) MHz . (12.18)

The dominant contribution to the uncertainty of this theoretical resultis generated by the uncertainty of the muon-electron mass ratio in (12.4).


Table 12.5. 1S − 2S Transition in Muonium

∆E (MHz)

Danzman, Fee, Chu, et al. (1989) [52] 2 455 527 936 (120) (140)

Jungmann, Baird, Barr, et al. (1991) [53] 2 455 528 016 (58) (43)

Maas, Braun, Geerds, et al. (1994) [54] 2 455 529 002 (33) (46)

Meyer, Bagaev, Baird, et al. (1999) [55] 2 455 528 941.0 (9.8)

Theory 2 455 528 934.9 (0.3)

All other contributions to the uncertainty of the theoretical prediction: un-certainty of the Rydberg constant, uncertainty of the theoretical expression,etc., are at least an order of magnitude smaller.

There is a complete agreement between the experimental and theoreticalresults for the 1S−2S transition frequency in (12.17) and (12.18), but clearlyfurther improvement of the experimental data is warranted.

12.1.10 Light Muonic Atoms

There are very few experimental results on the energy levels in light hydro-genlike muonic atoms. The classic 2P 1

2 ( 32 ) − 2S 1

2Lamb shift in muonic helium

ion (µ 4He)+ was measured at CERN many years ago [56, 57, 58, 59] andthe experimental data was found to be in agreement with the existing theo-retical predictions. A comprehensive theoretical review of these experimentalresults was given in [60], and we refer the interested reader to this review.It is necessary to mention, however, that a recent new experiment [61] failedto confirm the old experimental results. This leaves the problem of the ex-perimental measurement of the Lamb shift in muonic helium in an uncertainsituation, and further experimental efforts in this direction are clearly war-ranted. The theoretical contributions to the Lamb shift were discussed abovein Chap. 7 mainly in connection with muonic hydrogen, but the respectiveformulae may be used for muonic helium as well. Let us mention that some ofthese contributions were obtained a long time after publication of the review[60], and should be used in the comparison of the results of the future heliumexperiments with theory.

There also exists a proposal on measurement of the hyperfine splittingin the ground state of muonic hydrogen with the accuracy about 10−4 [62].Inspired by this proposal the hadronic vacuum polarization contribution ofthe ground state hyperfine splitting in muonic hydrogen was calculated in[63], where it was found that it gives a relative contribution of 2 × 10−5 tohyperfine splitting. We did not include this correction in our discussion ofhyperfine splitting in muonic hydrogen mainly because it is smaller than thetheoretical errors due to the polarizability contribution.

The current surge of interest in muonic hydrogen is mainly inspired bythe desire to obtain a new more precise value of the proton charge radius asa result of measurement of the 2P − 2S Lamb shift [64]. As we have seen


in Chap. 7 the leading proton radius contribution is about 2% of the total2P − 2S splitting, to be compared with the case of electronic hydrogen wherethis contribution is relatively two orders of magnitude smaller, about 10−4 ofthe total 2P − 2S. Any measurement of the 2P − 2S Lamb shift in muonichydrogen with relative error comparable with the relative error of the Lambshift measurement in electronic hydrogen is much more sensitive to the valueof the proton charge radius.

The natural linewidth of the 2P states in muonic hydrogen and respectivelyof the 2P − 2S transition is determined by the linewidth of the 2P − 1Stransition, which is equal hΓ = 0.077 meV. It is planned [64] to measure2P − 2S Lamb shift with an accuracy at the level of 10% of the naturallinewidth, or with an error about 0.008 meV, which means measuring the2P − 2S transition with relative error about 4 × 10−5.

The total 2P −2S Lamb shift in muonic hydrogen calculated according tothe formulae in Table 7.1 for rp = 0.895 (18) fm, is

∆E(2P − 2S) = 201.880 (167) meV , (12.19)

where the uncertainty is completely determined by the uncertainty of theproton charge radius.

We can write the 2P − 2S Lamb shift in muonic hydrogen as a differenceof a theoretical number and a term proportional to the proton charge radiussquared

∆E(2P − 2S) = 206.065 (4) − 5.2250 〈r2〉 meV . (12.20)

We see from this equation that when the experiment achieves the plannedaccuracy of about 0.008 meV [64] this would allow determination of the protoncharge radius with relative accuracy about 0.1% which is about an order ofmagnitude better than the accuracy of the available experimental results.

