Dissertation submitted to the Combined Faculties of the Natural Sciences and Mathematics of the Ruperto-Carola-University of Heidelberg, Germany for the degree of Doctor of Natural Sciences put forward by Bastian Sikora born in Munich, Germany Oral examination: April 18th, 2018 a Quantum field theory of the g-factor of bound systems Referees: Honorarprof. Dr. Christoph H. Keitel Prof. Dr. Maurits Haverkort a Abstract In this thesis, the theory of the g-factor of bound electrons and muons is presented. For light muonic ions, we include one-loop self-energy as well as one- and two-loop vacuum polarization corrections with the interaction with the strong nuclear potential taken into account to all orders. Furthermore, we include effects due to nuclear structure and mass. We show that our theory for the bound-muon g-factor, combined with possible future bound-muon experiments, can be used to improve the accuracy of the muon mass by one order of magnitude. Alternatively, our approach constitutes an independent access to the controversial anomalous magnetic moment of the free muon. Furthermore, two-loop self-energy corrections to the bound-electron g-factor are investigated theoretically to all orders in the nuclear coupling strength parameter Zα. Formulas are derived in the framework of the two-time Green's function method, and the separation of divergences is performed by dimensional regularization. Our numerical evaluation by treating the nuclear Coulomb interaction in the intermediate-state propagators to zero and first order show that such two-loop terms are mandatory to take into account in stringent tests of quantum electrodynamics with the bound-electron g-factor, and in projected near-future determinations of fundamental constants. Zusammenfassung In dieser Arbeit wird die Theorie des g -Faktors von gebundenen Elektronen und Myonen pr¨asentiert. F¨urleichte myonische Ionen betrachten wir die Ein-Schleifen-Selbstenergie- sowie Ein-und Zwei-Schleifen-Vakuumpolarisationskorrekturen, wobei die Wechselwir- kung mit dem starken Kernpotential zu allen Ordnungen ber¨ucksichtigt wird. Dar¨uber hinaus ber¨ucksichtigen wir die Effekte aufgrund von Kernstruktur und Masse. Wir zeigen, dass unsere Theorie f¨urden g -Faktor des gebundenen Myons, kombiniert mit m¨oglichen zuk¨unftigenExperimenten mit gebundenen Myonen, genutzt werden kann, um die Genauigkeit der Myon-Masse um eine Gr¨oßenordnung zu verbessern. Alter- nativ stellt unser Ansatz einen unabh¨angigenZugang zu dem umstrittenen anoma- len magnetischen Moment des freien Myons dar. Desweiteren werden Zwei-Schleifen- Selbstenergiekorrekturen f¨urden g-Faktor von gebundenen Elektronen zu allen Ord- nungen im Atomkern-Kopplungsst¨arke-Parameter Zα theoretisch untersucht. Formeln werden im Rahmen der two-time Green's function-Methode hergeleitet, und die Sepa- ration von Divergenzen erfolgt durch dimensionale Regularisierung. Unsere numerische Berechnung, bei der die Coulomb-Wechselwirkung in internen Propagatoren zu nullter und erster Ordnung ber¨ucksichtigt wird, zeigt, dass solche Zwei-Schleifen-Beitr¨agef¨ur strenge Tests der Quantenelektrodynamik mittels des g-Faktors des gebundenen Elek- trons ber¨ucksichtigt werden m¨ussen, sowie in naher Zukunft f¨urdie geplante Bestimmung von Naturkonstanten. 5 a Within the framework of this thesis, the following article has been published in peer- reviewed journals: • \Extraction of the electron mass from g factor measurements on light hydrogenlike ions" [1], J. Zatorski, B. Sikora, S. G. Karshenboim, S. Sturm, F. K¨ohler-Langes,K. Blaum, C. H. Keitel, and Z. Harman , Phys. Rev. A, 96, 012502, (2017). The following articles have been submitted or are in preparation: • \Access to improve the muon mass and magnetic moment anomaly via the bound- muon g factor" [2], B. Sikora, H. Cakir, N. Michel, V. Debierre, N. S. Oreshkina, N. A. Belov, V. A. Yerokhin, C. H. Keitel, and Z. Harman, submitted (2017); arXiv:1801.02501v1. • \Theory of the two-loop self-energy correction to the g-factor in non-perturbative Coulomb fields" [3], B. Sikora, N. S. Oreshkina, H. Cakir, Z. Harman, V. A. Yerokhin, and C. H. Keitel, in preparation • \Muonic vacuum polarization correction to the bound-electron g-factor" [4], N. A. Belov, B. Sikora, R. Weis, V. A. Yerokhin, S. Sturm, K. Blaum, C. H. Keitel, and Z. Harman, submitted (2016); arXiv:1610.01340v1. • \Light-by-light scattering-modified self-energy correction to the bound electron g factor" [5], V. Debierre, B. Sikora, H. Cakir, N. Oreshkina, Z. Harman, and C. H. Keitel, in preparation 7 a Contents 1 Introduction 11 2 Free-fermion g-factor and Zα expansion 17 3 Non-perturbative treatment of the nuclear interaction 21 3.1 Furry picture . 21 3.2 Solution of the Dirac equation - wavefunction . 22 3.3 Solution of the Dirac equation - Green's function . 25 3.4 Two-time Green's function method . 25 3.5 Dirac value of the bound-electron g-factor . 