Theory of Light Hydrogenic Bound States

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Michael I. Eides Howard Grotch Valery A. Shelyuto

Theory of Light Hydrogenic Bound States With 108 Figures

ABC

Michael I. Eides Howard Grotch

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

University of Kentucky Department of Physics and Astronomy Lexington, KY 40506 U.S.A. E-mail: [email protected] [email protected] [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 ISSN print edition: 0081-3869 ISSN electronic edition: 1615-0430 ISBN-10 3-540-45269-9 Springer Berlin Heidelberg New York ISBN-13 978-3-540-45269-0 Springer Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable for prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springer.com c Springer-Verlag Berlin Heidelberg 2007
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Preface

Light one-electron atoms are a classical subject of quantum physics. The very discovery and further progress of quantum mechanics is intimately connected to the explanation of the main features of hydrogen energy levels. Each step in the development of quantum physics led to a better understanding of the bound state physics. The Bohr quantization rules of the old quantum theory were created in order to explain the existence of the stable discrete energy levels. The nonrelativistic quantum mechanics of Heisenberg and Schr¨ odinger provided a self-consistent scheme for description of bound states. The relativistic spin one half Dirac equation quantitatively described the main experimental features of the hydrogen spectrum. Discovery of the Lamb shift [1], a subtle discrepancy between the predictions of the Dirac equation and the experimental data, triggered development of modern relativistic quantum electrodynamics, and subsequently the Standard Model of modern physics. Despite its long and rich history the theory of atomic bound states is still very much alive today. New importance to the bound state physics was given by the development of quantum chromodynamics, the modern theory of strong interactions. It was realized that all hadrons, once thought to be the elementary building blocks of matter, are themselves atom-like bound states of elementary quarks bound by the color forces. Hence, from a modern point of view, the theory of atomic bound states could be considered as a theoretical laboratory and testing ground for exploration of the subtle properties of the bound state physics, free from further complications connected with the nonperturbative effects of quantum chromodynamics which play an especially important role in the case of light hadrons. The quantum electrodynamics and quantum chromodynamics bound state theories are so intimately intertwined today that one often finds theoretical research where new results are obtained simultaneously, say for positronium and also heavy quarkonium. The other powerful stimulus for further development of the bound state theory is provided by the spectacular experimental progress in precise measurements of atomic energy levels. It suffices to mention that in about a decade the relative uncertainty of measurement of the frequency of the 1S −2S

VI

Preface

transition in hydrogen was reduced by four orders of magnitude from 3 · 10−10 [2] to 1.8 × 10−14 [3]. The relative uncertainty in measurement of the muonium hyperfine splitting was reduced by the factor 3 from 3.6 × 10−8 [4] to 1.2 × 10−8 [5]. This experimental development was matched by rapid theoretical progress, and the comparison and interplay between theory and experiment has been important in the field of metrology, leading to higher precision in the determination of the fundamental constants. We feel that now is a good time to review modern bound state theory. The theory of hydrogenic bound states is widely described in the literature. The basics of nonrelativistic theory are contained in any textbook on quantum mechanics, and the relativistic Dirac equation and the Lamb shift are discussed in any textbook on quantum electrodynamics and quantum field theory. An excellent source for the early results is the classic book by Bethe and Salpeter [6]. A number of excellent reviews contain more recent theoretical results, and a representative, but far from exhaustive, list of these reviews includes [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17]. This book is an attempt to present a coherent state of the art discussion of the theory of the Lamb shift and hyperfine splitting in light hydrogenlike atoms. It is based on our earlier review [14]. The spin independent corrections are discussed below mainly as corrections to the hydrogen and/or muonic hydrogen energy levels, and the theory of hyperfine splitting is discussed in the context of the hyperfine splitting in the ground state of muonium. These simple atomic systems are singled out for practical reasons, because high precision experimental data either exists or is expected in these cases, and the most accurate theoretical results are also obtained for these bound states. However, almost all formulae below are also valid for other light hydrogenlike systems, and some of these other applications will be discussed as well. We will try to present all theoretical results in the field, with emphasis on more recent results. Our emphasis on the theory means that, besides presenting an exhaustive compendium of theoretical results, we will also try to present a qualitative discussion of the origin and magnitude of different corrections to the energy levels, to give, when possible, semiquantitative estimates of expected magnitudes, and to describe the main steps of the theoretical calculations and the new effective methods which were developed in recent years. We will not attempt to present a detailed comparison of theory with the latest experimental results, leaving this task to the experimentalists. We will use the experimental results only for illustrative purposes. The book is organized as follows. In the introductory part we briefly discuss the main theoretical approaches to the physics of weakly bound two-particle systems. A detailed discussion then follows of the nuclear spin independent corrections to the energy levels. First, we discuss corrections which can be calculated in the external field approximation. Second, we turn to the essentially two-particle recoil and radiative-recoil corrections. Consideration of the spinindependent corrections is completed with discussion of the nuclear size and structure contributions. A special section is devoted to the spin-independent

Preface

VII

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

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

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tools, which in many respects are quite different from the physics of low Z systems. We discuss below the numerical results obtained in the high Z calculations only when they have a direct relevance for the low Z atoms. The reader can find a detailed discussion of the high Z physics in a number of reviews (see, e.g., [18]). In trying to preserve a reasonable size of this text we decided to omit discussion of positronium, even though many theoretical expressions below are written in such form that for the case of equal masses they turn into respective corrections for the positronium energy levels. Positronium is qualitatively different from hydrogen and muonium not only due to the equality of the masses of its constituents, but because unlike the other light atoms there exists a whole new class of corrections to the positronium energy levels generated by the annihilation channel which is absent in other cases. For many years, numerous friends and colleagues have discussed with us the bound state problem, have collaborated on different projects, and have shared with us their vision and insight. We are especially deeply grateful to the late D. Yennie and M. Samuel, to G. Adkins, E. Borie, M. Braun, A. Czarnecki, M. Doncheski, G. Drake, R. Faustov, U. Jentschura, K. Jungmann, S. Karshenboim, I. Khriplovich, T. Kinoshita, L. Labzowsky, P. Lepage, A. Martynenko, K. Melnikov, A. Milshtein, P. Mohr, D. Owen, K. Pachucki, V. Pal’chikov, J. Sapirstein, V. Shabaev, B. Taylor, A. Yelkhovsky, and V. Yerokhin. This work was supported by the NSF grants PHY-0138210 and PHY-0456462.

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

W. E. Lamb, Jr. and R. C. Retherford, Phys. Rev. 72, 339 (1947). M. G. Boshier, P. E. G. Baird, C. J. Foot et al, Phys. Rev. A 40, 6169 (1989). M. Niering, R. Holzwarth, J. Reichert et al, Phys. Rev. Lett. 84, 5496 (2000). F. G. Mariam, W. Beer, P. R. Bolton et al, Phys. Rev. Lett. 49, 993 (1982). W. Liu, M. G. Boshier, S. Dhawan et al, Phys. Rev. Lett. 82, 711 (1999). H. A. Bethe and E. E. Salpeter, Quantum Mechanics of One- and Two-Electron Atoms, Springer, Berlin, 1957. J. R. Sapirstein and D. R. Yennie, in Quantum Electrodynamics, ed. T. Kinoshita (World Scientific, Singapore, 1990), p. 560. V. V. Dvoeglazov, Yu. N. Tyukhtyaev, and R. N. Faustov, Fiz. Elem. Chastits At. Yadra 25 144 (1994) [Phys. Part. Nucl. 25, 58 (1994)]. T. Kinoshita, Rep. Prog. Phys. 59, 3803 (1996). J. Sapirstein, in Atomic, Molecular and Optical Physics Handbook, ed. G. W. F. Drake, AIP Press, 1996, p. 327. P. J. Mohr, in Atomic, Molecular and Optical Physics Handbook, ed. G. W. F. Drake, AIP Press, 1996, p. 341. K. Pachucki, D. Leibfried, M. Weitz, A. Huber, W. K¨ onig, and T. W. H¨ anch, J. Phys. B 29, 177 (1996); 29, 1573(E) (1996). T. Kinoshita, hep-ph/9808351, Cornell preprint, 1998. M. I. Eides, H. Grotch, and V. A. Shelyuto, Phys. Rep. C 342, 63 (2001). H. Grotch and D. A. Owen, Found. Phys. 32, 1419 (2002).

References

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

Lexington, Kentucky, USA & Saint-Petersburg, Russia Lexington, Kentucky, USA Saint-Petersburg, Russia August 2006

Michael Eides Howard Grotch Valery Shelyuto

Contents

1

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

2

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

13 13 15 17 18

3

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

19 19 22 22 27 29 31 32 36 36 38 40 47

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

48 48 58 68

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

81 81 83 85 87 89 89 92

68 73 76 77

93 94 94 95 98

5

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

6

Nuclear Size and Structure Corrections . . . . . . . . . . . . . . . . . . . . 109 6.1 Main Proton Size Contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 6.1.1 Spin One-Half Nuclei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 6.1.2 Nuclei with Other Spins . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 6.1.3 Empirical Nuclear Form Factor and the Contributions to the Lamb Shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

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

Lamb Shift in Light Muonic Atoms . . . . . . . . . . . . . . . . . . . . . . . . 131 7.1 Closed Electron-Loop Contributions of Order αn (Zα)2 m . . . . . 133 7.1.1 Diagrams with One External Coulomb Line . . . . . . . . . . 133 7.1.2 Diagrams with Two External Coulomb Lines . . . . . . . . . 137 7.2 Relativistic Corrections to the Leading Polarization Contribution with Exact Mass Dependence . . . . . . . . . . . . . . . . . 138 7.3 Higher Order Electron-Loop Polarization Contributions . . . . . . 141 7.3.1 Wichmann-Kroll Electron-Loop Contribution of Order α(Zα)4 m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 7.3.2 Light by Light Electron-Loop Contribution of Order α2 (Zα)3 m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 7.3.3 Diagrams with Radiative Photon and Electron-Loop Polarization Insertion in the Coulomb Photon. Contribution of Order α2 (Zα)4 m . . . . . . . . . . . . . . . . . . . 144 7.3.4 Electron-Loop Polarization Insertion in the Radiative Photon. Contribution of Order α2 (Zα)4 m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 7.3.5 Insertion of One Electron and One Muon Loops in the same Coulomb Photon. Contribution of Order α2 (Zα)2 (me /m)2 m . . . . . . . . . . . . . . . . . . . . . . . 146 7.4 Hadron Loop Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 7.4.1 Hadronic Vacuum Polarization Contribution of Order α(Zα)4 m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 7.4.2 Hadronic Vacuum Polarization Contribution of Order α(Zα)5 m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 7.4.3 Contribution of Order α2 (Zα)4 m Induced by Insertion of the Hadron Polarization in the Radiative Photon . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 7.4.4 Insertion of One Electron and One Hadron Loops in the Same Coulomb Photon . . . . . . . . . . . . . . . . . . . . . . 150

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

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

9

Nonrecoil Corrections to HFS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 9.1 Relativistic (Binding) Corrections to HFS . . . . . . . . . . . . . . . . . . 165 9.2 Electron Anomalous Magnetic Moment Contributions (Corrections of Order αn EF ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 9.3 Radiative Corrections of Order αn (Zα)EF . . . . . . . . . . . . . . . . . . 169 9.3.1 Corrections of Order α(Zα)EF . . . . . . . . . . . . . . . . . . . . . 169 9.3.2 Corrections of Order α2 (Zα)EF . . . . . . . . . . . . . . . . . . . . 173 9.3.3 Corrections of Order α3 (Zα)EF . . . . . . . . . . . . . . . . . . . . 179 9.4 Radiative Corrections of Order αn (Zα)2 EF . . . . . . . . . . . . . . . . . 180 9.4.1 Corrections of Order α(Zα)2 EF . . . . . . . . . . . . . . . . . . . . 180 9.4.2 Corrections of Order α2 (Zα)2 EF . . . . . . . . . . . . . . . . . . . 184 9.5 Radiative Corrections of Order α(Zα)3 EF and of Higher Orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 9.5.1 Corrections of Order α(Zα)3 EF . . . . . . . . . . . . . . . . . . . . 187 9.5.2 Corrections of Order α2 (Zα)3 EF and of Higher Orders in α . . . . . . . . . . . . . . . . . . . . . . . . . 190 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190

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

Contents

XV

10.2.3 Electron-Line Nonlogarithmic Contributions
F . . . . . . . . . . . . . . . . . . . . . . . . . 201 of Order α(Zα)(m/M )E
F 202 10.2.4 Muon-Line Contribution of Order (Z 2 α)(Zα)(m/M )E 10.2.5 Leading Photon-Line Double Logarithmic
F . . . . . . . . . . . . . 203 Contribution of Order α(Zα)(m/M )E 10.2.6 Photon-Line Single-Logarithmic and Nonlogarithmic
F . . . . . . . . . . . . 204 Contributions of Order α(Zα)(m/M )E 10.2.7 Heavy Particle Polarization Contributions
F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 of Order α(Zα)E 10.2.8 Leading Logarithmic Contributions
F . . . . . . . . . . . . . . . . . . . . . . . . 206 of Order α2 (Zα)(m/M )E 10.2.9 Leading Four-Loop Contribution
F . . . . . . . . . . . . . . . . . . . . . . . . 209 of Order α3 (Zα)(m/M )E 10.2.10 Corrections of Order α(Zα)(m/M )n EF . . . . . . . . . . . . . 209 10.2.11 Corrections of Orders α(Zα)2 (m/M )EF and Z 2 α(Zα)2 (m/M )EF . . . . . . . . . . . . . . . . . . . . . . . . . . 210 10.3 Weak Interaction Contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 11 Hyperfine Splitting in Hydrogen . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 11.1 Nuclear Size, Recoil and Structure Corrections of Orders (Zα)EF and (Zα)2 EF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218 11.1.1 Corrections of Order (Zα)EF . . . . . . . . . . . . . . . . . . . . . . 218 11.1.2 Recoil Corrections of Order (Zα)2 (m/M )EF . . . . . . . . . 226 11.1.3 Correction of Order (Zα)2 m2 r2 EF . . . . . . . . . . . . . . . . . 226 11.1.4 Correction of Order (Zα)3 (m/Λ)EF . . . . . . . . . . . . . . . . 227 11.2 Radiative Corrections to Nuclear Size and Recoil Effects . . . . . . 227 11.2.1 Radiative-Recoil Corrections of Order α(Zα)(m/Λ)EF 227 11.2.2 Radiative-Recoil Corrections of Order α(Zα)(m/M )EF 228 11.2.3 Heavy Particle Polarization Contributions . . . . . . . . . . . 229 11.3 Weak Interaction Contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 12 Notes on Phenomenology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 12.1 Lamb Shifts of the Energy Levels . . . . . . . . . . . . . . . . . . . . . . . . . . 233 12.1.1 Values of Some Physical Constants . . . . . . . . . . . . . . . . . 233 12.1.2 Theoretical Accuracy of S-State Lamb Shifts . . . . . . . . 234 12.1.3 Theoretical Accuracy of P -State Lamb Shifts . . . . . . . . 235 12.1.4 Theoretical Accuracy of the Interval ∆n = n3 L(nS) − L(1S) . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 12.1.5 Classic Lamb Shift 2S 12 − 2P 12 . . . . . . . . . . . . . . . . . . . . . 236 12.1.6 1S Lamb Shift and the Rydberg Constant . . . . . . . . . . . 238 12.1.7 Isotope Shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 12.1.8 Lamb Shift in Helium Ion He+ . . . . . . . . . . . . . . . . . . . . . 246 12.1.9 1S − 2S Transition in Muonium . . . . . . . . . . . . . . . . . . . . 247

XVI

Contents

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

1 Theoretical Approaches to the Energy Levels of Loosely Bound Systems

1.1 Nonrelativistic Electron in the Coulomb Field In the first approximation, energy levels of one-electron atoms (see Fig. 1.1) are described by the solutions of the Schr¨ odinger equation for an electron in the field of an infinitely heavy Coulomb center with charge Z in terms of the proton charge1   Zα ∆ − (1.1) − ψ(r) = En ψ(r) , 2m r r , ψnlm (r) = Rnl (r)Ylm r En = −

m(Zα)2 , 2n2

n = 1, 2, 3 . . . ,

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

1

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

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


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   2  p  (Zα)2 2
n|v |n = n 2 n = , (1.2) m n2     2  Zα  n = m(Zα) ,  n  2 r n   2  2 p  n = m(Zα) . n  2 2m 2n We see that due to the smallness of the fine structure constant α a oneelectron atom is a loosely bound nonrelativistic system2 and all relativistic effects may be treated as perturbations. There are three characteristic scales 2

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

1.2 Dirac Electron in the Coulomb Field

3

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

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

p 2 + m2 ≈ m +

p4 p2 − + ··· . 2m 8m3

(1.3)

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

4

1 Theoretical Approaches to the Energy Levels of Loosely Bound Systems

(binding) corrections are described by the expansion in even powers of Zα, as in the naive expansion of the classical relativistic square root in (1.1). Only the coefficients in this expansion change due to presence of spin. A proper description of all relativistic corrections to the energy levels is given by the Dirac equation with a Coulomb source. All relativistic corrections may easily be obtained from the exact solution of the Dirac equation in the external Coulomb field (see, e.g., [1, 2]) Enj = mf (n, j) , where

(1.4) − 12

 2

 f (n, j) = 1 + 

(Zα)

 2 

− (Zα)2 + n − j −   1 (Zα)2 (Zα)4 3 ≈ 1− − − 2n2 2n3 4n j + 12   1 3 5 6 (Zα)6 + + − + · · · , (1.5) − 8n3 2n3 (j + 12 )3 n(j + 12 )2 n2 (j + 12 ) (j +

1 2 2)

1 2

and j = 1/2, 3/2, . . . , n − 1/2 is the total angular momentum of the state. In the Dirac spectrum, energy levels with the same principal quantum number n but different total angular momentum j are split into n components of the fine structure, unlike the nonrelativistic Schr¨ odinger spectrum where all levels with the same n are degenerate. However, not all degeneracy is lifted in the spectrum of the Dirac equation: the energy levels corresponding to the same n and j but different l = j ± 1/2 remain doubly degenerate. This degeneracy is lifted by the corrections connected with the finite size of the Coulomb source, recoil contributions, and by the dominating QED loop contributions. The respective energy shifts are called the Lamb shifts (see exact definition in Sect. 3.1) and will be one of the main subjects of discussion below. We would like to emphasize that the quantum mechanical (recoil and finite nuclear size) effects alone do not predict anything of the scale of the experimentally observed Lamb shift which is thus essentially a quantum electrodynamic (field-theoretical) effect. One trivial improvement of the Dirac formula for the energy levels may easily be achieved if we take into account that, as was already discussed above, the electron motion in the Coulomb field is essentially nonrelativistic, and, hence, all contributions to the binding energy should contain as a factor the reduced mass of the electron-nucleus nonrelativistic system rather than the electron mass. Below we will consider the expression with the reduced mass factor (1.6) Enj = m + mr [f (n, j) − 1] , rather than the naive expression in (1.4), as a starting point for calculation of corrections to the electron energy levels. In order to provide a solid starting point for further calculations the Dirac spectrum with the reduced mass

1.3 Bethe-Salpeter Equation and the Effective Dirac Equation

5

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

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

Fig. 1.2. Bethe-Salpeter equation

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

Fig. 1.3. Kernel of the Bethe-Salpeter equation

6

1 Theoretical Approaches to the Energy Levels of Loosely Bound Systems

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

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

(1.8)

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

1.3 Bethe-Salpeter Equation and the Effective Dirac Equation

7

The free propagator in (1.8) determines other building blocks and the form of a two-body equation equivalent to the BS equation, and the regular perturbation theory formulae in this case were obtained in [9, 10]. In order to derive these formulae let us first write the BS equation in (1.7) in an explicit form   d4 k d4 q   l, E) , G(p, l, E) = S0 (p, l, E) + S0 (p, k, E)KBS (k, q, E)G(q, (2π)4 (2π)4 (1.9) where i i (2π)4 δ (4) (p − l) . (1.10) S0 (p, k, E) = p/ − M E/ − /l − m The amputated two-particle Green function GT satisfies the equation GT = KBS + KBS S0 GT .

(1.11)

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

(1.12)

Comparing (1.11) and (1.12) one easily obtains the diagrammatic series for the new kernel K (see Fig. 1.4) K(q, l, E) = [I − KBS (S0 − ΛS)]−1 KBS   i i d4 r = KBS (q, l, E) + K (q, r, E) BS (2π)4 /r − M E/ − /r − m  /r + M − 2πiδ (+) (r2 − M 2 ) KBS (r, l, E) + · · · . (1.13) E/ − /r − m The new bound state equation is constructed for the two-particle Green function defined by the relationship G = ΛS + ΛSGT ΛS .

(1.14)

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

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

8

1 Theoretical Approaches to the Energy Levels of Loosely Bound Systems

G = ΛS + ΛSKG ,

(1.15)

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

G(p, l, E) = 2πiδ (+) (p2 − M 2 )

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

p/ + M E/ − p/ − m   3 (3) × (2π) δ (p − l) +

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

where all four-momenta are on the mass shell p2 = l2 = q 2 = M 2 , Eq =
is deq 2 + M 2 , and the three-dimensional two-particle Green function G fined as follows
l, E)2πiδ (+) (l2 − M 2 ) . G(p, l, E) = 2πiδ (+) (p2 − M 2 )G(p,

(1.18)

Taking the residue at the bound state pole with energy En we obtain a homogeneous equation  d3 q (E/n − p/ − m)φ(p, En ) = (p/ + M ) iK(p, q, En )φ(q, En ) . (1.19) (2π)3 2Eq Due to the presence of the heavy particle mass shell projector on the right hand side the wave function in (1.19) satisfies a free Dirac equation with respect to the heavy particle indices (p/ − M )φ(p, En ) = 0 .

(1.20)

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

  Ep + M I U (p) = p·σ . Ep − M |p|

(1.22)

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

1.3 Bethe-Salpeter Equation and the Effective Dirac Equation

9

Fig. 1.5. Effective Dirac equation

spinor of the heavy particle) satisfies the effective Dirac equation (see Fig. 1.5)  d3 q
(E/n − p/ − m)ψ(p, En ) = iK(p, q, En )ψ(q, En ) , (1.23) (2π)3 2Eq where

¯ (p)K(p, q, En )U (q) U
K(p, q, En ) = , 4Ep Eq

(1.24)

k = (En − p0 , −p) is the electron momentum, and the crosses on the heavy line in Fig. 1.5 mean that the heavy particle is on its mass shell. The inhomogeneous equation (1.17) also fixes the normalization of the wave function. Even though the total kernel in (1.23) is unambiguously defined, we still have freedom to choose the zero-order kernel K0 at our convenience, in order to obtain a solvable lowest order approximation. It is not difficult to obtain a regular perturbation theory series for the corrections to the zero-order approximation corresponding to the difference between the zero-order kernel K0 and the exact kernel K0 + δK   En = En0 + (n|iδK(En0 )|n) 1 + (n|iδK  (En0 )|n) + (n|iδK(En0 )Gn0   × (En0 )iδK(En0 )|n) 1 + (n|iδK  (En0 )|n) + · · · , (1.25) where the summation of intermediate states goes with the weight d3 p/ [(2π)3 2Ep ] and is realized with the help of the subtracted free Green function of the EDE with the kernel K0 Gn0 (E) = G0 (E) −

|n)(n| , E − En0

(1.26)

conjugation is understood in the Dirac sense, and δK  (En0 ) ≡ (dK/dE)|E=En0 . The only apparent difference of the EDE (1.23) from the regular Dirac equation is connected with the dependence of the interaction kernels on energy. Respectively the perturbation theory series in (1.25) contain, unlike the regular nonrelativistic perturbation series, derivatives of the interaction kernels over energy. The presence of these derivatives is crucial for cancellation of the ultraviolet divergences in the expressions for the energy eigenvalues. A judicious choice of the zero-order kernel (sum of the Coulomb and Breit potentials, for more detail see, e.g, [6, 7, 10]) generates a solvable unperturbed

10

1 Theoretical Approaches to the Energy Levels of Loosely Bound Systems

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 this equation may be found exactly in the form of the Dirac-Coulomb wave functions (see, e.g, [10]). For practical purposes it is often sufficient to approximate these exact wave functions by the product of the Schr¨ odinger-Coulomb wave functions with the reduced mass and the free electron spinors which depend on the electron mass and not on the reduced mass. These functions are very convenient for calculation of the high order corrections, and while below we will often skip some steps in the derivation of one or another high order contribution from the EDE, we advise the reader to keep in mind that almost all calculations below are done with these unperturbed wave functions.

1.4 Nonrelativistic Quantum Electrodynamics A weakly bound state is necessarily nonrelativistic, v ∼ Zα (see discussion of the electron in the field of a Coulomb center above). Hence, there are two small parameters in a weakly bound state, namely, the fine structure constant α and nonrelativistic velocity v ∼ Zα. In the leading approximation weakly bound states are essentially quantum mechanical systems, and do not require quantum field theory for their description. But a nonrelativistic quantum mechanical description does not provide an unambiguous way for calculation of higher order corrections, when recoil and many particle effects become important. On the other hand the Bethe-Salpeter equation provides an explicit quantum field theory framework for discussion of bound states, both weakly and strongly bound. Just due to generality of the Bethe-Salpeter formalism separation of the basic nonrelativistic dynamics for weakly bound states becomes difficult, and systematic extraction of high order corrections over α and v ∼ Zα becomes prohibitively complicated. Nonrelativistic quantum electrodynamics (NRQED) [11] is an attempt to combine the simplicity of the quantum mechanical description with the power and rigor of field theory. The idea is to write ordinary relativistic quantum electrodynamics in the form of a nonrelativistic expansion with a Lagrangian containing vertices with arbitrary powers of fields. This is useful if we want to consider essentially nonrelativistic processes, like nonrelativistic bound states and threshold phenomena. In such a physical situation the dominant dynamics is nonrelativistic, and the calculations could be in principle simplified if 3

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

References

11

we could organize our expansions not only in terms of α but also in terms of the nonrelativistic velocity v ≈ p/m. In the scattering processes this velocity is determined by kinematics and is in our hands. In the nonrelativistic bound states it is connected with the coupling constant if the perturbation theory works, but even if the perturbation theory fails, velocity still remains a useful small parameter. The original idea of [11] was to introduce a finite cutoff Λ, and expand in p/Λ. Since the cutoff is finite we can choose it for convenience. For bound states it is natural to choose Λ ∼ m. Then we have two small parameters α and p/Λ ∼ p/m ∼ v. Next we write all vertices which are compatible with the symmetries of relativistic quantum electrodynamics. Then we calculate coefficients before the nonrelativistic vertices, comparing results for the complete relativistic QED and for the NRQED. It is crucial to realize that we can compare coefficients up to arbitrary order in α at the fixed v. There are no problems with ultraviolet convergence since we consider a theory with an explicit finite cutoff. Hence, if we want to calculate up to a particular order in v we can do it with arbitrary accuracy in α. If, as in the bound state applications, we need certain overall accuracy in α (Zα) we simply take the required order in v as to achieve the desired accuracy. There are now numerous implementations of these basic ideas in the vast literature, in the framework of schemes with an explicit cutoff as well as in the framework of dimensional regularization. We will skip technical details of NRQED here, referring the reader to papers [12, 13, 14, 15, 16] and references therein. In our discussions of high order corrections to the energy levels we will act in the general spirit of nonrelativistic QED, and use effective operators of NRQED for calculations of different contributions to the energy levels. In many cases the form and origin of these effective operators are intuitively clear and are more transparent than lengthy formal derivations.

References 1. J. D. Bjorken and S. D. Drell, Relativistic Quantum Mechanics, McGraw-Hill Book Co., NY, 1964. 2. V. B. Berestetskii, E. M. Lifshitz, and L. P. Pitaevskii, Quantum electrodynamics, 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 12. 13. 14. 15. 16.

1 Theoretical Approaches to the Energy Levels of Loosely Bound Systems T. Kinoshita and M. Nio, Phys. Rev. D 53, 4909 (1996). G. P. Lepage, preprint nucl-th/9706029, February 1997. P. Labelle, Phys. Rev. D 58, 093013 (1998). A. Pineda and J. Soto, Phys. Rev. D 59, 016005 (1999). A. H. Hoang, preprint hep-ph/0204299, April 2002.

