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

Springer Tracts in Modern Physics Volume 222 Managing Editor: G Höhler, Karlsruhe Editors: A Fujimori, Chiba J Kühn, Karlsruhe Th Müller, Karlsruhe F Steiner, Ulm J Trümper, Garching C Varma, California P Wölfle, Karlsruhe Starting with Volume 165, Springer Tracts in Modern Physics is part of the [SpringerLink] service For all customers with standing orders for Springer Tracts in Modern Physics we offer the full text in electronic form via [SpringerLink] free of charge Please contact your librarian who can receive a password for free access to the full articles by registration at: springerlink.com If you not have a standing order you can nevertheless browse online through the table of contents of the volumes and the abstracts of each article and perform a full text search There you will also find more information about the series www.pdfgrip.com Springer Tracts in Modern Physics Springer Tracts in Modern Physics provides comprehensive and critical reviews of topics of current interest 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http://wyvern.phys.s.u-tokyo.ac.jp/welcome_en.html Elementary Particle Physics, Editors Johann H Kühn C Varma Editor for The Americas Institut für Theoretische Teilchenphysik Universität Karlsruhe Postfach 69 80 76128 Karlsruhe, Germany Phone: +49 (7 21) 08 33 72 Fax: +49 (7 21) 37 07 26 Email: johann.kuehn@physik.uni-karlsruhe.de www-ttp.physik.uni-karlsruhe.de/∼jk Department of Physics University of California Riverside, CA 92521 Phone: +1 (951) 827-5331 Fax: +1 (951) 827-4529 Email: chandra.varma@ucr.edu www.physics.ucr.edu Thomas Müller Institut für Experimentelle Kernphysik Fakultät für Physik Universität Karlsruhe Postfach 69 80 76128 Karlsruhe, Germany Phone: +49 (7 21) 08 35 24 Fax: +49 (7 21) 07 26 21 Email: thomas.muller@physik.uni-karlsruhe.de www-ekp.physik.uni-karlsruhe.de Fundamental Astrophysics, Editor Joachim Trümper Max-Planck-Institut für Extraterrestrische Physik Postfach 13 12 85741 Garching, Germany Phone: +49 (89) 30 00 35 59 Fax: +49 (89) 30 00 33 15 Email: jtrumper@mpe.mpg.de www.mpe-garching.mpg.de/index.html Peter Wölfle Institut für Theorie der Kondensierten Materie Universität Karlsruhe Postfach 69 80 76128 Karlsruhe, Germany Phone: +49 (7 21) 08 35 90 Fax: +49 (7 21) 69 81 50 Email: woelfle@tkm.physik.uni-karlsruhe.de www-tkm.physik.uni-karlsruhe.de Complex Systems, Editor Frank Steiner Abteilung Theoretische Physik Universität Ulm Albert-Einstein-Allee 11 89069 Ulm, Germany Phone: +49 (7 31) 02 29 10 Fax: +49 (7 31) 02 29 24 Email: frank.steiner@uni-ulm.de www.physik.uni-ulm.de/theo/qc/group.html www.pdfgrip.com Michael I Eides Howard Grotch Valery A Shelyuto Theory of Light Hydrogenic Bound States With 108 Figures ABC www.pdfgrip.com Michael I Eides Howard Grotch Valery A Shelyuto Mendeleev Institute for Metrology Moskovsky Pr 19 190005 St Petersburg Russia E-mail: shelyuto@vniim.ru University of Kentucky Department of Physics and Astronomy Lexington, KY 40506 U.S.A E-mail: eides@pa.uky.edu hgrotch@pa.uky.edu hgrotch@adelphia.net 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 The use of general descriptive names, registered names, trademarks, etc in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use Typesetting: by the authors using a Springer LATEX macro package Cover production: WMXDesign GmbH, Heidelberg Printed on acid-free paper SPIN: 10786030 56/techbooks www.pdfgrip.com 543210 Preface Light one-electron atoms are a classical subject of quantum physics The very discovery and further progress of quantum mechanics is intimately connected to the explanation of the main features of hydrogen energy levels Each step in the development of quantum physics led to a better understanding of the bound state physics The Bohr quantization rules of the old quantum theory were created in order to explain the existence of the stable discrete energy levels The nonrelativistic quantum mechanics of Heisenberg and Schră odinger provided a self-consistent scheme for description of bound states The relativistic spin one half Dirac equation quantitatively described the main experimental features of the hydrogen spectrum Discovery of the Lamb shift [1], a subtle discrepancy between the predictions of the Dirac equation and the experimental data, triggered development of modern relativistic quantum electrodynamics, and subsequently the Standard Model of modern physics Despite its long and rich history the theory of atomic bound states is still very much alive today New importance to the bound state physics was given by the development of quantum chromodynamics, the modern theory of strong interactions It was realized that all hadrons, once thought to be the elementary building blocks of matter, are themselves atom-like bound states of elementary quarks bound by the color forces Hence, from a modern point of view, the