erutceL Notes scisyhP in Edited by .J Ehlers, Menchen, .K Hepp, Z~irich .R Kippenhahn, Menchen, .H A. Weidenmeller, Heidelberg and .J Zittartz, n16K Managing Editor: W. Beiglb6ck, Heidelberg 341 Present Status dna Aims fo Quantum scimanydortcelE sgnideecorP of eht Symposion Held ta Mainz ytisrevinU yaM 9-10, 0891 detidE yb .G ,ff~rG .E Klempt, dna .G Werth galreV-regnirpS nilreB Heidelberg New York 1891 Editors Gernot GrUff Eberhard Klempt 3fG nter Werth Institut for Physik, Johannes Gutenberg-Universit~t Jakob-Welder-Weg ,11 D-6500 Mainz ISBN 3-540-10847-5 Springer-Verlag Berlin Heidelberg New York ISBN 0-38740847-5 Springer-Verlag New York Heidelberg Berlin Library of Congress Cataloging in Publication Data Main entry under title: Present status and aims of quantum electrodynamics. (Lecture notes in physics; 143) Bibliography: p. Includes index. .1 Quantum electrodynamics--Congresses. .I Gr~ff, Gernot, 1929-. .1I Klempt, .E (Eberhard), 1939-. .1II Werth, G. (G(Jnter), 1938-. .VI Series. QC679.P73 537.6 1619-18 ISBN 0-387-10847-5 (U.S.) AACR2 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to "Verwertungsgesellschaft Wort", Munich. © by Springer-Verlag Berlin Heidelberg 1981 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2153/3140-543210 CONTENT H. Pietschmann QUANTUM ELECTRODYNAMICS WITHIN THE FRAMEWORK OF UNIFIED FIELD THEORIES - I F. Scheck UNIVERSALITY OF LEPTON INTERACTIONS - 11 V. Hepp TEST OF QUANTUM ELECTRODYNAMICS AT HIGH MOMENTUM TRANSFERS - 35 O. Steinmann SOME BASIC PROBLEMS OF QUANTUM ELECTRODYNAMICS - 58 E. Borie/ J. Calmet THE ANOMALOUS MAGNETIC MOMENT OF THE LEPTONS: THEORY AND NUMERICAL METHODS - 68 H. Dehmelt REFINED DATA ON ELECTRON STRUCTURE FROM INVARIANT FREQUENCY RATIOS IN ELECTRON AND POSITRON GEONIUM SPECTRA - 75 E. Klempt THE MUON MAGNETIC MOMENT - 77 F.J.M. Farley THE MEASUREMENT OF G-2 FOR THE MUON - 91 E. Borie QUANTUM ELECTRODYNAMICS IN BOUND SYSTEMS - 97 P.W. Zitzewitz GROUND STATE POSITRONIUM: HYFERFINE STRUCTURE AND DECAY RATES -118 E.W. Weber POSITRONIUM IN EXCITED STATES -146 H. Orth MUONIUM AND NEUTRAL MUONIC HELIUM -173 L.Tauscher VACUUM POLARIZATION IN MUONIC ATOMS -201 P.S. Farago THE DETERMINATION OF LAMB SHIFT FROM THE ANISOTROPY OF THE QUENCHING RADIATION FROM METASTABLE HYDROGENIC ATOMS -215 R. Wallenstein THE LAMB SHIFT OF THE HYDROGEN ATOM AND HYDROGENIC IONS -230 R. Neumann HELIUM AND HELIUM-LIKE SYSTEMS -251 B. Fricke COMPARISON BETWEEN EXPERIMENT AND THEORY IN HEAVY ELECTRONIC SYSTEMS -267 H. Backe POSITRONENERZEUGUNG BEI SCHWERIONENSTOSSEN -277 Foreword Since the first measurements of the Lamb shift and the anomalous magnetic moment of the electron,tests of quantum electrodynamics have become a continuous challenge for many experimental physicists. Several times during these years discrepancies between the predictions of the theory and the experimental data have been published, stimulating intense discussions about the physical grounds and mathematical methods of QED. Further improvements of experimental accuracy combined with more careful analysis of the experiments and calculation of higher- order contributions have led again and again to an agreement between the experimental data and the predictions of QED. From the small discrepancies still present at this time, nobody would deduce a breakdown of QED theory. However, regarding the fundamental importance and the model character of QED, further tests with larger momentum transfers and higher precision that check on the validity of higher-order contributions seem highly desirable. Therefore we felt that the time has come for a discussion of the following topics: physical ground and mathematical methods of QED, - mutual relations between theory and experiment, - - analysis of experimental data as being presented today, possible improvements of tests of QED regarding experimental - aspects and - contributions of other interactions. We were very pleased that so many experts actively engaged in this field supported our suggestion to hold a Symposion on the Fresent Status and Aims of Quantum Electrodynamics at Mainz. As far as the theory is concerned, the contributions discuss fundamental problems of QED, aspects of unified field theories, relations between theory and experiment, and examples of numerical calculations of QED interactions at large momentum transfer and corrections of higher order in ~ and Za. However, the major part of the contributions assesses the QED tests at high energies and represents the current status of precision experiments on bound systems and free particles at low energies. Of course, within the time allotted the total spectrum of QED could not be covered. Several important topics, such as the interactions with real photons or macroscopic QED, had to be omitted. IV As a result, the symposion revealed some general trends and problems. At h~gh energies new experiments with an even larger momentum transfer may be realised in the future. In contrast, however, further improvement of accuracy in precision