Currently uncertainty in the sum of all theoretical contributions whichare not proportional to the proton charge radius squared in (12.20) is deter-mined by the uncertainties of the purely electrodynamic contributions and bythe uncertainty of the nuclear polarizability contribution of order (Zα)5m.Purely electrodynamic uncertainties are introduced by the uncalculated non-logarithmic contribution of order α2(Zα)4 corresponding to the diagrams withradiative photon insertions in the graph for leading electron polarization inFig. 7.8, and by the uncalculated light by light contributions in Fig. 3.11(e),and may be as large as 0.004 meV. Calculation of these contributions andelimination of the respective uncertainties is the most immediate theoreticalproblem in the theory of muonic hydrogen.

After calculation of these corrections, the uncertainty in the sum of alltheoretical contributions except those which are directly proportional to theproton radius squared will be determined by the uncertainty of the protonpolarizability contribution of order (Zα)5. This uncertainty of the protonpolarizability contribution is currently about 0.002 meV, and it will be difficultto reduce it in the near future. If the experimental error of measurement


2P − 2S Lamb shift in hydrogen will be reduced to a comparable level, itwould be possible to determine the proton radius with relative error smallerthat 3× 10−4 or with absolute error about 2× 10−4 fm, to be compared withthe current accuracy of the proton radius measurements producing the resultswith error on the scale of 0.01 fm.

12.2 Hyperfine Splitting

12.2.1 Hyperfine Splitting in Hydrogen

Hyperfine splitting in the ground state of hydrogen was measured preciselymore than thirty years ago [65, 66]

∆EHFS(H) = 1 420 405.751 766 7 (9) kHz δ = 6 × 10−13 . (12.21)

For many years, this hydrogen maser measurement remained the mostaccurate experiment in modern physics. Only recently the accuracy of theDoppler-free two-photon spectroscopy achieved comparable precision [34] (seethe result for the 1S − 2S transition frequency in (12.7)).

The theoretical situation for the hyperfine splitting in hydrogen alwaysremained less satisfactory due to the uncertainties connected with the protonstructure.

The scale of hyperfine splitting in hydrogen is determined by the Fermienergy in (8.2)

EF (H) = 1 418 840.101 (2) kHz , (12.22)

where the uncertainty is predominately determined by the uncertainty of theproton anomalous magnetic moment κ measured in nuclear magnetons.

The sum of all nonrecoil corrections to hyperfine splitting collected inTables 9.1, 9.2, 9.2, 9.3, 9.4, and 9.5 is equal to

∆EHFS(H) = 1 420 452.04 (2) kHz , (12.23)

where again the error of κ determines the uncertainty of the sum of all non-recoil contributions to the hydrogen hyperfine splitting.

The theoretical error of the sum of all nonrecoil contributions is about 1Hz, at least an order of magnitude smaller than the uncertainty introducedby the proton anomalous magnetic moment κ, and we did not write it ex-plicitly in (12.23). In relative units this theoretical error is about 2 × 10−10,to be compared with the estimate of the same error 1.2 × 10−7 made in [67].Reduction of the theoretical error by three orders of magnitude emphasizesthe progress achieved in calculations of nonrecoil corrections during the lastyears.

Until recently the stumbling block on the road to a more precise theory ofhydrogen hyperfine splitting was the inability to calculate the polarizability

12.2 Hyperfine Splitting 251

contribution. As we discussed above it was first calculated in [68] and improvedin [69]. After these calculations, the theoretical uncertainty of the total recoilcontribution to hydrogen hyperfine splitting was reduced to 0.8 kHz. Thecurrent theoretical result for hydrogen hyperfine splitting is

∆EHFS(H)th = 1 420 403.1 (8) kHz . (12.24)

There is a discrepancy between theory and the experimental result in (12.21)which is more than three standard deviations. Clearly, this situation is quiteunsatisfactory, and further theoretical and experimental efforts are requiredto rectify it.