theory of atomic bound states could be considered as a theoretical laboratory and testing ground for exploration of the subtle properties of the bound state physics, free from further complications connected with the nonperturbative effects of quantum chromodynamics which play an especially important role in the case of light hadrons The quantum electrodynamics and quantum chromodynamics bound state theories are so intimately intertwined today that one often finds theoretical research where new results are obtained simultaneously, say for positronium and also heavy quarkonium The other powerful stimulus for further development of the bound state theory is provided by the spectacular experimental progress in precise measurements of atomic energy levels It suffices to mention that in about a decade the relative uncertainty of measurement of the frequency of the 1S −2S www.pdfgrip.com VI Preface transition in hydrogen was reduced by four orders of magnitude from · 10−10 [2] to 1.8 × 10−14 [3] The relative uncertainty in measurement of the muonium hyperfine splitting was reduced by the factor 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 www.pdfgrip.com 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 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 We will return to a more detailed discussion of the role of different small parameters below www.pdfgrip.com VIII Preface tools, which in many respects are quite different from the physics of low Z systems We discuss below the numerical results obtained in the high Z calculations only when they have a direct relevance for the low Z atoms The reader can find a detailed discussion of the high Z physics in a number of reviews (see, e.g., [18]) In trying to preserve a reasonable size of this text we decided to omit discussion of positronium, even though many theoretical expressions below are written in such form that for the case of equal masses they turn into respective corrections for the positronium energy levels Positronium is qualitatively different from hydrogen and muonium not only due to the equality of the masses of its constituents, but because unlike the other light atoms there exists a whole new class of corrections to the positronium energy levels generated by the annihilation channel which is absent in other cases For many years, numerous friends and colleagues have discussed with us the bound state problem, have collaborated on different projects, and have shared with us their vision and insight We are especially deeply grateful to the late D Yennie and M Samuel, to G Adkins, E Borie, M Braun, A Czarnecki, M Doncheski, G Drake, R Faustov, U Jentschura, K Jungmann, S Karshenboim, I Khriplovich, T Kinoshita, L Labzowsky, P Lepage, A Martynenko, K Melnikov, A Milshtein, P Mohr, D Owen, K Pachucki, V Pal’chikov, J Sapirstein, V Shabaev, B Taylor, A Yelkhovsky, and V Yerokhin This work was supported by the NSF grants PHY-0138210 and PHY-0456462 References 10 11 12 13 14 15 W E Lamb, Jr and R C Retherford, Phys Rev 72, 339 (1947) M G Boshier, P E G Baird, C J Foot et al, Phys Rev A 40, 6169 (1989) M Niering, R Holzwarth, J Reichert et al, Phys Rev Lett 84, 5496 (2000) F G Mariam, W Beer, P R Bolton et al, Phys Rev Lett 49, 993 (1982) W Liu, M G Boshier, S Dhawan et al, Phys Rev Lett 82, 711 (1999) H A Bethe and E E Salpeter, Quantum Mechanics of One- and Two-Electron Atoms, Springer, Berlin, 1957 J R Sapirstein and D R Yennie, in Quantum Electrodynamics, ed T Kinoshita (World Scientific, Singapore, 1990), p 560 V V Dvoeglazov, Yu N Tyukhtyaev, and R N Faustov, Fiz Elem Chastits At Yadra 25 144 (1994) [Phys Part Nucl 25, 58 (1994)] T Kinoshita, Rep Prog Phys 59, 3803 (1996) J Sapirstein, in Atomic, Molecular and Optical Physics Handbook, ed G W F Drake, AIP Press, 1996, p 327 P J Mohr, in Atomic, Molecular and Optical Physics Handbook, ed G W F Drake, AIP Press, 1996, p 341 K Pachucki, D Leibfried, M Weitz, A Huber, W Kă onig, and T W Hă anch, J Phys B 29, 177 (1996); 29, 1573(E) (1996) T Kinoshita, hep-ph/9808351, Cornell preprint, 1998 M I Eides, H Grotch, and V A Shelyuto, Phys Rep C 342, 63 (2001) H Grotch and D A Owen, Found Phys 32, 1419 (2002) www.pdfgrip.com References IX 16 S G Karshenboim, Phys Rep 422, (2005) 17 P J Mohr and B N Taylor, Rev Mod Phys 77, (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 www.pdfgrip.com 12.1 Lamb Shifts of the Energy Levels 247 Lamb shift obtained in [51] changes to L(2S 12 − 2P 12 , He+ ) = 14 041.12 (17) MHz Theoretical calculation of the He+ Lamb shift is straightforward with all the formulae given above It is only necessary to recall that all contributions scale with the power of Z, and the terms with high power of Z are enhanced in comparison with the hydrogen case Theoretical uncertainty is estimated by scaling with Z the uncertainty of the hydrogen formulae After calculation we obtain Lth (2S − 2P, He+ ) = 