2 General Features of the Hydrogen Energy Levels

2.1 Classification of Corrections The zero-order effective Dirac equation with a Coulomb source provides only an approximate description of loosely bound states in QED, but the spectrum of this Dirac equation may serve as a good starting point for obtaining more precise results. The magnetic moment of the heavy nucleus is completely ignored in the Dirac equation with a Coulomb source, and, hence, the hyperfine splitting (HFS) of the energy levels is missing in its spectrum. Notice that the magnetic interaction between the nucleus and the electron may be easily described even in the framework of the nonrelativistic quantum mechanics, and the respective calculation of the leading contribution to the hyperfine splitting was done a long time ago by Fermi [1]. Other corrections to the Dirac energy levels do not arise in the quantum mechanical treatment with a potential, and for their calculation, as well as for calculation of the corrections to the hyperfine splitting, field-theoretical methods are necessary. All electrodynamic corrections to the energy levels may be written in the form of the power series expansion over three small parameters α, Zα and m/M which determine the properties of the bound state. Account for the additional corrections of nonelectromagnetic origin induced by the strong and weak interactions introduces additional small parameters, namely, the ratio of the nuclear radius and the Bohr radius, the Fermi constant, etc. It should be noted that the coefficients in the power series for the energy levels might themselves be slowly varying functions (logarithms) of these parameters. Each of the small parameters above plays an important and unique role. In order to organize further discussion of different contributions to the energy levels it is convenient to classify corrections in accordance with the small parameters on which they depend. Corrections which depend only on the parameter Zα will be called relativistic or binding corrections. Higher powers of Zα arise due to deviation of M.I. Eides et al.: Theory of Light Hydrogenic Bound States, STMP 222, 13–18 (2007) c Springer-Verlag Berlin Heidelberg 2007 DOI 10.1007/3-540-45270-2 2


14

2 General Features of the Hydrogen Energy Levels

the theory from a nonrelativistic limit, and thus represent a relativistic expansion. All such contributions are contained in the spectrum of the effective Dirac equation in the external Coulomb field. Contributions to the energy which depend only on the small parameters α and Zα are called radiative corrections. Powers of α arise only from the quantum electrodynamics loops, and all associated corrections have a quantum field theory nature. Radiative corrections do not depend on the recoil factor m/M and thus may be calculated in the framework of QED for a bound electron in an external field. In respective calculations one deals only with the complications connected with the presence of quantized fields, but the two-particle nature of the bound state and all problems connected with the description of the bound states in relativistic quantum field theory still may be ignored. Corrections which depend on the mass ratio m/M of the light and heavy particles 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 recoil corrections. They describe contributions to the energy levels which cannot be taken into account with the help of the reduced mass factor. The presence of 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 be taken into account with the help of the effective Dirac equation in the external field since these corrections are induced by the one-photon exchange. This is impossible for the higher order recoil terms which reflect the truly relativistic two-body nature of the bound state problem. Technically, respective contributions are induced by the Bethe-Salpeter kernels with at least twophoton exchanges and the whole machinery of relativistic QFT is necessary for their calculation. Calculation of the recoil corrections is simplified by the absence of ultraviolet divergences, connected with the purely radiative loops. Radiative-Recoil corrections are the expansion terms in the expressions for the energy levels which depend simultaneously on the parameters α, m/M and Zα. Their calculation requires application of all the heavy artillery of QED, since we have to account both for the purely radiative loops and for the relativistic two-body nature of the bound states. The last class of corrections contains nonelectromagnetic corrections, effects of weak and strong interactions. The largest correction induced by the strong 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 the Dirac energy spectrum. Discussion of their calculations is our main topic below.

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 charge is spread over a finite volume instead of being pointlike1
r2  = −6

2α dF1 (−k2 ) 2α m ≈ ≈ ln ln(Zα)−2 . 2 2 2 2 πm ρ πm dk |k =0

(2.2)

In order to obtain this estimate of the electron radius we have taken into account 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 by its virtuality ρ = (m2 − p2 )/m which is of order of the nonrelativistic binding energy 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 Coulomb potential (see, e.g., [2]) δV =

1 2 2π
r ∆V = Zα
r2 δ(r) , 6 3

(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 size of the electron produced by the QED radiative corrections leads to a shift of the hydrogen energy levels. Moreover, since this perturbation is nonvanishing only at the source of the Coulomb potential, it influences quite differently the energy levels with different orbital angular momenta, and, hence, leads to splitting of the levels with the same total angular momenta but different orbital momenta. This splitting lifts the degeneracy in the spectrum of the Dirac equation in the Coulomb field, where the energy levels depend only on the 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 (−k 2 ) = 1 −

1 2 2
r k . 6

(2.1)

16

2 General Features of the Hydrogen Energy Levels

It is very easy to estimate this splitting (shift of the S level energy) ∆E =
nS|δV |nS ≈ |Ψn (0)|2 ≈

2π(Zα) 2
r  3

4m(Zα)4 α ln[(Zα)−2 ]δl0 n3 3π

|n=2

≈ 1330 MHz .

(2.4)

This result should be compared with the experimental number of about 1040 MHz and the agreement is satisfactory for such a crude estimate. There are two qualitative features of this result to which we would like to attract the reader’s attention. First, the sign of the energy shift may be obtained without calculation. Due to the finite radius of the electron its charge in the S state is on the average more spread out around the Coulomb source than in the case of the pointlike electron. Hence, the binding is weaker than in the case of the pointlike electron and the energy of the level is higher. Second, despite the presence of nonlogarithmic contributions missing in our crude calculation, their magnitude is comparatively small, and the logarithmic term above is responsible for the main contribution to the Lamb shift. This property is due to the would be infrared divergence of the considered contribution, which is cutoff by the small (in comparison with the electron mass) binding energy. As we will see below, whenever a correction is logarithmically enhanced, the respective logarithm gives a significant part of the correction, as is the case above. Another obvious contribution to the Lamb shift of the same leading order is connected with the polarization insertion in the photon propagator (see Fig. 2.2). This correction also induces a correction to the Coulomb potential δV = −

α 4 α(Zα) Π(−k2 ) ∆V = ∆V = − δ(r) , 4 2 2 15πm 15 m2 k |k =0

(2.5)

and the respective correction to the S-level energy is equal to ∆E =
nS|δV |nS = −|Ψn (0)|2 =−

4m(Zα)4 α δl0 n3 15π

|n=2

4α(Zα) 15m2

(2.6)

≈ −30 MHz .

Once again the sign of this correction is evident in advance. The polarization 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 finite distance.2 This means that the electron, which has penetrated in the polarization cloud, sees effectively a larger charge and experiences a stronger binding force, which lowers the energy level. Experimental observation of this contribution to the Lamb shift played an important role in the development of modern quantum electrodynamics since it explicitly confirmed the very existence of the closed electron loops. Numerically the vacuum polarization contribution is much less important than the contribution connected with the electron spreading due to quantum corrections, and the total shift of the level is positive.

2.3 Natural Magnitudes of Corrections to the Lamb Shift Let us emphasize that the main contribution to the Lamb shift is a radiative correction itself (compare (2.4), (2.6)) and contains a logarithmic enhancement factor. This is extremely important when one wants to get a qualitative understanding of the magnitude of the higher order corrections to the Lamb shift discussed below. Due to the presence of this accidental logarithmic enhancement it is impossible to draw conclusions about the expected magnitude of higher order corrections to the Lamb shift simply by comparing them to the magnitude of the leading order contribution. It is more reasonable to extract from this leading order contribution the term which can be called the skeleton factor and to use it further as a normalization factor. Let us write the leading order contributions in (2.4), (2.6) obtained above in the form 4m(Zα)4 × radiative correction , n3

(2.7)

where the radiative correction is either the slope of the Dirac form factor, roughly speaking equal to m2 dF1 (−k2 )/dk2 |k2 =0 = α/(3π) ln(Zα)−2 , or the polarization correction m2 Π(−k2 )/k4 |k2 =0 = α/(15π). It is clear now that the scale setting factor which should be used for qualitative estimates of the high order corrections to the Lamb shift is equal to 4m(Zα)4 /n3 . Note the characteristic dependence on the principal quantum number 1/n3 which originates from the square of the wave function at the origin |ψ(0)|2 ∼ 1/n3 . All corrections induced at small distances (or at high virtual momenta) have this characteristic dependence and are called stateindependent. Even the coefficients before the leading powers of the low energy logarithms ln(Zα)2 are state-independent since these leading logarithms originate from integration over the wide intermediate momenta region from m(Zα)2 to m, and the respective factor before the logarithm is determined 2

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.

18

2 General Features of the Hydrogen Energy Levels

by the high momenta part of the integration region. Estimating higher order corrections to the Lamb shift it is necessary to remember, as mentioned above, that unlike the case of radiative corrections to the scattering amplitudes, in the bound state problem factors Zα are not accompanied by an extra factor π in the denominator. This well known feature of the Coulomb problem provides one more reason to preserve Z in analytic expressions (even when Z = 1), since in this way one may easily separate powers of Zα not accompanied by π from powers of α which always enter formulae in the combination α/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 arise in the external field approximation. These are leading relativistic corrections with exact mass dependence and radiative corrections.

3.1 Leading Relativistic Corrections with Exact Mass Dependence We are considering a loosely bound two-particle system. Due to the nonrelativistic nature of this bound state it is clear that the main (Zα)2 contribution to the binding energy depends only on one mass parameter, namely, on the nonrelativistic reduced mass, and does not depend separately on the masses of the constituents. Relativistic corrections to the energy levels of order (Zα)4 , describing the fine structure of the hydrogen spectrum, are missing in the nonrelativistic Schr¨ odinger equation approach. The correct description of the fine structure for an infinitely heavy Coulomb center is provided by the relativistic Dirac equation, but it tells us nothing about the proper mass dependence of these corrections for the nucleus of finite mass. There are no reasons to expect that in the case of a system of two particles with finite masses relativistic corrections of order (Zα)4 will depend only on the reduced mass. The dependence of these corrections on the masses of the constituents should be more complicated. The solution of the problem of the proper mass dependence of the relativistic corrections of order (Zα)4 may be found in the effective Hamiltonian framework. In the center of mass system the nonrelativistic Hamiltonian for a system of two particles with Coulomb interaction has the form H0 =

p2 Zα p2 + − . 2m 2M r

(3.1)

In a nonrelativistic loosely bound system expansion over (Zα)2 corresponds to the nonrelativistic expansion over v 2 /c2 . Hence, we need an effective M.I. Eides et al.: Theory of Light Hydrogenic Bound States, STMP 222, 19–80 (2007) c Springer-Verlag Berlin Heidelberg 2007 DOI 10.1007/3-540-45270-2 3


20

3 External Field Approximation

Hamiltonian including the terms of the first order in v 2 /c2 for proper description of the corrections of relative order (Zα)2 to the nonrelativistic energy levels. Such a Hamiltonian was first considered by Breit [1], who realized that all corrections to the nonrelativistic two-particle Hamiltonian of the first order in v 2 /c2 may be obtained from the sum of the free relativistic Hamiltonians of each of the particles and the relativistic one-photon exchange. This conjecture is intuitively obvious since extra exchange photons lead to at least one extra factor 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 the one-photon exchange amplitude with the help of the Foldy-Wouthuysen transformation1     1 πZα 1 Zα r(r · p) · p 3 2 VBr = + (r) − + δ p 2 m2 M2 2mM r r2   1 Zα 1 + 3 + [r × p] · σ . (3.2) 2 r 4m 2mM A simplified derivation of the Breit interaction potential may be found in many textbooks (see, e.g., [3]). All contributions to the energy levels up to order (Zα)4 may be calculated from the total Breit Hamiltonian HBr = H0 + VI ,

(3.3)

where the interaction potential is the sum of the instantaneous Coulomb and Breit 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 of the Breit interaction between the Coulomb-Schr¨odinger eigenfunctions of the Coulomb Hamiltonian H0 in (3.1). The mass dependence of the Breit interaction is known exactly, and the same is true for its matrix elements. These matrix elements and, hence, the exact mass dependence of the contributions to the energy levels of order (Zα)4 , beyond the reduced mass, were first obtained a long time ago [2] 1

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

3.1 Leading Relativistic Corrections with Exact Mass Dependence

21

  1 mr mr (Zα)2 mr (Zα)4 3 + − − 2n2 2n3 4n 4n(m + M ) j + 12   4 3 1 1 (Zα) mr − (3.4) (1 − δl0 ) . + 2n3 M 2 j + 12 l + 12

tot Enj = (m + M ) −

Note the emergence of the last term in (3.4) which lifts the characteristic degeneracy 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) already predicts a nonvanishing contribution to the classical Lamb shift E(2S 12 ) − 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 Lamb shift, induced by the QED radiative correction, is much larger. In the Breit Hamiltonian in (3.2) we have omitted all terms which depend on spin variables of the heavy particle. As a result the corrections to the energy levels in (3.4) do not depend on the relative orientation of the spins of the heavy and light particles (in other words they do not describe hyperfine splitting). Moreover, almost all contributions in (3.4) are independent not only of the mutual orientation of spins of the heavy and light particles but also of the magnitude of the spin of the heavy particle. The only exception is the small contribution proportional to the term δl0 , called the Darwin-Foldy contribution. This term arises in the matrix element of the Breit Hamiltonian only for the spin one-half nucleus and should be omitted for spinless or spin one nuclei. This contribution combines naturally with the nuclear size correction, and we postpone its discussion to Subsect. 6.1.2 dealing with the nuclear size contribution. In the framework of the effective Dirac equation in the external Coulomb field2 (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 m2r tot [f (n, j) − 1]2 Enj = (m + M ) + mr [f (n, j) − 1] − 2(m + M )   1 (Zα)4 m3r 1 + − (1 − δl0 ) . 2n3 M 2 j + 12 l + 12

(3.5)

This equation has the same contributions of order (Zα)4 as in (3.4), but formally this expression also contains nonrecoil and recoil corrections of order (Zα)6 and higher. The nonrecoil part of these contributions is definitely correct since the Dirac energy spectrum is the proper limit of the spectrum of a two-particle system in the nonrecoil limit m/M = 0. As we will discuss later the first-order mass ratio contributions in (3.5) correctly reproduce recoil corrections of higher orders in Zα generated by the Coulomb and Breit exchange photons. 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 a transverse contribution.

22

3 External Field Approximation

will be calculated below. Recoil corrections of order (Zα)8 were never calculated and at the present stage the mass dependence of these terms should be considered 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 exchange necessarily depend on the even powers of Zα. Hence, to calculate recoil corrections of order (Zα)5 one has to consider the nontrivial contribution of the 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 Lamb shift. In the early days of the Lamb shift studies, experimentalists measured not a shift but the classical Lamb splitting E(2S 12 ) − E(2P 12 ) between the energy levels which are degenerate according to the naive Dirac equation in the Coulomb field. This splitting is an experimental observable defined independently of any theory. Modern experiments now produce high precision experimental data for the nondegenerate 1S energy level, and the very notion of the Lamb shift in this case, as well as in the case of an arbitrary energy level, does not admit an unambiguous definition. It is most natural to call the Lamb shift the sum of all contributions to the energy levels which lift the double degeneracy of the Dirac-Coulomb spectrum with respect to l = j ± 1/2 (see Sect. 1.2). There emerged an almost universally adopted convention to call the Lamb shift the sum of all contributions to the energy levels beyond the first three terms in (3.5) and excluding, of course, all hyperfine splitting contributions. This means that we define the Lamb shift by the relationship tot = (m + M ) + mr [f (n, j) − 1] Enjl

−

m2r DR [f (n, j) − 1]2 + Lnjl ≡ Enj + Lnjl . 2(m + M )

(3.6)

We will adopt this definition below.

3.2 Radiative Corrections of Order αn(Zα)4 m Let us turn now to the discussion of radiative corrections which may be calculated in the external field approximation. 3.2.1 Leading Contribution to the Lamb Shift 3.2.1.1 Radiative Insertions in the Electron Line and the Dirac Form Factor Contribution The main contribution to the Lamb shift was first estimated in the nonrelativistic approximation by Bethe [8], and calculated by Kroll and Lamb [9], and by French and Weisskopf [10]. We have already discussed above qualitatively the nature of this contribution. In the effective Dirac equation framework the

3.2 Radiative Corrections of Order αn (Zα)4 m

23

Fig. 3.2. Kernels with many spanned Coulomb photons

apparent perturbation kernels to be taken into account are the diagrams with the radiative photon spanning any number of the exchanged Coulomb photons in Fig. 2.1 and Fig. 3.2. The dominant logarithmic contribution to the Lamb shift is produced by the slope of the Dirac form factor F1 , but superficially all these kernels can lead to corrections of order α(Zα)4 and one cannot discard any of them. An additional problem is connected with the infrared divergence of the kernels on the mass shell. There cannot be any true infrared divergence in the bound state problem since all would be infrared divergences are cut off either at the inverse Bohr radius or by the electron binding energy. Nevertheless such spurious on-shell infrared divergences can complicate the calculations. An important step which greatly facilitates treatment of all these problems consists in separation of the radiative photon integration region with the help of auxiliary parameter σ in such way that m(Zα)2  σ  m(Zα). It is easy to see that in the high momentum region each additional Coulomb photon produces an extra factor Zα, so it is sufficient to consider only the kernel with one Coulomb photon in this region. Moreover, the auxiliary parameter σ provides an infrared cutoff for the vertex graph and thus solves the problem of 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 factor F1 (k 2 ) − 1 which is proportional to the transferred momentum squared at small momentum transfer. This transferred momentum squared factor exactly cancels the Coulomb photon propagator attached to the Dirac form factor, and momentum space integrations over wave function momenta factorize, thereby producing the wave function squared at the origin in the coordinate space. It is clear that if one would take into account the small virtuality of either external electron line, expanding the integrand in this virtuality, it would lead to an extra factor of momentum squared in the integrand, and after integration with the wave function would lead to an extra factor (Zα)2 in the contribution to the energy shift. Hence, since we are interested in the contribution of order α(Zα)4 , we may freely ignore the virtuality of the electron line in the kernel in the high momentum region. It is also clear even at this stage, that the high momentum region does not produce any contribution for the non-S states because the wave function vanishes at the origin for such states, and, hence, the logarithmic contribution is missing for the non-S states.

24

3 External Field Approximation

Of course, all approximations made above are invalid in the low momentum integration region, where one cannot consider only the kernel with the radiative photon spanning only one Coulomb exchange. For soft radiative photons, with characteristic momenta of order m(Zα)2 , graphs with any number of spanned exchanged photons in Fig. 3.2 have the same order of magnitude and one has to take these graphs into account simultaneously. This means that one has to calculate the matrix element of the exact self-energy operator in the external Coulomb field between Dirac-Coulomb wave functions. This problem may seem formidable at first sight, but it can be readily solved with the help of old-fashioned perturbation theory by inserting a complete set of intermediate states and performing calculations in the dipole approximation, which is adequate to accuracy α(Zα)4 m. It should be mentioned that the magnitude of the upper boundary of the interval for the auxiliary parameter σ was chosen in order to provide validity of the dipole approximation. The most important fact about the auxiliary parameter σ is that one can use different approximations for calculations of the high- and low-momentum contributions. 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 effects as small corrections. In the low-momentum region k/(mZα) ≤ σ/(mZα)  1 one may use the nonrelativistic multipole expansion, and the main contribution in this region is given by the dipole contribution. The crucial point is that for k ∼ σ both expansions are valid simultaneously and one can match them without loss in accuracy. Matching the high- and low-momentum contributions one obtains the classical result for the shift of the energy level generated by the slope of the Dirac form factor    4α(Zα)4  mr 3 1 m(Zα)−2 11 1 ln + m, δl0 − ln k0 (n, l) ∆E = 3 mr 72 3 πn3 m (3.7) where mr = mM/(m + M ) is the reduced mass and ln k0 (n, l) is the so called Bethe logarithm. The factor m/mr arises in the argument of the would be infrared divergent logarithm ln(m/λ) since in the nonrelativistic approximation the energy levels of an atom depend only on the reduced mass and, hence, the infrared 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 reduced mass factor is properly described by the expression in (3.7) as was proved in [11, 12]. In the same way as for the case of the leading relativistic correction 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 earlier papers the reduced mass factors in (3.7) were expanded up to first order in the small mass ratio m/M . Nowadays it is important to preserve an exact mass dependence in (3.7) because current experiments may be able to detect quadratic mass corrections (about 2 kHz for the 1S level in hydrogen) to the leading nonrecoil Lamb shift contribution.

3.2 Radiative Corrections of Order αn (Zα)4 m

25

The Bethe logarithm is formally defined as a certain normalized infinite sum of matrix elements of the coordinate operator over the Schr¨odingerCoulomb wave functions. It is a pure number which can in principle be calculated with arbitrary accuracy, and high accuracy results for the Bethe logarithm 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 Lamb shift. This form factor does not produce any contribution if one neglects the lower components of the unperturbed wave functions, since the respective matrix element is identically zero between the upper components in the standard representation for the Dirac matrices which we use everywhere. Taking into account lower components in the nonrelativistic approximation we easily obtain an explicit expression for the respective perturbation   m 1 [∇V × p] F2 (0) , (3.8) δV = − 2 ∆V + 2 σ · 4m mr where V = −Zα/r is the Coulomb potential. Note the appearance of an extra factor m/mr in the coefficient before the second term. This is readily obtained from an explicit consideration of the radiatively corrected one photon exchange. In momentum space the term with the Laplacian of the Coulomb potential depends only on the exchanged momentum, while the second term contains explicit dependence on the electron momentum. Since the Pauli form factor depends explicitly on the electron momentum and not on the relative momentum of the electron-proton system, the transition to relative 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 may be written in terms of the spin-orbit interaction   Zαπ 3 Zαπ δ (r) + 3 (s · l) F2 (0) , (3.9) δV = m2 r mmr where

σ , l=r×p. 2 The respective contributions to the Lamb shift are given by  m 3 (Zα)4 m α(Zα)4 m  mr 3 r ∆E|l=0 = F2 (0) = , 3 n m 2πn3 m (Zα)4 m j(j + 1) − l(l + 1) − 3/4  mr 2 ∆E|l=0 = F (0) 2 n3 l(l + 1)(2l + 1) m α(Zα)4 m j(j + 1) − l(l + 1) − 3/4  mr 2 = , 2πn3 l(l + 1)(2l + 1) m s=

(3.10)

(3.11)

26

3 External Field Approximation Table 3.1. Bethe Logarithms for Lower Levels [14] 4

ln k0 (n, l)

r 3 ∆E = − 43 ln k0 (n, l) α(Zα) (m ) m kHz m πn3

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

3.2 Radiative Corrections of Order αn (Zα)4 m

27

where we have used the Schwinger value [15] of the anomalous magnetic moment F2 (0) = α/(2π). Correct reduced mass factors have been retained in this expression instead of expanding in m/M . 3.2.1.3 Polarization Operator Contribution The leading polarization operator contribution to the Lamb shift in Fig. 2.2 was already calculated above in (2.6). Restoring the reduced mass factors which were omitted in that qualitative discussion, we easily obtain ∆E = 4π(Zα)|Ψn (0)|2

Π(−k2 ) 4α(Zα)4 m  mr 3 = − δl0 . 15πn3 m k4 |k2 =0

(3.12)

This result was originally obtained in [16] long before the advent of modern QED, and was the only known source for the 2S − 2P splitting. There is a certain historic irony that for many years it was the common wisdom “that this effect is . . . much too small and has, in addition, the wrong sign” [8] to explain the 2S − 2P splitting. 3.2.2 Radiative Corrections of Order α2 (Zα)4 m From the theoretical point of view, calculation of the corrections of order α2 (Zα)4 contains nothing fundamentally new in comparison with the corrections of order α(Zα)4 . The scale for these corrections is provided by the factor 4α2 (Zα)4 /(π 2 n3 )m, as one may easily see from the respective discussion above in Sect. 2.3. Corrections depend only on the values of the form factors and their derivatives at zero transferred momentum and the only challenge is to calculate 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 radiative photon insertions in the electron line is even simpler than the respective calculation of the leading order contribution. The point is that the second and higher order contributions to the slope of the Dirac form factor are infrared finite, and hence, the total contribution of order (Zα)4 to the Lamb shift is given by the slope of the Dirac form factor. Hence, there is no need to sum an infinite number of diagrams. One readily obtains for the respective contribution (2)

dF1 (−k2 ) dk2 |k2 =0 2 4  4α (Zα) mr 3 = 0.469 941 4 . . . m δl0 , π 2 n3 m

∆EF1 = −4π(Zα)|Ψn (0)|2

(3.13)

28

3 External Field Approximation

where we have used the second order slope of the Dirac form factor generated by the diagrams in Fig. 3.3   (2) 3 dF1 (−k2 ) π2 49 2 4 819 1  α 2 ζ(3) − ln 2 + π = + 4 2 432 5 184 m2 π dk2 |k2 =0 0.469 941 4 . . .  α 2 ≈− . (3.14) m2 π The two-loop slope was considered in the early pioneer works [17, 18], and for the first time the correct result was obtained numerically in [19]. This last work triggered a flurry of theoretical activity [20, 21, 22, 23], followed by the first completely analytical calculation in [24]. The same analytical result for the slope of the Dirac form factor was derived in [25] from the total e+ e− cross section and the unitarity condition. 3.2.2.2 Pauli Form Factor Contribution Calculation of the Pauli form factor contribution follows closely the one which was performed in order α(Zα)4 , the only difference being that we have to employ 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)    α 2 3 π2 π2 197  α 2 (2) ζ(3) − ln 2 + + ≈ −0.328 478 9 . . . . F2 (0) = 4 2 12 144 π π (3.15) Then one readily obtains for the Lamb shift contribution

Fig. 3.3. Two-loop electron formfactor

3.2 Radiative Corrections of Order αn (Zα)4 m

29

Fig. 3.4. Insertions of two-loop polarization operator

α2 (Zα)4 m  mr 3 , π 2 n3 m α2 (Zα)4 m j(j + 1) − l(l + 1) − 3/4  mr 2 = −0.328 478 9 . . . . π 2 n3 l(l + 1)(2l + 1) m (3.16)

∆E|l=0 = −0.328 478 9 . . . ∆E|l=0

3.2.2.3 Polarization Operator Contribution Here we use well known low momentum asymptotics of the second order polarization operator [29, 30, 31] in Fig. 3.4 Π(−k2 ) 41  α 2 = − , 162m2 π k4 |k2 =0 and obtain [29] ∆E = −

82 α2 (Zα)4 m  mr 3 δl0 . 81 π 2 n3 m

(3.17)

(3.18)

3.2.3 Corrections of Order α3 (Zα)4 m 3.2.3.1 Dirac Form Factor Contribution Calculation of the corrections of order α3 (Zα)4 is similar to calculation of the contributions of order α2 (Zα)4 . Respective corrections depend only on the values of the three-loop form factors or their derivatives at vanishing transferred momentum. The three-loop contribution to the slope of the Dirac form factor (Fig. 3.5) was calculated analytically [32]  (2) 25 17 217 2929 217 4 dF1 (−k2 ) ζ(5) − π 2 ζ(3) − ζ(3) − a4 − ln 2 = − 2 8 24 288 9 216 dk |k2 =0 103 2 2 41671 2 3899 4 − π ln 2 + π ln 2 + π 1080 2160 25920  3  454979 2 77513 1 α π − − 38880 186624 m2 π 0.171 72 . . .  α 3 ≈− , (3.19) m2 π

30

3 External Field Approximation

where a4 =

∞ 

1 . n n4 2 n=1

(3.20)

The respective contribution to the Lamb shift is equal to ∆EF1 = 0.171 72 . . .

4α3 (Zα)4  mr 3 m δl0 . π 3 n3 m

(3.21)

3.2.3.2 Pauli Form Factor Contribution For calculation of the Pauli form factor contribution to the Lamb shift the third order contribution to the Pauli form factor (Fig. 3.5), calculated numerically in [33], and analytically in [34] is used    83 2 100 215 1 4 1 (3) π ζ(3) − ζ(5) + ln 2) − π 2 ln2 2 F2 (0) = (a4 + 72 24 3 24 24  239 4 139 298 2 17 101 2 28 259  α 3 − π + ζ(3) − π ln 2 + π + 2 160 18 9 810 5 184 π  α 3 ≈ 1.181 241 456 . . . . (3.22) π Then one obtains for the Lamb shift α3 (Zα)4 m  mr 3 , π 3 n3 m 3 4 α (Zα) m j(j + 1) − l(l + 1) − 3/4  mr 2 = 1.181 241 456 . . . . π 3 n3 l(l + 1)(2l + 1) m (3.23)

∆E|l=0 = 1.181 241 456 . . . ∆E|l=0

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

3.2.3.3 Polarization Operator Contribution In this case the analytic result for the low frequency asymptotics of the third order polarization operator (see Fig. 3.6) [35] is used

3.2 Radiative Corrections of Order αn (Zα)4 m

31

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

Π(−k2 ) =− k4 |k2 =0



8 135 π 2 ln 2 23π 2 325 805 ζ(3) − + − 9 216 15 360 373 248 0.362 654 440 . . .  α 3 ≈− , m2 π



1  α 3 m2 π (3.24)

and one obtains [36] ∆E = −1.450 617 763 . . .