12.2.2 Hyperfine Splitting in Deuterium

The hyperfine splitting in the ground state of deuterium was measured withvery high accuracy a long time ago [70, 71]

∆EHFS(D) = 327 384.352 521 9 (17) kHz δ = 5.2 × 10−12 . (12.25)

The expression for the Fermi energy in (8.2), besides the trivial substitu-tions similar to the ones in the case of hydrogen, should also be multiplied byan additional factor 3/4 corresponding to the transition from a spin one halfnucleus in the case of hydrogen and muonium to the spin one nucleus in thecase of deuterium. The final expression for the deuterium Fermi energy hasthe form

EF (D) =49α2µd

m

Mp

(

1 +m

Md

)−3

ch R∞ , (12.26)

where µd = 0.857 438 2329 (92) [1] is the deuteron magnetic moment innuclear magnetons, Md is the deuteron mass, and Mp is the proton mass.Numerically

EF (D) = 326 967.681 (4) kHz , (12.27)

where the main contribution to the uncertainty is due to the uncertainty ofthe deuteron anomalous magnetic moment measured in nuclear magnetons.

As in the case of hydrogen, after trivial modifications, we can use all non-recoil corrections in Tables 9.1, 9.2, 9.2, 9.3, 9.4, and 9.5 for calculations indeuterium. The sum of all nonrecoil corrections is numerically equal to

∆Enrec(D) = 327 339.147 (4) kHz . (12.28)

Unlike the proton, the deuteron is a weakly bound system so one cannotsimply use the results for the hydrogen recoil and structure corrections fordeuterium. What is needed in the case of deuterium is a completely newconsideration. Only one minor nuclear structure correction [72, 73, 74, 75]was discussed in the literature for many years, but it was by far too small toexplain the difference between the experimental result in (12.25) and the sumof nonrecoil corrections in (12.28)


∆EexpHFS(D) − ∆Enrec(D) = 45.2 kHz . (12.29)

A breakthrough was achieved a few years ago when it was realized thatan analytic calculation of the deuterium recoil, structure and polarizabilitycorrections is possible in the zero range approximation [76, 77]. An analyticresult for the difference in (12.29), obtained as a result of a nice calculation in[77], is numerically equal 44 kHz, and within the accuracy of the zero rangeapproximation perfectly explains the difference between the experimental re-sult and the sum of the nonrecoil corrections. More accurate calculations ofthe nuclear effects in the deuterium hyperfine structure beyond the zero rangeapproximation are feasible, and the theory of recoil and nuclear correctionswas later improved in a number of papers [78, 79, 80, 81, 82]. Comparisonof the results of these works with the experimental data on the deuteriumhyperfine splitting may be used as a test of the deuteron models and state ofthe art of the nuclear calculations.

12.2.3 Hyperfine Splitting in Muonium

Being a purely electrodynamic bound state, muonium is the best system forcomparison between the hyperfine splitting theory and experiment. Unlikethe case of hydrogen the theory of hyperfine splitting in muonium is freefrom uncertainties generated by the hadronic nature of the proton, and isthus much more precise. The scale of hyperfine splitting is determined by theFermi energy in (8.2)

EF (Mu) = 4 459 031.936 (518) (30) kHz , (12.30)

where the uncertainty in the first brackets is due to the uncertainty ofthe best direct experimental value of muon-electron mass ratio M/m =206.768 277 (24) in [83], and the uncertainty in the second brackets is dueto the uncertainty of the fine structure constant in (12.2). We used the directexperimental value of muon-electron mass ratio in this calculation instead ofthe CODATA value in (12.4) because as we will see the CODATA value itselfis mainly determined by the measurement of hyperfine splitting in muonium.

The theoretical accuracy of hyperfine splitting in muonium is deter-mined by the still uncalculated terms which include single-logarithmic andnonlogarithmic radiative-recoil corrections of order α2(Zα)(m/M)EF , aswell as by the nonlogarithmic contributions of orders (Zα)3(m/M)EF andα(Zα)2(m/M)EF . We estimate all these unknown corrections to hyperfinesplitting in muonium as about 70 Hz. Calculation of all these contributionsand reduction of the theoretical uncertainty of the hyperfine splitting in muo-nium below 10 Hz is the current task of the theory.