14 041.46 (3) MHz We see that the theoretical and experimental results for the classic Lamb shift in helium differ by about two standard deviations of the experimental result Clearly, further reduction of the experimental error is desirable, and the reason for this mild discrepancy should be clarified 12.1.9 1S − 2S Transition in Muonium Starting with the pioneering work [52] Doppler-free two-photon laser spectroscopy was also applied for measurements of the gross structure interval in muonium Experimental results [52, 53, 54, 55] are collected in Table 12.5, where the error in the first brackets is due to statistics and the second error is due to systematic effects The highest accuracy was achieved in the latest experiment [55] ∆E = 455 528 941 (9.8) MHz (12.17) Theoretically, muonium differs from hydrogen in two main respects First, the nucleus in the muonium atom is an elementary structureless particle unlike the composite proton which is a quantum chromodynamic bound state of quarks Hence nuclear size and structure corrections in Table 6.1 not contribute to the muonium energy levels Second, the muon is about ten times lighter than the proton, and recoil and radiative-recoil corrections are numerically much more important for muonium than for hydrogen In almost all other respects, muonium looks exactly like hydrogen with a somewhat lighter nucleus, and the theoretical expression for the 1S − 2S transition frequency may easily be obtained from the leading external field contribution in (3.6) and different contributions to the energy levels collected in Tables 3.2, 3.3, 3.7, 3.9, 4.1, and 5.1, after a trivial substitution of the muon mass Unlike the case of hydrogen, for muonium we cannot ignore corrections in the two last lines of Table 3.2, and we have to substitute the classical elementary particle contributions in (5.6) and (5.7) instead of the composite proton contribution in the fourth line in Table 5.1 After these modifications we obtain a theoretical prediction for the frequency of the 1S − 2S transition in muonium ∆E = 455 528 935 (0.3) MHz (12.18) The dominant contribution to the uncertainty of this theoretical result is generated by the uncertainty of the muon-electron mass ratio in (12.4) www.pdfgrip.com 248 12 Notes on Phenomenology Table 12.5 1S − 2S Transition in Muonium Danzman, Fee, Chu, et al (1989) [52] Jungmann, Baird, Barr, et al (1991) [53] Maas, Braun, Geerds, et al (1994) [54] Meyer, Bagaev, Baird, et al (1999) [55] Theory ∆E (MHz) 455 527 936 (120) (140) 455 528 016 (58) (43) 455 529 002 (33) (46) 455 528 941.0 (9.8) 455 528 934.9 (0.3) All other contributions to the uncertainty of the theoretical prediction: uncertainty of the Rydberg constant, uncertainty of the theoretical expression, etc., are at least an order of magnitude smaller There is a complete agreement between the experimental and theoretical results for the 1S − 2S transition frequency in (12.17) and (12.18), but clearly further improvement of the experimental data is warranted 12.1.10 Light Muonic Atoms There are very few experimental results on the energy levels in light hydrogenlike muonic atoms The classic 2P 12 ( 32 ) − 2S 12 Lamb shift in muonic helium ion (µ He)+ was measured at CERN many years ago [56, 57, 58, 59] and the experimental data was found to be in agreement with the existing theoretical predictions A comprehensive theoretical review of these experimental results was given in [60], and we refer the interested reader to this review It is necessary to mention, however, that a recent new experiment [61] failed to confirm the old experimental results This leaves the problem of the experimental measurement of the Lamb shift in muonic helium in an uncertain situation, and further experimental efforts in this direction are clearly warranted The theoretical contributions to the Lamb shift were discussed above in Chap mainly in connection with muonic hydrogen, but the respective formulae may be used for muonic helium as well Let us mention that some of these contributions were obtained a long time after publication of the review [60], and should be used in the comparison of the results of the future helium experiments with theory There also exists a proposal on measurement of the hyperfine splitting in the ground state of muonic hydrogen with the accuracy about 10−4 [62] Inspired by this proposal the hadronic vacuum polarization contribution of the ground state hyperfine splitting in muonic hydrogen was calculated in [63], where it was found that it gives a relative contribution of × 10−5 to hyperfine splitting We did not include this correction in our discussion of hyperfine splitting in muonic hydrogen mainly because it is smaller than the theoretical errors due to the polarizability contribution The current surge of interest in muonic hydrogen is mainly inspired by the desire to obtain a new more precise value of the proton