α3 (Zα)4 m  mr 3 δl0 . π 3 n3 m

(3.25)

3.2.4 Total Correction of Order αn (Zα)4 m The total contribution of order αn (Zα)4 m is given by the sum of corrections in (3.7), (3.11), (3.12), (3.13), (3.16), (3.18), (3.21), (3.23), (3.25). It is equal to   4 m(Zα)−2 4 38 ln − ln k0 (n, 0) + ∆E|l=0 = 3 mr 3 45   3 10 2 179  α  9 + − ζ(3) + π 2 ln 2 − π 2 − 4 2 27 648 π  85 121 2 71 4 84 071 239 2 2 ζ(5) − π ζ(3) − ζ(3) − ln 2 − π ln 2 + 24 72 2 304 27 135   568 1591 4 252 251 2 679 441  α 2 4 787 2 π ln 2 − a4 + π − π + + 108 9 3 240 9 720 93 312 π   4 3 α(Zα) m mr × 3 m  πn  α 4 m(Zα)−2 4 38 ln = − ln k0 (n, 0) + + 0.538 95 . . . 3 mr 3 45 π  α 2  α(Zα)4 m  m 3 r , (3.26) + 0.417 504 . . . π πn3 m

32

3 External Field Approximation

for the S-states, and    m 3  1  3 π2 π2 197  α  4 r + ζ(3) − ln 2 + + ∆E|l=0 = − ln k0 (n, l) + 3 m 2 4 2 12 144 π  83 2 100 215 25 4 25 2 2 239 4 π ζ(3)− ζ(5) + a4 + ln 2 − π ln 2 − π + 72 24 3 18 18 2 160    298 2 α 2 139 17 101 2 28 259 ζ(3) − π ln 2 + π + + 18 9 810 5 184 π  j(j + 1) − l(l + 1) − 3/4  mr 2 α(Zα)4 m × l(l + 1)(2l + 1) m πn3    α  1 4 mr 3 − 0.328 478 9 . . . = − ln k0 (n, l) + 3 m 2 π  α 2  + 1.181 241 456 . . . π  j(j + 1) − l(l + 1) − 3/4  mr 2 α(Zα)4 m (3.27) × l(l + 1)(2l + 1) m πn3 for the non-S-states. Numerically corrections of order αn (Zα)4 m 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)4 m are suppressed by an extra factor α/π in comparison with the corrections of order α3 (Zα)4 m. Their expected magnitude 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 Contributions of Order α(Zα)4 m We have considered above only radiative corrections containing virtual photons and electrons. However, at the current level of accuracy one has to consider also effects induced by the virtual muons and lightest strongly interacting particles. The respective corrections to the electron anomalous magnetic moment are well known [33] and are still too small to be of any practical interest for the Lamb shift calculations. Heavy particle contributions to the polarization operator numerically have the same magnitude as polarization corrections of order α3 (Zα)4 . Corrections to the low-frequency asymptotics of the polarization operator are generated by the diagrams in Fig. 3.7. The muon loop contribution to the polarization operator

3.2 Radiative Corrections of Order αn (Zα)4 m

33

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

Π(−k2 ) α =− 4 2 15πm2µ k |k =0

(3.29)

immediately leads (compare (3.12)) to an additional contribution to the Lamb shift [37, 38] 2  α(Zα)4 4 mr ∆E = − mr δl0 . (3.30) 15 mµ πn3 The hadronic polarization contribution to the Lamb shift was discussed in a number of papers [37, 38, 39]. The light hadron contribution to the polarization operator may easily be estimated with the help of vector dominance  4πα Π(−k2 ) =− , 4 fv2i m2vi k |k2 =0 i

(3.31)

where mvi are 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 hadronic vacuum polarization contribution to the Lamb shift in the form [38]   α(Zα)4 1 4π 2 2 m δl0 . (3.32) ∆E = −4 Σvi 2 2 + fvi mvi 3 1 GeV2 πn3 Numerically this correction is −3.18 kHz for the 1S-state and −0.40 kHz for the 2S-state in hydrogen. Only the value of the leading coefficient in the low energy expansion of the hadronic vacuum polarization is needed for calculation of the hadronic contribution to the Lamb shift (see the LHS of (3.32)). A model independent value of this coefficient may be obtained for the analysis of the experimental data on the low energy e+ e− annihilation. Respective contribution to the 1S Lamb shift [39] is −3.40(7) kHz. This value is compatible but more accurate than 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 be included in the phenomenological analysis of the Lamb shift measurements, since experimentally it is indistinguishable from an additional contribution to the proton charge radius. We will return to this problem below in Sect. 6.1.3.

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

Bethe (1947) [8]



1 3

ln

m(Zα)−2 mr

+

11 72



δl0 −

1 3

kHz ∆E(1S) kHz

∆E(2S) kHz

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

Kroll, Lamb (1949) [9] One-loop Pauli FF l = 0

1 8

One-loop Pauli FF l = 0

j(j+1)−l(l+1)−3/4 m 8l(l+1)(2l+1) mr

One-loop Vacuum Polarization Uehling (1935) [16] Two-loop Dirac FF Appelquist, Brodsky (1970) [19] Barbieri, Mignaco, Remiddi (1971) [24] Two-loop Pauli FF l = 0 Sommerfield (1957) [28] Peterman (1957) [27] Pauli FF l = 0 Sommerfield (1957) [28] Peterman (1957) [27] Two-loop Vacuum Polarization Baranger, Dyson, Salpeter (1952) [29]

1 − 15 δl0

[− 34 ζ(3) +

π2 2

ln 2 −

49 2 π 432

−

π2 8

ln 2 +

π2 48

+

−216 675.85

−27 084.48

3 547.82

443.48

−619.96

−77.50

−1 910.67

−238.83

197 α ) 576 π

≈ −0.082 119 7 . . . 2

50 783.40

4 819 α ] δ 5 184 π l0

≈ 0.469 941 4 . . . α δ π l0 3 ( 16 ζ(3) −

406 267.21

α π

2

3 ( 16 ζ(3) − π8 ln 2 + π48 + 197 ) 576 α m × j(j+1)−l(l+1)−3/4 l(l+1)(2l+1) π mr m α ≈ −0.082 119 7 . . . j(j+1)−l(l+1)−3/4 l(l+1)(2l+1) mr π

41 α − 162 ( π )δl0

3 External Field Approximation

French, Weisskopf (1949) [10]

3 250 137.65(4) n3

34

4

r 3 4 α(Zα) (m ) m≈ m πn3

Table 3.2. (Continued) Three-loop Dirac FF Melnikov, van Ritbergen (1999) [32]

Three-loop Vacuum Polarization Baikov, Broadhurst (1995) [35] Eides, Grotch (1995) [36] Muonic Polarization Karshenboim (1995) [37] Eides, Shelyuto (1995) [38] Hadronic Polarization Friar, Martorell, Sprung (1999) [39]

3.01

0.38

5.18

0.65

805 −( 89 135 ζ(3) − π 15ln 2 + 23π − 325 ) 360  α216 2 373  248 α 2 × π δl0 ≈ −0.362 654 4 . . . π δl0

−6.36

−0.79

1 − 15 ( mmµ )2

−5.07

−0.63

−3.40

−0.40

83 1 { 288 π ζ(3) − 215 ζ(5) + 25 [(a4 + 24 ln 2) 96 3 4 1 2 139 − 24 π ln2 2] − 8239 π + ζ(3) 640 72 2 17 101 2 28 259 α 2 − 149 π ln 2 + π + }( ) 18 3 240 20 736 π ≈ 0.295 310 3 . . . ( α )2 π 4 83 2 25 1 { 288 π ζ(3) − 215 ζ(5) + [(a + ln 2) 4 96 3 24 1 2 − 24 π ln2 2] − 8239 π 4 + 139 ζ(3) − 149 π 2 ln 2 640 72 18 17 101 2 28 259 j(j+1)−l(l+1)−3/4 m α 2 + 3 240 π + 20 736 } l(l+1)(2l+1) ( ) mr π j(j+1)−l(l+1)−3/4 m α 2 ≈ 0.295 310 3 . . . ( ) l(l+1)(2l+1) mr π 2

4

2

2

3.2 Radiative Corrections of Order αn (Zα)4 m

Three-loop Pauli FF l = 0 Kinoshita (1990) [33] Laporta, Remiddi (1996) [34] Three-loop l = 0 Pauli FF Kinoshita (1990) [33] Laporta, Remiddi (1996) [34]

[ 25 ζ(5) − 17 π 2 ζ(3) − 2929 ζ(3) − 217 a4 8 24 288 9 4 2 2 217 103 2 − 216 ln 2 − 1080 π ln 2 + 41671 π ln 2 2160 3899 4 77513 + 25920 π − 454979 π 2 − 186624 ]( α )2 δl0 38880 π ≈ 0.171 72 . . .( α )2 δl0 π

35

36

3 External Field Approximation

3.3 Radiative Corrections of Order αn(Zα)5 m 3.3.1 Skeleton Integral Approach to Calculations of Radiative Corrections We have seen above that calculation of the corrections of order αn (Zα)4 m (n > 1) reduces to calculation of higher order corrections to the properties of a free electron and to the photon propagator, namely to calculation of the slope of the electron Dirac form factor and anomalous magnetic moment, and to calculation of the leading term in the low-frequency expansion of the polarization operator. Hence, these contributions to the Lamb shift are independent of any features of the bound state. A nontrivial interplay between radiative corrections and binding effects arises first in calculation of contributions of order α(Zα)5 m, and in calculations of higher order terms in the combined expansion over α and Zα. Calculation of the contribution of order αn (Zα)5 m to the energy shift is even simpler than calculation of the leading order contribution to the Lamb shift because the scattering approximation is sufficient in this case [40, 41, 42]. Formally this correction is induced by kernels with at least two-photon exchanges, and in analogy with the leading order contribution one could also anticipate the appearance of irreducible kernels with higher number of exchanges. This does not happen, however, as can be proved formally, but in fact no formal proof is needed. First one has to realize that for high exchanged momenta expansion in Zα is valid, and addition of any extra exchanged photon always produces an extra power of Zα. Hence, in the high-momentum region only diagrams with two exchanged photons are relevant. Treatment of the low-momentum region is greatly facilitated by a very general feature of the Feynman diagrams, namely that the infrared behavior of any radiatively corrected Feynman diagram (or more accurately any gauge invariant sum of Feynman diagrams) is milder than the behavior of the skeleton diagram. Consider the matrix element in momentum space of the diagram in Fig. 3.8 with two exchanged Coulomb photons between the Schr¨ odinger-Coulomb wave functions. We will take the external electron momenta to be on-shell and to have vanishing space components. It is then easy to see that the contribution of such a diagram to the Lamb shift is given by the infrared divergent integral  ∞ dk 16(Zα)5  mr 3 m δl0 , (3.33) − 3 πn m k4 0

Fig. 3.8. Skeleton diagram with two exchanged Coulomb photons

3.3 Radiative Corrections of Order αn (Zα)5 m

37

Fig. 3.9. Radiative insertions in the electron line

where k is the dimensionless momentum of the exchanged photon measured in the units of the electron mass. This divergence has a simple physical interpretation. If we do not ignore small virtualities of the external electron lines and the external wave functions this two-Coulomb exchange adds one extra rung to the Coulomb wave function and should simply reproduce it. The naive infrared divergence above would be regularized at the characteristic atomic scale mZα. Hence, it is evident that the kernel with two-photon exchange is already taken into account in the effective Dirac equation above and there is no need to try to consider it as a perturbation. Let us consider now radiative photon insertions in the electron line (see Fig. 3.9). Account of these corrections effectively leads to insertion of an additional factor L(k) in the divergent integral above, and while this factor has at most a logarithmic asymptotic behavior at large momenta and does not spoil the ultraviolet convergence of the integral, in the low momentum region it behaves as L(k) ∼ k 2 (again up to logarithmic factors), and improves the low frequency behavior of the integrand. However, the integrand is still divergent even after inclusion of the radiative corrections because the two-photon-exchange box diagram, even with radiative corrections, contains a contribution of the previous order in Zα, namely the main contribution to the Lamb shift induced by the electron form factor. This spurious contribution may be easily removed by subtracting the leading low momentum term from L(k)/k 4 . The result of the subtraction is a convergent integral which is responsible for the correction of order α(Zα)5 . As an additional bonus of this approach one does not need to worry about the ultraviolet divergence of the one-loop radiative corrections. The subtraction automatically eliminates any ultraviolet divergent terms and the result is both ultraviolet and infrared finite. Due to radiative insertions low integration momenta (of atomic order mZα) are suppressed in the exchange loops and the effective integration momenta are of order m. Hence, one may neglect the small virtuality of external fermion lines and calculate the above diagrams with on-mass-shell external momenta. Contributions to the Lamb shift are given by the product of the square of the Schr¨ odinger-Coulomb wave function at the origin |ψ(0)|2 and the diagram. Under these conditions the diagrams in Fig. 3.9 comprise a gauge invariant set and may easily be calculated. Contributions of the diagrams with more than two exchanged Coulomb photons are of higher order in Zα. This is obvious for the high exchanged momenta integration region. It is not difficult to demonstrate that in the Yennie gauge [43, 44, 45] contributions from the low exchanged momentum

38

3 External Field Approximation

region to the matrix element with the on-shell external electron lines remain infrared 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 gauge invariant this is true in any gauge. It is also clear that small virtuality of the external electron lines would lead to an additional suppression of the matrix element under consideration, and, hence, it is sufficient to consider only twophoton exchanges for calculation of all corrections of order α(Zα)5 . The magnitude of the correction of order α(Zα)5 may be easily estimated before the calculation is carried out. We need to take into account the skeleton factor 4m(Zα)4 /n3 discussed above in Sect. 2.3, and multiply it by an extra factor 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 photons always produces an extra factor π in the numerator, thus compensating the factor π in the denominator generated by the radiative correction. Hence, calculation of the correction of order α(Zα)5 should lead to a numerical factor of order unity multiplied by 4mα(Zα)5 /n3 . 3.3.2 Radiative Corrections of Order α(Zα)5 m 3.3.2.1 Correction Induced by the Radiative Insertions in the Electron Line This correction is generated by the sum of all possible radiative insertions in the electron line in Fig. 3.9. In the approach described above one has to calculate the electron factor corresponding to the sum of all radiative corrections in the electron line, make the necessary subtraction of the leading infrared asymptote, insert the subtracted expression in the integrand in (3.33), and then integrate over the exchanged momentum. This leads to the result   α(Zα)5  mr 3 1 11 − ln 2 ∆E = 4 1 + m δl0 , (3.34) 128 2 n3 m which was first obtained in [40, 41, 42] in other approaches. Note that numerically 1 + 11/128 − 1/2 ln 2 ≈ 0.739 in excellent agreement with the qualitative considerations above. 3.3.2.2 Correction Induced by the Polarization Insertions in the External Photons The correction of order α(Zα)5 induced by the polarization operator insertions in the external photon lines in Fig. 3.10 was obtained in [40, 41, 42] and may again be calculated in the skeleton integral approach. We will use the simplicity of the one-loop polarization operator, and perform this calculation in more detail in order to illustrate the general considerations above. For calculation of the respective contribution one has to insert the polarization operator in the skeleton integrand in (3.33)

3.3 Radiative Corrections of Order αn (Zα)5 m

39

Fig. 3.10. Polarization insertions in the Coulomb lines

1 α → I1 (k) , 2 k π where

 I1 (k) =

1

dv 0

v 2 (1 − v 2 /3) . 4 + (1 − v 2 )k 2

(3.35)

(3.36)

Of course, the skeleton integral still diverges in the infrared after this substitution since 1 . (3.37) I1 (0) = 15 This linear infrared divergence dk/k 2 is effectively cut off at the characteristic atomic scale mZα, it lowers the power of the factor Zα, respective would be divergent contribution turns out to be of order α(Zα)4 , and corresponds to the polarization part of the leading order contribution to the Lamb shift. We carry out the subtraction of the leading low frequency asymptote of the polarization operator insertion, which corresponds to the subtraction of the leading low frequency asymptote in the integrand for the contribution to the energy shift  k 2 1 v 2 (1 − v 2 )(1 − v 2 /3) ˜ dv , (3.38) I1 (k) ≡ I1 (k) − I1 (0) = − 4 0 4 + (1 − v 2 )k 2 and substitute the subtracted expression in the formula for the Lamb shift in (3.33). We also insert an additional factor 2 in order to take into account possible insertions of the polarization operator in both photon lines. Then  ∞  m 3 α(Zα)5 I˜1 (k) 32 r ∆E = − m dk δl0 2 m m π 2 n3 (1 − M k2 0 2)  1  ∞  m 3 α(Zα)5 8 v 2 (1 − v 2 )(1 − v 2 /3) r = m dv dk δl0 2 2 3 m m π n (1 − M 2 ) 0 4 + (1 − v 2 )k 2 0 5 α(Zα)5  mr 3 = m δl0 . (3.39) 48 n3 m We have restored in (3.39) the characteristic factor 1/(1 − m2 /M 2 ) which was 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 this factor is only about 0.02 kHz even for the electron-line contribution to the 1S level shift, and, hence, this correction may be safely omitted at the present level of experimental accuracy.

40

3 External Field Approximation

3.3.2.3 Total Correction of Order α(Zα)5 m The total correction of order α(Zα)5 m is given by the sum of contributions in (3.34), (3.39)   α(Zα)5  mr 3 5 1 11 + − ln 2 ∆E = 4 1 + m δl0 128 192 2 n3 m α(Zα)5  mr 3 = 3.061 622 . . . m δl0 n3 m = 57 030.70 kHz|n=1 , = 7 128.84 kHz|n=2 .

(3.40)

3.3.3 Corrections of Order α2 (Zα)5 m Corrections of order α2 (Zα)5 have the same physical origin as corrections of order α(Zα)5 , and the scattering approximation is sufficient for their calculation [46]. We consider now corrections of higher order in α than in the previous section and there is a larger variety of relevant graphs. All six gauge invariant sets of diagrams [46] which produce corrections of order α2 (Zα)5 are presented in Fig. 3.11. The blob called “2 loops” in Fig. 3.11 (f ) means the gauge invariant sum of diagrams with all possible insertions of two radiative photons in the electron line. All diagrams in Fig. 3.11 may be obtained from the skeleton diagram in Fig. 3.8 with the help of different two-loop radiative insertions. As in the case of the corrections of order α(Zα)5 , corrections to the energy shifts are given by the matrix elements of the diagrams in Fig. 3.11 calculated between free electron spinors with all external electron lines on the mass shell, projected on the respective spin states, and multiplied by the square of the Schr¨ odinger-Coulomb wave function at the origin [46]. It should be mentioned that some of the diagrams under consideration contain contributions of the previous order in Zα. These contributions are produced by the terms proportional to the exchanged momentum squared in the low-frequency asymptotic expansion of the radiative corrections, and are connected with integration over external photon momenta of characteristic atomic scale mZα. The scattering approximation is inadequate for their calculation. In the skeleton integral approach these previous order contributions arise as powerlike infrared divergences in the final integration over the exchanged momentum. We subtract leading low-frequency terms in the lowfrequency asymptotic expansions of the integrands, when necessary, and thus remove 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 two insertions of the one-loop vacuum polarization in the external photon lines.

3.3 Radiative Corrections of Order αn (Zα)5 m

41

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

The contribution to the Lamb shift is given by the insertion of the one-loop polarization operator squared I12 (k) in the skeleton integral in (3.33), and taking into account the multiplicity factor 3 one easily obtains [46, 47, 48]  ∞ 48α2 (Zα)5  mr 3 23 α2 (Zα)5  mr 3 ∆E = − m dkI12 (k)δl0 = − mδl0 . 3 3 π n m 378 πn3 m 0 (3.41) 3.3.3.2 Insertions of the Irreducible Two-Loop Polarization in the Coulomb Lines The naive insertion 1/k 2 → I2 (k) of the irreducible two-loop vacuum polarization operator I2 (k) [30, 31] in the skeleton integral in (3.33) would lead to an infrared divergent integral for the diagrams in Fig. 3.11 (b). This divergence reflects the existence of the correction of the previous order in Zα connected with the two-loop irreducible polarization. This contribution of order α2 (Zα)4 m was discussed in Subsect. 3.2.2.3, and as we have seen the respective contribution to the Lamb shift is given simply by the product of the Schr¨ odinger-Coulomb wave function squared at the origin and the leading

42

3 External Field Approximation

low-frequency term of the function I2 (0). In terms of the loop momentum integration this means that the relevant loop momenta are of the atomic scale mZα. Subtraction of the value I2 (0) from the function I2 (k) effectively removes the previous order contribution (the low momentum region) from the loop integral and one obtains the radiative correction of order α2 (Zα)5 m generated by the irreducible two-loop polarization operator [46, 47, 48]  ∞ dk 32α2 (Zα)5  mr 3 m [I2 (k) − I2 (0)] δl0 ∆E = − π 3 n3 m k2 0   2 52 25 15 647 α (Zα)5  mr 3 ln 2 − π + = m δl0 . (3.42) 63 63 13 230 πn3 m 3.3.3.3 Insertion of One-Loop Electron Factor in the Electron Line and of the One-Loop Polarization in the Coulomb Lines The next correction of order α2 (Zα)5 is generated by the gauge invariant set of diagrams in Fig. 3.11 (c). The respective analytic expression is obtained from the skeleton integral by simultaneous insertion in the integrand of the one-loop polarization function I1 (k) and of the expressions corresponding to all possible insertions of the radiative photon in the electron line. It is simpler first to obtain an explicit analytic expression for the sum of all these radiative insertions 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 integral. It is easy to check explicitly that the resulting integral for the radiative correction is both ultraviolet and infrared finite. The infrared finiteness nicely correlates with the physical understanding that for these diagrams there is no correction of order α2 (Zα)4 generated at the atomic scale. The respective integral for the radiative correction was calculated both numerically and analytically [49, 47, 51], and the result has the following elegant form  ∞ 32α2 (Zα)5  mr 3 m dkL(k)I1 (k) δl0 ∆E = − π 3 n3 m 0   √ √ 2π 2 67 282 8 2 1 + 5 872 √ 1 + 5 628 ln − + ln 2 − + = 5 ln 3 2 63 2 63 9 6 615 5 α2 (Zα)  mr 3 × m δl0 . (3.43) πn3 m 3.3.3.4 One-Loop Polarization Insertions in 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. The respective radiatively corrected electron factor is given by the expression [50]

3.3 Radiative Corrections of Order αn (Zα)5 m

 L(k) = 0

1

43

2

v 2 (1 − v3 ) dv L(k, λ) , 1 − v2

(3.44)

where L(k, λ) is just the one-loop electron factor used in (3.43) but with a finite photon mass λ2 = 4/(1 − v 2 ). Direct substitution of the radiatively corrected electron factor L(k) in the skeleton integral in (3.33) would lead to an infrared divergence. This divergence reflects existence in this case of the correction of the previous order in Zα generated by the two-loop insertions in the electron line. The magnitude of this previous order correction is determined by the nonvanishing value of the electron factor L(k) at zero 1 L(0) = −2F1 (0) − F2 (0) , 2

(3.45)

which is simply a linear combination of the slope of the two-loop Dirac form factor and the two-loop contribution to the electron anomalous magnetic moment. Subtraction of the radiatively corrected electron factor removes this previous order contribution which was already considered above, and leads to a finite integral for the correction of order α2 (Zα)5 [50, 47]  ∞ 16α2 (Zα)5  mr 3 L(k) − L(0) ∆E = − m dk δl0 (3.46) π 3 n3 m k2 0 = −0.072 90 . . .

5 α2 (Zα)  mr 3 m δl0 . πn3 m

3.3.3.5 Light by Light Scattering Insertions in the External Photons The diagrams in Fig. 3.11 (e) with the light by light scattering insertions in the external photons do not generate corrections of the previous order in Zα. They are both ultraviolet and infrared finite and respective calculations are in principle quite straightforward though technically involved. Only numerical results were obtained for the contributions to the Lamb shift [47, 52] ∆E = −0.122 9 . . .

5 α2 (Zα)  mr 3 m δl0 . πn3 m

(3.47)

3.3.3.6 Diagrams with Insertions of Two Radiative Photons in the Electron Line As we have already seen, contributions of the diagrams with radiative insertions in the electron line always dominate over the contributions of the diagrams with radiative insertions in the external photon lines. This property

44

3 External Field Approximation

of the diagrams is due to the gauge invariance of QED. The diagrams (radiative insertions) with the external photon lines should be gauge invariant, and as a result transverse projectors correspond to each external photon. These projectors are rational functions of external momenta, and they additionally suppress low momentum integration regions in the integrals for energy shifts. Respective projectors are of course missing in the diagrams with insertions in the electron line. The low momentum integration region is less suppressed in such diagrams, and hence they generate larger contributions to the energy shifts. This general property of radiative corrections clearly manifests itself in the case of six gauge invariant sets of diagrams in Fig. 3.11. By far the largest contribution of order α2 (Zα)5 to the Lamb shift is generated by the last gauge invariant set of diagrams in Fig. 3.11 (f), which consists of nineteen topologically different diagrams [53] presented in Fig. 3.12. These nineteen graphs may be obtained from the three graphs for the two-loop electron selfenergy by insertion of two external photons in all possible ways. Graphs in Fig. 3.12 (a–c) are obtained from the two-loop reducible electron self-energy diagram, graphs in Fig. 3.12 (d − k) are the result of all possible insertions of two external photons in the rainbow self-energy diagram, and diagrams in Fig. 3.12 (l–s) are connected with the overlapping two-loop self-energy graph. Calculation of the respective energy shift was initiated in [53, 54], where contributions 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 was first calculated in [55]. In the framework of the skeleton integral approach the calculation was completed in [56, 38] with the result ∆E = −7.725(1) . . .

5 α2 (Zα)  mr 3 m δl0 , 3 πn m

(3.48)

which confirmed the one in [55] but is about two orders of magnitude more precise than the result in [55, 57]. A few comments are due on the magnitude of this important result. It is sometimes claimed in the literature that it has an unexpectedly large magnitude. A brief glance at Table 3.3 is sufficient to convince oneself that this is not the case. For the reader who followed closely the discussion of the scales of different contributions above, it should be clear that the natural scale for the correction under discussion is set by the factor 4α2 (Zα)5 /(πn3 )m. The coefficient before this factor obtained in (3.48) is about −1.9 and there is nothing unusual in its magnitude for a numerical factor corresponding to a radiative correction. It should be compared with the respective coefficient 0.739 before the factor 4α(Zα)5 /n3 m in the case of the electron-line contribution of the previous order in α. The misunderstanding about the magnitude of the correction of order α2 (Zα)5 m has its roots in the idea that the expansion of energy in a series over the parameter Zα at fixed power of α should have coefficients of order one. As is clear from the numerous discussions above, however natural such

3.3 Radiative Corrections of Order αn (Zα)5 m

45

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

expansion might seem from the point of view of calculations performed without expansion over Zα, there are no real reasons to expect that the coefficients would be of the same order of magnitude in an expansion of this kind. We have already seen that quite different physics is connected with the different terms in expansion over Zα. The terms of order αn (Zα)4 (and αn (Zα)6 , as we will see below) are generated at large distances (exchanged momenta of order of the atomic scale mZα) while terms of order αn (Zα)5 originate from the small 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

46

3 External Field Approximation Table 3.3. Radiative Corrections of Order αn (Zα)5 m 4

One-Loop Electron-Line Insertions Karplus, Klein, Schwinger (1951) [40, 41] Baranger, Bethe, Feynman (1951) [42]

α(Zα)5 n3

(1 +

−

11 128

One-Loop Polarization Contribution Karplus, Klein, Schwinger (1951) [40, 41] Baranger, Bethe, Feynman (1951) [42]

Pachucki (1993) [47] Eides, Grotch, Shelyuto (1997) [51]

ln 2)δl0

( 13 63 ln 2 −

( 23 ln2

25 252 π

√ 1+ 5 2

+ 157 63 ln 2 −

− π2 18

∆E(1S)

∆E(2S)

kHz

kHz

55 090.31 6 886.29

1 940.38

242.55

α π δl0

−2.63

−0.33

15 647 α 52 920 ) π δl0

21.99

2.75

√ 1+ 5 2 33 641 α 13 230 ) π δl0

26.45

3.31

−0.018 2 α π δl0

−3.15

−0.39

−0.030 7 α π δl0

−5.31

−0.66

− 1 23 512

One-Loop Polarization and Electron Factor Eides, Grotch (1993) [49]

1 2

5 192 δl0

Two One-Loop Polarizations Eides, Grotch, Owen (1992) [46] Pachucki; Laporta (1993) [47, 48] Two-Loop Polarization Eides, Grotch, Owen (1992) [46] Pachucki; Laporta (1993) [47, 48]

r 3 (m m ) m

+

218 63

+

√

5 ln

Polarization insertion in the Electron Factor Eides, Grotch (1993) [50] Pachucki (1993) [47] Light by Light Scattering Pachucki (1993) [47], Eides, Grotch, Pebler (1994) [52]

Insertions of Two Radiative Photons in the Electron Line Pachucki (1994) [55], Eides, Shelyuto (1995) [56, 38]

−1.931 2(3) α π δl0 −334.24(5) −41.78

Reducible Three-Loop Polarization Insertions Eides, Shelyuto (2003) [58]

2 2.651 9 (6)( α π ) δl0

Reducible Three-Loop Radiative Insertions Eides, Shelyuto (2004) [59]

2 −5. 321 93 (1)( α π ) δl0

0.27

0.03

−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 the physics is the same independent of the power of α, and respective coefficients are all of order one, as in the series for the radiative corrections in scattering problems.