Current theoretical prediction for the hyperfine splitting interval in theground state in muonium may easily be obtained collecting all contributionsto HFS displayed in Tables 9.1, 9.2, 9.2, 9.3, 9.4, 9.5, 10.1, and 10.2

12.2 Hyperfine Splitting 253

∆EHFS(Mu) = 4 463 302.904 (518) (30) (70) kHz , (12.31)

where the first error comes from the experimental error of the electron-muonmass ratio m/M , the second comes from the error in the value of the finestructure constant α, and the third is an estimate of the yet unknown theo-retical contributions. We see that the uncertainty of the muon-electron massratio gives by far the largest contributions both in the uncertainty of the Fermienergy and the theoretical value of the ground state hyperfine splitting.

On the experimental side, hyperfine splitting in the ground state of muo-nium admits very precise determination due to its small natural linewidth.The lifetime of the higher energy hyperfine state with the total angular mo-mentum F = 1 with respect to the M1-transition to the lower level state withF = 0 is extremely large τ = 1×1013 s and gives negligible contribution to thelinewidth. The natural linewidth Γµ/h = 72.3 kHz is completely determinedby the muon lifetime τµ ≈ 2.2 × 10−6 s.

A high precision value of the muonium hyperfine splitting was obtainedmany years ago [84]

∆EHFS(Mu) = 4 463 302.88 (16) kHz δ = 3.6 × 10−8 . (12.32)

In the latest measurement [83] this value was improved by a factor of three

∆EHFS(Mu) = 4 463 302.776 (51) kHz, δ = 1.1 × 10−8 , (12.33)

The new value has an experimental error which corresponds to measuringthe hyperfine energy splitting at the level of ∆νexp/(Γµ/h) ≈ 7× 10−4 of thenatural linewidth. This is a remarkable experimental achievement.

The agreement between theory and experiment is excellent. However, theerror bars of the theoretical value are apparently about an order of magnitudelarger than respective error bars of the experimental result. This is a deceptiveimpression. The error of the theoretical prediction in (12.31) is dominated bythe experimental error of the value of the electron-muon mass ratio. As aresult of the latest experiment [83] this error was reduced threefold but it isstill by far the largest source of error in the theoretical value for the muoniumhyperfine splitting.

The estimate of the theoretical uncertainty is only marginally larger thanthe experimental error. The largest source of theoretical error is connectedwith the yet uncalculated theoretical contributions to hyperfine splitting,mainly with the unknown recoil and radiative-recoil corrections. As we havealready mentioned, reducing the theoretical uncertainty by an order of mag-nitude to about 10 Hz is now a realistic aim for the theory.

One can use high accuracy of the hyperfine splitting theory, and highlyprecise experimental result in (12.33) in order to obtain the value of the Fermienergy much more precise than the one in (12.30). But according to (8.2) theFermi energy is proportional to the electron-muon mass ratio, and we canextract this mass ratio from the experimental value of HFS and the mostprecise value of α


M

m= 206.768 282 9 (23) (14) (32) , (12.34)

where the first error comes from the experimental error of the hyperfine split-ting measurement, the second comes from the error in the value of the finestructure constant α, and the third from an estimate of the yet unknowntheoretical contributions.

Combining all errors we obtain the mass ratio

M

m= 206.768 282 9 (41) δ = 2.0 × 10−8 , (12.35)

which is almost six times more accurate than the best direct experimentalvalue [83].

The largest contribution to the uncertainty of the indirect mass ratio in(12.34) is supplied by the unknown theoretical contributions to hyperfine split-ting. This sets a clear task for the theory to reduce the contribution of thetheoretical uncertainty in the error bars in (12.34) to the level below two othercontributions to the error bars. It is sufficient to this end to calculate all con-tributions to HFS which are larger than 10 Hz. This would lead to furtherreduction of the uncertainty of the indirect value of the muon-electron massratio. There is thus a real incentive for improvement of the theory of HFSto account for all corrections to HFS of order 10 Hz, created by the recentexperimental and theoretical achievements.

Another reason to improve the HFS theory is provided by the perspectiveof reducing the experimental uncertainty of hyperfine splitting below the weakinteraction contribution in (10.38). In such a case, muonium could become thefirst atom where a shift of atomic energy levels due to weak interaction wouldbe observed [85].