charge radius as a result of measurement of the 2P − 2S Lamb shift [64] As we have seen www.pdfgrip.com 12.1 Lamb Shifts of the Energy Levels 249 in Chap the leading proton radius contribution is about 2% of the total 2P − 2S splitting, to be compared with the case of electronic hydrogen where this contribution is relatively two orders of magnitude smaller, about 10−4 of the total 2P − 2S Any measurement of the 2P − 2S Lamb shift in muonic hydrogen with relative error comparable with the relative error of the Lamb shift measurement in electronic hydrogen is much more sensitive to the value of the proton charge radius The natural linewidth of the 2P states in muonic hydrogen and respectively of the 2P − 2S transition is determined by the linewidth of the 2P − 1S transition, which is equal h ¯ Γ = 0.077 meV It is planned [64] to measure 2P − 2S Lamb shift with an accuracy at the level of 10% of the natural linewidth, or with an error about 0.008 meV, which means measuring the 2P − 2S transition with relative error about × 10−5 The total 2P − 2S Lamb shift in muonic hydrogen calculated according to the formulae in Table 7.1 for rp = 0.895 (18) fm, is ∆E(2P − 2S) = 201.880 (167) meV , (12.19) where the uncertainty is completely determined by the uncertainty of the proton charge radius We can write the 2P − 2S Lamb shift in muonic hydrogen as a difference of a theoretical number and a term proportional to the proton charge radius squared (12.20) ∆E(2P − 2S) = 206.065 (4) − 5.2250 r2 meV We see from this equation that when the experiment achieves the planned accuracy of about 0.008 meV [64] this would allow determination of the proton charge radius with relative accuracy about 0.1% which is about an order of magnitude better than the accuracy of the available experimental results Currently uncertainty in the sum of all theoretical contributions which are not proportional to the proton charge radius squared in (12.20) is determined by the uncertainties of the purely electrodynamic contributions and by the uncertainty of the nuclear polarizability contribution of order (Zα)5 m Purely electrodynamic uncertainties are introduced by the uncalculated nonlogarithmic contribution of order α2 (Zα)4 corresponding to the diagrams with radiative photon insertions in the graph for leading electron polarization in Fig 7.8, and by the uncalculated light by light contributions in Fig 3.11(e), and may be as large as 0.004 meV Calculation of these contributions and elimination of the respective uncertainties is the most immediate theoretical problem in the theory of muonic hydrogen After calculation of these corrections, the uncertainty in the sum of all theoretical contributions except those which are directly proportional to the proton radius squared will be determined by the uncertainty of the proton polarizability contribution of order (Zα)5 This uncertainty of the proton polarizability contribution is currently about 0.002 meV, and it will be difficult to reduce it in the near future If the experimental error of measurement www.pdfgrip.com 250 12 Notes on Phenomenology 2P − 2S Lamb shift in hydrogen will be reduced to a comparable level, it would be possible to determine the proton radius with relative error smaller that × 10−4 or with absolute error about × 10−4 fm, to be compared with the current accuracy of the proton radius measurements producing the results with error on the scale of 0.01 fm 12.2 Hyperfine Splitting 12.2.1 Hyperfine Splitting in Hydrogen Hyperfine splitting in the ground state of hydrogen was measured precisely more than thirty years ago [65, 66] ∆EHF S (H) = 420 405.751 766 (9) kHz δ = × 10−13 (12.21) For many years, this hydrogen maser measurement remained the most accurate experiment in modern physics Only recently the accuracy of the Doppler-free two-photon spectroscopy achieved comparable precision [34] (see the result for the 1S − 2S transition frequency in (12.7)) The theoretical situation for the hyperfine splitting in hydrogen always remained less satisfactory due to the uncertainties connected with the proton structure The scale of hyperfine splitting in hydrogen is determined by the Fermi energy in (8.2) (12.22) EF (H) = 418 840.101 (2) kHz , where the uncertainty is predominately determined by the uncertainty of the proton anomalous magnetic moment κ measured in nuclear magnetons The sum of all nonrecoil corrections to hyperfine splitting collected in Tables 9.1, 9.2, 9.2, 9.3, 9.4, and 9.5 is equal to ∆EHF S (H) = 420 452.04 (2) kHz , (12.23) where again the error of κ determines the uncertainty of the sum of all nonrecoil contributions to the hydrogen hyperfine splitting The theoretical error of the sum of all nonrecoil contributions is about Hz, at least an order of magnitude smaller than the uncertainty introduced by the proton anomalous magnetic moment κ, and we did not write it explicitly in (12.23) In