3.3 Radiative Corrections of Order αn (Zα)5 m

47

Fig. 3.13. Reducible three-loop diagrams

3.3.3.7 Total Correction of Order α2 (Zα)5 m The total contribution of order α2 (Zα)5 is given by the sum of contributions in (3.41), (3.42), (3.43), (3.46), (3.47), (3.48) [51]  √ √ 2π 2 25π 1 + 5 680 8 2 1 + 5 872 √ ln − + ln 2 − − 5 ln ∆E = 3 2 63 2 63 9 63  2 5  α (Zα) mr 3 24 901 − 7.921(1) m δl0 + 2 205 πn3 m 5 α2 (Zα)  mr 3 = −6.862(1) m δl0 . (3.49) πn3 m 5 α2 (Zα)  mr 3 ∆E = −6.862(1) m δl0 3 πn m = −296.92 (4) kHz|n=1 , = −37.115 (5) kHz|n=2 .

(3.50)

3.3.4 Corrections of Order α3 (Zα)5 m 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 discussion above their natural scale is determined by the factor 4α3 (Zα)5 /(π 2 n3 ) m. Allowing for an additional numerical factor about 1 − 2 we conclude that a fair estimate of these corrections is about 1 kHz for the 1S-state and about 0.1 kHz for the 2S-state. Corrections of order α3 (Zα)5 are generated by three-loop radiative insertions in the skeleton diagram in Fig. 3.8. All such corrections connected with the diagrams containing at least one one-loop or two-loop polarization insertion were obtained in [58] ∆E = 2.651 9 (6)

α3 (Zα)5  mr 3 m δl0 . π 2 n3 m

(3.51)

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

48

3 External Field Approximation

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 radiative photons in the electron line in Fig. 3.13 were considered thus far. The contribution of these diagrams in the Yennie gauge is equal to [59] ∆E = −5. 321 93 (1)

α3 (Zα)5  mr 3 m δl0 . π 2 n3 m

(3.52)

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

3.4 Radiative Corrections of Order αn(Zα)6 m 3.4.1 Radiative Corrections of Order α(Zα)6 m 3.4.1.1 Logarithmic Contribution Induced by the Radiative Insertions in the Electron Line Unlike the corrections of order αn (Zα)5 , corrections of order αn (Zα)6 depend on the large distance behavior of the wave functions. Roughly speaking this happens because in order to produce a correction containing six factors of Zα one needs at least three exchange photons like in Fig. 3.14. The radiative photon responsible for the additional factor of α does not suppress completely the low-momentum region of the exchange integrals. As usual, long distance contributions 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 order contribution to the Lamb shift. One can understand the appearance of the logarithm squared factor qualitatively. In the leading order contribution to the Lamb shift the logarithm was completely connected with the logarithmic infrared singularity of the electron form factor. Now we have two exchanged loops and one should anticipate the emergence of a logarithm generated by these loops. Note that the diagram with one exchange loop (e.g, relevant for the correction of order α(Zα)5 ) cannot produce a logarithm, since in the external field approximation the loop integration measure d3 k is odd in the

3.4 Radiative Corrections of Order αn (Zα)6 m

49

exchanged momentum, while all other factors in the exchanged integral are even in the exchanged momentum. Hence, in order to produce a logarithm which can only arise from the dimensionless integrand it is necessary to consider an even number of exchanged loops. These simple remarks may also be understood in another way if one recollects that in the relativistic corrections to the Schr¨ odinger-Coulomb wave function each power of logarithm is multiplied by the factor (Zα)2 (this is evident if one expands the exact Dirac wave function near the origin). The logarithm squared term is, of course, state-independent since the coefficient before this term is determined by the high momentum integration region, where the dependence on the principal quantum number may enter only via the value of the wave function at the origin squared. Terms linear in the large logarithm are already state dependent. Logarithmic terms were first calculated in [60, 61, 62, 63]. For the S-states the logarithmic contribution is equal to     4 2 1 2 m(Zα)−2 ln 2 + ln + ψ(n + 1) − ψ(1) + ∆Elog |l=0 = − ln 4 mr 3 n    m(Zα)−2 4α(Zα)6  mr 3 601 77 − − m , (3.53) ln 720 180n2 mr πn3 m where ψ(n) =

n−1  1

1 + ψ(1) , k

(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 disappears and the single-logarithmic contribution has the form     1 1 1 6 − 2l(l + 1)/n2 + δj, 12 δl1 + 1− 2 ∆Elog |l=0 = n 30 12 3(2l + 3)l(l + 1)(4l2 − 1)     m(Zα)−2 4α(Zα)6 mr 3 m. (3.55) × ln mr πn3 m Calculation of the state-dependent nonlogarithmic contribution of order α(Zα)6 is a difficult task, and has not been done for an arbitrary principal quantum number n. The first estimate of this contribution was made in [63]. Next the problem was attacked from a different angle [64, 65]. Instead of calculating corrections of order α(Zα)6 an exact numerical calculation of all contributions with one radiative photon, without expansion over Zα, was performed for comparatively large values of Z (n = 2), and then the result was extrapolated to Z = 1. In this way an estimate of the sum of the contribution of order α(Zα)6 and higher order contributions α(Zα)7 was obtained (for n = 2 and Z = 1). We will postpone discussion of the results obtained in this

50

3 External Field Approximation

way up to Subsect. 3.5.1, dealing with corrections of order α(Zα)7 , and will consider 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 the form (3.56) ∆E =
n|Σ (2) |n , where Σ (2) is an “exact” second-order self-energy operator for the electron in the Coulomb field (see Fig. 3.15), and hence contains the unmanageable exact Dirac-Coulomb Green function. The real problem with this formula is to extract useful information from it despite the absence of a convenient expression for the Dirac-Coulomb Green function. Numerical calculation without expansion over Zα, mentioned in the previous paragraph, was performed directly with 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 order α(Zα)6 for the 1S-state was obtained in [66, 67], with the help of a specially developed ”perturbation theory” for the Dirac-Coulomb Green function which expressed this function in terms of the nonrelativistic Schr¨ odinger-Coulomb Green function [68, 69]. But the real breakthrough was achieved in [70, 71], where a new very effective method of calculation was suggested and very precise values of the nonlogarithmic corrections of order α(Zα)6 for the 1Sand 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-Momentum Contributions. Nonlogarithmic Corrections Starting with the very first nonrelativistic consideration of the main contribution to the Lamb shift [8] separation of the contributions of high- and low-frequency radiative photons became a characteristic feature of the Lamb shift calculations. The main idea of this approach was already explained in Subsect. 3.2.1.1, but we skipped over two obstacles impeding effective implementation of this idea. Both problems are connected with the effective realization of the matching procedure. In real calculations it is not always obvious how to separate the two integration regions in a consistent way, since in the high-momenta region one uses explicitly relativistic expressions, while the starting point of the calculation in the low-momenta region is the nonrelativistic dipole approximation. The problem is aggravated by the inclination to use different gauges in different regions, since the explicitly covariant Feynman gauge is the simplest one for explicitly relativistic expressions in the

3.4 Radiative Corrections of Order αn (Zα)6 m

51

high-momenta region, while the Coulomb gauge is the gauge of choice in the nonrelativistic region. In order to emphasize the seriousness of these problems it suffices to mention that incorrect matching of high- and low-frequency contributions in the initial calculations of Feynman and Schwinger led to a significant delay in the publication of the first fully relativistic Lamb shift calculation of French and Weisskopf [10]4 ! It was a strange irony of history that due to these difficulties it became common wisdom in the sixties that it is better to try to avoid the separation of the contributions coming from different momenta regions (or different distances) than to try to invent an accurate matching procedure. A few citations are appropriate here. Bjorken and Drell [73] wrote, having in mind the separation procedure: “The reader may understandably be unhappy with this procedure . . . we recommend the recent treatment of Erickson and Yennie [62, 63], which avoids the division into soft and hard photons”. Schwinger [31] wrote: “. . . there is a moral here for us. The artificial separation of high and low frequencies, which are handled in different ways, must be avoided.” All this was written even though it was understood that the separation of the large and small distances was physically quite natural and the contributions coming from large and small distances have a different physical nature. However, the distrust of the methods used for separation of the small and large distances was well justified by the lack of a regular method of separation. Apparently different methods were used for calculation of the high and low frequency contributions, high frequency contributions being commonly treated in a covariant four-dimensional approach, while old-fashioned nonrelativistic perturbation theory was used for calculation of the low-frequency contributions. Matching these contributions obtained in different frameworks was an ambiguous and far from obvious procedure, more art than science. As a result, despite the fact that the methods based on separation of long and short distance contributions had led to some spectacular results (see, e.g., [74, 75]), their self-consistency remained suspect, especially when it was necessary to calculate the contributions of higher order than in the classic works. It seemed more or less obvious that in order to facilitate such calculations one needed to develop uniform methods for treatment of both small and large distances. The actual development took, however, a different direction. Instead of rejecting the separation of high and low frequencies, more elaborate methods of matching respective contributions were developed in the last decade, and the general attitude to separation of small and large distances radically changed. Perhaps the first step to carefully separate the long and short distances was done in [5], where the authors had rearranged the old-fashioned perturbation theory in such a way that one contribution emphasized the small momentum contributions and led to a Bethe logarithm, while in the other the small momentum integration region was naturally suppressed. Matching of both contributions in this approach was more natural and automatic. However, the 4

See fascinating description of this episode in [72].

52

3 External Field Approximation

price for this was perhaps too high, since the high momentum contribution was to be calculated in a three-dimensional way, thus losing all advantages of the covariant four-dimensional methods. Almost all new approaches, the skeleton integral approach described above in Subsect. 3.3.1 ([38] and references there), -method described in this section [70, 71], nonrelativistic approach by Khriplovich and coworkers [76], nonrelativistic QED of Caswell and Lepage [77]) not only make separation of the small and large distances, but try to exploit it most effectively. In some cases, when the whole contribution comes only from the small distances, a rather simple approach to this problem is appropriate (like in the calculation of corrections of order α2 (Zα)4 , α3 (Zα)4 , α(Zα)5 and α2 (Zα)5 above, more examples below) and the scattering approximation is often sufficient. In such cases, would-be infrared divergences are powerlike. They simply indicate the presence of the contributions of the previous order in Zα and may safely be thrown away. In other cases, when one encounters logarithms which get contributions both from the small and large distances, a more accurate approach is necessary such as the one described below. In any case “the separation of low and high frequencies, which are handled in different ways” not only should not be avoided but turns out to be a very convenient calculational tool and clarifies the physical nature of the corrections under consideration. An effective method to separate contributions of low- and high-momenta avoiding 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 of all corrections of orders α(Zα)n m (n ≥ 4) generated by the insertion of one radiative photon in the electron line  d4 k 2 Dµν (k)
n|γµ G(p − k; p − k)γν |n , ∆E = e (3.57) (2π)4 i where G(p − k; p − k) is the exact electron Green function in the external Coulomb field. As was noted in [70, 71] one can rotate the integration contour over the frequency of the radiative photon in such a way that it encloses singularities along the positive real axes in the ω (k 0 ) plane. Then one considers separately the region Re ω ≤ σ (region I) and Re ω ≥ σ (region II), where m(Zα)2  σ  m(Zα). It is easy to see that due to the structure of the singularities of the integrand, integration over k in the region I also goes only over the momenta smaller than σ (|k| ≤ σ), while in the region II the final integration over ω cuts off all would be infrared divergences of the integral. Hence, effective separation of high- and low-momenta integration regions is achieved in this way and, as was explained above, due to the choice of the magnitude of the parameter σ all would be divergences should exactly cancel in the sum of contributions of these regions. This cancellation provides an additional effective method of control of the accuracy of all calculations. It was also shown in [71] that a change of gauge in the low-frequency region changes the result of the calculations by a term linear in σ. But anyway one

3.4 Radiative Corrections of Order αn (Zα)6 m

53

should discard such contributions matching high- and low-frequency contributions. The matrix element of the self-energy operator between the exact Coulomb-Dirac wave functions is gauge invariant with respect to changes of gauge of the radiative photon [78]. Hence, it is possible to use the simple Feynman gauge for calculation of the high-momenta contribution, and the physical Coulomb gauge in the low-momenta part. It should be clear now that this method resolves all problems connected with the separation of the high- and low-momenta contributions and thus provides an effective tool for calculation of all corrections with insertion of one radiative photon in the electron line. The calculations originally performed in [70, 71, 79] successfully reproduced all results of order α(Zα)4 and α(Zα)5 and produced a high precision result for the constant of order α(Zα)6 . Results for nonlogarithmic contributions were later obtained also for excited S states in [80, 81], where accuracy of calculation was improved even further. The most recent results for the 1S and 2S states are [80, 81] α(Zα)6  mr 3 m, (3.58) π m α(Zα)6  mr 3 m. ∆Enon−log (2S) = −31.840 465 09 (1) 8π m We would like to emphasize two features of these results. First, the state dependence of the constant is very weak, and second, the scale of the constant is just of the magnitude one should expect. In order to make this last point more transparent let us write the total electron-line contribution of order α(Zα)6 to the 1S energy shift in the form       m(Zα)−2 m(Zα)−2 + 5.42 ln − 30.92 ∆E(1S) ≈ − ln2 mr mr  6  3 α(Zα) mr m. (3.59) × π m ∆Enon−log (1S) = −30.924 149 46 (1)

Now we see that the ratio of the nonlogarithmic term and the coefficient before the single-logarithmic term is about 31/5.4 ≈ 5.7 ≈ 0.6π 2 . It is well known that the logarithm squared terms in QED are always accompanied by the single-logarithmic and nonlogarithmic terms, and the nonlogarithmic terms are of order π 2 (in relation with the current problem see, e.g., [62, 63]). This is just what happens in the present case. Nonlogarithmic contributions of order α(Zα)6 to the energies of the nonS-states induced by the radiative photon insertions in the electron line were obtained in the same framework in [82, 83, 84, 85]. The nonlogarithmic coefficients of order α(Zα)6 are similar to the nonlogarithmic coefficients of order α(Zα)4 in Table 3.1 (Bethe logarithms), and are called relativistic Bethe logarithms in [84, 85]. However, unlike the ordinary Bethe logarithms the nonlogarithmic coefficients of order α(Zα)6 depend not only on the principal quantum number n and the orbital momentum l, but also on the total angular

54

3 External Field Approximation

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 coefficient A60 [62] which is defined by the relationship ∆E = A60

α(Zα)6  mr 3 m. πn3 m

(3.60)

3.4.1.3 Correction Induced by the Radiative Insertions in the External Photons There are two kernels with radiative insertions in the external photon lines which produce corrections of order α(Zα)6 to the Lamb shift. First is our old acquaintance – one-loop polarization insertion in the Coulomb line in Fig. 2.2. Its Fourier transform is called the Uehling potential [16, 86]. The second kernel contains the light-by-light scattering diagrams in Fig. 3.16 with three external photons originating from the Coulomb source. The sum of all closed electron loops in Fig. 3.17 with one photon connected with the electron line and an arbitrary number of Coulomb photons originating from the Coulomb source may be considered as a radiatively corrected Coulomb potential V . It generates a shift 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 energy shift generated by the Wichmann-Kroll potential increases for large Z. For practical reasons the effects of the Uehling and Wichmann-Kroll potentials were investigated mainly numerically and without expansion in Zα, since only such results could be compared with the experiments. Now there exist many 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

3.4 Radiative Corrections of Order αn (Zα)6 m Table 3.4. Nonlogarithmic Coefficient A60 α(Zα)6 mr 3 (m) m πn3

1S 1 2 2S 1 2 2P 1 2 2P 3 2 3S 1 2 3P 1 2 3P 3 2 3D 3 2 3D 5 2 4S 1 2 4P 1 2 4P 3 2 4D 3 2 4D 5 2 4F 5 2 4F 7 2 5S 1 2 5P 1 2 5P 3 2 5D 3 2 5D 5 2 5F 5 2 5F 7 2 5G 7 2 5G 9 2 6S 1 2 6P 1 2 6P 3 2 6D 3 2 6D 5 2 7S 1 2 7P 1 2 7P 3 2 7D 3 2 7D 5 2 8S 1 2 8D 3 2 8D 5 2

−30.924 149 46 −31.840 465 09 −0.998 904 402 −0.503 373 465 −31.702 501 −1.148 189 956 −0.597 569 388 0.005 551 573 0.027 609 989 −31.561 922 −1.195 688 142 −0.630 945 795 0.005 585 985 0.031 411 862 0.002 326 988 0.007 074 961 −31.455 393 −1.216 224 512 −0.647 013 508 0.006 152 175 0.033 077 570 0.002 403 158 0.008 087 020 0.000 814 415 0.002 412 929 −31.375 130 −1.226 702 391 −0.656 154 893 0.006 749 745 0.033 908 493 −31.313 224 −1.232 715 957 −0.662 027 568 0.007 277 403 0.034 355 926 −31.264 257 0.007 723 850 0.034 607 492

(1) (1) (1) (1) (1) (1) (1) (1) (1) (1) (1) (1) (1) (1) (1) (1) (1) (1) (1) (1) (1) (1) (1) (1) (1) (1) (1) (1) (1) (1) (1) (1) (1) (1) (1) (1) (1) (1)

kHz −1338.0434 −172.2114 −5.4026 −2.7225 −50.8045 −1.8400 −0.9576 0.0089 0.0442 −21.3381 −0.8083 −0.4266 0.0038 0.0212 0.0016 0.0048 −10.8882 −0.4210 −0.2240 0.0021 0.0114 0.0008 0.0028 0.0003 0.0008 −6.2850 −0.2457 −0.1314 0.0014 0.0068 −3.9501 −0.1555 −0.0835 0.0009 0.0043 −2.6421 0.0007 0.0029

55

56

3 External Field Approximation

with our emphasis on the analytic results we will discuss here only analytic contributions of order α(Zα)6 , and will return to numerical results in Subsect. 3.5.2. Uehling Potential Contribution It is not difficult to present an exact formula containing all corrections produced by the Uehling potential in Fig. 2.2 (compare with the respective expression for the self-energy operator above)      Π(k2 )  n . (3.62) ∆E = 4π(Zα) n  k4  We have already seen that the matrix element of the first term of the low-momentum expansion of the one-loop polarization operator between the nonrelativistic Schr¨ odinger-Coulomb wave functions produces a correction of order α(Zα)4 . The next term in the low-momentum expansion of the polarization operator pushes characteristic momenta in the integrand to relativistic values, where the very nonrelativistic expansion is no longer valid, and even makes the integral divergent if one tries to calculate it between the nonrelativistic wave functions. Due to this effect we prefer to calculate the correction of order α(Zα)5 induced by the one-loop polarization insertion (as well as the correction of order α2 (Zα)5 induced by the two-loop polarization) in the skeleton integral approach in Sect. 3.3. Note that all these corrections contribute only to the S-states. It is useful to realize that both these calculations are, from another point of view, simply results of approximate calculation of the integral in (3.62) with accuracy (Zα)4 and (Zα)5 . Our next task is to calculate this integral with accuracy (Zα)6 . In this order both small atomic (∼ mZα) and large relativistic (∼ m) momenta produce nonvanishing contributions to the integral, and as a result we get nonvanishing contributions to the energy shifts also for states with nonvanishing angular momenta. Consider first corrections to the energy levels with nonvanishing angular momenta. The respective wave functions vanish at the origin in coordinate space, hence only small photon momenta contribute to the integral, and one can use the first two terms in the nonrelativistic expansion of the polarization operator α αk2 Π(k2 ) ≈− + (3.63) 4 2 15πm 35m4 k for calculation of these contributions [88, 83] (compare (3.12) above). Corrections of order α(Zα)6 turn out to be nonvanishing only for l ≤ 1. For 2P -states these corrections were first calculated in [65], and the result for arbitrary P -states [89] has the form    1 α(Zα)6  mr 3 1 4 1 + δj 12 ∆E(nPj ) = − m. (3.64) 1− 2 15 n 14 4 πn3 m

3.4 Radiative Corrections of Order αn (Zα)6 m

57

The respective correction to the energy levels of S-states originates both from the large and small distances since the Schr¨ odinger-Coulomb wave function in the S-states does not vanish at small distances. Hence, one cannot immediately apply low-momenta expansion of the polarization operator for calculation of the matrix element in (3.62). The leading logarithmic stateindependent contribution to the energy shift is still completely determined by the first term in the low-momentum expansion of the polarization operator in (3.63), but one has to consider the exact expression for the polarization operator in order to obtain the nonlogarithmic contribution. The logarithmic term originates from the logarithmic correction to the Schr¨ odinger-Coulomb wave function which arises when one takes into account the Darwin (δ-function) potential which arises in the nonrelativistic expansion of 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 corrections to the Schr¨ odinger-Coulomb wave function, one expands the singular factor in the Dirac-Coulomb wave function over Zα. The correction to the Schr¨ odinger-Coulomb wave function Ψ at the distances of order 1/m has the form [44] 1 (3.65) δΨ = − (Zα)2 ln(Zα) Ψ , 2 and substituting this correction in (3.62) one easily obtains the leading logarithmic contribution to the energy shift [60, 62, 63]   m(Zα)−2  mr 3 2 α(Zα)6 ln m. (3.66) ∆Elog (nS) = − 15 πn3 mr m Note that the numerical factor before the leading logarithm here is simply the product of the respective numerical factors in the correction to the wave function 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 that both wave functions in the matrix element in (3.62) have to be corrected. Calculation of the nonlogarithmic contributions requires more effort. Complete analytic results for the lowest states were first obtained by P. Mohr [65]     m(Zα)−2 1 1289 4α(Zα)6  mr 3 1 ∆E(1S) = m, + ln 2 − − ln 15 2 mr 420 π m     m(Zα)−2 1 743 4α(Zα)6  mr 3 1 ∆E(2S) = m. (3.67) − − ln 15 2 mr 240 8π m As was mentioned above, short distance contributions are state independent and always cancel in the differences of the form ∆n ≡ n3 ∆E(nS) − ∆E(1S). This means that such state-dependent differences of energies contain only contributions of large distances and are much easier to calculate,

58

3 External Field Approximation

since one may employ a nonrelativistic approximation.5 The Uehling potential contributions to the difference of level shifts n3 ∆E(nS) − ∆E(1S) were calculated in [88] with the help of the nonrelativistic expansion of the polarization operator in (3.63). The result of this calculation, in conjunction with the Mohr result in (3.67), leads to an analytic expression for the Uehling potential contribution to the Lamb shift    m(Zα)−2 2(n − 1) 431 1 1 + (ψ(n + 1) − ψ(1)) − − ∆E(nS) = − ln 15 2 mr 105 n2  1 n 4α(Zα)6  mr 3 + − ln m, (3.68) 2 28n 2 πn3 m 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 radiative insertions in the external photons is produced by the term trilinear in Zα in the Wichmann-Kroll potential in Fig. 3.16. One may easily check that the first term in the small momentum expansion of the Wichmann-Kroll potential has the form [87, 92]   19 π 2 α(Zα)3 − . (3.69) VW K (k) = 45 27 m2 This potential generates the energy shift [93, 92]   19 π 2 α(Zα)6  mr 3 ∆E = − m δl0 , 45 27 πn3 m

(3.70)

which is nonvanishing only for the S-states. 3.4.2 Corrections of Order α2 (Zα)6 m 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 calculated in the same way. The total contribution of order α2 (Zα)6 is a polynomial in ln(Zα)−2 , starting with the cube of the large logarithm. The factor before the leading logarithm cubed and the contribution of the logarithm squared terms to the difference ∆EL (1S) − 8∆EL (2S) were obtained first. Calculation of these contributions is relatively simple because large logarithms always originate from the wide region of large virtual momenta (mZα  k  m) and the respective matrix elements of the perturbation potentials depend only on the value of the Schr¨ odinger-Coulomb wave function or its derivative at the 5

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.

3.4 Radiative Corrections of Order αn (Zα)6 m

59

origin, as we have already seen above in discussion of the main contribution to the Lamb shift and corrections of order α(Zα)6 . The other contributions of order α2 (Zα)6 depend on all distances, and their calculation is much more involved. All terms in the polynomial in ln(Zα)−2 besides the constant are known 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 irreducible electron two-loop self-energy operator in Fig. 3.18 does not generate a ln3 (Zα)−2 contribution to the Lamb shift. The only way to obtain the highest power of the logarithm is to consider effective potentials corresponding to the one-loop electron self-energy operator in the external Coulomb field, and to use the ordinary second-order perturbation theory expression (see Fig. 3.19) ∆E = 2


n|V1 |m
m|V2 |n , En − E m

(3.71)

m,m=n

where V1 and V2 are the perturbation operators, and the factor 2 is due to two possible orders of the perturbation operators; it is not present when V1 = V2 . Summation over the intermediate states  above includes integration over the continuous spectrum with the weight d3 k/(2π)3 .

Fig. 3.18. Two-loop self-energy operator

A logarithmic matrix element of the first-order electron self-energy operator (like the one producing the leading contribution to the Lamb shift) may be considered in the framework of perturbation theory as a matrix element of an almost local (it depends on the momentum transfer only logarithmically) perturbation theory operator since it is induced by the diagram with relativistic virtual momenta. We may use such a local operator in order to calculate the higher order perturbation theory contributions [76]. The maximum power of the large logarithm arises if we consider as perturbation (V1 = V2 = V ) the first order quasilocal effective potential which generates the leading contribution to the Lamb shift (compare (2.3)), which in momentum space has the form k 8α(Zα) . (3.72) ln V =− 2 3m m

60

3 External Field Approximation

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

It is evident that this potential leads to a logarithm squared contribution of order α2 (Zα)6 after substitution in (3.71). One may obtain one more logarithm from the continuous spectrum contribution in (3.71). Due to locality of the potential, matrix elements reduce to the products of the values of the respective wave functions at the origin and the potentials in (3.72). The value of the continuous spectrum Coulomb wave function at the origin is well known (see, e.g., [94]), and    γr 2 2πγr πγr + O , (3.73) ≈ 1 + |ψk (0)|2 = 2πγr − k k k(1 − e k ) where γr = mr Zα. The leading term in the large momentum expansion in (3.73) generates an apparently linearly ultraviolet divergent contribution to the energy shift, but this ultraviolet divergence is due to our nonrelativistic approximation, and it would be cut off at the electron mass in a truly relativistic 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 in the high momentum expansion in (3.73)  (mr Zα)4 mr
n|V1 |m
m|V2 |n . (3.74) ∆E = −2 dk π 2 n2 k With the help of this formula one immediately obtains [95]   8 α2 (Zα)6 3 m(Zα)−2  mr 5 ∆E = − ln m. 27 π 2 n3 mr m

(3.75)

Note that the scale of this contribution is once again exactly of the expected magnitude, namely, this contribution is suppressed by the factor (α/π) ln(Zα)−2 in comparison with the leading logarithm squared contribution of order α(Zα)6 . Of course, the additional numerical suppression factor 8/27 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 diagrams with two-loop radiative insertions in the electron line were calculated numerically without expansion in Zα. The results of these considerations were controversial, and the authors of [96, 98] even came to the conclusion that the leading logarithm contribution should be different from the result in (3.75).