12.3 Theoretical Perspectives

High precision experiments with hydrogenlike systems have achieved a newlevel of accuracy in recent years and further dramatic progress is still expected.The experimental errors of measurements of many energy shifts in hydrogenand muonium were reduced by orders of magnitude. This rapid experimentalprogress was matched by theoretical developments as discussed above. The ac-curacy of the quantum electrodynamic theory of such classical effects as Lambshift in hydrogen and hyperfine splitting in muonium has increased in manycases by one or two orders of magnitudes. This was achieved due to intensivework of many theorists and development of new ingenious original theoreticalapproaches which can be applied to the theory of bound states, not only inQED but also in other field theories, such as quantum chromodynamics. Fromthe phenomenological point of view recent developments opened new perspec-tives for precise determination of many fundamental constants (the Rydberg

References 255

constant, electron-muon mass ratio, proton charge radius, deuteron structureradius, etc.), and for comparison of the experimental and theoretical resultson the Lamb shifts and hyperfine splitting.

Recent progress also poses new theoretical challenges. Reduction of thetheoretical error in prediction of the value of the 1S Lamb shift in hydrogensignificantly below the level of 1 kHz (and, respectively, of the 2S Lamb shiftsignificantly below tenth of kHz) should be considered as a next stage of thetheory. The theoretical error of the hyperfine splitting in muonium should bereduced to about 10 Hz. Achievement of these goals will require hard workand considerable resourcefulness, but results which years ago hardly seemedpossible are now within reach.

References

1. P. J. Mohr and B. N. Taylor, Rev. Mod. Phys. 77, 1 (2005).2. S. G. Karshenboim, Phys. Rep. 422, 1 (2005).3. K. Pachucki and U. Jentschura, Phys. Rev. Lett. 91, 113005 (2003).4. U. D. Jentschura, Phys. Rev. A 70, 052108 (2004).5. V. A. Yerokhin, P. Indelicato, and V. M. Shabaev, Phys. Rev. A 71, 040101(R)

(2005).6. K. Pachucki and S. G. Karshenboim, Phys. Rev. A 60, 2792 (1999).7. K. Melnikov and A. S. Yelkhovsky, Phys. Lett. B 458, 143 (1999).8. V. M. Shabaev, A. N. Artemyev, T. Beier et al, Phys. Rev. A 57, 4235 (1998).9. V. M. Shabaev, in Lecture Notes in Physics, v.570, ed. S. G. Karshenboim et

al, Springer-Verlag, Berlin, Heidelberg, 2001, p.714.10. U. Jentschura, P. J. Mohr, and G. Soff, Phys. Rev. Lett. 82, 53 (1999).11. U. D. Jentschura, P. J. Mohr, and G. Soff, Phys. Rev. A 63, 042512 (2001).12. U. D. Jentschura and P. J. Mohr, Phys. Rev. A 69, 064103 (2004).13. U. D. Jentschura and P. J. Mohr, Phys. Rev. A 72, 014103 (2005).14. A. Czarnecki, U. D. Jentschura, and K. Pachucki, Phys. Rev. Lett. 95, 180404

(2005).15. E. W. Hagley and F. M. Pipkin, Phys. Rev. Lett. 72, 1172 (1994).16. Yu. L. Sokolov and V. P. Yakovlev, Zh. Eksp. Teor. Fiz. 83, 15 (1982) [Sov.Phys.-

JETP 56, 7 (1982)]; V. G. Palchikov, Yu. L. Sokolov, and V. P. Yakovlev, Pis’maZh. Eksp. Teor. Fiz. 38, 347 (1983) [JETP Letters 38, 418 (1983)].

17. S. G. Karshenboim, Zh. Eksp. Teor. Fiz. 106, 414 (1994) [JETP 79, 230 (1994)].18. V. G. Palchikov, Yu. L. Sokolov, and V. P. Yakovlev, Physica Scripta 55, 33

(1997).19. S. G. Karshenboim, Physica Scripta 57, 213 (1998).20. U. Jentschura and K. Pachucki, Phys. Rev. A 54, 1853 (1996).21. U. Jentschura and K. Pachucki, J. Phys. A 35, 1927 (2002).22. V. W. Hughes and T. Kinoshita, Rev. Mod. Phys. 71, S133 (1999).23. G. Newton, D. A. Andrews, and P. J. Unsworth, Phil. Trans. R. Soc. London

290, 373 (1979).24. S. R. Lundeen and F. M. Pipkin, Phys. Rev. Lett. 46, 232 (1981); Metrologia

22, 9 (1986).