relative units this theoretical error is about × 10−10 , to be compared with the estimate of the same error 1.2 × 10−7 made in [67] Reduction of the theoretical error by three orders of magnitude emphasizes the progress achieved in calculations of nonrecoil corrections during the last years Until recently the stumbling block on the road to a more precise theory of hydrogen hyperfine splitting was the inability to calculate the polarizability www.pdfgrip.com 12.2 Hyperfine Splitting 251 contribution As we discussed above it was first calculated in [68] and improved in [69] After these calculations, the theoretical uncertainty of the total recoil contribution to hydrogen hyperfine splitting was reduced to 0.8 kHz The current theoretical result for hydrogen hyperfine splitting is ∆EHF S (H)th = 420 403.1 (8) kHz (12.24) There is a discrepancy between theory and the experimental result in (12.21) which is more than three standard deviations Clearly, this situation is quite unsatisfactory, and further theoretical and experimental efforts are required to rectify it 12.2.2 Hyperfine Splitting in Deuterium The hyperfine splitting in the ground state of deuterium was measured with very high accuracy a long time ago [70, 71] ∆EHF S (D) = 327 384.352 521 (17) kHz δ = 5.2 × 10−12 (12.25) The expression for the Fermi energy in (8.2), besides the trivial substitutions similar to the ones in the case of hydrogen, should also be multiplied by an additional factor 3/4 corresponding to the transition from a spin one half nucleus in the case of hydrogen and muonium to the spin one nucleus in the case of deuterium The final expression for the deuterium Fermi energy has the form −3 m m ch R∞ , (12.26) EF (D) = α2 µd 1+ Mp Md where µd = 0.857 438 2329 (92) [1] is the deuteron magnetic moment in nuclear magnetons, Md is the deuteron mass, and Mp is the proton mass Numerically (12.27) EF (D) = 326 967.681 (4) kHz , where the main contribution to the uncertainty is due to the uncertainty of the deuteron anomalous magnetic moment measured in nuclear magnetons As in the case of hydrogen, after trivial modifications, we can use all nonrecoil corrections in Tables 9.1, 9.2, 9.2, 9.3, 9.4, and 9.5 for calculations in deuterium The sum of all nonrecoil corrections is numerically equal to ∆Enrec (D) = 327 339.147 (4) kHz (12.28) Unlike the proton, the deuteron is a weakly bound system so one cannot simply use the results for the hydrogen recoil and structure corrections for deuterium What is needed in the case of deuterium is a completely new consideration Only one minor nuclear structure correction [72, 73, 74, 75] was discussed in the literature for many years, but it was by far too small to explain the difference between the experimental result in (12.25) and the sum of nonrecoil corrections in (12.28) www.pdfgrip.com 252 12 Notes on Phenomenology exp ∆EHF S (D) − ∆Enrec (D) = 45.2 kHz (12.29) A breakthrough was achieved a few years ago when it was realized that an analytic calculation of the deuterium recoil, structure and polarizability corrections is possible in the zero range approximation [76, 77] An analytic result for the difference in (12.29), obtained as a result of a nice calculation in [77], is numerically equal 44 kHz, and within the accuracy of the zero range approximation perfectly explains the difference between the experimental result and the sum of the nonrecoil corrections More accurate calculations of the nuclear effects in the deuterium hyperfine structure beyond the zero range approximation are feasible, and the theory of recoil and nuclear corrections was later improved in a number of papers [78, 79, 80, 81, 82] Comparison of the results of these works with the experimental data on the deuterium hyperfine splitting may be used as a test of the deuteron models and state of the art of the nuclear calculations 12.2.3 Hyperfine Splitting in Muonium Being a purely electrodynamic bound state, muonium is the best system for comparison between the hyperfine splitting theory and experiment Unlike the case of hydrogen the theory of hyperfine splitting in muonium is free from uncertainties generated by the hadronic nature of the proton, and is thus much more precise The scale of hyperfine splitting is determined by the Fermi energy in (8.2) EF (M u) = 459 031.936 (518) (30) kHz , (12.30) where the uncertainty in the first brackets is due to the uncertainty of the best direct experimental value of muon-electron mass ratio M/m = 206.768 277 (24) in [83], and the uncertainty in the second brackets is due to the uncertainty of the fine structure constant in (12.2) We used the direct experimental value of muon-electron mass ratio in this calculation instead of the CODATA value in (12.4) because as we will see the CODATA value itself is mainly determined by the measurement of hyperfine splitting in muonium The theoretical accuracy of hyperfine splitting