3.4 Radiative Corrections of Order αn (Zα)6 m

61

The leading perturbation theory contribution in (3.75) was reproduced with the help of the renormalization group equations in the approach based on nonrelativistic QED [99]. But this result should be considered more as a test of the validity of the renormalization group equations in a specific version of NRQED, than as an independent confirmation of the value of the coefficient in (3.75). On the other hand an additional logarithm cubed contribution was found in a later paper [100]. However, only an incomplete set of diagrams was considered in that paper. All doubts about validity of (3.75) were put to rest in [101], where among other results a complete calculation of the logarithm cubed in the Coulomb gauge was presented. Later the relationship between the results obtained with the help of the effective potentials and the results of 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 the power of the large logarithm ln(Zα)−2 are needed for any meaningful comparison with the experimental results. Even the effective potential calculation of logarithm squared contributions to the energy shift of S-levels is impeded by the fact that such contributions arise both from the discrete and continuous spectrum intermediate states in (3.71), and a complicated interplay of contributions from the different regions occurs. Hence, in such a calculation it is necessary to consider the contributions of the one-loop electron self-energy operators more accurately and the local approximation used above becomes inappropriate. It is easier to use other methods for calculation of the coefficients before the nonleading logarithms. All corrections of order α2 (Zα)6 are usually written as an expansion over the powers of the large logarithm  α2 (Zα)6 ∆E = B63 ln3 (Zα)−2 + B62 ln2 (Zα)−2 + B61 ln(Zα)−2 + B60 m. π 2 n3 (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 high frequency contributions developed earlier for calculation of the corrections of order α(Zα)6 , and described above in Subsect. 3.4.1.2 was generalized for the case of corrections of order α2 (Zα)6 . This generalization is far from being straightforward because due to insertions of two radiative photons in the electron 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

62

3 External Field Approximation

independent photon integration momenta and a nontrivial interplay between their magnitudes should be taken into account. The contribution to the logarithm squared coefficient generated by the diagrams in Fig. 3.18 and Fig. 3.20 was obtained in [101]. We write it in the form (see also [103])   16 n − 1 n2 − 1 + B62 (nS) = B62 (1S) − , (3.77) ln n − ψ(n) + ψ(1) − 9 n 4n2 where

16 16 ln 2 + . (3.78) 9 27 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 contributions cancel in the difference B62 (1S) − B62 (nS) which corresponds to the energy shifts combination ∆n = n3 ∆E(nS) − ∆E(1S). This difference B62 (1S) − B62 (nS) was independently calculated earlier in the framework of the effective potential approach [104, 105, 106, 107], and we will briefly describe this calculation. B62 (1S) = −

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

The logarithm squared contributions to the difference of energies are generated in the second order of perturbation theory by two one-loop vertex operators, and in the first order of perturbation theory by the one two-loop vertex (see diagrams in Fig. 3.21). Due to cancellation of the short-distance state-independent terms in the difference of energy levels only intermediate continuous spectrum states with momenta of the atomic scale mZα give contributions in the second order of perturbation theory [106, 107]. Then the local approximation for the one-loop vertices and the nonrelativistic approximation for the wave functions is sufficient for calculation of the logarithm squared contribution to the energy difference generated by the first diagram in Fig. 3.21. Calculation of the contribution induced by the second order vertex operator (second diagram in Fig. 3.21) is quite straightforward. Both contributions were calculated in a series of papers [104, 105, 106, 107], where the 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 in Fig. 3.18 and Fig. 3.20 has the form [101, 103]

3.4 Radiative Corrections of Order αn (Zα)6 m

63

   80 32 n − 1 n2 − 1 − ln 2 ln n − ψ(n) + ψ(1) − + B61 (nS) = B61 (1S)− 27 9 n 4n2 4 + [N (nS) − N (1S)] , (3.79) 3 where [101, 103, 81] 1039 2 379913 4 2 40 2 152 B61 (1S) = ζ(3)− π 2 ln 2+ ln 2− ln 2+ π + + N (1S) , 3 9 27 432 64800 3 (3.80) and the additive terms N (nS) introduced in [101] and calculated with high accuracy in [108, 81] are collected in Table 3.5. Table 3.5. Functions N (nS) and N (nP ) nS 1S 2S 3S 4S 5S 6S 7S 8S

N (nS) 17.855 672 03 (1) 12.032 141 58 (1) 10.449 809 (1) 9.722 413 (1) 9.304 114 (1) 9.031 832 (1) 8.840 123 (1) 8.697 639 (1)

nP 2P 3P 4P 5P 6P 7P 8P

N (nP ) 0.003 0.003 −0.000 −0.004 −0.007 −0.010 −0.011

300 572 394 303 496 014 999

635 084 332 806 998 614 223

(1) (1) (1) (1) (1) (1) (1)

Corrections of order α2 (Zα)6 are also generated by the insertions of the one-loop polarization operator in the one-loop electron self-energy operator in the external field in Fig. 3.22. It was shown in [101] that the respective logarithm squared contribution vanishes. There are two contributions to the single-logarithmic coefficient. One is due to the effective potential generated by the insertions of the one-loop electron polarization operator in the oneloop electron form factors in the same way as we obtained above the leading logarithmic contribution of order α(Zα)6 generated by the Uehling potential in (3.66). The respective contribution to the coefficient B61 (nS) is equal to (2) (2) 8F  1 (0) + 2F2 (0) [101] 5 2 7 π − . (3.81) 432 162 Later a second contribution to the single-logarithmic coefficient generated by the diagram in Fig. 3.22 which cannot be accounted for by the form factors was discovered in [81] (1)

B61 (nS) =

61 2 493 π + . (3.82) 864 648 Finally the total single-logarithmic coefficient generated by the diagram in Fig. 3.22 is equal to (2)

B61 (nS) = −

64

3 External Field Approximation

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

(1)

(2)

B61 (nS) = B61 (nS) + B61 (nS) = −

17 2 155 π + . 288 216

(3.83)

Let us turn to the nonlogarithmic coefficients B60 . These coefficients in the case of corrections of order α2 (Zα)6 m are similar to the nonlogarithmic terms in the leading contribution to the Lamb shift of order α(Zα)4 m in (3.7). The dominant contribution to the nonlogarithmic term in (3.7) is given by the Bethe logarithm, which describes the effect of the emission and absorption of soft virtual photons in the dipole approximation. Similarly one should expect that the dominant low-energy contributions to the nonlogarithmic coefficients B60 is given by the emission and absorption of two soft virtual photons in the dipole approximation. Respective contributions to the coefficients B60 for the 1S and 2S states were obtained in [109, 110], where they were called two-loop Bethe logarithms. The authors of [109, 110] estimate the magnitude of other uncalculated contributions to the coefficients B60 as about 15% of the twoloop Bethe logarithms. We collected the results of [109, 110] in Table 3.6. The 15% uncertainty in the value of the coefficients B60 determines the theoretical uncertainty of the total α2 (Zα)6 contribution to the energy shift. Numerically it constitutes 0.9 kHz for the 1S-state and 0.1 kHz for the 2S-state. Table 3.6. Two-Loop Bethe Logarithms nS 1S 2S 3S 4S 5S 6S

B60 (nS) −61.6 −53.2 −51.9 −51.0 −50.3 −49.8

(3) ± 15% (3) ± 15% (6) ± 15% (8) ± 15% (8) ± 15% (8) ± 15%

The values of the coefficients B6n obtained above lead to a quite peculiar behavior of different contributions to the energy shifts. To get a better idea about 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 α2 (Zα)6 m ≈ [−282 − 62 + 476 − 61] m, π2 π2

(3.84)

3.4 Radiative Corrections of Order αn (Zα)6 m

65

Fig. 3.23. Leading polarization operator logarithm squared contributions of order α2 (Zα)6

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

α2 (Zα)6 α2 (Zα)6 m ≈ [−282 + 45 + 403 − 53] m. 2 8π 8π 2

(3.85)

We observe that contrary to our expectations the contributions generated by the consecutive terms in the expansion over the powers of the large logarithm do not decrease. The term linear in the large logarithm numerically dominates and even reverses the sign of the leading logarithmic contribution. The correction of order α2 (Zα)6 m plays an important role in the analysis of the experimental results on the Lamb shift, and an independent calculation of the coefficients in (3.76) is clearly warranted. We could also expect that, as in the case of corrections of order α(Zα)6 m and α(Zα)7 m, contributions of order α2 (Zα)7 m may be phenomenologically relevant. Therefore calculation of the corrections generated by the diagrams in Fig. 3.18 and Fig. 3.20 without expansion 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 polarization insertions in the external photon lines in Fig. 3.23. The general expression for these corrections may be written as in (3.76). It is easy to see that the leading term generated by the diagrams in Fig. 3.23 is quadratic in the large logarithm, and hence, respective B63 = 0. These logarithm squared contributions corresponding to the diagrams with at least one polarization insertion may be calculated with the help of the effective potentials. The leading logarithm squared term in Fig. 3.23 is generated when we combine the perturbation potential in (3.72) which corresponds to the one-loop electron vertex and the perturbation potential in (2.5) which corresponds to the polarization operator contribution to the Lamb shift. Then the standard second order perturbation theory (3.71) immediately leads to the expression [104] B62 =

8 . 45

(3.86)

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

66

3 External Field Approximation

Coulomb photons was obtained in [101]. We write it in the form (see also [103])   32 n − 1 n2 − 1 B61 (nS) = B61 (1S) − − , (3.87) − ln n + ψ(n) − ψ(1) + 45 n 4n2 where

401 16 ln 2 − . (3.88) 15 2025 In this notation we have separated the state-independent contribution B61 (1S) originating exclusively from the short distances, first obtained in [101] from the other contributions to this coefficient. The short distance contributions cancel in the difference B61 (1S) − B61 (nS) which corresponds to the energy shifts combination ∆n = n3 ∆E(nS) − ∆E(1S). This difference B61 (1S) − B61 (nS) was independently calculated earlier in the framework of the effective potential approach [104]. As in the previous section, to get a better idea about relative magnitude of different polarization contributions of order α2 (Zα)6 let us write explicitly contributions to 1S and 2S states B61 (1S) =

 α2 (Zα)6 ∆E(1S) = 0.178 ln2 (Zα)−2 + 0.541 ln(Zα)−2 + B60 m π2 α2 (Zα)6 ≈ [17.2 + 5.33 + B60 ] m, (3.89) π2  α2 (Zα)6 ∆E(2S) = 0.178 ln2 (Zα)−2 + 0.101 ln(Zα)−2 + B60 m. 8π 2 α2 (Zα)6 ≈ [17.2 + 0.993 + B60 ] m. (3.90) 8π 2 Unlike the case of radiative insertions in the electron line, this time the coefficients before the logarithms do not demonstrate any peculiarities. The respective correction to the energy shift is about 2 kHz for the 1S state in hydrogen. Recent progress in calculation of nonlogarithmic coefficients B60 is reported in [80, 81]. The authors of these papers derived a general NRQED expression for the contribution of all diagrams (both with radiative insertions in the electron line and in the external photons) to the difference of the coefficients B60 (1S) − B60 (nS), which is necessary for calculation of the phenomenologically important energy difference ∆n = n3 ∆E(nS)−∆E(1S). The accuracy 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 with nonvanishing angular momenta is much simpler than for the S levels [104, 105].

3.4 Radiative Corrections of Order αn (Zα)6 m

67

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

The second order perturbation theory term with two one-loop self-energy operators does not generate any logarithm squared contribution for the state with nonzero angular momentum since the respective nonrelativistic wave function vanishes at the origin. Only the two-loop vertex in Fig. 3.24 produces a logarithm squared term in this case. The respective perturbation potential determined by the second term in the low-momentum expansion of the twoloop Dirac form factor [111] has the form V2 = −

2α2 (Zα)k 2 2 ln (Zα)−2 . 9πm4

(3.91)

Calculation of the matrix element of this effective perturbation with the nonrelativistic wave functions for the P -states yields [104, 105] (see also [102]) α2 (Zα)6 m 2 4(n2 − 1) α2 (Zα)6 m 2 −2 ln (Zα) = ln (Zα)−2 , ∆E(nP ) ≡ B62 (nP ) π 2 n3 27n2 π 2 n3 (3.92) while for l > 1 there are no logarithm squared contributions. Terms linear in the large logarithm were recently obtained in [80, 81]   n2 − 1 8 ln 2 166 4 B61 (nP 12 ) = + (3.93) − + N (nP ) , n2 27 405 5   31 n2 − 1 8 ln 2 4 3 B61 (nP 2 ) = + (3.94) − + N (nP ) , n2 27 405 5 where the additive terms N (nP ), calculated with high accuracy in [108], are collected in Table 3.5. It was shown in [80, 81] that all logarithmic terms of order α2 (Zα)6 vanish for states with orbital momenta l > 1. A general NRQED expression for the nonlogarithmic terms for P and D states also was obtained in [80, 81]. 3.4.2.4 Corrections of Order α3 (Zα)6 m The analysis of the contributions of order α2 (Zα)n confirms once again, as also emphasized in [96], that there is no regular rule for the magnitude of the

68

3 External Field Approximation

coefficients before the successive terms in the series over Zα at fixed α. This happens because the terms, say of relative order Zα and (Zα)2 , correspond to completely different physics at small and large distances and, hence, there is no reason to expect a regular law for the coefficients in these series. This should be compared with the series over α at fixed Zα. As we have shown above, different terms in these series correspond to the same physics and hence the coefficients in these series change smoothly and may easily be estimated. This is why we have organized the discussion in terms of such series. Note that the best way to estimate an unknown correction of order, say α2 (Zα)6 m, which corresponds to the long distance physics, is to compare it with the long distance correction of order α(Zα)6 m, and not with the correction of order α2 (Zα)5 m which corresponds to the short distance physics. Of course, such logic contradicts the spirit of the numerical calculations made without expansion over Zα but it reflects properly the physical nature of different contributions at small Z. Corrections of order α3 (Zα)6 were never considered in the literature. They are suppressed in comparison to contributions of order α2 (Zα)6 by at least an additional factor α/π and are too small to be of any phenomenological interest now.

3.5 Radiative Corrections of Order α(Zα)7 m and of Higher Orders Only partial results are known for corrections of order α(Zα)7 m. However, recent achievements [112] in the numerical calculations without expansion in Zα completely solve the problem of the corrections of order α(Zα)7 m and of higher orders in Zα. 3.5.1 Corrections Induced by the Radiative Insertions in the Electron Line Consider first corrections of order α(Zα)7 induced by the radiative photon insertions in the electron line. Due to the Layzer theorem [60] the diagram with the radiative photon spanning four Coulomb photons does not lead to a logarithmic contribution. Hence, all leading logarithmic contributions of this order may be calculated with the help of second order perturbation theory in (3.71). It is easy to check that the leading contribution is linear in the large logarithm and arises when one takes as the first perturbation the local potential corresponding to the order α(Zα)5 m contribution to the Lamb shift (3.34)   πα(Zα)2 1 11 − ln 2 , (3.95) V1 = 4 1 + 128 2 m2 and the second perturbation corresponds to the Darwin potential

Table 3.7. Radiative Corrections of Order αn (Zα)6 m

Logarithmic Electron-Line Contribution (l = 0) Layzer (1960) [60] Fried, Yennie (1960) [61] Erickson, Yennie (1965) [62, 63]

173.074 n3

kHz ∆E(1S) kHz ∆E(2S) kHz

m − 14 ln2 [ m (Zα)−2 ] + [ 43 ln 2 + ln n2 r 77 +ψ(n + 1) − ψ(1) − 601 − 180n 2] 720 m × ln[ mr (Zα)−2 ]

−1 882.77

−208.16

−1 338.04

−172.21

−56.77

−7.10

Logarithmic Electron-Line Contribution (l = 0) Erickson, Yennie (1965) [62, 63]

Nonlogarithmic Electron-Line Contribution Pachucki (1993) (1S, 2S) [70, 71] [79], Jentschura, Pachucki (1996) (2P 1 , 2P 3 ) [82] 2 2 Jentschura, Soff, Mohr (1997) (3P 1 , 3P 3 , 4P 1 , 4P 3 ) [83] 2

2

2

1 1 [(1 − n12 )( 30 + 12 δj, 1 ) 2 6−2l(l+1)/n2 m + 3(2l+3)l(l+1)(4l2 −1) ] ln[ mr (Zα)−2 ]

A60 4

2

Logarithmic Polarization Operator Contribution Layzer (1960) [60] Erickson, Yennie (1965) [62, 63]

1 m − 30 ln[ m (Zα)−2 ]δl0 r

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

6

r 3 4 α(Zα) (m ) m≈ m πn3

69

1 [− 431 + ψ(n + 1) − ψ(1) 15 105 1 n − 2(n−1) + 28n 2 − ln 2 ]δl0 n2

−27.41

−4.47

2.45

0.31

−28.38

−3.55

−6.23

0.56

Nonlogarithmic Polarization Operator Contribution (l = 1) 1 − 15 (1 −

Mohr (1975) [65] Manakov, Nekipelov, Fainstein (1989) [89]

1 1 )[ 14 n2

Wichmann-Kroll Contribution Wichmann, Kroll (1956) [87] Mohr (1976) [93, 92]

+ 14 δj, 1 ]

19 ( 180 −

2

π2 )δ 108 l0

Leading Logarithmic Two-Loop Electron-Line Contribution 2 α − 27 ( π ) ln3 (Zα)−2 δl0

Karshenboim (1993) [95] Electron-Line Two-Loop Log Squared Term ∆E(nS) Pachucki (2001) [101] Karshenboim (1996) [104, 105]

!

−

+ψ(1) −

4 9

ln 2 +

n−1 n

+

2

4 27

n −1 4n2

−

"

4 9



ln n − ψ(n)

(α ) ln2 (Zα)−2 π

[106, 107] (Continued)

3 External Field Approximation

Nonlogarithmic Polarization Operator Contribution (l = 0) Mohr (1975) [65] Ivanov, Karshenboim (1997) [88]

70

Table 3.7. (Continued)

Table 3.7. (Continued) Single Logarithmic Two-Loop

!

1 ζ(3) 4

− 16 π 2 ln 2 +

Pachucki (2001) [101] Jentschura (2003) [108]

+ 2027 π2 + 3456

426413 259200

−

Jentschura, Czarnecki Pachucki (2005) [81]

+ψ(1) −

n−1 n

+

n2 −1 4n2





20 27

−

8 9

10 9

ln2 2 −

 ln 2

B60 α (π) 4

5.06

−6.2 (9)

−0.67 (10)

2 α ( ) ln2 (Zα)−2 45 π

1.73

0.22

!

4 15

ln 2 −

−ψ(1) +

401 8100

n−1 n

−

−

8 45

n2 −1 4n2



0.54

0.01

− ln n + ψ(n)

"

(α ) ln(Zα)−2 π

B61 (nPj ) α ( π ) ln(Zα)−2 4

71

Single Logarithmic Term ∆E(nPj ) Czarnecki, Jentschura, Pachucki (2005) [80]

47.89

n2 −1 α ( ) ln2 (Zα)−2 27n2 π

Log Squared Two-Loop Polarization Term ∆E(nS) Karshenboim (1996) [104]

Karshenboim (1996) [104]

ln n − ψ(n)

"

Electron-Line Two-Loop Log squared term ∆E(nP ) Karshenboim (1996) [104, 105]

Pachucki (2001) [101]

ln 2

+ 13 N (nS) ( α ) ln(Zα)−2 π

Nonlogarithmic Two-Loop Term ∆E(nS) Pachucki, Jentschura (2003) [109]

Single Logarithmic Two-Loop Polarization Term ∆E(nS)

38 27

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

Electron-Line Term ∆E(nS)

72

3 External Field Approximation

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

V2 = −

πZα , 2m2r

(3.96)

where both potentials are written in momentum space (see Fig. 3.25). Substituting these potentials in (3.71) one easily obtains [107]     m(Zα)−2 α(Zα)7  mr 3 11 − ln 2 ln m δl0 . (3.97) ∆E = 2 + 64 mr n3 m The Darwin potential generates the logarithmic correction to the nonrelativistic Schr¨ odinger-Coulomb wave function in (3.65), and the result in (3.97) could be obtained by taking into account this correction to the wave function in calculation of the contribution to the Lamb shift of order α(Zα)5 m. This logarithmic correction is numerically equal 14.43 kHz for the 1S-level in hydrogen, and 1.80 kHz for the 2S level. The nonlogarithmic contributions of order α(Zα)7 were never calculated directly. These corrections together with contributions in (3.60), (3.97) and the corrections of higher orders in Zα are usually parametrized with the help of an auxiliary function GSE (Zα) ∆E = GSE (Zα)

α(Zα)6  mr 3 m, πn3 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 Z without expansion in Zα. For years the values of the function GSE (Zα) for small Z were obtained from the numerical results for larger Z [65, 113, 114] with the help of extrapolation. Many such extrapolations made under slightly different assumptions are known in the literature [115, 116, 117, 118, 82, 83]. A spectacular success was achieved when the function GSE (Zα) was directly calculated numerically for low Z [112, 119, 120, 121]. Using powerful 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 contribution was obtained in this work with the numerical uncertainty 0.8 Hz for the 1S-level in hydrogen, and 1.0 Hz for the 2P -level in hydrogen. From the practical point of view the results [112, 119, 120, 121] completely solve all problems with calculation of the higher order corrections in Zα of the form

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

73

α(Zα)n in the foreseeable future. We will use these results collected in Table 3.8 in the numerical calculations of the positions of energy levels in hydrogen and helium below. The values of the function GSE (Zα) in Table 3.8 were calculated for α = 1/137.036, but it was explicitly checked that variation of the inverse fine structure constant in the interval from α = 1/137.035 999 5 to α = 1/137.036 000 5 does not change the entries in Table 3.8 at the current level of accuracy [112, 119, 120, 121]. However, a number of theoretical questions connected with the electronline corrections of order α(Zα)n remain unresolved. Comparing the perturbative results in Tables 3.4 and 3.7 with the numerical results in Table 3.8 we immediately see that the difference between the respective contributions to the Lamb shift is about 13 kHz for the 1S state in hydrogen and about 1600 kHz for He+ . This means that consistency between the analytic and numerical results requires existence of relatively large nonlogarithmic contributions of order α(Zα)7 and/or large logarithmic contributions of order α(Zα)8 . Clearly this problem deserves further investigation. An independent numerical calculation without expansion in Zα as well as an analytic calculation of nonlogarithmic corrections of order α(Zα)7 and corrections of order α(Zα)8 is also warranted. 3.5.2 Corrections Induced by the Radiative Insertions in the Coulomb Lines There are two contributions of order α(Zα)7 m to the energy shift induced by the 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 case of the Coulomb-line corrections of order α(Zα)6 considered above. Uehling Potential Contribution The logarithmic contribution is induced only by the Uehling potential in Fig. 3.10, and may easily be calculated exactly in the same way as the logarithmic contribution induced by the radiative photon in (3.97). The only difference is that now the role of the perturbation potential is played by the kernel which corresponds to the polarization contribution to the Lamb shift of order α(Zα)5 m 5 πα(Zα)2 . (3.99) V1 = 48 m2 Then we immediately obtain [65]   m(Zα)−2 α(Zα)7  mr 3 5 ln ∆E = m δl0 . (3.100) 96 mr n3 m It is not difficult to calculate analytically nonlogarithmic corrections of order α(Zα)7 generated by the Uehling potential. Using the formulae from [65] one obtains for a few lower levels (see also [122] for the case of 1S-state)

74

1S 1 2

2S 1 2

2P 1 2

2P 3 2

3S 1 2

3P 1 2

3P 3 2

4S 1 2

4P 1 2

4P 3 2

GSE (Zα) ∆E kHz GSE (Zα) ∆E kHz GSE (Zα) ∆E kHz GSE (Zα) ∆E kHz GSE (Zα) ∆E kHz GSE (Zα) ∆E kHz GSE (Zα) ∆E kHz GSE (Zα) ∆E kHz GSE (Zα) ∆E kHz GSE (Zα) ∆E kHz

Z=1

Z=2

Z=3

Z=4

Z=5

−30.290 24 (2) −1310.615 0 (9) −31.185 15 (9) −168.667 1 (5) −0.973 5 (2) −5.265 (1) −0.486 5 (2) −2.631 (1) −31.047 7 (9) −49.755 (1) −1.118 5 (9) −1.792 (1) −0.577 5 (9) −0.925 (1) −30.912 (4) −20.899 (3) −1.164 (2) −0.787 (1) −0.609 (2) −0.412 (1)

−29.770 967 (5) −82 441.40 (1) −30.644 66 (5) −10 607.60 (2) −0.949 40 (5) −328.63 (2) −0.470 94 (5) −163.02 (2) −30.512 6 (2) −3129.45 (2) −1.089 7 (2) −114.71 (9) −0.559 3 (2) −59.23 (9) −30.380 0 (9) −1314.50 (4) −1.134 1 (5) −49.07 (2) −0.590 4 (5) −25.55 (2)

−29.299 170 (2) −924 177.23 (6) −30.151 93 (2) −118 884.46 (8) −0.926 37 (2) −3 652.54 (8) −0.456 65 (2) −1 800.50 (8) −30.022 7 (2) −35 074.1 (2) −1.062 5 (1) −1307 (1) −0.542 1 (1) −675 (1) −29.892 4 (4) −14 732.6 (2) −1.105 5 (2) −544.8 (1) −0.572 9 (2) −282.4 (1)

−28.859 222 (1) −5 114 662.6 (2) −29.691 27 (1) −657 765.6 (2) −0.904 12 (1) −20 029.4 (2) −0.443 13 (1) −9 816.9 (2) −29.564 53 (6) −194 061.6 (4) −1.036 30 (6) −7 342 (6) −0.526 06 (6) −3 791 (6) −29.437 1 (2) −81 516.9 (6) −1.078 1 (1) −2 985.5 (3) −0.555 5 (1) −1 538.3 (3)

−28.443 372 3 (8) −19 229 745.3 (5) −29.255 033 (8) −2 472 310.7 (7) −0.882 478 (8) −74 577.2 (7) −0.430 244 (8) −36 359.4 (7) −29.130 61 (4) −729 420 (1) −1.010 84 (4) −28 007 (23) −0.510 76 (4) −14 460 (23) −29.006 0 (2) −306 408 (2) −1.051 20 (8) −11 104.5 (8) −0.539 41 (8) −5 698.1 (8)

3 External Field Approximation

Table 3.8. Values of the Function GSE (Zα)

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

75

     m 3 m(Zα)−2 5 23 r m, + 2 ln 2 + ln α(Zα)7 96 mr 15 m     m(Zα)−2 5 841 α(Zα)7  mr 3 ∆E(2S) = m, + 4 ln 2 + ln 96 mr 480 23 m   41 α(Zα)7  mr 3 ∆E 2P 12 = m, 3 072 23 m   7 α(Zα)7  mr 3 ∆E 2P 32 = m. (3.101) 1 024 23 m There are no obstacles to exact numerical calculation of the Uehling potential contribution to the energy shift without expansion over Zα and such calculations have been performed with high accuracy (see [65, 117] and references therein). The results of these calculations may be conveniently presented with the help of an auxiliary function GU,7 (Zα) defined by the relationship     m(Zα)−2 5 α(Zα)7  mr 3 1 ln ∆E(l, n) = (Zα) m . (3.102) δl0 + Gl,n U,7 96 mr π n3 m ∆E(1S) =

For the case of atoms with low Z (hydrogen and helium), values of the function GU,7 (Zα) for the states with n = 1, 2, 4 are tabulated in [117] and respective contributions may easily be calculated for other states when needed. These numerical results may be used for comparison of the theory and experiment instead of the results of order α(Zα)7 given above. We may also use the results of numerical calculations in order to make an estimate of uncalculated contributions of the Uehling potential of order α(Zα)8 and higher. According to [117] Gl=0,n=1 (α) = 0.428 052 . (3.103) U Comparing this value with the order α(Zα)7 m result in (3.101) we see that the difference between the exact numerical result and analytic calculation up to order α(Zα)7 is about 0.015 kHz for the 1S-level in hydrogen, and, taking into account the accuracy of experimental results, one may use analytic results for comparison of the theory and experiment without loss of accuracy. A similar conclusion is valid for other hydrogen levels. Wichmann-Kroll Potential Contribution Contribution of the Wichmann-Kroll potential in Fig. 3.16 may be calculated in the same way as the respective contribution of order α(Zα)6 m in (3.70) by taking the next term in Zα in the small momentum expansion of the Wichmann-Kroll potential in (3.69). One easily finds [93, 92]   1 31π 2 α(Zα)7  mr 3 − m δl0 . (3.104) ∆E = 16 2 880 n3 m This contribution is very small and it is clear that at the present level of experimental accuracy calculation of higher order contributions of the Wichmann-Kroll potential is not necessary.

76

3 External Field Approximation Table 3.9. Radiative Corrections of Order αn (Zα)7 m 7

r 3 4 α(Zα) (m ) m ∆E(1S) kHz ∆E(2S) kHz m n3

Logarithmic Electron-Line Contribution Karshenboim (1994) [107]

( 139 − 256

1 4

ln 2) ln(Zα)−2 δl0

Nonlogarithmic Electron-Line Contribution Mohr (1992) [114] Karshenboim (1995) [118] Logarithmic Polarization Operator Contribution Mohr (1975) [65]

5 384

ln(Zα)−2 δl0

Nonlogarithmic Polarization Operator Contribution ∆E(nS, P ), Mohr (1975) [65] Wichmann, Kroll (1956) [87] Mohr (1976) [93, 92]

1 ( 64 −

31π 2 )δ 11 520 l0

Corrections of order α2 (Zα)7

(±) α π

14.43

1.80

−0.76(5)

−0.09(1)

0.51

0.06

0.15

0.03

−0.04

−0.01

±1

±0.1

3.5.3 Corrections of Order α2 (Zα)7 m Corrections of order α2 (Zα)7 m were never considered in the literature. They should be suppressed in comparison with the corrections of order α(Zα)7 by at least 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-state and 0.1 kHz for 2S-state in hydrogen. This estimate means that calculation of these corrections is not only a challenging theoretical problem but is also interesting from the phenomenological point of view. The first results of calculation of all corrections of order α2 (Zα)n m without expansion in Zα were reported in [123]. According to the results of this paper the effective value of the coefficient B60 (see (3.76)) for the 1S level in hydrogen which takes into account also all contributions α2 (Zα)n m 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, and taken 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 data to (Zα) → 0 leads to the value of B60 twice as large as the analytic result for B60 in [109, 110]. Clearly any phenomenological conclusions in this situation are premature, and the problem of corrections of order α2 (Zα)n m requires

References

77

more work. In view of this discrepancy we will assume that uncertainty of the contribution of α2 (Zα)n is about 4 kHz for 1S state and about 0.5 kHz for 2S state in hydrogen. Concluding our discussion of the purely radiative corrections to the Lamb shift let us mention once more that the main sources of the theoretical uncertainty in these contributions is connected with the nonlogarithmic corrections of order α2 (Zα)n and uncalculated contributions of orders α3 (Zα)5 , which may be as large as a few kHz for 1S-state and a few tenths of kHz for the 2S-state in hydrogen. All other unknown purely radiative contributions to the Lamb 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 electrodynamics, 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. Kinoshita (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).