25. A. van Wijngaarden, F. Holuj, and G. W. F. Drake, Can. J. Phys. 76, 95 (1998);Errata 76, 993 (1998).

26. C. Schwob, L. Jozefovski, B. de Beauvoir et al, Phys. Rev. Lett. 82, 4960 (1999).27. I. Sick, Phys. Lett. B 576, 62 (2003).28. D. J. Drickey and L. N. Hand, Phys. Rev. Lett. 9, 521 (1962); L. N. Hand, D.

J. Miller, and R. Wilson, Rev. Mod. Phys. 35, 335 (1963).29. G. G. Simon, Ch. Schmidt, F. Borkowski, and V. H. Walther, Nucl. Phys. A

333, 381 (1980).30. B. de Beauvoir, F. Nez, L. Julien et al, Phys. Rev. Lett. 78, 440 (1997).31. M. Weitz, A. Huber, F. Schmidt-Kaler et al., Phys. Rev. A 52, 2664 (1995).32. D. J. Berkeland, E. A. Hinds, and M. G. Boshier, Phys. Rev. Lett. 75, 2470

(1995).33. S. Bourzeix, B. de Beauvoir, F. Nez et al, Phys. Rev. Lett. 76, 384 (1996).34. Th. Udem, A. Huber, B. Gross et al, Phys. Rev. Lett. 79, 2646 (1997).35. J. Reichert, M. Niering, R. Holzwarth et al, Phys. Rev. Lett. 84, 3232 (2000).36. M. Niering, R. Holzwarth, J. Reichert et al, Phys. Rev. Lett. 84, 5496 (2000).37. H. A. Bethe and E. E. Salpeter, Quantum Mechanics of One- and Two-Electron

Atoms, Springer, Berlin, 1957.38. V. B. Berestetskii, E. M. Lifshitz, and L. P. Pitaevskii, Quantum electrodynam-

ics, 2nd Edition, Pergamon Press, Oxford, 1982.39. R. Rosenfelder, Phys. Lett. B 479, 381 (2000).40. V. V. Ezhela and B. V. Polishcuk, Protvino preprint IHEP 99-48, hep-

ph/9912401.41. R. G. Beausoleil and T. W. Hancsh, Phys. Rev. A 33, 1661 (1986).42. A. Huber, Th. Udem, B. Gross et al., Phys. Rev. Lett. 80, 468 (1998).43. S. Klarsfeld, J. Martorell, J. A. Oteo et al., Nucl. Phys. A 456, 373 (1986).44. J. L. Friar, J. Martorell, and D. W. L. Sprung, Phys. Rev. A 56, 4579 (1997).45. I. Sick and D. Trautman, Nucl. Phys. A637, 559 (1998).46. T. Herrmann and R. Rosenfelder, Eur. Phys. Journ. A 2, 29 (1998).47. J. L. Friar, Can. J. Phys. 80, 1337 (2002).48. M. G. Boshier, P. E. G. Baird, C.J. Foot et al, Phys. Rev. A 40, 6169 (1989).49. F. Schmidt-Kaler, D. Leibfried, M. Weitz et al., Phys. Rev. Lett. 70, 2261 (1993).50. A. van Wijngaarden, J. Kwela, and G. W. F. Drake, Phys. Rev. A 43, 3325

(1991).51. A. van Wijngaarden, F. Holuj, and G. W. F. Drake, Phys. Rev. A 63, 012505

(2001).52. S. Chu, A. P. Mills, Jr., A. G. Yodth et al, Phys. Rev. Lett. 60, 101 (1988); K.

Danzman, M. S. Fee and S. Chu, Phys. Rev. A 39, 6072 (1989).53. K. Jungmann, P. E. G. Baird, J. R. M. Barr et al, Z. Phys. D 21, 241 (1991).54. F. Maas, B. Braun, H. Geerds et al, Phys. Lett. A 187, 247 (1994).55. V. Meyer, S. N. Bagaev, P. E. G. Baird et al, Phys. Rev. Lett. 84, 1136 (2000).56. A. Bertin, G. Carboni, J. Duclos et al, Phys. Lett. B 55 (1975).57. G. Carboni, U. Gastaldi, G. Neri et al, Nuovo Cimento A 34, 493 (1976).58. G. Carboni, G. Gorini, G. Torelli et al, Nucl. Phys. A 278, 381 (1977).59. G. Carboni, G. Gorini, E. Iacopini et al, Phys. Lett. B 73, 229 (1978).60. E. Borie and G. A. Rinker, Rev. Mod. Phys. 54, 67 (1982).61. P. Hauser, H. P. von Arb, A. Biancchetti et al, Phys. Rev. A 46, 2363 (1992).62. D. Bakalov et al, Proc. III International Symposium on Weak and Electromag-

netic Interactions in Nuclei (WEIN-92) Dubna, Russia, 1992, p.656-662.