in muonium is determined by the still uncalculated terms which include single-logarithmic and nonlogarithmic radiative-recoil corrections of order α2 (Zα)(m/M )EF , as well as by the nonlogarithmic contributions of orders (Zα)3 (m/M )EF and α(Zα)2 (m/M )EF We estimate all these unknown corrections to hyperfine splitting in muonium as about 70 Hz Calculation of all these contributions and reduction of the theoretical uncertainty of the hyperfine splitting in muonium below 10 Hz is the current task of the theory Current theoretical prediction for the hyperfine splitting interval in the ground state in muonium may easily be obtained collecting all contributions to HFS displayed in Tables 9.1, 9.2, 9.2, 9.3, 9.4, 9.5, 10.1, and 10.2 www.pdfgrip.com 12.2 Hyperfine Splitting ∆EHF S (M u) = 463 302.904 (518) (30) (70) kHz , 253 (12.31) where the first error comes from the experimental error of the electron-muon mass ratio m/M , the second comes from the error in the value of the fine structure constant α, and the third is an estimate of the yet unknown theoretical contributions We see that the uncertainty of the muon-electron mass ratio gives by far the largest contributions both in the uncertainty of the Fermi energy and the theoretical value of the ground state hyperfine splitting On the experimental side, hyperfine splitting in the ground state of muonium admits very precise determination due to its small natural linewidth The lifetime of the higher energy hyperfine state with the total angular momentum F = with respect to the M 1-transition to the lower level state with F = is extremely large τ = 1×1013 s and gives negligible contribution to the linewidth The natural linewidth Γµ /h = 72.3 kHz is completely determined by the muon lifetime 2.2 ì 106 s A high precision value of the muonium hyperfine splitting was obtained many years ago [84] ∆EHF S (M u) = 463 302.88 (16) kHz δ = 3.6 × 10−8 (12.32) In the latest measurement [83] this value was improved by a factor of three ∆EHF S (M u) = 463 302.776 (51) kHz, δ = 1.1 × 10−8 , (12.33) The new value has an experimental error which corresponds to measuring the hyperfine energy splitting at the level of exp /(à /h) ì 104 of the natural linewidth This is a remarkable experimental achievement The agreement between theory and experiment is excellent However, the error bars of the theoretical value are apparently about an order of magnitude larger than respective error bars of the experimental result This is a deceptive impression The error of the theoretical prediction in (12.31) is dominated by the experimental error of the value of the electron-muon mass ratio As a result of the latest experiment [83] this error was reduced threefold but it is still by far the largest source of error in the theoretical value for the muonium hyperfine splitting The estimate of the theoretical uncertainty is only marginally larger than the experimental error The largest source of theoretical error is connected with the yet uncalculated theoretical contributions to hyperfine splitting, mainly with the unknown recoil and radiative-recoil corrections As we have already mentioned, reducing the theoretical uncertainty by an order of magnitude to about 10 Hz is now a realistic aim for the theory One can use high accuracy of the hyperfine splitting theory, and highly precise experimental result in (12.33) in order to obtain the value of the Fermi energy much more precise than the one in (12.30) But according to (8.2) the Fermi energy is proportional to the electron-muon mass ratio, and we can extract this mass ratio from the experimental value of HFS and the most precise value of α www.pdfgrip.com 254 12 Notes on Phenomenology M = 206.768 282 (23) (14) (32) , m (12.34) where the first error comes from the experimental error of the hyperfine splitting measurement, the second comes from the error in the value of the fine structure constant α, and the third from an estimate of the yet unknown theoretical contributions Combining all errors we obtain the mass ratio M = 206.768 282 (41) m δ = 2.0 × 10−8 , (12.35) which is almost six times more accurate than the best direct experimental value [83] The largest contribution to the uncertainty of the indirect mass ratio in (12.34) is supplied by the unknown theoretical contributions to hyperfine splitting This sets a clear task for the theory to reduce the contribution of the theoretical uncertainty in the error bars in (12.34) to the level below two other contributions to the error bars It is sufficient to this end to calculate all contributions to HFS which are larger than 10 Hz This would lead to further reduction of the uncertainty of the indirect value of the muon-electron mass ratio There is thus a real incentive for improvement of the theory of HFS to account for all corrections to HFS of order 10 Hz, created by the recent experimental and theoretical achievements