78

3 External Field Approximation

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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 dependence were discussed above in Sect. 3.1 in the framework of the Breit equation and the effective Dirac equation in the external field in Fig. 1.6. The exact mass dependence of these corrections could be easily calculated because all these corrections are induced by the one-photon exchange. The effective Dirac equation in the external field produces leading relativistic corrections with correct mass dependence because the one-photon exchange kernel is properly taken into account in this equation. Some other recoil corrections of higher orders in Zα are also partially generated by the effective Dirac equation with the external source. All such corrections are necessarily of even order in Zα since all expansions for the energy levels of the Dirac equation are effectively nonrelativistic expansions over v 2 ; they go over (Zα)2 , and, hence, the next recoil correction produced by the effective Dirac equation in the external field is 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 we will consider the even larger recoil correction of order (Zα)5 (m/M ), which is completely missed in the spectrum of the Breit equation or of the effective Dirac equation with the Coulomb potential, and which can be calculated only by taking into account the two-particle nature of the QED bound-state problem. The external field approximation is clearly inadequate for calculation of the recoil corrections and, in principle, one needs the machinery of the relativistic two-particle equations to deal with such contributions to the energy levels. The first nontrivial recoil corrections are generated by kernels with two-photon exchanges. Naively one might expect that all corrections of order (Zα)5 (m/M )m are generated only by the two-photon exchanges in Fig. 4.1. However, the situation is more complicated. More detailed consideration shows that the two-photon kernels are not sufficient and irreducible kernels in Fig. 4.2 with arbitrary number of the exchanged Coulomb phoM.I. Eides et al.: Theory of Light Hydrogenic Bound States, STMP 222, 81–98 (2007) c Springer-Verlag Berlin Heidelberg 2007 DOI 10.1007/3-540-45270-2 4


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Fig. 4.2. Irreducible kernels with arbitrary number of the exchanged Coulomb photons

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 radiative correction of order α(Zα)4 considered in Subsubsect. 3.2.1.1 when, due to a would-be infrared divergence, diagrams in Fig. 3.2 with any number of the external Coulomb photons spanned by a radiative photon give contributions of one and the same order since the apparent factor Zα accompanying each extra external photon is compensated by a small denominator connected with the small virtuality of the bound electron. Exactly the same effect arises in the case of the leading recoil corrections. All kernels with any number of exchanged Coulomb photons spanned by an exchanged transverse photon generate contributions to the leading recoil correction. Let us describe this similarity between the leading contribution to the Lamb shift and the leading recoil correction in more detail following a nice physical interpretation which was given in [1]. The leading contribution to the Lamb shift in (2.4) is proportional to the mean square of the electron radius which may be understood as a result of smearing of the fluctuating electron coordinate due to its interaction with the fluctuating electromagnetic field [2]. In the considerations leading to (2.4) we considered the proton as an infinitely heavy source of the Coulomb field. If we take into account the finiteness 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 particles proceeds exactly as in the case of the electron in the Coulomb field in (2.2) and generates the leading contribution to the Lamb shift and recoil correction of relative order (m/M )2 . This recoil factor arises because the average fluctuation of the coordinate squared equal to the average radius squared of the particle is inversely proportional to mass squared of this particle. Hence, it is clear that the average
∆r1 ∆r2  generates a recoil correction of the first order in the recoil factor m/M . Note that the correlator
∆r1 ∆r2  is different from zero only 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 mZα and smaller. For smaller distances (or larger momenta) fluctuations of the coordinates of two particles are completely uncorrelated and the correlator of two coordinates is equal to zero. Hence, the logarithmic contribution to the recoil correction originates from the momentum integration region m(Zα)2  k  m(Zα), unlike the leading logarithmic contribution to the Lamb shift which originates from a wider region m(Zα)2  k  m. A new feature of the leading recoil correction is that the upper cutoff to the logarithmic integration is determined by the inverse size of the atom. We will see below how all these qualitative features are reproduced in the exact calculations. Complete formal analysis of the recoil corrections in the framework of the relativistic two-particle equations, with derivation of all relevant kernels, perturbation theory contributions, and necessary subtraction terms may be performed along the same lines as was done for hyperfine splitting in [3]. However, these results may also be understood without a cumbersome formalism by starting with the simple scattering approximation. We will discuss recoil corrections below using this less rigorous but more physically transparent approach. As we have already realized from the qualitative discussion above, the leading recoil correction is generated at large distances, and small exchanged momenta are relevant for its calculation. The choice of gauge of the exchanged photons is, in such a case, determined by the choice of gauge in the effective Dirac equation with the one photon potential. This equation was written in the Coulomb gauge and, hence, we have to use the Coulomb gauge also in the kernels with more than one exchanged photon. Since the Coulomb and transverse propagators have different form in the Coulomb gauge it is natural to consider separately diagrams with Coulomb-Coulomb, transversetransverse and Coulomb-transverse exchanges. 4.1.1 Coulomb-Coulomb Term Coulomb exchange is already taken into account in the construction of the zero-order effective Dirac equation, where the Coulomb source plays the role of the external potential. Hence, additional contributions of order (Zα)5 could be connected only with the high-momentum Coulomb exchanges. Let us start by calculating the contribution of the skeleton Coulomb-Coulomb diagrams with on-shell external electron lines in Fig. 4.3, with the usual hope that the integrals would tell us themselves about any possible inadequacy of such an approximation. Direct calculation of the two Coulomb exchange photon contribution leads to the integral  (Zα)5 m  mr 3 ∞ dk 4 [f (µk) − µf (k)] , (4.1) ∆E = − (1 − µ2 ) πn3 m k4 0 where

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Fig. 4.3. Coulomb-Coulomb two-photon exchanges

1 f (k) = 3 1 + k 2 + √ , (4.2) 1 + k2 and µ = m/M . The apparent asymmetry of the expression in (4.1) with respect to masses of the heavy and light particle emerged because the dimensionless momentum k in this formula is measured in terms of the electron mass. At small momenta the function f (k) behaves as f (µk) − µf (k) ≈ 4(1 − µ) − µ(1 − µ)k 2 + O(k 6 ) , (4.3)  and the skeleton integral in (4.1) diverges as dk/k 4 in the infrared region. The physical meaning of this low-momenta infrared divergence is clear; it corresponds to the Coulomb exchange contribution to the Schr¨ odinger-Coulomb wave function. The Coulomb wave function graphically includes a sum of Coulomb ladders and the addition of an extra rung does not change the wave function. However, if one omits the binding energy, as we have effectively done above, one would end up with an infrared divergent integral instead of the self-reproducing Schr¨ odinger-Coulomb wave function. A slightly different way to 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 of the heavy proton pole in the box diagram. Once again these heavy particle pole contributions build the Coulomb wave function and we have to subtract them not only to avoid an apparent divergence in the approximation when we neglect the binding energy, but in order to avoid double counting. We would like to emphasize here that, even if one would forget about the threat of double counting, an emerging powerlike infrared divergence would remind us of its necessity. Any powerlike infrared divergence is cutoff by the binding energy, and has a well defined order in the parameter Zα. It is most important that the integral in (4.1) does not contain any logarithmic infrared divergence at small momenta. In such a case one can unambiguously subtract in the integrand the powerlike infrared divergent terms and the remaining integral will be completely convergent. Then only high intermediate momenta of the order of the electron mass contribute to the subtracted integral, the respective diagram is effectively local in coordinate space, and the contribution to the energy shift of order (Zα)5 is simply given by the product of this integral and the nonrelativistic Schr¨ odinger-Coulomb wave function squared at the origin. Any attempt to take into account small virtuality of the external electron lines (equivalent to taking into account nonlocality of the diagram in the coordinate space) would lead to additional factors of Zα, which we do

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

85

Fig. 4.4. Transverse-transverse two-photon exchanges

not consider yet. Direct calculation, after subtraction, of the first two leading low-frequency terms in the integrand in (4.1) immediately gives  (Zα)5 m  mr 3 ∞ dk 4 ∆Esub = − {f (µk) − µf (k) (4.4) (1 − µ2 ) πn3 m k4 0 # 4µ (Zα)5 m  mr 3 , −[4(1 − µ) − µ(1 − µ)k 2 ] = − 3 πn3 m 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 contribution with the help of the skeleton integral turned out to be feasible due to the absence of the low-frequency logarithmic divergence. For the logarithmically divergent 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 background of the logarithm in the skeleton approximation. In such cases more accurate treatment 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 contribution to the energy shift in the scattering approximation  2µ (Zα)5 m  mr 3 ∞ dk(f (k) − µ3 f (µk)) , (4.5) ∆E = − 1 − µ2 πn3 m 0 where f (k) =

1 1 −√ . k 1 + k2

(4.6)

This integral diverges only logarithmically at small momenta. Hence, this contribution does not contain either corrections of the previous order or the non-recoil corrections. The main low-frequency logarithmic divergence produces ln Zα and the factor before this logarithm may easily be calculated in the scattering approximation. This approximation is insufficient for calculation of the nonlogarithmic contribution, and the respective calculation requires a 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 exchanged momenta higher than the electron mass produce a nonvanishing contribution.

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This new integration region from the electron to the proton mass, which was discovered 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, these high momenta are responsible for a number of important contributions to the energy shifts. The high-momentum contribution to the Lamb shift is suppressed by the second power of the recoil factor (m/M )2 , and is rather small. Let us note that the result we will obtain below in this section is literally valid only for an elementary proton, since for the integration momenta comparable with the proton mass one cannot ignore the composite nature of the proton and has to take into account its internal structure as it is described by the phenomenological form factors. It is also necessary to take into account inelastic contributions in the diagrams with the two exchanged photons. We will consider these additional contributions later in Sect. 6.2 dealing with the nonelectromagnetic contributions to the Lamb shift. The state-independent high-frequency contribution as well as the lowfrequency logarithmic term are different from zero only for the S-states and may easily be calculated with the help of (4.5)   (Zα)5 m  mr 3 2µ2 ln µ − 2 ln(1 + µ) + (2 ln Zα + 2 ln 2) µ δl0 , ∆E = − 1 − µ2 πn3 m (4.7) in complete accord with the well known result [4, 5, 6]. Note that despite its appearance this result is symmetric under permutation of the heavy and light particles, as expected beforehand, since the diagrams with two transverse exchanges are symmetric. In order to preserve this symmetry we cut the integral from below at momenta of order mr Zα, calculating the contribution in (4.7). In order to obtain the state-dependent low-frequency contribution of the double transverse exchange it is necessary to restore the dependence of the graphs with two exchanged photons on the external momenta and calculate the matrix elements of these diagrams between the momentum dependent wave functions. Respective momentum integrals should be cut off from above at mr Zα. The wave function momenta provide an effective lower cutoff for the loop integrals and one may get rid of the upper cutoff by matching the lowand high-frequency contributions. The calculation for an arbitrary principal quantum number is rather straightforward but tedious [4, 5, 7, 6, 8, 9] and leads to the result $% n − 1 8(1 − ln 2) 2Zα + 2[ψ(n + 1) − ψ(1)] + + ∆E = 2 ln n n 3 ' & m M M 2 ln m − m2 ln m m (Zα)5 m  mr 3 1 − δl0 r r −2 − . δ l0 M 2 − m2 l(l + 1)(2l + 1) M πn3 m (4.8) Let us emphasize that the total contribution of the double transverse exchange is given by the matrix element of the two-photon exchanges between

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

87

the Schr¨ odinger-Coulomb wave functions, and no kernels with higher number of exchanges arise in this case, unlike the case of the main contribution to the Lamb shift discussed in Subsubsect. 3.2.1.1 and the case of the transverseCoulomb recoil contribution which we will discuss next. 4.1.3 Transverse-Coulomb Term One should expect that the contribution of the transverse-Coulomb diagrams in Fig. 4.5 would vanish in the scattering approximation because, in this approximation, there are no external vectors which are needed in order to contract the transverse photon propagator. The only available vector in the scattering approximation is the exchanged momentum itself, which turns into zero after contraction with the transverse photon propagator. It is easy to check that this is just what happens, the electron and proton traces are proportional to the exchanged momentum ki in the scattering approximation and vanish, 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 exchange do not contribute to the energy shift. We still have to explore if any contributions could be generated by the exchange of the transverse photon, with a small momentum between mr (Zα) and mr (Zα)2 , when one clearly cannot neglect the momenta of the external wave functions which are of the same order of magnitude. Hence, we have to consider all kernels in Fig. 4.2 with a transverse exchanged photon spanning an arbitrary number of Coulomb exchanges. As we have already discussed in the beginning of this section one might expect that when the momentum of the transverse photon is smaller than the characteristic atomic momentum mr Zα (in other words when the wavelength of the transverse quantum is larger than the size of the atom) the contribution to the Lamb shift generated by such a photon would only differ by an additional factor m/M from the leading contribution to the Lamb shift of order α(Zα)4 m, simply because such a photon cannot tell the electron from the proton. The extra factor m/M is due simply to the smaller velocity of the heavy particle in the atom (we remind the reader that the transverse photon interaction vertex with a charged particle in the nonrelativistic approximation is proportional to the velocity of the particle). Old-fashioned perturbation theory is more suitable for exploration of such small intermediate momenta contributions. Correction due to the exchange of the transverse photon is described in this framework simply as a second order perturbation theory

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4 Essentially Two-Particle Recoil Corrections

contribution where the role of the perturbation potential plays the transverse photon emission (absorption) vertex. In this framework the Coulomb potential plays the role of the unperturbed potential, so the simple second order contribution which we just described takes into account all kernels of the relativistic two-body equation in Fig. 4.2 with any number of the Coulomb exchanges spanned by the transverse photon. Summation over intermediate states in the nonrelativistic perturbation theory in our case means integration over all intermediate momenta. It is clear that for momenta larger than the characteristic atomic momentum mr Zα integration over external wave function momenta decouples (and we obtain instead of the wave functions their value at the origin) and one may forget about the binding energies in the intermediate states. Then the contribution of this high (larger than mr Zα) region of momenta reduces to the matrix element of the Breit interaction (transverse quanta exchange). As we have explained above, this matrix element does not give any contribution to the Lamb shift (but it gives the main contribution to hyperfine splitting, see below). All this means that the total recoil correction of order (Zα)5 (m/M )m may be calculated in the nonrelativistic approximation. Calculations go exactly in the same way as calculation of the leading low energy contribution to the Lamb shift in Subsubsect. 3.2.1.1. Due to validity of the nonrelativistic approximation the matrix elements of the Dirac matrices corresponding to the emission of transverse quanta by the constituent particles reduce to the velocities αi → pi /mi , where pi and mi are the momenta of the constituents (of the same magnitude and with opposite directions in the center of mass system) and their masses. Note that the recoil factor arises simply as a result of kinematics. Again, as in the case of the main contribution to the Lamb shift in Subsubsect. 3.2.1.1 one introduces an auxiliary parameter 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-frequency contribution of order α(Zα)4 m, and we may simply borrow the result from that calculation (compare (3.7))   m (Zα)5 m  mr 3 8 2σ < δ − ln k (n, l) . (4.9) ∆E = ln l0 0 3 mr (Zα)2 M πn3 m In the region where k ≥ σ we may safely neglect the binding energies in the denominators of the second-order perturbation theory and thus simplify the integrand. After integration one obtains  1 5 8 mr (Zα) + [ψ(n + 1) − ψ(1)] − + + ln 2 ln ∆E > = 3 nσ 2n 6  1 − δl0 m (Zα)5 m  mr 3 − δl0 . (4.10) 2l(l + 1)(2l + 1) M πn3 m Let us emphasize once more that, as was discussed above, the “high frequencies” in this formula are effectively cut at the characteristic bound state

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

89

momenta mr (Zα)/n. This leads to two specific features of the formula above. First, this expression contains the characteristic logarithm of the principal quantum 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)4 m in (3.7). The source of this difference is easy to realize. In the nonrecoil case the effective upper cutoff was supplied by the electron mass m, while at low frequency only the reduced mass enters all expressions. This mismatch between masses leads to appearance of the logarithm of the recoil factor. In the present case, the effective upper cutoff mr (Zα)/n also depends only on the reduced mass, and, hence, an extra factor under the logarithm does not arise. Matching both contributions we obtain [4, 5, 7, 6, 8]   1 5 8 2 + [ψ(n + 1) − ψ(1)] − + + ln 2 δl0 − ln k0 (n, l) ∆E = ln 3 nZα 2n 6  m (Zα)5 m  mr 3 1 − δl0 − . (4.11) 2l(l + 1)(2l + 1) M πn3 m The total recoil correction of order (Zα)5 (m/M )m is given by the sum of the expressions in (4.4), (4.8), and (4.11)    2 2n − 1 14 2 1 −1 ln(Zα) + ∆E = ln + ψ(n + 1) − ψ(1) + − 3 3 n 2n 9 ' & m M 2 2 M ln mr − m ln mr 8 7(1 − δl0 ) −2 δl0 − ln k0 (n, l) − M 2 − m2 3 3l(l + 1)(2l + 1) m (Zα)5 m  mr 3 × . (4.12) M πn3 m

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 consideration of the kernels with three exchanged photons. As in the case of recoil corrections of order (Zα)5 low exchange momenta produce nonvanishing contributions, external wave functions do not decouple, and exact calculations in the direct diagrammatic framework are rather tedious and cumbersome. There is a long and complicated history of theoretical investigation on this correction. The program of diagrammatic calculation was started in [9, 10, 11]. Corrections obtained in these works contained logarithm Zα as well as a constant term. However, completely independent calculations [12, 13] of both recoil and nonrecoil logarithmic contributions of order (Zα)6 showed that somewhat miraculously all logarithmic terms cancel in the final result. This observation required complete reconsideration of the whole problem. The breakthrough

90

4 Essentially Two-Particle Recoil Corrections

was achieved in [14], where one and the same result was obtained in two apparently different frameworks. The first, more traditional approach, used earlier in [15, 9, 10, 11], starts with an effective Dirac equation in the external field. Corrections to the Dirac energy levels are calculated with the help of a systematic diagrammatic procedure. The other logically independent calculational framework, also used in [14], starts with an exact expression for all recoil corrections of the first order in the mass ratio of the light and heavy particles m/M . This remarkable expression, which is exact in Zα, was first discovered by M. A. Braun [16], and rederived and refined later in a number of papers [17, 18, 14]. A particularly transparent representation of the Braun formula was obtained in [18]  dωd3 k 1  
n|(p−D(ω, k))G(E+ω)(p−D(ω, k))|n , (4.13) ∆Erec = − Re M (2π)4 i 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 the Coulomb gauge αk  D(ω, k) = −4πZα 2 , (4.14) ω − k2 + i0 and k(α · k) , α = γ0 γ . (4.15) αk = α − k2 Before returning to the recoil corrections of order (Zα)6 we will digress to 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 will however present below some physically transparent semiquantative arguments which make the existence and even the exact appearance of the Braun formula very natural. Let us return to the original Bethe-Salpeter equation (see (1.7)). As we have already discussed there are many ways to organize the Feynman diagrams which comprise the kernel of this equation. However, in all common perturbation theory considerations of this kernel the main emphasis is on presenting the kernel in an approximate form sufficient for calculation of corrections 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 to reject such an approach completely, and instead to organize the perturbation theory with respect to another small parameter, namely, the mass ratio of the light and heavy particles. To this end an expansion of the heavy particle propagator over 1/M was considered in [16]. It is well known (see, e.g., [16]) that in the leading order of this expansion the Bethe-Salpeter equation reduces to the Dirac equation for the light particle in the external Coulomb field created by the heavy particle. This is by itself nontrivial, but well understood by now, since to restore the Dirac equation in the external field one has to take into account irreducible kernels of the Bethe-Salpeter equation with arbitrary

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

91

Fig. 4.6. Irreducible kernels with crossed exchange photon lines

number of crossed exchange photon lines (see Fig. 4.6). Unlike the solutions of the effective Dirac equation, considered above in Sect. 1.3, the solutions of the Dirac equation obtained in this way contain as a mass parameter the mass of the light particle, and not the reduced mass of the system. This is the price one has to pay in the Braun approach for summation of all corrections in the expansion over Zα. The zero-order Green function in this approach is simply the Coulomb-Dirac Green function. The next step in the derivation of the Braun formula is to consider all kernels of the Bethe-Salpeter equation which produce corrections of order m/M . The crucial observation which immediately leads to the closed expression for the recoil corrections of order m/M , is that all corrections linear in the mass ratio are generated by the kernels where all but one of the heavy particle propagators are replaced by the leading terms in their large mass expansion, and this remaining propagator is replaced by the next term in the large mass expansion of the heavy particle propagator. Respective kernels with the minimum number of exchanged photons are the box and the crossed box diagrams in Fig. 4.1 where the heavy particle propagator is replaced by the second term in its large mass expansion. All diagrams obtained from these two by insertions of any number of external Coulomb photons between the two exchanges in Fig. 4.1 and/or of any number of the radiative photons in the electron line also generate linear in the mass ratio corrections. It is not difficult to figure out that these are the only kernels which produce corrections linear in the small mass ratio, all other kernels generate corrections of higher order in m/M , and, hence, are not interesting in this context. Then the linear in the small mass ratio contribution to the energy shift is equal to the matrix element of the two graphs in Fig. 4.1 with the total electron Green function in the external Coulomb field instead of the upper electron line. This matrix element which should be calculated between the unperturbed Dirac-Coulomb wave functions reduces after simple algebraic transformations to the Braun formula in (4.13). All terms in the Braun formula have a transparent physical sense. The term containing product pp (first obtained in [19])  originates from the exchange of two Coulomb photons, the terms with pD  and Dp correspond to the exchange of Coulomb and magnetic (transverse) D  is connected with the double transverse exchange. quanta, and the term D 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 ratio m/M are associated with the matrix element of the nonrelativistic proton

92

4 Essentially Two-Particle Recoil Corrections

Hamiltonian

(p − eA)2 . (4.16) 2M There is a clear one-to one correspondence between the terms in this nonrelativistic Hamiltonian and the respective terms in (4.13). The latter could be obtained as matrix elements of the operators which enter the Hamiltonian in (4.16) [18, 14]. H=

4.2.2 Lower Order Recoil Corrections and the Braun Formula Being exact in the parameter Zα and an expansion in the mass ratio m/M the Braun formula in (4.13) should reproduce with linear accuracy in the 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 CoulombCoulomb and Coulomb-transverse entries in (4.13). The main part of the Coulomb-Coulomb contribution in Eq. (4.13) may be written in the form (1)

∆ECoul =

1
n|p2 |n , 2M

(4.17)

while the Breit (nonretarded) part of the magnetic contribution has the form ∆EBr = −

1  k) + D(0,  k) · p|n .
n|p · D(0, 2M

(4.18)

Calculation of the matrix elements in (4.17) and (4.18) is greatly simplified by the use of the virial relations (see, e.g., [20, 21]), and one obtains the sum of the contributions in (4.17) and (4.18) in a very nice form [17] (compare (3.5)) 2 " ! m m2 − Enj m 2 = − [f (n, j) − 1] − [f (n, j) − 1] m , 2M M 2M (4.19) where Enj and f (n, j) are defined in (1.4) and (1.5), respectively. This representation again emphasizes the simple physical idea behind the Braun formula that the recoil corrections of the first order in the small mass ratio m/M are given by the matrix elements of the heavy particle kinetic energy. The recoil correction in (4.19) is the leading order (Zα)4 relativistic contribution to the energy levels generated by the Braun formula. All other contributions to the energy levels produced by the remaining terms in the Braun formula 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 potentials which were taken in account in the construction of the effective Dirac equation which produced (3.5). The exact mass dependence of the terms of order (Zα)2 (m/M )m and (Zα)4 (m/M )m is contained in (3.5), and, hence, (1)

∆ECoul + ∆EBr =

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

93

terms linear in the mass ratio in (4.19) give nothing new. It is important to realize 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 the help 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 states generated by the Braun formula was first performed in [14]. Separation of the high- and low-frequency contributions was made with the help of the -method [22] described above in Subsubsect. 3.4.1.2. Hence, not only were contributions of order (Zα)6 (m/M )m obtained in [14], but also parts of recoil corrections of order (Zα)5 linear in m/M , discussed in Sect. 4.1, were reproduced. The older methods of Sect. 4.1 produce the recoil corrections of order (Zα)5 with a correct nonrelativistic dependence on the reduced mass. The Braun formula in principle cannot reproduce this reduced mass dependence because it is constructed as a first term in a rigorous expansion over the light-heavy mass ratio. However, the results for the recoil corrections of order (Zα)5 obtained in [14] served as an important check of self-consistency of calculations. On the other hand calculations in [14], while in principle very straightforward, turned out to be rather lengthy just because all corrections of previous orders in Zα were recovered. The agreement on the magnitude of (Zα)6 (m/M )m contribution for the 1S and 2S states obtained in the diagrammatic approach and in the framework of the Braun formula achieved in [14] seemed to put an end to all problems 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 results of [14, 23] was even more confusing since the calculation in [23] was also performed with the help of the Braun formula. It was observed in [23] that due to the absence of the logarithmic contributions of order (Zα)6 (m/M )m proved earlier in [13], the calculations may be organized in a more compact way than in [14]. The main idea of [23] is that it is possible to make some approximations which are inadequate for calculation of the contributions of the previous orders in Zα, significantly simplifying calculation of the correction of order (Zα)6 (m/M )m. Due to absence of the logarithmic contributions of order (Zα)6 (m/M )m proved in [13], infrared divergences connected with these crude approximations would be powerlike and can be safely thrown away. Next, absence of logarithmic corrections of order (Zα)6 (m/M )m means that it is not necessary to worry too much about matching the low- and highfrequency (long- and short-distance in terms of [23]) contributions, since each region will produce only nonlogarithmic contributions and correction terms would be suppressed as powers of the separation parameter. Of course, such an 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

94

4 Essentially Two-Particle Recoil Corrections

the logarithm, since the exact value of the integration cutoff would not be known. Despite all these nice features of the approach of [23] its result (as well as a corrected result of a later paper [24]) contradicted the result in [14]. The discrepancy was resolved in [25], where a new logically independent calculation of the recoil corrections of order (Zα)6 (m/M )m was performed. A subtle error in dealing with cutoffs in [23] was discovered and the result of [14] was confirmed. The recoil correction of order (Zα)6 (m/M )m for S states with arbitrary principal quantum number n beyond that which is already contained in (3.5) has the form [14, 25]   7 (Zα)6 m m. (4.20) ∆Etot = 4 ln 2 − 2 n3 M This result was additionally confirmed in [26] where the recoil correction of the first order in the mass ratio was calculated without expansion over Zα for 1S and 2S states in hydrogen. 4.2.4 Higher Order in Mass Ratio Recoil Correction of Order (Zα)6 (m/M )n m to the S Levels Corrections of order (Zα)6 (m/M )n m, n ≥ 2, are generated by the diagrams in Fig. 4.6. As in the case of recoil corrections of relative order (Zα)5 both high frequency and low frequency integration regions produce nonzero contributions for n ≥ 2. Quadratic in mass ratio high frequency contribution (together with higher order terms in the mass ratio expansion) was obtained in [35] in the NRQED framework     2 (Zα)6 m m3r 8π M 8π 2 − 4 ln − 12ζ(3) + +3 . (4.21) ∆E = − 3 m 3 π 2 n3 M mM The respective low frequency contribution was never calculated, so one can use the result in (4.21) only to get an idea of the scale of the correction of order (Zα)6 (m/M )n m. Numerically the contribution in (4.21) is about −0.09 kHz for the 1S level in hydrogen, and calculation of the respective low frequency part is warranted. 4.2.5 Recoil Correction of Order (Zα)6 (m/M )m to the Non-S Levels Recoil corrections of order (Zα)6 (m/M )m to the energy levels with nonvanishing orbital angular momentum may also be calculated with the help of the Braun formula [27]. We would prefer to discuss briefly another approach, which was used in the first calculation of the recoil corrections of order (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 v 2 /c2 to the next order in the nonrelativistic expansion, and also take into account the corrections of order v 4 /c4 . Contrary to the common wisdom, such an approach turns out to be quite feasible and effective, and it was worked out in a number of papers [29, 12], and references therein. This nonrelativistic approach is limited to the calculation of the large distance (small intermediate momenta) contributions since any short distance correction leads effectively to an ultraviolet divergence in this framework. Powerlike ultraviolet divergences demonstrate the presence of the corrections of the lower order in Zα (in contrast to the scattering approximation where the presence of such corrections reveals itself in the form of powerlike infrared divergences), and are not under control in this approach. However, the logarithmic ultraviolet divergences are well under control and produce logarithms of the fine structure constant. A number of logarithmic contributions to the energy levels and decay widths were calculated in this approach [29, 12]. In the case of states with nonvanishing angular momenta the small distance contributions are effectively suppressed by the vanishing of the wave function at the origin, and the perturbation theory becomes convergent in the nonrelativistic region. Then this nonrelativistic approach leads to an exact result for the recoil correction of order (Zα)6 (m/M )m for the P states [30]   (Zα)6 m 2 2 ∆E(nP ) = m. (4.22) 1− 2 5 3n n3 M Again, this expression contains only corrections not taken into account in (3.5). The approach of [30] may be generalized for calculation of the recoil corrections to the energy levels with even higher orbital angular momenta. The general expression for the recoil corrections of order (Zα)6 (m/M )m to the energy level with an arbitrary nonvanishing angular momentum was obtained in [27]   3 l(l + 1) (Zα)6 m m. (4.23) ∆E(nL) = 1 − 3n2 n3 M 4(l − 12 )(l + 12 )(l + 32 )

4.3 Recoil Correction of Order (Zα)7 (m/M ) Recoil corrections of order (Zα)7 (m/M ) are connected only with the diagrams with multiphoton exchanges similar to the diagrams in Figs. 4.1–4.5. Following the approach developed in [1] one can classify all these diagrams and extract 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 methods in [32, 31]

96

4 Essentially Two-Particle Recoil Corrections Table 4.1. Recoil Corrections to Lamb Shift 3 (Zα)5 mr πn3 mM

Coulomb-Coulomb Term Salpeter (1952) [4] Fulton, Martin (1954) [5] Transverse-Transverse Term Bulk Contribution

− 34 δl0

!

