Quantum Electrodynamic Bound–State Calculations and Large–Order Perturbation Theory based on a Habilitation Thesis Dresden University of Technology by Ulrich Jentschura st arXiv:hep-ph/0306153v2 27 Apr 2004 1 Edition June 2002 2nd Edition April 2003 [with hypertext references and updates] 3rd Edition April 2004 E–Mail: [email protected] and [email protected] 1 This manuscript is also available – in the form of a book – from Shaker Verlag GmbH, Postfach 101818, 52018 Aachen, Germany, world-wide web address: http://www.shaker.de, electronic-mail address: [email protected]. It has been posted in the online archives with permission of the publisher. 2 Contents I Quantum Electrodynamic Bound–State Calculations 7 1 Numerical Calculation of the One–Loop Self Energy (Excited States) 8 1.1 Orientation ............................................ 8 1.2 Introduction to the Numerical Calculation of Radiative Corrections ............. 8 1.3 Method of Evaluation ...................................... 9 1.3.1 Status of Analytic Calculations ............................. 9 1.3.2 Formulation of the Numerical Problem ......................... 12 1.3.3 Treatment of the divergent terms ............................ 15 1.4 The Low-Energy Part ...................................... 17 1.4.1 The Infrared Part .................................... 17 1.4.2 The Middle-Energy Subtraction Term ......................... 21 1.4.3 The Middle-Energy Remainder ............................. 22 1.5 The High-Energy Part ...................................... 25 1.5.1 The High-Energy Subtraction Term .......................... 25 1.5.2 The High-Energy Remainder .............................. 27 1.5.3 Results for the High-Energy Part ............................ 29 1.6 Comparison to Analytic Calculations .............................. 29 2 Analytic Self–Energy Calculations for Excited States 34 2.1 Orientation ............................................ 34 2.2 The epsilon-Method ....................................... 35 2.3 Results for High– and Low–Energy Contributions ....................... 37 2.3.1 P 1/2 states (kappa = 1) ................................ 37 2.3.2 P 3/2 states (kappa = -2) ................................ 38 2.3.3 D 3/2 states (kappa = 2) ................................ 39 2.3.4 D 5/2 states (kappa = -3) ................................ 39 2.3.5 F 5/2 states (kappa = 3) ................................ 40 2.3.6 F 7/2 states (kappa = -4) ................................ 40 2.3.7 G 7/2 states (kappa = 4) ................................ 41 2.3.8 G 9/2 states (kappa = -5) ................................ 41 2.4 Summary of Results ....................................... 41 2.5 Typical Cancellations ...................................... 41 2.6 Observations ........................................... 42 3 3 The Two–Loop Self–Energy 45 3.1 Orientation ............................................ 45 3.2 Introduction to the Two–Loop Self–Energy .......................... 45 3.3 Two-loop Form Factors ..................................... 47 3.4 High–Energy Part ........................................ 50 3.5 Low–Energy Part ......................................... 52 3.6 Results for the Two–Loop Corrections ............................. 55 4 Spinless Particles in Bound–State Quantum Electrodynamics 56 4.1 Orientation ............................................ 56 4.2 Introduction to Spinless QED and Pionium .......................... 56 4.3 Breit hamiltonian for Spinless Particles ............................. 56 4.4 Vacuum Polarization Effects ................................... 60 4.5 Effects due to Scalar QED .................................... 60 5 QED Calculations: A Summary 63 II Convergence Acceleration and Divergent Series 67 6 Introduction to Convergence Acceleration and Divergent Series in Physics 68 7 Convergence Acceleration 71 7.1 The Concept of Convergence Acceleration ........................... 71 7.1.1 A Brief Survey ...................................... 71 7.1.2 The Forward Difference Operator ............................ 72 7.1.3 Linear and Logarithmic Convergence .......................... 72 7.1.4 The Standard Tool: Pad´eApproximants ........................ 73 7.1.5 Nonlinear Sequence Transformations .......................... 74 7.1.6 The Combined Nonlinear–Condensation Transformation (CNCT) .......... 77 7.2 Applications of Convergence Acceleration Methods ...................... 79 7.2.1 Applications in Statistics and Applied Biophysics ................... 79 7.2.2 An Application in Experimental Mathematics ..................... 80 7.2.3 Other Applications of the CNCT ............................ 84 7.3 Conclusions and Outlook for Convergence Acceleration .................... 