The determination of the electron mass from Penning-trap measurements with 12 C 5+ ions and from theoretical results for the bound-electron g factor is described in detail. Some recently calculated contributions slightly shift the extracted mass value. Prospects of a further improvement of the electron mass are discussed both from the experimental and from the theoretical point of view. Measurements with 4 He + ions will enable a consistency check of the electron mass value, and in future an improvement of the 4 He nuclear mass and a determination of the fine-structure constant.

The procedure for the calculation of the self-energy-corrected wave function of the bound electron in the field of the nucleus is discussed. We present the related formulas and discuss the numerical difficulties and the methods used to overcome them. The results of the calculation are presented for a wide range of ions. Possible applications of the numerically obtained wave functions are discussed.

QED corrections to the g factor of Li-like and B-like ions in a wide range of nuclear charges are presented. Many-electron contributions as well as radiative effects on the one-loop level are calculated. Contributions resulting from the interelectronic interaction, the self-energy effect, and most of the terms of the vacuumpolarization effect are evaluated to all orders in the nuclear coupling strength Zα. Uncertainties resulting from nuclear size effects, numerical computations, and uncalculated effects are discussed.

Two-loop self-energy corrections to the bound-electron g factor are investigated theoretically to all orders in the nuclear binding strength parameter Zα. The separation of divergences is performed by dimensional regularization, and the contributing diagrams are regrouped into specific categories to yield finite results. We evaluate numerically the loop-after-loop terms, and the remaining diagrams by treating the Coulomb interaction in the electron propagators up to first order. The results show that such two-loop terms are mandatory to take into account for projected near-future stringent tests of quantum electrodynamics and for the determination of fundamental constants through the g factor. PACS numbers: 06.20.Jr, 21.10.Ky, 31.30.jn, 31.15.ac, 32.10.Dk The g factor of one-electron ions can be measured and calculated with an exceptional accuracy [1-10]. Its theoretical and experimental values in 28 Si 13+ were found to be in excellent agreement [1]. Since then, the experimental uncertainty decreased by an order of magnitude [2]. Such measurements also allowed an improved determination of the electron mass [11] (see also [12,13]). It is anticipated that boundelectron g factor measurements will also enable in the foreseeable future an independent determination of the fine-structure constant α [14,15].To push forward the boundaries of theory, quantum electrodynamic (QED) corrections at the one-and two-loop level need to be calculated with increasing accuracy. One-loop corrections have been evaluated both as a power series in Zα (with Z being the atomic number) and non-perturbatively in this parameter (see e.g. [16][17][18]). Two-loop corrections were evaluated up to fourth order in Zα [16,19]. Contributions of order α 2 (Zα) 5 were completed very recently [10]. At high nuclear charges, where Zα ≈ 1, an expansion in Zα is not applicable. So far, the two-loop diagrams with two electric vacuum polarization (VP) loops and those with one electric VP and one self-energy (SE) loop were evaluated nonperturbatively in Zα [20].For a broad range of Z, the two-loop SE corrections, which are by far the hardest to calculate, constitute the largest source of uncertainty. This holds true even at Z = 6, after a recent high-precision evaluation of the one-loop SE corrections [4,18]. We thus see that higher-order terms in Zα are also necessary at lower nuclear charges, if an ultimate precision is required. Therefore, in the current Letter we present the theoretical framework for the non-perturbative evaluation of the two-loop SE terms.There are three two-loop SE diagrams contributing to the binding energy of a hydrogenlike ion, namely, the loopafter-loop (LAL), the nested loops (N) and the overlapping loops (O) diagrams. Their calculation has been presented in detail in Refs. [21][22][23][24][25][26]. The corresponding diagrams for the g factor can be generated by magnetic vertex insertions into the Lamb shift diagrams, yielding three nonequivalent diagrams in each of the above classes, shown in Fig. 1.Basic analysis.-We derived ...

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