References 257

63. R. N. Faustov and A. P. Martynenko, Yad. Phys. 61, 534 (1998) [Phys. Atom.Nuclei 61, 471 (1998)].

64. F. Kottmann et al, Hyperfine Interactions 138, 55 (2001).65. H. Hellwig, R. F. C. Vessot, M. W. Levine et al, IEEE Trans. IM-19, 200 (1970).66. L. Essen, R. W. Donaldson, M. J. Bangham et al, Nature 229, 110 (1971).67. G. T. Bodwin and D. R. Yennie, Phys. Rev. D 37, 498 (1988).68. R. N. Faustov, A. P. Martynenko, and V. A. Saleev, Yad. Phys. 62, 2280 (1999)

[Phys. Atom. Nuclei 62, 2099 (1999)].69. V. Nazaryan, C. E. Carlson, and K. A. Griffion, Phys. Rev. Lett 96, 163001

(2006).70. D. J. Wineland and N. F. Ramsey, Phys. Rev. A 5, 821 (1972).71. N. F. Ramsey, in Quantum Electrodynamics, ed. T. Kinoshita (World Scientific,

Singapore, 1990), p.673.72. A. Bohr, Phys. Rev. 73, 1109 (1948).73. F. E. Low, Phys. Rev. 77, 361 (1950).74. F. E. Low and E. E. Salpeter, Phys. Rev. 83, 478 (1951).75. D. A. Greenberg and H. M. Foley, Phys. Rev. 120, 1684 (1960).76. I. B. Khriplovich, A. I. Milstein, and S. S. Petrosyan, Phys. Lett. B 366 13

(1996).77. A. I. Milshtein, I. B. Khriplovich, and S. S. Petrosyan, Zh. Eksp. Teor. Fiz. 109,

1146 (1996) [JETP 82, 616 (1996)].78. R. N. Faustov and A. P. Martynenko, Phys. Rev. A 67, 052506 (2003); A. P.

Martynenko and R. N. Faustov, Yad. Phys. 67, 477 (2004) [Phys. Atom. Nuclei67, 457 (2004)].

79. I. B. Khriplovich and A. I. Milstein, Zh. Eksp. Teor. Fiz. 125, 205 (2004) [JETP98, 181 (2004)].

80. J. L. Friar and I. Sick, Phys. Lett. B 579, 285 (2004).81. J. L. Friar and G. L. Payne, Phys. Lett. B 618, 68 (2005).82. J. L. Friar and G. L. Payne, Phys. Rev. C 72, 014002 (2005).83. W. Liu, M. G. Boshier, S. Dhawan et al, Phys. Rev. Lett. 82, 711 (1999).84. F. G. Mariam, W. Beer, P.R. Bolton et al, Phys. Rev. Lett. 49, 993 (1982).85. K. P. Jungmann, “Muonium,” preprint physics/9809020, September 1998.

Index

Bethe logarithm 24, 64relativistic 53two-loop 64

Bethe-Salpeter equation 5, 10, 90Breit

equation 81interaction 20, 95potential 20, 139, 194

correctionsbinding 3, 4, 13, 163, 165, 166nonelectromagnetic 14radiative 3, 14, 15, 17, 18, 21, 22, 27,