experiments is now often limited by the finite lifetime of the system under investigation {e.g., positronium); also the comparison between experimental data and theory is becoming more difficult due to the not precisely predictab].e contribution of other interactions (consider, e.g., the anomalous magnetic moment of the muon). Therefore the general trend is characterised by the measurement of the QED properties of stabile systems (electron, positron}, by investigations of (n'S - nS) transitions, or by the exploration of systems with high Z values, especially hydrogen-like ions and muonic atoms. The reader will find these trends in several articles. The editors thank all contributors for their engagement. We gratefully acknowledge the hospitality of the Akademie der Wissenschaften und der Literatur zu Mainz. We are indebted to the Johannes-Gutenberg- Universit~t Mainz, the Verband der Freunde der Universit~t Mainz and the Regionalverband Rheinhessen der Deutschen Physikalischen Gesellschaft for their financial support. Mainz, May 1981 G. GrUff E. Klempt G. Werth MUTNAUQ ELECTRODYNAMICS WITHIN EHT KROWEMARF FO UNIFIED FIELD THEORIES* Herbert Pietschmann Institut fQr Theoretische Physik Universit~t Wien .I Historical Background In 1930~ P.A.M. Dirac found the relativistically invariant equation of motion for an electron in an electromagnetic Field A (i y~ 6 - e X~ A - m) ~(x) = 0 (i) or (i~ (z,) - m) .--~ (x) = e ~ ~(x) . (Eqs. (i) and (i') should also serve to define the notation used sub- sequently.) Together with Maxwell's equations I~ A = e j~(x) (2) they mroF the Fundamental set of equations for the theory of photons and electrons. From them, Dirac derived his Famous hole theory I) which he first interpreted as a theory for electrons and protons. After it was shown that this leads to an unacceptable instability of matter, the theory was re-interpreted as one for electrons and positrons. The notion of antiparticles was thus created and the discovery of a positive electron by Anderson 2) made independently of theoretical developments led our understanding of the elements of matter to one of its greatest triumphs. * Supported in part by "Fonds zur FSrderung der wissenschaftlichen For- schung in ~sterreich", Projeot Nr. 3800. In spite of these exciting discoveries in the old days, the birth- day of Quantum Electrodynamics is usually associated with the first successful calculation of a higher order correction. In 1948, 3) J. Schwinger computed the anomalous magnetic moment of the electron to be eWA a Be " be 2x (3) Today we understand Quantum Electrodynamics to be the theory of charged leptons and the photon. It is defined by the Lagrangian (4) DEQL = 1 Z {~l(i~ - m) ~1 + e A" jl} _ 1 F vFpV with the electromagnetic current of leptons .1 ~J = l@ ~x ~I" (5) The sum goes over the 3 known charged leptons e, p and T. Table i summarizes those static properties of these 3 leptons, which are not equal for all three, namely mass and lifetime. Table i: Mass dna lifetime of the charged leptons 1 m I (MeV) 1T (sec) e 0.511 003 4 (14) 105.659 46 (24) 2,197 134 (77)'10 -6 + 3 < 2.3.10-13 T 1782 - 4 (theor: 2.8"10 -13 ) It can be seen from eq. (4) that the mass is indeed the only basic quantity in which the 3 leptons differ. (Since there are no transitions between leptons in eq. (4), lepton number is a good quantum number and we could say that they also differ in this quantity.) There is a uni- versality principle, called "~-e-T universality", which is only broken by the difference in masses. The difference in lifetime is a direct consequence of this mass difference. Indeed, the theoretical prediction of the lifetime of the T in Table i is based on e-p-T universality. The point-like nature of lepton-photon interactions as predicted from eq. (4) is today tested to the breathtaking limit of about 4"10 -16 cm for all leptons. A special section of this conference will give more information on this point. 2. Limits of Applicability of Quantum Electrodynamics The different lifetime of leptons is not the only consequence of mass differences. There are more subtle effects also, all of which can be computed from eq. (4). Schwinger's correction to the magnetic moment as given in eq. (3) holds for all three types of leptons because it is the lowest order correction in which no mass ratios enter. But if we go to the next order, differences do show up. 1 {o 0.328 48 (~.)2 + ... (6e) Be - 2 T1 - 1 a ap - 2 TI" + 0.765 78 (~)2 + ... (6p.) The difference is due to contributions from a class of Feynman-graphs, a typical one being shown in Fig. i. When the lepton of the close loop differs from the external lepton whose magnetic moment is measured, the mass ratio enters, causing differences in the contribution to a I for different 1. But this class of graphs also leads us to the first limit of Quantum Eleotrodynamics. For the closed loop does not have to be a lepton; it can also be a hadron (or a quark). In this case, the contri- bution can no longer be derived