Salpeter (1952) [4]

2 ln

+ n−1 + n

Erickson, Yennie (1965) [7] Grotch, Yennie (1969) [6] Erickson (1977) [8] Erickson, Grotch (1988) [9] Transverse-Transverse Term Very high momentum

"

8(1−ln 2) δl0 3 1−δl0 − l(l+1)(2l+1)



2 − M 2 −m M 2 ln 2

Contribution

m mr

Fulton, Martin (1954) [5] m2 ln

Erickson (1977) [8] Coulomb-Transverse Term Salpeter (1952) [4]

8 3

! ln

2 nZα

−73.75

M mr

−

−2 494.01 −305.46

−0.48

−0.06

5 494.03

720.56

−7.38

−0.92

−0.09

−0.01

−0.42

−0.05

 δl0

+ [ψ(n + 1) − ψ(1)] 1 − 2n +

Fulton, Martin (1954) [5] Erickson, Yennie (1965) [7]

−590.03

+ 2[ψ(n + 1) − ψ(1)]

2Zα n

Fulton, Martin (1954) [5]

∆E(1S) ∆E(2S) kHz kHz

− ln k0 (n, l) −



5 6

+ ln 2 δl0

1−δl0 2l(l+1)(2l+1)

"

Grotch, Yennie (1969) [6] Erickson (1977) [8] Erickson, Grotch (1988) [9] ∆E(nS) Pachucki, Grotch (1993) [14] Eides, Grotch (1997) [25] ∆E(nS), second order in mass ratio, high frequency Blokland, Czarnecki, Melnikov (2002) [35] ∆E(nL)(l
= 0) Golosov, Elkhovski, Milshtein, Khriplovich (1995) [30]

(4 ln 2 − 72 )(πZα)δl0

  −

8π 2 3



− 4 ln 2

M m

+ 8π3 + 3



− 12ζ(3) Zα m δ π M l0 3

 4(l− 2 )(l+2 )(l+ 2 ) 1

× 1−

l(l+1) 3n2

1

3

(πZα)

Jentschura, Pachucki (1996) [27] ∆E(nS) Pachucki, Karshenboim (1999) [32] Melnikov, Yelkhovsky (1999) [31]

11 − 15 (Zα)2 ln2 (Zα)δl0

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

97

11 (Zα)7 m 2 m  mr 3 ln (Zα) δl0 . (4.24) 15 πn3 M m Note that this expression is symmetric with respect to the masses of the constituents, this result was obtained without expansion in the small mass ratio, and it is valid for arbitrary mass ratio. Numerically the contribution in (4.24) is below 1 kHz. Due to linear dependence of the recoil correction on the electron-nucleus mass ratio, the respective contribution to the hydrogen-deuterium isotope shift (see Subsect. 12.1.7 below) is phenomenologically much more important, it is larger than the experimental uncertainty, and should be taken into account in comparison between theory and experiment at the current level of experimental uncertainty. Further progress was also achieved in numerical calculations of higher order recoil corrections without expansion over Zα in the framework of the approach based 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 the form (note absence of the characteristic factor (mr /m)3 which cannot be reproduced in the Braun formula framework) ∆E = −

∆E = PSE (Zα)

(Zα)5  m  m, πn3 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 in hydrogenlike atoms with Z = 1 ∼ 100 was calculated numerically in [33, 34]. The results for hydrogen are as follows     PSE 1S 12 = −0.016 16 (3), PSE 2S 12 = −0.016 17 (5) ,   (4.26) PSE 2P 12 = 0.007 72 . The difference between the contribution to the energy shift calculated using the numerical result in (4.26) and the sum of analytic results in Table 4.1 may be considered as an estimate of the single logarithmic and nonlogarithmic recoil corrections of order (Zα)7 . This difference is equal 0.64(1) kHz for the 1S level in hydrogen, and exceeds total logarithm squared contribution. Hence, the situation with the recoil corrections of order (Zα)7 resembles the case with the nonrecoil corrections of order α2 (Zα)6 . As in the case of the nonrecoil corrections the numerical result in (4.26) indicates that the nonleading corrections of order (Zα)7 are larger than the leading logarithm squared term. Clearly, an analytic calculation of the nonleading contributions of order (Zα)7 is warranted. Meanwhile, it is preferable to use apparently more precise 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 all analytic results for these corrections in Table 4.1.

98

4 Essentially Two-Particle Recoil Corrections

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 electrodynamics, 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 the recoil corrections to pure radiative effects, i.e., for corrections of the form αm (Zα)n (m/M )k . Let us start systematic discussion of such corrections with the recoil corrections to the leading contribution to the Lamb shift. The most important observation 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 additional mass dependence beyond the one already present in (3.7)–(3.24). This conclusion resembles the similar conclusion about the exact mass dependence of the contributions to the energy levels of order (Zα)4 m discussed above, and it is valid essentially for the same reason. The high frequency part of these corrections is generated only by the one photon exchanges, for which we know the 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 have already discussed the nonrecoil contribution of this order in Subsect. 3.3.2. Due to the wave function squared factor this correction naturally contained an explicit factor (mr /m)3 . Below we will discuss radiative-recoil corrections of order α(Zα)5 with mass ratio dependence beyond this factor (mr /m)3 . 5.1.1 Corrections Generated by the Radiative Insertions in the Electron Line Over the years different methods were applied for calculation of the radiativerecoil correction of order α(Zα)5 . It was first considered in the diagrammatic approach [1, 3, 2]. Later it was reconsidered on the basis of the Braun formula [4]. The Braun formula depends on the total electron Green function M.I. Eides et al.: Theory of Light Hydrogenic Bound States, STMP 222, 99–107 (2007) c Springer-Verlag Berlin Heidelberg 2007 DOI 10.1007/3-540-45270-2 5


100

5 Radiative-Recoil Corrections

in the external Coulomb field and automatically includes all radiative corrections to the electron line. Only one-loop insertions of the radiative photon in the electron line should be preserved in order to obtain corrections of order α(Zα)5 . At first sight calculation of these corrections with the help of the Braun formula may seem to be overcomplicated because, as we already mentioned above, the Braun formula produces a total correction of the first order in the mass ratio. Hence, exact calculation of the radiative-recoil correction of order α(Zα)5 with its help would produce not only the contributions we called radiative-recoil above, but also the first term in the expansion over the mass ratio of the purely radiative contribution in (3.34). This contribution should be omitted in order to avoid double counting. It is not difficult to organize the calculations based on the Braun formula in such a way that the reduced mass correction connected with the nonrecoil contribution would not show up and calculation of the remaining terms would be significantly simplified [4]. The idea is as follows. Purely radiative corrections of order α(Zα)5 , together with the standard (mr /m)3 factor were connected with the nonrelativistic heavy particle pole in the two photon exchange diagrams which corresponds to the zero order term in the proton propagator expansion over 1/M . On the other hand, the Braun formula explicitly picks up the first order term in the proton propagator heavy mass expansion. This means that the Braun formula produces the term corresponding to the reduced mass dependence of the nonrecoil contribution only when the high integration momentum goes through one of the external wave functions. New radiative-recoil contributions, which do not reduce to the tail of the mass ratio cubed factor in (3.34) are generated only by the integration region where the high momentum goes through the loop described by an explicit operator in the Braun formula. For calculation of the matrix element in this regime, it is sufficient to ignore external virtualities and to approximate the external wave functions by their values at the origin. The respective calculation reduces therefore to a variant of the scattering approximation calculation, the only difference being that the form of the skeleton structure is now determined by the Braun formula. As usual, in the scattering approximation approach the integral under consideration contains powerlike infrared divergence, which corresponds to the recoil contribution of the previous order in Zα and should simply be subtracted. After an explicit calculation of the Braun formula contribution one obtains [4] ∆E ≈ −1.324

α(Zα)5 m m δl0 . n3 M

(5.1)

This result contradicts the result of earlier calculations in the diagrammatic approach [1, 3, 2]. Later two groups of authors [5, 6, 7] realized that the radiative-recoil correction of order α(Zα)5 is generated by the high integration momenta region in the two-photon exchange kernels. Hence, there is no need to use the Braun formula for its calculation, instead it is sufficient to calculate the contribution 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 organized in completely different ways, but both groups obtained one and the same analytic expression for the radiative-recoil correction of order α(Zα)5 [5, 6, 7]   α(Zα)5 m  mr 3 3π 2 2 − 14 m δl0 , (5.2) ∆E = 6ζ(3) − 2π ln 2 + 4 π 2 n3 M m 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 quadratic in mass ratio radiative-recoil corrections of order α(Zα)5   127 α(Zα)5  m 2  mr 3 ∆E = 8 ln 2 − m δl0 , (5.3) 32 n3 M m as well as corrections of even higher orders in the small mass ratio. Due to the small magnitude of electron-proton mass ratio these higher order corrections are not phenomenologically relevant in the case of hydrogen, but could be important for other quantum electrodynamic bound systems with larger mass ratio of the constituents, like muonic hydrogen and muonium. 5.1.2 Corrections Generated by the Polarization Insertions in the Photon Lines Calculation of the radiative-recoil correction generated by the one-loop polarization insertions in the exchanged photon lines in Fig. 5.2 follows the same path as calculation of the correction induced by the insertions in the electron line. The respective correction was independently calculated analytically both in the skeleton integral approach [8] and with the help of the Braun formula [4]. Due to the simplicity of the photon polarization operator the calculation based on the scattering approximation [8] is so straightforward that we can

Fig. 5.2. Photon-line radiative-recoil corrections

102

5 Radiative-Recoil Corrections

present here all relevant formulae without making the text too technical. The skeleton integral for the recoil corrections corresponding to the diagrams with two exchanged photons in Fig. 4.1 has the form $ (    kdk k2 1 k3 16(Zα)2 |ψ(0)|2 ∞ + 1 + µ ∆E = (5.4) m2 (1 − µ2 ) (k 2 + λ2 )2 4 k 8 0 (

µ2 k 2 − 1+ 4



1 µ4 k 3 + k 8



µk 2 − 8



k2 1+ 2



µ3 k 2 + 8



µ2 k 2 1+ 2



1 + k

' ,

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 particle pole contribution (3.33) in the two photon exchange. Next, we insert the polarization operator in the integrand in (5.4) according to the rule in (3.35). The skeleton integrand in (5.4) behaves as µ/k 4 at small momenta and naive substitution in (3.35) leads to a linear infrared divergence. This divergence  dk/k 2 would be cut off at the atomic scale 1/(mZα) by the wave function momenta in an exact calculation. The low-momentum contribution would clearly be of order α(Zα)4 and we may simply omit it since we already know this correction. Thus, to obtain the recoil correction of order α(Zα)5 m it is sufficient to subtract the leading low-frequency asymptote in the radiatively corrected skeleton integrand. The subtracted integral for the radiative-recoil correction (the integral in (5.4) with inserted polarization operator and the low-frequency asymptote subtracted) has to be multiplied by an additional factor 2 needed in order to take into account that the polarization may be inserted 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 higher order in the small mass ratio [8]   2 70 α(Zα)5 m  mr 3 2π − m δl0 . (5.5) ∆E = 9 27 π 2 n3 M m Calculation of the same contribution with the help of the Braun formula was made in [4]. In the Braun formula approach one also makes the substitution in (3.35) in the propagators of the exchange photons, factorizes external wave functions as was explained above (see Subsect. 5.1.1), subtracts the infrared divergent part of the integral corresponding to the correction of previous order 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 polarization insertions in the exchanged photons was also calculated in [9]. The result of that work contradicts the results in [8, 4]. The calculations in [9] are made in the same way as the calculation of the recoil correction of order (Zα)6 (m/M )m in [10], and lead to a wrong result for the same reason.

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

103

5.1.3 Corrections Generated by the Radiative Insertions in the Proton Line We have discussed above insertions of radiative corrections either in the electron 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 particle line also should be considered. The leading order correction generated by such insertions is a radiative-recoil contribution of order (Z 2 α)(Zα)4 (m/M )m. Note that this correction contains one less power of the parameter α than the first nontrivial radiative-recoil correction of order α(Zα)5 (m/M )m generated by the radiative insertions in the electron line considered above. There is nothing enigmatic about this apparent asymmetry, since the mass dependence of the leading order contribution to the Lamb shift of order α(Zα)4 m in (3.7) is known exactly, and thus the would-be radiative correction of order α(Zα)4 (m/M )m is hidden in the leading order contribution to the Lamb shift. No new calculation is needed to obtain the correction of order (Z 2 α)(Zα)4 (m/M )m generated by the radiative insertions in the proton line. It is almost obvious that to order (Zα)4 the contributions to the energy level generated by the radiative insertions in the fermion lines are symmetric with respect to interchange of the light and heavy lines [11]. Then in the case of an elementary proton radiative-recoil correction generated by the radiative photon insertions in the heavy line may be obtained from the leading order contributions to the Lamb shift in (3.7) and (3.11) by the substitutions m → M and α → (Z 2 α). Both substitutions are obvious, the first one arises because the leading term in the infrared expansion of the first order radiative corrections to the fermion vertex contains the mass of the particle under consideration, and the second simply reminds us that the charge of the heavy particle is Ze. Hence, the Dirac form factor contribution is equal to    4(Z 2 α)(Zα)4 m3r 1 M (Zα)−2 11 1 ln + , ∆E = δl0 − ln k0 (n, l) 3 mr 72 3 πn3 M2 (5.6) and the Pauli form factor leads to the correction (Z 2 α)(Zα)4 m3r , 2πn3 M2 j(j + 1) − l(l + 1) − 3/4 (Z 2 α)(Zα)4 m2r . = l(l + 1)(2l + 1) 2πn3 M

δE|l=0 = δE|l=0

(5.7)

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

104

5 Radiative-Recoil Corrections

mass but by the size of the proton. The usual way to account for the proton structure is to substitute the proton form factor in the loop integral. After calculation we obtain instead of (5.6)  & $%  2 2 1 7π Λ2 Λ 11 2 1 Λ(Zα)−2 ln + − − + + + ··· δl0 ∆E = 3 mr 72 24 32 4M 2 3 4M 2 ' 1 4(Z 2 α)(Zα)4 m3r − ln k0 (n, l) , (5.8) 3 πn3 M2 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 upper cutoff for the logarithmic loop integration. Note also that we have obtained an additional constant in (5.8). The anomalous magnetic moment contribution in (5.7) also would be modified by inclusion of a nontrivial form factor, but the contribution to the proton magnetic moment should be considered together with the nonelectromagnetic contributions to the proton magnetic moment. The anomalous magnetic moment of the nucleus determined experimentally includes the electromagnetic contribution and, hence, even modified by the nontrivial form factor contribution in (5.7) should be ignored in the phenomenological analysis. Usually the total contribution of the proton anomalous magnetic moment is hidden in the main proton charge radius contribution defined via the Sachs electric form 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 of experimental precision the distinctions between the expressions in (5.6) and (5.8) may be ignored in the discussion of the Lamb shift measurements. These distinctions should, however, be taken into account in the discussion of the hydrogen-deuterium isotope shift (see below Subsect. 12.1.7). An alternative treatment of the correction of order (Z 2 α)(Zα)4 (m/M )m was given in [4]. The idea of this work was to modify the standard definition of the proton charge radius, and include the first order quantum electrodynamic radiative correction into the proton radius determined by the strong interactions. From the practical point of view for the nS levels in hydrogen the recipe of [4] reduces to elimination of the constant 11/72 in (5.6) and omission of the Pauli correction in (5.7). Numerically such a modification reduces the contribution to the 1S energy level in hydrogen by 0.14 kHz in comparison with the naive result in (5.6), and increases it by 0.03 kHz in comparison with the result in (5.8). Hence, for all practical needs at the current level of experimental precision there are no contradictions between our result above in (5.8), and the result in [4]. However, the approach of [4] from our point of view is unattractive; we prefer to stick with the standard definition of the intrinsic characteristics 2

This topic will be discussed in more detail below in Subsect. 6.1.1.

5.2 Corrections of Order α(Zα)6 (m/M )m

105

of the proton as determined by the strong interactions. Of course, in such an approach one has to extract the values of the proton parameters from the experimental data, properly taking into account quantum electrodynamic corrections. Another advantage of the standard approach advocated above is that in the case of the absence of a nontrivial nuclear form factor (as, for example, 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 the graphs with multiphoton exchanges. These graphs are similar to the graphs generating 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 these corrections were first obtained by the same methods and in the same works [12, 13]. The approximations used in these works did not include expansion in the small mass ratio, and therefore the obtained results are more general and are valid for arbitrary mass ratio   α(Zα)6 2 8 m3r ln (Zα)−2 δl0 . (5.9) ∆E = −mr + 3 mM πn3 The first term here is of nonrecoil nature and was discussed earlier (see the logarithm squared term in (3.53)). Note however, that the logarithm squared term in (3.53) is accompanied by the factor m3r /(mM ) instead of the factor mr in (5.9). Hence, only the leading terms in the expansions of these coefficients over the small mass ratio m/M coincide. One could insist that the mass dependence in (5.9) is “natural” because the calculations leading to it are done without expansion over the mass ratio, and are therefore exact. On the other hand, the factor m3r /(mM ) symmetric with respect to the masses naturally arises in all apparently nonrecoil calculations just from the Schr¨ odinger-Coulomb wave function squared. According to the tradition we preserve the coefficient before the logarithm squared term in the form given in (3.53). Then the new contribution contained in (5.9) has the form [12, 13] ∆E =

3 2 α(Zα)6 2 −2 mr δl0 . ln (Zα) 3 πn3 mM

(5.10)

Due to linear dependence on the electron-nucleus mass ratio, the double logarithm contribution in (5.10) is nowadays very important for phenomenological analysis of the hydrogen-deuterium isotope shift (see Subsect. 12.1.7). Single logarithmic and nonlogarithmic radiative-recoil contributions of order α(Zα)6 (m/M )m may be estimated as one half of the leading logarithm

106

Table 5.1. Radiative-Recoil Corrections

Radiative insertions in the electron line Pachucki (1995) [4]



Czarnecki, Melnikov (2001) [5, 6]

6ζ(3) − 2π 2 ln 2



2

+ 3π4 − 14

Eides, Grotch, Shelyuto (2001) [7] Radiative insertions in the electron line, second order in mass ratio



Czarnecki, Melnikov (2001) [5, 6]

8 ln 2 −

127 32

3 α(Zα)5 mr δ π 2 n3 mM l0

−13.43

−1.68

3 m mr δ M mM l0

0.009

0.001

m3 r δ mM l0

−0.41

−0.05

m3 r M2

4.58

0.58

r ln2 (Zα)−2 mM δl0

1.52

0.19

 α(Zα)5 n3

Polarization insertions Eides, Grotch (1995) [8] 2

( 2π9 −

Pachucki (1995) [4] Dirac FF insertions in the heavy line

!

1 3

ln

Λ(Zα)−2 mr

70 α(Zα) ) π2 n3 27

5

1 + (− 24 −

2

2 Λ + 23 ( 4M 2 ) + · · ·) +

"

− 13 ln k0 (n)

2

7π Λ2 32 4M 2 11 72

4(Z α)(Zα) πn3



4

δl0

Radiative insertions in the electron line Pachucki, Karshenboim (1999) [12] Melnikov, Yelkhovsky (1999) [13]

6 2 α(Zα) 3 πn3

m3

5 Radiative-Recoil Corrections

∆E(1S) kHz ∆E(2S) kHz

References

107

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

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

G. Bhatt and H. Grotch, Phys. Rev. A 31, 2794 (1985). G. Bhatt and H. Grotch, Ann. Phys. (NY) 178, 1 (1987). G. Bhatt and H. Grotch, Phys. Rev. Lett. 58, 471 (1987). K. Pachucki, Phys. Rev. A 52, 1079 (1995). A. Czarnecki and K. Melnikov, Phys. Rev. Lett. 87, 013001 (2001). I. Blokland, A. Czarnecki, and K. Melnikov, Phys. Rev. D 65, 073015 (2002). M. I. Eides, H. Grotch, and V. A. Shelyuto, Phys. Rev. A 63, 052509 (2001). M. I. Eides and H. Grotch, Phys. Rev. A 52, 1757 (1995). A. S. Yelkhovsky, preprint Budker INP 97-80, hep-ph/9710377 (1997). A. S. Elkhovskii, Zh. Eksp. Teor. Fiz. 110, 431 (1996); JETP 83, 230 (1996). T. Fulton and P. C. Martin, Phys. Rev. 95, 811 (1954). K. Pachucki and S. G. Karshenboim, Phys. Rev. A 60, 2792 (1999). 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 bound by electromagnetic forces. The characteristic size of the atom is of the order of the Bohr radius 1/(mZα), and this scale is too large for effects of other interactions (weak and strong, to say nothing about gravitational) to play a significant role. Nevertheless, in high precision experiments effects connected with the composite nature of the nucleus can become observable. By far the most important nonelectomagnetic contributions are connected with the finite size of the nucleus and its structure. Both the finite radius of the proton and its structure constants do not at present admit precise calculation from first principles in the framework of quantum chromodynamics (QCD) – the modern theory of strong interactions. Fortunately, the main nuclear parameters affecting the atomic energy levels may be either measured directly, or admit almost model independent calculation in terms of other experimentally measured parameters. Besides the strong interaction effects connected with the nucleus, strong interactions affect the energy levels of atoms via nonleptonic contributions to the photon polarization operator. Once again, these contributions admit calculation in terms of experimental data, as we have already discussed above in Subsect. 3.2.5. A minor contribution to the energy shift is also generated by 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 state properties of the proton (or other nuclei) from first principles, since the QCD perturbation expansion does not work at large (from the QCD point of view) distances which are characteristic for the proton structure, and the nonperturbative 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) c Springer-Verlag Berlin Heidelberg 2007 DOI 10.1007/3-540-45270-2 6


110

6 Nuclear Size and Structure Corrections

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

Fortunately, the characteristic scales of the strong and electromagnetic interactions are vastly different, and at the large distances which are relevant for the atomic problem the influence of the proton (or nuclear) structure may be taken into account with the help of a few experimentally measurable proton properties. The largest and by far the most important correction to the atomic energy levels connected with the proton structure is induced by its finite size. The leading nuclear structure contribution to the energy shift is completely determined by the slope of the nuclear form factor in Fig. 6.1 (compare (2.1)) k2 2
r  . (6.1) 6 The respective perturbation potential is given by the form factor slope insertion in the external Coulomb potential (see (2.3)) F (−k2 ) ≈ 1 −

−

4π(Zα) 2π(Zα) 2
r  , → 2 k 3

(6.2)

and we immediately obtain 2π(Zα) 2 2(Zα)4 3 2
r |ψ(0)|2 = mr
r δl0 . (6.3) 3 3n3 We see that the correction to the energy level induced by the finiteness of the proton charge radius shifts the energy level upwards, and is nonvanishing only for the S states1 Physically the finite radius of the proton means that the proton charge is smeared over a finite volume, and the electron spends some time inside the proton charge cloud and experiences a smaller attraction than in the case of the pointlike nucleus (Compare similar arguments in relation with the finite radius of the electron below (2.4)). The result in (6.3) needs some clarification. In the derivation above it was implicitly assumed that the photon-nucleus vertex is determined by the expression in (6.1). However, for a nucleus with spin this interaction depends on more than one form factor, and the effective slope of the photon-nucleus vertex contains in the general case some additional terms besides the nucleus radius. We will consider the real situation for nuclei of different spins below. ∆E =

1

Note the similarity of this discussion to the consideration of the level shift induced by the polarization insertion in the external Coulomb photon in Sect. 2.2. However, unlike the present case the polarization insertion leads to a negative contribution to the energy levels since the polarization cloud screens the source charge.

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 ) and Pauli (F2 ) form factors γµ F1 (k 2 ) −

1 σµν k ν F2 (k 2 ) , 2M

(6.4)

where at small momentum transfers F1 (−k2 ) ≈ 1 −


r2 F 2 k , 6

(6.5)

g−2 . 2 Hence, at low momenta the photon-nucleus interaction vertex (after the FoldyWouthuysen transformation and transition to the two-component nuclear spinors) is described by the expression    2 1 g−2
r2 F 2 1 + 8F1 M + 2F2 (0) 2 + 1−k =1−k + . (6.6) 8M 2 8M 2 6 8M 2 F2 (0) =

For an elementary proton
r2 F = 0, g = 2, and only the first term in the square brackets survives. This term leads to the well known local Darwin term in the electron-nuclear effective potential (see, e.g., [1]) and generates the 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 of the same order produced by the term proportional to the anomalous magnetic moment in (6.6). However, this is not yet the end of the story, since the proton charge radius is usually defined via the Sachs electric form factor GE , rather than the Dirac form factor F1
r2 G 2 k . GE (−k2 ) ≈ 1 − (6.7) 6 The Sachs electric and magnetic form factors are defined as (see, e.g. [3]) k2 F2 (−k2 ) , 4M 2 GM (−k2 ) = F1 (−k2 ) + F2 (−k2 ) . GE (−k2 ) = F1 (−k2 ) −

(6.8)

In terms of this new charge radius the photon-nucleus vertex above has the form   1
r2 G 1 − k2 + . (6.9) 8M 2 6 We now see that for a real proton the charge radius contribution has exactly the form in (6.3), where the charge radius is defined in (6.7). The only other term linear in the momentum transfer in the photon-nucleus vertex in (6.9)

112

6 Nuclear Size and Structure Corrections

generates the δl0 term in (3.4). Hence, if one uses the proton charge radius defined via the Sachs form factor, the net contribution of order (Zα)4 m3r /M 2 has exactly the same form as if the proton were an elementary particle with g = 2. Numerically the proton size correction is about 1100 kHz for the 1S state in hydrogen and about 140 kHz for the 2S state, and is thus much larger than the experimental accuracy of the Lamb shift measurements. Unfortunately, precise experimental determination of the proton root mean square (rms) charge radius is a tricky business. Early measurements produced contradictory results [4, 5]. For many years the value rp = 0.862 (12) fm [5] was widely accepted as a standard value of the proton rms radius. All recent analysis indicates that the real value of the proton rms radius is larger. An analysis of the low momentum transfer electron scattering data with account of the Coulomb and recoil 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 even larger value of the proton charge radius [7] rp = 0.897 (2) (1) (3) fm, where the error in the first brackets is due to statistics, the second error is due to normalization effects, and the third error reflects the model dependence. Comparing results of these two analysis one has to remember that the Coulomb corrections which played the most important role in [6] were ignored in [7]. The results of [7] depend also on specific parametrization of the nucleon form factors. The larger values of the rms charge radius were also confirmed in a comprehensive analysis in [8] which produced rp = 0.895 (18) fm. We will use this 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 contribution for a nucleus with arbitrary spin was obtained in [9]. It was shown there that one may write a universal formula for the sum of these contributions irrespective of the spin of the nucleus if the nuclear charge radius is defined with the help of the same form factor for any spin. However, for historic reasons, the definitions of the nuclear charge radius are not universal, and respective formulae have different appearances for different spins. We will discuss here only 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 elementary scalar particle the low momentum nonrelativistic expansion of the photonscalar vertex starts with the k4 /M 4 term, and, hence, the respective contribution to the energy shift is suppressed by an additional factor 1/M 2 in comparison to the spin one-half case. Hence, in the case of the scalar nucleus there is no Darwin term δl0 in (3.4) [10, 11]. Interaction of the composite scalar nucleus with photons is described only by one form factor, and the slope of this form factor is called the charge radius squared. Hence, in the scalar case the charge radius contribution is described by (6.3), and the Darwin term is absent in (3.4).