32, 36–38, 40, 42, 44, 46, 77, 100,103, 104, 114, 125–127, 131, 133,149, 153–155, 163, 165, 167, 169,170, 173, 182, 188, 189, 193, 198,200, 201, 204, 211, 227, 228

radiative-recoil 14, 99–103, 114, 163,173, 182, 195, 196, 198, 200–204,206, 209–211, 217, 226, 228, 247,252, 253

recoil 14, 21, 22, 81–83, 88–95,139, 140, 151, 163, 172, 193–198,220, 222–224, 226, 234, 235, 247,251–253

relativistic 3, 4, 13, 19, 24, 49, 81, 95,138–140, 163, 165, 167

state-independent 17, 183, 236

Darwin-Foldy contribution 21, 112, 113potential 57, 68, 72term 111, 112, 183

deuteron 113, 117, 119–121, 124, 221,245, 246, 251, 252, 255

dipoleapproximation 24, 50, 64, 144, 194,

238contribution 24fit 221form factor 151, 152, 154, 221

Dirac-Coulomb wave function 10, 24, 53,

57, 91, 132, 138, 165–167equation 4–6, 8, 13, 15, 19, 22, 81, 90,

165equation effective 6, 9, 13, 14, 21, 22,

24, 37, 81, 83, 165form factor 17, 23, 24, 27–29, 36, 43,

67, 103, 111, 114, 132, 146, 149spectrum 4, 14, 21, 22

external field approximation 19, 22,48, 81, 151, 165, 229

Fermi energy 162, 165, 167, 182, 193,194, 197, 210, 217, 218, 220, 222,226, 250–253

Foldy-Wouthuysen transformation 20,111

Green function 5–9, 50, 52, 90, 91, 99,138, 220

hyperfine splitting (HFS) 13, 161–163,165–178, 182, 193, 194, 197, 200,

260 Index

211, 218, 220, 222, 224, 227, 229,248, 250–254

Kallen-Sabry potential 141, 142

Lamb shift 4, 16, 17, 21, 22, 50, 51, 234

muonicatoms 118, 131, 133, 139, 143, 154,

248helium 131, 132, 143, 248hydrogen 101, 131–133, 135, 137,

139, 143–145, 148–151, 153–155,233, 243, 248, 249

NRQED 10, 61, 66, 67, 94, 178, 185,188, 196, 197, 202, 210

Pauli form factor 25, 28, 30, 103, 111,114

perturbationpotential 58, 65, 67, 73, 88, 110, 134,

135, 147, 182, 187, 188, 210, 226theory 6, 7, 9, 11, 24, 50, 51, 59, 62,

65, 67, 68, 87, 88, 90, 95, 113, 122,123, 137, 184, 187, 205, 226

theory contribution 59, 61, 83, 88,140

propagatorelectron 6, 183heavy particle 90, 91, 208nonrelativistic 183photon 16, 23, 36, 87, 90, 125, 134,

139, 162proton 100two-particle 6, 7

protoncharge radius 33, 104, 110–112, 116,

117, 122, 124, 149, 151, 154, 155,220, 221, 238, 242, 243, 245, 246,248–250, 255

form factor 104, 114, 117, 125, 126,149, 151, 152, 217, 218, 222–224,227–229

magnetic moment 104, 217, 218,220, 222–224, 226, 250

polarizability contribution 119, 121,122, 124, 223, 225, 248, 249, 251

size contribution 112, 116, 117, 122,125, 131, 132, 149, 151, 219, 220,222–224, 226, 236, 245, 249

Sachs form factor 104, 111, 112, 218,221

scaleatomic 37, 39, 40, 42, 45, 62, 102,

114, 123, 127, 147, 154, 171, 174,180, 194, 217

characteristic 2, 110, 206, 218natural 44, 47, 165, 179

scattering approximation 36, 40, 83,85, 87, 95, 125, 137, 144, 151, 169,173, 178, 180, 194, 198, 200, 201

Schrodinger-Coulomb wave function 10, 25,

36, 37, 40, 41, 49, 56–58, 72, 84, 87,105, 123, 134, 138, 147, 162, 167,169, 197, 220, 227

equation 1, 5, 6skeleton

diagram 36, 40, 47, 48, 83, 102, 103,114, 117, 169, 171, 179, 180, 198,200, 208, 218, 222

factor 17, 38integral 39, 41–43, 84, 85, 102, 114,

115, 118, 125, 126, 170–175, 178,194, 201, 203–205, 207, 220, 223,229

integral approach 38, 40, 44, 52, 56,101, 169, 171, 174, 178, 226, 228

Uehling potential 54, 56, 58, 63, 73, 75,139, 141, 142

Wichmann-Kroll potential 54, 58, 73,75, 141–143

Zemachcorrection 219–224, 226–229moment first 219, 221moment third 116–118, 151, 152,

154, 155, 219, 220radius 122, 219


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Theory of Light Hydrogenic Bound States

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