84 8 Divergent Series 86 8.1 Introduction to Divergent Series in Physics .......................... 86 8.2 The Stark Effect: A Paradigmatic Example .......................... 87 8.2.1 Perturbation Series for the Stark Effect ........................ 87 8.2.2 Borel–Pad´eResummation ................................ 88 8.2.3 Doubly–Cut Borel Plane ................................. 90 8.2.4 Numerical Calculations for the Stark Effect ...................... 92 8.3 Further Applications of Resummation Methods ........................ 95 8.3.1 Zero–Dimensional Theories with Degenerate Minima ................. 95 8.3.2 The Effective Action as a Divergent Series ....................... 98 8.3.3 The Double–Well Problem ................................ 103 8.4 Divergent Series: Some Conclusions ............................... 108 9 Conclusions 111 4 Abstract This Thesis is based on quantum electrodynamic (QED) bound-state calculations [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], as well as on investigations related to divergent series, convergence acceleration and applications of these concepts to physical problems [23, 24, 25, 26, 27, 28, 29, 30, 31]. The subjects which are discussed in this Thesis include: the self energy of a bound electron and the spin-dependence of QED corrections in bound systems (Chs. 1 – 5), convergence acceleration techniques (Chs. 6 and 7), and resummation methods for divergent series with an emphasis on physical applications (Ch. 8). In Chapter 1, we present numerical results [12], accurate to the level of 1 Hz for atomic hydrogen, for the energy correction to the K and L shell states due to the one-loop self energy. We investigate hydrogenlike systems with low nuclear charge number Z = 1–5. Calculations are carried out using on-mass shell renormalization, which guarantees that the final result is written in terms of the physical electron charge. The purpose of the calculation is twofold: first, we provide accurate theoretical predictions for the one-loop self energy of K and L shell states. Second, the comparison of the analytic and numerical approaches to the Lamb shift calculations, which have followed separate paths for the past few decades, is demonstrated by comparing the numerical values with analytic data obtained using the Zα-expansion [1, 2, 32]. Our calculation essentially eliminates the uncertainty due to the truncation of the Zα-expansion, and it demonstrates the consistency of the numerical and analytic approaches which have attracted attention for more than five decades, beginning with the seminal paper [33]. The most important numerical results are summarized in Table 1.5. In Chapter 2, we investigate higher-order analytic calculations of the one-photon self energy for excited atomic states. These calculations rely on mathematical methods described in Sec. 2.2 which, in physical terms, lead to a separation of the calculation into a high- and a low-energy part for the virtual photon. This separation does not only permit adequate simplifications for the two energy regions [1, 2], but also an adequate treatment of the infrared divergences which plague all bound-state calculations (see also Ch. 7 of [34]). The investigation represents a continuation of previous work on the subject [1, 2, 32]. The calculations are relevant for transitions to highly excited states, which are relevant for the extraction of fundamental constants from the high-precision measurements in atomic hydrogen [35], and for the estimation of QED effects in more complex atomic systems where some of the electrons occupy highly excited states. Chapter 3. We investigate the two-loop self energy. The calculation is based on a generalization of the methods introduced in Sec. 2.2. Historically, the two-loop self energy for bound states has represented a major task for theoretical evaluations. We present an analytic calculation [16] of the fine-structure differ- 2 6 2 ence of the two-loop self energy for P states in atomic hydrogen in the order α (Zα) me c . This energy difference can be parameterized by two analytic coefficients, known as B61 and B60 (see Sec. 3.2). These coefficients describe the two-loop self energy in the sixth order in Zα, with an additional enhancement −2 due to a large logarithm ln[(Zα) ] (in the case of the B61-coefficient). The calculations are relevant for an improved theoretical understanding of the fine-structure in hydrogenlike systems. They are also in part relevant for current experiments on atomic helium [36, 37, 38, 39], whose motivation is the determination of the fine-structure constant with improved accuracy. Finally, it is hoped that numerical calculations of the two-loop effect will be carried out in the near future for which the current analytic evaluation will be an important consistency check.