Another reason to improve the HFS theory is provided by the perspective of reducing the experimental uncertainty of hyperfine splitting below the weak interaction contribution in (10.38) In such a case, muonium could become the first atom where a shift of atomic energy levels due to weak interaction would be observed [85] 12.3 Theoretical Perspectives High precision experiments with hydrogenlike systems have achieved a new level of accuracy in recent years and further dramatic progress is still expected The experimental errors of measurements of many energy shifts in hydrogen and muonium were reduced by orders of magnitude This rapid experimental progress was matched by theoretical developments as discussed above The accuracy of the quantum electrodynamic theory of such classical effects as Lamb shift in hydrogen and hyperfine splitting in muonium has increased in many cases by one or two orders of magnitudes This was achieved due to intensive work of many theorists and development of new ingenious original theoretical approaches which can be applied to the theory of bound states, not only in QED but also in other field theories, such as quantum chromodynamics From the phenomenological point of view recent developments opened new perspectives for precise determination of many fundamental constants (the Rydberg www.pdfgrip.com References 255 constant, electron-muon mass ratio, proton charge radius, deuteron structure radius, etc.), and for comparison of the experimental and theoretical results on the Lamb shifts and hyperfine splitting Recent progress also poses new theoretical challenges Reduction of the theoretical error in prediction of the value of the 1S Lamb shift in hydrogen significantly below the level of kHz (and, respectively, of the 2S Lamb shift significantly below tenth of kHz) should be considered as a next stage of the theory The theoretical error of the hyperfine splitting in muonium should be reduced to about 10 Hz Achievement of these goals will require hard work and considerable resourcefulness, but results which years ago hardly seemed possible are now within reach 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139, 140, 151, 163, 172, 193–198, 220, 222–224, 226, 234, 235, 247, 251–253 relativistic 3, 4, 13, 19, 24, 49, 81, 95, 138–140, 163, 165, 167 state-independent 17, 183, 236 Darwin -Foldy contribution 21, 112, 113 potential 57, 68, 72 term 111, 112, 183 deuteron 113, 117, 119–121, 124, 221, 245, 246, 251, 252, 255 dipole approximation 24, 50, 64, 144, 194, 238 contribution 24 fit 221 form factor 151, 152, 154, 221 Dirac -Coulomb wave function 10, 24, 53, 57, 91, 132, 138, 165–167 equation 4–6, 8, 13, 15, 19, 22, 81, 90, 165 equation effective 6, 9, 13, 14, 21, 22, 24, 37, 81, 83, 165 form factor 17, 23, 24, 27–29, 36, 43, 67, 103, 111, 114, 132, 146, 149 spectrum 4, 14, 21, 22 external field approximation 48, 81, 151, 165, 229 19, 22, Fermi energy 162, 165, 167, 182, 193, 194, 197, 210, 217, 218, 220, 222, 226, 250–253 Foldy-Wouthuysen transformation 20, 111 Green function 138, 220 5–9, 50, 52, 90, 91, 99, hyperfine splitting (HFS) 13, 161–163, 165–178, 182, 193, 194, 197, 200, www.pdfgrip.com 260 Index 211, 218, 220, 222, 224, 227, 229, 248, 250254 Kă allen-Sabry potential Lamb shift 141, 142 4, 16, 17, 21, 22, 50, 51, 234 muonic atoms 118, 131, 133, 139, 143, 154, 248 helium 131, 132, 143, 248 hydrogen 101, 131–133, 135, 137, 139, 143–145, 148–151, 153–155, 233, 243, 248, 249 NRQED 10, 61, 66, 67, 94, 178, 185, 188, 196, 197, 202, 210 Pauli form factor 25, 28, 30, 103, 111, 114 perturbation potential 58, 65, 67, 73, 88, 110, 134, 135, 147, 182, 187, 188, 210, 226 theory 6, 7, 9, 11, 24, 50, 51, 59, 62, 65, 67, 68, 87, 88, 90, 95, 113, 122, 123, 137, 184, 187, 205, 226 theory contribution 59, 61, 83, 88, 140 propagator electron 6, 183 heavy particle 90, 91, 208 nonrelativistic 183 photon 16, 23, 36, 87, 90, 125, 134, 139, 162 proton 100 two-particle 6, proton charge radius 33, 104, 110–112, 116, 117, 122, 124, 149, 151, 154, 155, 220, 221, 238, 242, 243, 245, 246, 248–250, 255 form factor 104, 114, 117, 125, 126, 149, 151, 152, 217, 218, 222–224, 227–229 magnetic moment 104, 217, 218, 220, 222–224, 226, 250 polarizability contribution 119, 121, 122, 124, 223, 225, 248, 249, 251 size contribution 112, 116, 117, 122, 125, 131, 132, 149, 151, 219, 220, 222–224, 226, 236, 245, 249 Sachs form factor 104, 111, 112, 218, 221 scale atomic 37, 39, 40, 42, 45, 62, 102, 114, 123, 127, 147, 154, 171, 174, 180, 194, 217 characteristic 2, 110, 206, 218 natural 44, 47, 165, 179 scattering approximation 36, 40, 83, 85, 87, 95, 125, 137, 144, 151, 169, 173, 178, 180, 194, 198, 200, 201 Schră odinger -Coulomb wave function 10, 25, 36, 37, 40, 41, 49, 56–58, 72, 84, 87, 105, 123, 134, 138, 147, 162, 167, 169, 197, 220, 227 equation 1, 5, skeleton diagram 36, 40, 47, 48, 83, 102, 103, 114, 117, 169, 171, 179, 