30 3.6 Nuclear effects on the bound-electron g factor . 32 3.6.1 Finite-size effect . 32 3.6.2 Finite-mass effect . 33 3.6.3 Nuclear deformation and polarization . 33 4 One-loop quantum electrodynamic corrections 35 4.1 One-loop Lamb shift . 35 4.2 One-loop self-energy correction to the g-factor . 37 4.2.1 One-loop self-energy diagrams . 38 4.2.2 Renormalization . 40 4.2.3 Evaluation of the individual terms . 43 4.2.4 Reference-state infrared divergences . 46 4.3 Energy-dependent perturbation theory - a consistency check . 47 4.4 Vacuum polarization . 48 4.4.1 Renormalization . 49 4.4.2 Uehling contribution . 50 4.4.3 Wichmann-Kroll contribution and magnetic loop . 52 5 Mass determinations by means of the bound-fermion g-factor 53 5.1 Determination of the electron mass . 53 5.2 Muon mass and magnetic moment anomaly from the bound-muon g-factor 54 5.2.1 Muonic systems . 55 5.2.2 Theory of the bound-muon g-factor . 56 5.2.3 Results and discussion . 58 6 Theory of the two-loop self-energy corrections to the g-factor 60 6.1 Two-loop self-energy diagrams . 60 6.1.1 Nested-loop and overlapping-loop diagrams . 61 6.1.2 The loop-after-loop diagrams . 63 6.2 Energy-dependent perturbation theory for two-loop SE diagrams . 67 6.3 Two-loop self-energy, renormalization . 68 6.3.1 Mass renormalization . 68 6.3.2 Charge renormalization . 69 6.4 Reference-state infrared divergences . 75 6.5 Separation of the two-loop terms into categories . 78 6.6 Loop-after-loop diagrams . 82 7 Evaluation of the F term 86 7.1 LAL, reducible contributions . 86 7.2 Evaluation of the zero-potential two-loop self-energy functions . 90 7.3 N and O irreducible, zero-potential contributions . 100 7.4 N and O irreducible, one-potential term . 101 7.5 Zero-potential, N and O reducible contribution . 101 7.6 Zero-potential, vertex contribution . 102 7.7 Alternative computation of the vertex type 1 contributions . 108 7.8 Remaining UV-divergent term . 110 7.9 Numerical results and the free-electron limit . 110 8 Conclusions and outlook 114 Appendix 117 A.1 Calculation of loop integrals . 117 A.2 The wave functions perturbed by a magnetic field . 118 A.3 Radial integrals . 120 1 Introduction The g-factor of the electron and the muon Quantum field theory (QFT) describes the fundamental quantum interactions between elementary particles. The Standard Model of particle physics which is based on QFT provides accurate descriptions of the electromagnetic, the strong and weak interactions [6{11]. Quantum electrodynamics (QED), which is the quantum theory of the elec- tromagnetic interaction, is a part of the Standard Model. It is, as of now, the most successful physical theory [6, 12, 13]. Among the most precisely determined quantities are the magnetic dipole moments (or, the associated g-factors) of the free electron and muon [6, 14{20]. An excellent 11-digit agreement between the theoretical and experimental g-factors of the free electron has been found (e.g. Ref. [19]). In fact, experimental measurements and theoretical calculations for the free electron have reached a level of precision that the g-factor measurement is used to determine the fine-structure constant α, assuming the correctness of QED [21]. The value of α extracted from the comparison of theory and experiment of the electron g-factor [22] was found to be in good agreement with α determined in independent methods such as atom recoil measurements [17]. Nowadays, the most precise value of α stems from a comparison between the experimental and theoretical g-factors of the free electron [19, 22]. For the g-factor of the free muon, however, a significant disagreement between the theoretical and experimental values has been found. This could be due to uncertainties of the hadronic vacuum polarization contribution, but the origin of this discrepancy is not known [19]. Another muon-related mystery is the proton radius puzzle [23{25]. Due to a larger overlap between the muon's wavefunction and the proton, a Lamb shift measurement on muonic hydrogen, i.e. the bound system formed by a proton and a negatively charged muon, allowed the determination of the proton radius with a better accuracy compared to the value accepted by the Committee on Data for Science and Technology (CODATA) [23]. The proton radius determined this way was found to be smaller by 5 standard deviations than the accepted CODATA value [23] which is determined from electron-proton scattering experiments and precision spectroscopy of the hydrogen atom [23]. A similar deviation was observed for the deuteron radius [26, 27]. However, a recent measurement using hydrogen atoms found a proton radius which is consistent with the proton radius from the muonic experiment [28]. Another discrepancy involving bound particles is the recently observed deviation be- tween the theoretical and experimental hyperfine splittings in hydrogenlike and lithi- umlike bismuth [29].