from eq. (4). Thus a natural limit is reached at a precision, in which these hadronic contributions become important. We can then either use measurements from hadron physics to compute the contributions or we can use precision measurement of QED to set limits on hadronic quantities. In either case, a comparison of theory and experiments bears no longer exclusively on QED. A similar type of graph gives rise to the second limit of QED which we shall discuss presently. The graph is shown in Fig. 2; it stems from weak interactions, in which leptons do participate also. Of course, its contribution is expected to be small, but a computation gives infinite result. As soon as we allow neutrinos into the picture, more difficulties arise. Within QED itself, all infinities can be buried into unobservable quantities such as bare masses or coupling constants. Radiative corrections to the weak coupling constants are - in general - infinite. True that in the purely leptonic case of p-decay or T-decay, radiative corrections are finite, but this is rather a coincidence than a deep phenomenon. It is due to the good fortune, that a Fierz transformation allows us to collect the charged particles into one current alone by yXCi + ys)v (7) 5p yl(l + y5)~ ~ yl(l + y5)Ve = ~ Yx(l + yS)~ Dp e (Thus the vertex correction of QED can be applied as the only radiative correction.) As soon as we take into account the finite mass of the intermediate boson or we turn to n-decay, infinities pop up. Thus we arrive at the second limit of pure QED: its connection to weak interactions, typically demonstrated by the contribution of Fig.2. 5. Unification of QED with Weak Interactions The second limit of pure QED has been overcome by the beautiful theory of unified electro-weak interactions of Glashow, Salam, Weinberg (and others). It is a renormalizable theory, so that no infinities occur in measurable quantities (except for electromagnetic mass differences of hadrons). Nothing is changed in QED proper, the Lagrangian (4) remains identically the same. Also, ordinary charged current weak interactions are taken over from the good old V-A theory. CL c = ~ _29_/ ~ {i XY (i + 75)~ i xw + h.e. (8) Due to the marriage with QED, however, g is now related to the electric charge e = g sin 8 W . (9) 8 W is the weak mixing angle and in writing eq. (9) we have merely re- placed g by another parameter, 8 W. But 8 W is - for all practical pur- poses - the only new free parameter to be determined once and for all by experiment. It will occur over and over again; thus eq. (9) may be taken as its definition so that consequent relations actually reduce the number of free parameters. A characteristic feature of electro-weak interactions (or Quantum Leptodynamics to be extended to Quantum Flavourdynamics when hadrons are incorporated) is the presence of the weak neutral current. Its pre- diction and verification was one of the corner stones on which the whole framework rests. The neutral current Lagrangian is LN c = ~ 9 ~v i TX (i + Y5)~u i ~i ~x(Cv + Y5)~i zx iO) i 4 cos 8 W with C V = i - 4 sin28w (ii) Z is the neutral equivalent of the charged intermediate boson W. From neutral current interactions, the value of the weak mixing angle can be determined and the best world value at present is 4) sin28w = 0.230 ± 0.009 (iz) It is precisely the existence of this additional part of weak interactions, which allows finite predictions. In our example, the infinite contribution to the anomalous magnetic moment from the charged intermediate bosch (as shown in Fig. 2), another graph contributes due to eq. (i0). It is shown in Fig. .3 The most divergent contributions are equal with Opposite signs and thus cancel. To render all predictions of physical processes completely finite (thus to possess a renormalizable theory.) needs yet another piece added to the Lagrangian; we will deal with it shortly. But before, let us understand the other aspect of electro-weak theory: the unification of electromagnetic and weak interactions. To see this in physical terms, let us look at the second limit of applicability of pure QED, as we have defined it in section 2. Electro-weak theory extends beyond that limit, containing QED as a special case in much the same way that special relativity extends beyond Newtonian mechanics. The limit is reached, whenever v/c approaches unity. In our case, deviations from pure QED will typically show up, when the ratio of typical energies over the mass of the neutral intermediate boson reaches unity or when the prediction of the experiments reaches distances comparable to the Compton wave length of the Z °. In spite of the very successful high energy experiments with neutral current neutrino interactions (leading to the result of eq. (12)), I personally think that the most direct way of approaching the limit of QED given by eq. (lO) is to find its effects in the atomic shell. Though contributions of eq. (iO) at low energies are of course exceedingly small, hope to find them lies in the