6.1 Main Proton Size Contribution

113

The spin one case is more complicated since for the vector nucleus its interaction with the photon is described in the general case by three form factors. The nonrelativistic limit of the photon-nucleus vertex in this case was considered in [12], where it was shown that with the standard definition of the deuteron charge radius (the case of the deuteron is the only phenomenologically interesting case of the spin one nucleus) the situation with the Darwin-Foldy and the charge radius contribution is exactly the same as in the case of the scalar particle. Namely, the Darwin-Foldy contribution is missing in (3.4) and the charge radius contribution is given by (6.3). It would be appropriate to mention here recent work [13], where a special choice of definition of the nuclear charge radius is advocated, namely it is suggested to include the Darwin-Foldy contribution in the definition of the nuclear charge radius. While one can use any consistent definition of the nuclear charge radius, this particular choice seems to us to be unattractive since in this case even a truly pointlike particle in the sense of quantum field theory (say an electron) would have a finite charge radius even in zero-order approximation. The phenomenological aspects of the deuteron charge radius contribution to the hydrogen-deuterium isotope shift will be discussed later in Subsect. 12.1.7. 6.1.3 Empirical Nuclear Form Factor and the Contributions to the Lamb Shift In all considerations above we have assumed the most natural theoretical definition of nuclear form factors, namely, the form factor was assumed to be an intrinsic property of the nucleus. Therefore, the form factor is defined via the effective nuclear-photon vertex in the absence of electromagnetic interaction. Such a form factor can in principle be calculated with the help of QCD. The electromagnetic corrections to the form factor defined in this way may be calculated in the framework of QED perturbation theory. Strictly speaking all formula above are valid with this definition of the form factor. However, in practice, form factors are measured experimentally and there is no way to switch off the electromagnetic interaction. Hence, in order to determine the form factor experimentally one has in principle to calculate electromagnetic corrections to the elastic electron-nucleus scattering which is usually used to measure the form factors [4, 5]. In the usual fit to the experimental data not all electromagnetic corrections to the scattering amplitude are usually taken into account (see e.g., discussion in [14, 15]). First, all vacuum polarization insertions, excluding the electron vacuum polarization, are usually ignored. This means that respective contributions to the energy shift in (3.32) are swallowed by the empirical value of the nuclear charge radius squared. They are effectively taken into account in the contribution to the energy shift in (6.3), and should not be considered separately. Next there are the

114

6 Nuclear Size and Structure Corrections

corrections of order (Z 2 α)(Zα)4 to the energy shift2 . The perturbative electromagnetic contributions to the Pauli form factor should be ignored, since they definitely enter the empirical value of the nuclear g-factor. The situation is a bit more involved with respect to the electromagnetic contribution to the Dirac nuclear form factor. The QED contribution to the slope of the Dirac form factor is infrared divergent, and, hence, one cannot simply include it in the 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 electronnucleus scattering if the soft photon radiation is properly taken into account (see, e.g., [10, 3]). This means that the proper determination of the empirical proton form factor, on the basis of the experimental data, requires account of the electromagnetic radiative corrections, and the measured value of the nuclear charge radius squared does not include the electromagnetic contribution. Hence, the radiative correction of order (Z 2 α)(Zα)4 in (5.6) should be included in the comparison of the theory with the experimental data on the energy shifts.

6.2 Nuclear Size and Structure Corrections of Order (Zα)5 m Corrections of relative order (Zα)5 connected with the nonelementarity of the nucleus are generated by the diagrams with two-photon exchanges. As usual all corrections of order (Zα)5 , originate from high (on the atomic scale) intermediate momenta. Due to the composite nature of the nucleus, besides intermediate elastic nuclear states, we also have to consider the contribution of the diagrams with inelastic intermediate states. 6.2.1 Nuclear Size Corrections of Order (Zα)5 m Let us consider first the contribution generated only by the elastic intermediate nuclear states. This means that we will treat the nucleus here as a particle which interacts with the photons via a nontrivial experimentally measurable form factor GE (k 2 ), i.e. the electromagnetic interaction of our nucleus is nonlocal, but we will temporarily ignore its excited states. As usual we start with the skeleton integral contribution in (3.33) corresponding to the two-photon skeleton diagram in Fig. 3.8. Insertion of the factor GE (−k 2 ) − 1 in the proton vertex corresponds to the presence of a nontrivial proton form factor3 . 2

3

We have already considered these corrections together with other radiative-recoil corrections above, in Subsect. 5.1.3. This discussion will be partially reproduced here in order to make the present section self-contained. Subtraction is necessary in order to avoid double counting since the subtracted term in the vertex corresponds to the pointlike proton contribution, already taken into account in the effective Dirac equation.

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

115

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

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

We have to consider diagrams in Fig. 6.2 with insertion of one factor GE (−k 2 ) − 1 in the proton vertex (there are two such diagrams, hence an extra factor two below)4   (Zα)5 ∞ dk  GE (−k 2 ) − 1 , ∆E = −32mm3r (6.10) 3 4 πn k 0 and the diagrams in Fig. 6.3 with insertion of two factors GE (−k 2 ) − 1 in two proton vertices  2 (Zα)5 ∞ dk  GE (−k 2 ) − 1 . ∆E = −16mm3r (6.11) 3 4 πn k 0 The low momentum integration region in the integral in (6.10) produces a linearly divergent infrared contribution, which simply reflects the presence of the correction of order (Zα)4 , calculated in Sect. 6.1. We need to subtract this divergent contribution, connected with the low energy expansion of the form factor in (6.7). Technically subtraction reduces to insertion of the term
r2 k 2 /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 by the 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 extra factor m3 in the numerators in (6.10), (6.11) and (6.12) in comparison with the factor in the skeleton integral (3.33). Notice also the minus sign before the momentum in the arguments of form factors; it arises because in the equations below k = |k|.

116

6 Nuclear Size and Structure Corrections

of the form factor scale, to be distinguished from high momenta in other chapters which often meant momenta of the scale of the electron mass. The characteristic momenta in the present case are much higher. The total contribution of order (Zα)5 is given by the sum of the subtracted expression in (6.10) and the contribution in (6.11)   5  ∞ dk
r2  2 2 2 3 (Zα) k ∆E = −16mmr GE (−k ) − 1 + . (6.12) πn3 0 k 4 3 This expression is usually written in the form [16, 17]5 ∆E = −

m(Zα)5 3 3 mr
r (2) , 3n3

(6.13)

where
r3 (2) is called the third Zemach moment [18], and is defined as
r (2) 3

48 = π

 0

∞

dk k4

 G2E (−k 2 )


r2  2 k −1+ 3

 .

(6.14)

Carrying out the Fourier transformation, one obtains the coordinate space representation for the third Zemach moment, in terms of the weighted convolution of two nuclear charge densities ρ(r) [18]  3
r (2) ≡ d3 r1 d3 r2 ρ(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 is the form factor scale. Hence, this correction is suppressed in comparison with the leading proton size contribution not only by an extra factor Zα but also by the 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 below Subsect. 6.3.2), since one factor m/Λ in (6.13) is traded for a much larger factor Zα in that logarithmically enhanced contribution. The result in (6.13) depends on the third Zemach moment, or in other words, on a nontrivial weighted integral of the product of two charge densities, and cannot be measured directly, like the rms proton charge radius. This means that the correction under consideration may only conditionally be called the proton size contribution. It depends on the fine details of the form factor momentum dependence, and not only on the directly measurable low-momentum behavior of the form factor. This feature of the result is quite natural taking into account high intermediate momenta characteristic for the integral in (6.12). In the practically important cases of hydrogen and deuterium, model dependent results for this contribution may be obtained with 5

The result in [17] has the factor m4r instead of mm3r before the integral in (6.13). This difference could become important only after calculation of a recoil correction to the contribution in (6.13).

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

117

the help of model dependent proton form factors and/or model independent experimental data on electron-proton scattering [17, 19]. The latest model independent analysis of the proton-electron scattering and application of the expression in (6.14) [19] produced the value
r3 (2) = 2.71 (13) fm3 for the third Zemach moment of the proton. The respective nuclear radius correction of order (Zα)5 is equal to −40.0 (1.9) Hz for the 1S state and −5.0 (2) Hz for the 2S state in hydrogen. These corrections are rather small. A much larger contribution 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 than the proton radius, and the respective correction to the energy levels is also larger. 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 in the 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 (−k 2 ) − 1) in the external field skeleton diagrams in Fig. 3.8. Technically the external field diagrams correspond to the heavy particle pole contribution in the sum of all skeleton diagrams with two exchanged photons in Fig. 4.3-Fig. 4.5. In the absence of form factors the nonpole contributions of the diagrams in Fig. 4.3-Fig. 4.5 were suppressed by the recoil factor m/M in comparison with the heavy pole contribution, and this justified their separate consideration. However, as we have seen, insertion of the form factors pushes the effective integration momenta in the high momenta region ∼ Λ even in the external field diagrams. Then even the external field contribution contains the recoil factor (m/Λ)3 . We might expect that, after insertion of the proton form factors, the nonpole contribution of the skeleton diagrams with two exchanged photons in Fig. 4.3-Fig. 4.5 would contain the recoil factor (m/M )(m/Λ)2 , and would not be parametrically suppressed in comparison with the pole contribution in (6.13). The total contribution of the skeleton diagrams with the proton form factor insertions was calculated in [22] for the 2S state, and the difference between this result and the nonrecoil result in (6.13) turned out to be −0.25 Hz. At the current level of theoretical and experimental accuracy we can safely ignore such tiny differences between the pole and total proton size contributions of order (Zα)5 . 6.2.2 Nuclear Polarizability Contribution of Order (Zα)5 m to S-Levels The description of nuclear structure corrections of order (Zα)5 m in terms of nuclear size and nuclear polarizability contributions is somewhat artificial. As we have seen above the nuclear size correction of this order depends not on the 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

118

6 Nuclear Size and Structure Corrections

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

generate corrections which are even smaller than those connected with the third Zemach moment, but this does not happen. In reality, the contribution of the inelastic intermediate states turns out to be even larger than the elastic contribution since the powerlike decrease of the form factor is compensated in this case by the summation over a large number of nuclear energy levels. The inelastic contributions to the energy shift were a subject of intensive study for a 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 of certain integrals of the inelastic nuclear structure functions, and the dominant contribution is produced by the nuclear electric and magnetic polarizabilities. The main feature of the polarizability contribution to the energy shift is its logarithmic enhancement [26, 30]. The appearance of the large logarithm may easily be understood with the help of the skeleton integral. The heavy particle factor in the two-photon exchange diagrams is now described by the photon-nucleus inelastic forward Compton amplitude [31] ¯ k2 )B · B ∗ , M =α ¯ (ω, k2 )E · E ∗ + β(ω,

(6.17)

¯ k2 ) are proton (nuclear) electric and magnetic polarwhere α ¯ (ω, k2 ) and β(ω, izabilities. In terms of this Compton amplitude the two-photon contribution has the form  ˆ j} 4α(mr Zα)3 d4 k Dim Djn T r{γi ((1 + γ0 )m − k)γ ∆E = − Mmn , 3 4 4 2 n (2π) i k k − 2mk0 (6.18) which reduces after elementary transformations to6  ∞ α(Zα)3 3 4 ∆E = − m m dk{[(k + 8) k 2 + 4 − k(k 4 + 2k 2 + 6)]¯ α(ω, k 2 ) r 4πn3 0 3 ¯ k 2 )} , + [k 3 (k 2 + 4) 2 − k(k 4 + 6k 2 + 6)]β(ω, (6.19) and we remind the reader that an extra factor Z 2 α is hidden in the definition of the polarizabilities. Ignoring the momentum dependence of the polarizabilities, one immediately comes to a logarithmically ultraviolet divergent integral [31]   α(Zα)3 3  Λ ¯ + O(1) , (6.20) ∆E = − mr m 5¯ α(0, 0) − β(0, 0) ln πn3 m 6

Integration in (6.19) goes over dimensionless momentum k measured in electron mass.

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

119

where Λ is an ultraviolet cutoff. In real calculations the role of the cutoff is played by the characteristic excitation energy of the nucleus. The sign of the energy shift is determined by the electric polarizability and has a clear physical origin. The electron polarizes the nucleus, an additional attraction between the induced dipole and the electron emerges, and shifts the energy level down. In the case of hydrogen the characteristic excitation energy is about 300 MeV, the logarithm is rather large, and the logarithmic approximation works very well. Using the proton polarizabilities [32] one easily obtains the polarizability contribution for the proton nS state [26, 31, 33, 34] ∆E(nS) = −

70 (11) (7) Hz , n3

(6.21)

where the error in the first parenthesis describes the accuracy of the logarithmic approximation, and the error in the second parenthesis is due to the experimental data on the polarizabilities. A slightly different numerical polarizability contribution ∆E(1S) = −

95 (7) Hz , n3

(6.22)

was obtained in [35]. Discrepancy between the results in (6.21) and (6.22) is preserved even when both groups of authors use one and the same data on proton polarizabilities from [36]. Technically the disagreement between the results in [26, 31, 33, 34] and [35] is due to the expression for the polarizability energy shift in the form of an integral of the total photoabsorption cross section, which was used in [35]. This expression was derived in [30] under the assumption that the invariant amplitudes for the forward Compton scattering satisfy the dispersion relations without subtractions. Without this subtraction term the dominant logarithmic contribution to the energy shift becomes pro¯ while portional to the sum of the electric and magnetic polarizabilities α ¯ + β, in [26, 31, 33, 34] this contribution is proportional to another linear combination of polarizabilities (see (6.20)). Restoring this necessary subtraction term the authors of [22, 37] obtained the polarizability contribution ∆E(1S) = −

82 (9) Hz , n3

(6.23)

which is compatible both with (6.21) and (6.22). In the other experimentally interesting case of deuterium, nuclear excitation energies are much lower and a more accurate account of the internal structure of the deuteron is necessary. As is well known, due to smallness of the binding energy the model independent zero-range approximation provides a very accurate description of the deuteron. The polarizability contributions to the energy shift in deuterium are again logarithmically enhanced and in the zero-range approximation one obtains a model independent result [38, 39]

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6 Nuclear Size and Structure Corrections

∆E = −

     1 α(Zα)3 3 8I 8I + − 1.24 , m m 5α ln ln − β r d d πn3 m 20 m

(6.24)

where κ = 45.7 MeV is the inverse deuteron size, I = κ2 /mp is the absolute value of the deuteron binding energy, and the deuteron electric and magnetic polarizabilities are defined as αd (ω) =

2 2  (En + I)|
0|r|n|2 (Z α) , 3 ω 2 − (En + I)2 n

βd (0) =

α(µp − µn )2 1 + 8mp κ2 1+

κ1 3κ κ1 κ

,

(6.25)

where κ1 = 7.9 MeV determines the position of the virtual level in the neutronproton 1 S0 state. Numerically the polarizability contribution to the deuterium 1S energy shift 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 electric polarizability, 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 contribution with an accuracy of about one percent. Experimental data on the deuterium-hydrogen isotope shift (see Table 12.4 below) have an accuracy of about 0.1 kHz and, hence, a more accurate theoretical result for the polarizability contribution is required. In order to obtain such a result it is necessary to go beyond the zero range approximation, and take the deuteron structure into account in more detail. Fortunately, there exist a number of phenomenological potentials which describe the properties of the deuteron in all details. Some calculations with realistic proton-neutron potentials were performed [40, 41, 42, 43]. The most precise results were obtained 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 the polarizability contribution of the proton in the hydrogen atom. Meanwhile, as may be seen from (6.21), proton polarizability contributions are comparable to the accuracy of the polarizability contribution in (6.28), and cannot be ignored. The deuteron is a weakly bound system and it is natural to assume that the deuteron polarizability is a sum of the polarizability due to

6.3 Nuclear Size and Structure Corrections of Order (Zα)6 m

121

relative motion of the nucleons and their internal polarizabilities. The nucleon polarizabilities in the deuteron coincide with the polarizabilities of the free nucleons well within the accuracy of the logarithmic approximation [33]. Therefore the proton polarizability contribution to the hydrogen-deuterium isotope shift cancels, and the contribution to this shift, which is due to the internal polarizabilities of the nucleons, is completely determined by the neutron polarizability. This neutron polarizability contribution to the hydrogendeuterium 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 obtained from the analysis of the experimental data on the proton and deuteron structure functions in [37].

6.3 Nuclear Size and Structure Corrections of Order (Zα)6 m 6.3.1 Nuclear Polarizability Contribution to P -Levels The leading polarizability contribution of order (Zα)5 obtained above is proportional to the nonrelativistic wave function at the origin squared, and hence, exists only for the S states. The leading polarizability contribution to the non S-states is of order (Zα)6 and may easily be calculated. Consider the Compton amplitude in (6.17) as the contribution to the bound state energy induced by the external field of the electron at the nuclear site. Then we calculate the matrix element in (6.17) between the electron states, considering the field strengths from the Coulomb field generated by the orbiting electron. We obtain [23] (the overall factor 1/2 is due to the induced nature of the nuclear dipole)     1 3n2 − l(l + 1) α(Zα)4 α ¯ m4r αα ¯ . n, l 4 n, l = − 5 ∆E = − 3 1 1 2 r 2 2n (l + 2 )(l + 1)(l + 2 )l(l − 2 ) (6.30) This energy shift is negative because when the electron polarizes the nucleus this leads to an additional attraction of the induced dipole to the electron. The contribution induced by the magnetic susceptibility may also be easily calculated [23], but it is even of higher order in Zα (order (Zα)8 ) since the magnetic field strength behaves as 1/r3 . This additional suppression of the magnetic effect is quite reasonable, since the magnetic field itself is a relativistic effect. Consideration of the P -state polarizability contribution provides us with a new perspective on the S-state polarizability contribution. One could try to consider the matrix element in (6.30) between the S states. Due to nonvanishing of the S-state wave functions at the origin this matrix element is

122

6 Nuclear Size and Structure Corrections

linearly divergent at small distances, which once more demonstrates that the S-state contribution is of a lower order in Zα than for the P -state, and that for its calculation one has to treat small distances more accurately than was done in (6.30). 6.3.2 Nuclear Size Correction of Order (Zα)6 m 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 contribution is due to the large magnitude of the characteristic momenta responsible for this correction. The nature of the suppression is especially clear in the case of the Zemach radius contribution in (6.13), which contains the small factor m3r
r3 (2) . The nuclear size correction of order (Zα)6 m contains an extra factor Zα in comparison with the nuclear size and polarizability corrections of order (Zα)5 , but its main part is proportional to the proton charge radius squared. Hence, we should expect that despite an extra power of Zα this correction is numerically larger than the nuclear size and polarizability contributions of the previous order in Zα. As we will see below, calculations confirm this expectation and, moreover, the contribution of order (Zα)6 m is additionally logarithmically enhanced. Nuclear size corrections of order (Zα)6 may be obtained in a quite straightforward way in the framework of the quantum mechanical third order perturbation theory. In this approach one considers the difference between the electric field generated by the nonlocal charge density described by the nuclear form 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 proportional to the nuclear charge radius squared may also be easily obtained in a simpler way, which clearly demonstrates the source of the logarithmic enhancement of this contribution. We will first discuss in some detail this simple-minded approach, which essentially coincides with the arguments used above to obtain the main contribution to the Lamb shift in (2.4), and the leading proton radius contribution in (6.3). The potential of an extended nucleus is given by the expression  ρ(r ) , (6.31) V (r) = −Zα d3 r |r − r
| where ρ(r ) is the nuclear charge density. Due to the finite size of the nuclear charge distribution, the relative distance between the nucleus and the electron is not constant but is subject to additional fluctuations with probability ρ(r). Hence, the energy levels experience an additional shift       3     (6.32) ∆E = n d r ρ(r )[V (r + r ) − V (r)]n .

6.3 Nuclear Size and Structure Corrections of Order (Zα)6 m

123

Taking into account that the size of the nuclear charge distribution is much smaller than the atomic scale, we immediately obtain  2π (Zα)
r2  d3 rρ(r)ψ(r)+ ψ(r) . (6.33) ∆E = 3 We will now discuss contributions contained in (6.33) for different special cases. 6.3.2.1 Correction to the nS-Levels In the Schr¨ odinger-Coulomb approximation the expression in (6.33) reduces to the leading nuclear size correction in (6.3). New results arise if we take into account Dirac corrections to the Schr¨odinger-Coulomb wave functions of relative order (Zα)2 . For the nS states the product of the wave functions in (6.33) has the form (see, e.g, [17])    9 11 1 2mrZα + ψ(n) + 2γ + 2 − − |ψ(r)|2 = |ψSchr (r)|2 1 − (Zα)2 ln , n 4n n 4 (6.34) and the additional contribution to the energy shift is equal to    11 2mrZα 1 2(Zα)6 3 2 9 − m
r  ln − ∆E = − + ψ(n) + 2γ + . r 3n3 n 4n2 n 4 (6.35) This expression nicely illustrates the main qualitative features of the order (Zα)6 nuclear size contribution. First, we observe a logarithmic enhancement connected with the singularity of the Dirac wave function at small distances. Due to the smallness of the nuclear size, the effective logarithm of the ratio of the atomic size and the nuclear size is a rather large number; it is equal to about −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] produces some additional state-independent terms with the net result being a few percent different from the naive result above. The additional state-independent contribution beyond the naive result above has the form [20]   2(Zα)6 3
r2 
r3 
1r  − + d3 rd3 r ρ(r)ρ(r )θ(|r| − |r |) m ∆E = r 3n3 2 3   r3 r3 r2 − r2 |r | − + + × (r2 + r2 ) ln |r| 3|r| 3|r | 3  + 6 d3 rd3 r d3 r ρ(r)ρ(r )ρ(r )θ(|r| − |r |)θ(|r | − |r |)  2   r |r | r4 r3 1 1 ln  e − + + × 3 |r | 45|r||r | 9 |r | |r|  r2 2r r2 r2 r2 + − + . (6.36) 36r2 9r 9

124

6 Nuclear Size and Structure Corrections

Note that, unlike the leading naive terms in (6.35), this additional contribution depends on more detailed features of the nuclear charge distribution than simply the charge radius squared. Detailed numerical calculations in the interesting cases of hydrogen and deuterium were performed in [20]. Nuclear size contributions of order (Zα)6 to the 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 polarizability contributions of the previous order in Zα. Respective corrections to the energy levels in deuterium are even much larger than in hydrogen due to the large radius of the deuteron. The nuclear size contribution of order (Zα)6 to the 2S − 1S splitting in deuterium is equal to (we have used in this calculation the value of the deuteron charge radius obtained in [44] from the analysis of all available experimental data) ∆E = −3.43 kHz ,

(6.37)

∆E = −0.61 kHz .

(6.38)

and in hydrogen We see that the difference of these corrections gives an important contribution to 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 charge radius squared contributions of lower order, unlike the case of S states, and we immediately obtain [17] ∆E(nPj ) =

(n2 − 1)(Zα)6 3 2 mr
r δj 12 . 6n5

(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 with the 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 r 2 m3r to the Finite Size Effect Due to the large magnitude of the leading nuclear size correction in (6.3) at the current level of experimental accuracy one also has to take into account 7

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 and 12.1.6.

6.4 Radiative Corrections to the Finite Size Effect

125

radiative corrections to this effect. These radiative corrections were first discussed and greatly overestimated in [45]. The problem was almost immediately clarified in [46], where it was shown that the contribution is generated by large intermediate momenta states and is parametrically a small correction of order α(Zα)5 m3r
r2 . On the basis of the estimate in [46] the authors of [14] expected the radiative correction to the leading nuclear charge radius contribution to be of order 10 Hz for the 1S-state in hydrogen. The large magnitude of the characteristic integration momenta [46] is quite clear. As we have seen above, in the calculation of the main proton charge radius contribution, the exchange momentum squared factor in the numerator connected with the proton radius cancels with a similar factor in the denominator supplied by the photon propagator. Any radiative correction behaves as k2 at small momenta, and the presence of such a correction additionally suppresses small integration momenta and pushes the characteristic integration momenta into the relativistic region of order of the electron mass. Hence, the corrections may be calculated with the help of the skeleton integrals in the scattering approximation. The characteristic integration momenta in the skeleton integral are of order of the electron mass, and are still much smaller than the scale of the proton form factor. As a result the respective contribution to the energy shift depends only on the slope of the form factor. The actual calculation essentially coincides with the calculation of the corrections of order α2 (Zα)5 to the Lamb shift in Subsect. 3.3.3 but is technically simpler 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 size effect, namely, the diagrams with one-loop radiative insertions in the electron line in Fig. 6.5, and the diagrams with one-loop polarization insertions in one of 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 an integral for the electron-line contribution which does not depend on any parameters, and can be easily calculated numerically with an arbitrary precision [49]. Like a more complicated integral for the corrections of order α2 (Zα)5 in [48] this integral also admits an analytic evaluation, and the analytic result was obtained in [50, 51]   8 ln 2 23 α(Zα)5 3 2 − mr
r δl0 . (6.40) ∆Ee−line = 3 6 n3 6.4.1.2 Polarization Correction Calculation of the diagrams with the polarization operator insertion proceeds exactly as in the case of the electron factor insertion. The only difference is

126

6 Nuclear Size and Structure Corrections

Fig. 6.5. Electron-line radiative correction to the nuclear size effect. Bold dot corresponds to proton form factor slope

Fig. 6.6. Coulomb-line radiative correction to the nuclear size effect. Bold dot corresponds to proton form factor slope

that one inserts an additional factor 4 in the skeleton integral to take into account all possible ways to insert the polarization operator and the slope of the proton form factor in the Coulomb photons. After an easy analytic calculation one obtains [49, 52, 53] ∆Epol =

α(Zα)5 3 2 mr
r δl0 . 2n3

(6.41)

6.4.1.3 Total Radiative Correction The total radiative correction to the proton size effect is given by the sum of contributions in (6.41) and (6.40)   8 ln 2 10 α(Zα)5 3 2 − ∆E = mr
r δl0 . (6.42) 3 3 n3 This correction without separation of the polarization operator and electron factor contributions was also considered in [54]. Correcting an apparent misprint 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 minor discrepancy is unclear, but the result in (6.42) was independently confirmed by at least two different groups [49, 50, 51] and is now firmly established. Numerically the total radiative contribution in (6.42) for hydrogen is equal to ∆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 the hydrogen-deuterium isotope shift.

6.5 Weak Interaction Contribution

127

6.4.2 Higher Order Radiative Corrections to 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 the large logarithm ln(1/(mr0 )) (r0 here is the radius of the nucleus) was obtained in [50, 51, 55]. Its relative magnitude with respect to the leading finite size contribution 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 and P states. These corrections originate from large distances and are enhanced by large infrared logarithms ln(Zα) squared. Radiative corrections for S states are enhanced by the large logarithm ln(Zα) squared, and are like the technically similar purely radiative correction to the Lamb shift of order α2 (Zα)6 m considered above. The coefficient before this logarithm squared can be easily obtained. 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 to imagine that these nice results would find any phenomenological applications for 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 Zboson exchange in Fig. 6.7, which may be described by the effective local low-energy Hamiltonian  2 1 mM 16πα 2 − sin θW H (L) = − 2 2 MZ2 sin θW cos θW 4     × d3 x ψ + (x)ψ(x) Ψ + (x)Ψ (x) , Z

(6.45)

where MZ is the Z-boson mass, θW is the Weinberg angle, and ψ and Ψ are the two-component wave functions of the light and heavy particles, respectively. Then we easily obtain the weak interaction contribution to the Lamb shift in hydrogen [56]

Fig. 6.7. Z-boson exchange diagram

128

6 Nuclear Size and Structure Corrections Table 6.1. Nuclear Size and Structure Corrections ∆E(1S) kHz ∆E(2S) kHz

Leading nuclear size contribution

2 (Zα)4 m3r r 2 δl0 3n3

1 253 (50)

157 (6)

m(Zα)5 m3r r 3 (2) δl0 3n3

−0.040 (19)

−0.005

Proton form factor contribution of order (Zα)5 Borisoglebsky, Trofimenko (1979) [16] Friar (1979) [17] −

Friar, Sick (2005) [19] Polarizability contribution ∆E(nS) Startsev, Petrun’kin, Khomkin (1976) [26]

α(Zα)3

− πn3 mr 3 m ¯ 0)] ln Λ −0.070(11)(7) −0.009(1)(1) ×[5α(0, ¯ 0) − β(0, m

Khriplovich, Sen’kov (1997) [31, 33, 34] Polarizability contribution ∆E(nP ) Ericson, Hufner (1972) [23]

α(Zα)4 αm ¯ 4 r 2 3n2 −l(l+1) × 5 2n (l+ 3 )(l+1)(l+ 1 )l(l− 1 ) 2 2 2

−

Nuclear size correction of order (Zα)6 ∆E(nS) Borisoglebsky, Trofimenko (1979) [16] Friar (1979) [17]