180, 198, 200, 208, 218, 222 factor 17, 38 integral 39, 41–43, 84, 85, 102, 114, 115, 118, 125, 126, 170–175, 178, 194, 201, 203–205, 207, 220, 223, 229 integral approach 38, 40, 44, 52, 56, 101, 169, 171, 174, 178, 226, 228 Uehling potential 54, 56, 58, 63, 73, 75, 139, 141, 142 Wichmann-Kroll potential 75, 141–143 54, 58, 73, Zemach correction 219–224, 226–229 moment first 219, 221 moment third 116–118, 151, 152, 154, 155, 219, 220 radius 122, 219 www.pdfgrip.com Springer Tracts in Modern Physics 180 Coverings of Discrete Quasiperiodic Sets Theory and Applications to Quasicrystals Edited by P Kramer and Z Papadopolos 2002 128 figs., XIV, 274 pages 181 Emulsion Science Basic Principles An Overview By J Bibette, F Leal-Calderon, V Schmitt, and P Poulin 2002 50 figs., IX, 140 pages 182 Transmission Electron Microscopy of Semiconductor Nanostructures An Analysis of Composition and Strain State By A Rosenauer 2003 136 figs., XII, 238 pages 183 Transverse Patterns in Nonlinear Optical Resonators By K Stali¯unas, V J Sánchez-Morcillo 2003 132 figs., XII, 226 pages 184 Statistical Physics and Economics Concepts, Tools and Applications By M Schulz 2003 54 figs., XII, 244 pages 185 Electronic Defect States in Alkali Halides Effects of 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Light Emitting Silicon for Microphotonics By S Ossicini, L Pavesi, and F Priolo 2003 206 figs., XII, 284 pages 195 Uncovering CP Violation Experimental Clarification in the Neutral K Meson and B Meson Systems By K Kleinknecht 2003 67 figs., XII, 144 pages 196 Ising-type Antiferromagnets Model Systems in Statistical Physics and in the Magnetism of Exchange Bias By C Binek 2003 52 figs., X, 120 pages 197 Electroweak Processes in External Electromagnetic Fields By A Kuznetsov and N Mikheev 2003 24 figs., XII, 136 pages 198 Electroweak Symmetry Breaking The Bottom-Up Approach By W Kilian 2003 25 figs., X, 128 pages 199 X-Ray Diffuse Scattering from Self-Organized Mesoscopic Semiconductor Structures By M Schmidbauer 2003 102 figs., X, 204 pages www.pdfgrip.com Springer Tracts in Modern Physics 200 Compton Scattering Investigating the Structure of the Nucleon with Real Photons By F Wissmann 2003 68 figs., VIII, 142 pages 201 Heavy Quark Effective Theory By A Grozin 2004 72 figs., X, 213 pages 202 Theory of Unconventional Superconductors By D Manske 2004 84 figs., XII, 228 pages 203 Effective Field Theories in Flavour Physics By T Mannel 2004 29 figs., VIII, 175 pages 204 Stopping of Heavy Ions By P Sigmund 2004 43 figs., XIV, 157 pages 205 Three-Dimensional X-Ray Diffraction Microscopy Mapping Polycrystals and Their Dynamics By H Poulsen 2004 49 figs., XI, 154 pages 206 Ultrathin Metal Films Magnetic and Structural Properties By M Wuttig and X Liu 2004 234 figs., XII, 375 pages 207 Dynamics of Spatio-Temporal Cellular Structures Henri Benard Centenary Review Edited by I Mutabazi, J.E Wesfreid, and E Guyon 2005 approx 50 figs., 150 pages 208 Nuclear Condensed Matter Physics with Synchrotron Radiation Basic Principles, Methodology and Applications By R Röhlsberger 2004 152 figs., XVI, 318 pages 209 Infrared Ellipsometry on Semiconductor Layer Structures Phonons, Plasmons, and Polaritons By M Schubert 2004 77 figs., XI, 193 pages 210 Cosmology By D.-E Liebscher 2005 Approx 100 figs., 300 pages 211 Evaluating Feynman Integrals By V.A Smirnov 2004 48 figs., IX, 247 pages 213 Parametric X-ray Radiation in Crystals By V.G Baryshevsky, I.D Feranchuk, and A.P Ulyanenkov 2006 63 figs., IX, 172 pages 214 Unconventional Superconductors Experimental Investigation of the Order-Parameter Symmetry By G Goll 2006 67 figs., XII, 172 pages 215 Control Theory in Physics and other Fields of Science Concepts, Tools, and Applications By M Schulz 2006 46 figs., X, 294 pages 216 Theory of the Muon Anomalous Magnetic Moment By K Melnikov, A Vainshtein 2006 33 figs., XII, 176 pages 217 The Flow Equation Approach to Many-Particle Systems By S Kehrein 2006 24 figs., XII, 170 pages 219 Inelastic Light Scattering of Semiconductor Nanostructures By C Schüller 2007 105 figs., XII, 178 pages 220 Precision Electroweak Physics at Electron-Positron Colliders By S Roth 2007 107 figs., X, 174 pages 221 Free Surface Flows under Compensated Gravity Conditions By M Dreyer 2007 128 figs., X, 272 pages 222 Theory of Light Hydrogenic Bound States By M.I Eides, H Grotch, and V.A Shelyuto 2007 108 figs., XVI, 260 pages www.pdfgrip.com ... 12.20.-m, 31.30.Jv, 32.10.Fn, 36.10.Dr ISSN print edition: 008 1-3 869 ISSN electronic edition: 161 5-0 430 ISBN-10 3-5 4 0-4 526 9-9 Springer Berlin Heidelberg New York ISBN-13 97 8-3 -5 4 0-4 526 9-0 Springer... www.pdfgrip.com Springer Tracts in Modern Physics Springer Tracts in Modern Physics provides comprehensive and critical reviews of topics of current interest in physics The following fields are emphasized:... 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