Two-loop corrections

The two-loop electron polarization contribution to the Lamb shift may be calculated exactly like the one-loop contribution, the only difference is that one has to use as a perturbation potential the two-loop correction to the Coulomb... 

Recent progress in the study of high- and medium-Z ions [26,27] make calculations of two-loop corrections an important problem and that involves more QED effects in the study and now the status of high-Z and low-Z physics is very similar both need to take into account a2 terms and both cannot only apply an expansion in Za. 

The latter presents the largest sources of uncertainty in the theory of the muo-nium hfs interval, positronium energy spectrum and the specific nuclear-structure-independent difference for the hfs in the helium ion. The former are crucially important for the theory of the Lamb shift in hydrogen and medium-Z ions, for the difference in Eq. (2) applied to the Lamb shift and hyperfine structure in hydrogen and helium ion, and for the bound electron (/-factor. In the case of high-Z, the Lamb shift, (/-factor and hyperfine structure require an exact treatment of the two-loop correction. 

The higher-order two-loop corrections are to be calculated within the so-called external filed approximation (i. e. neglecting by the nuclear motion), while the recoil effects require an essential two-body treatment. There are a few approaches to solve the two-body problem (see e.g. [31]). Most start with the Green function of the two-body system which has to have a pole at the energy of the bound state... 

Historically, measurements of the IS Lamb shift in hydrogen have constituted the most accurate tests of bound-state QED. However, these recently calculated terms are obscured in hydrogen by the experimental error in the proton charge radius. This is because a non-QED correction to the Dirac levels due to the finite size of the nucleus is included in the Lamb shift, and the uncertainty in this term for the proton is comparable to the two-loop correction in the IS state. In He+, the error introduced by the experimental uncertainty in the alpha particle radius is relatively much smaller [14,15], making Lamb shift measure-... 

Continuing to the two-loop corrections, all terms known to date are state-independent, so we give only the ground state result [18,19,20],... 

The scaling of the unknown higher-order two-loop corrections to the Lamb... 

Since the uncertainty of the QED calculations is determined for these two ions (He+ and N6+) by the higher-order two-loop terms, we are going to reduce the other sources of uncertainty. We present results appropriate to provide an interpretation of the experiments mentioned as a direct study of the higher-order two loop corrections. The results of the ions experiments should afterwards be useful for the hydrogen atom. 

Two-loop corrections have not yet been calculated exactly and only a few of the terms have been known up-to-date [29,30] ... 

An important feature of the study of the g factor of a bound electron at Z = 20 — 30 is also the possibility to learn about higher-order two-loop corrections, which are one of the crucial problems of bound state QED theory. Below we discuss in detail the present status of theory and experiment. We consider a new opportunity to precisely test bound state QED and to accurately determine two fundamental constants the electron-to-proton mass ratio and the fine structure constant. 

In the case of the recent experiment with hydrogen-like carbon the nontrivial QED effects contribute an observable amount (see Table 1). We need to mention that, due to some delay of the final publications of the experimental result [1] and theoretical calculations [10], no actual theoretical predictions have been published. Most of the presentations (conference and seminar talks and posters) dealt with unaccurate theoretical predictions because it was believed that nothing had been known on the two-loop corrections. However, that was not the case, because from the beginning of the theoretical calculations up to recent re-calculations it was clearly stated ed [6] that the (Za)2 term in Eq. (4) is of pure kinematic origin and so the result is valid in any order of the expansion in a for the anomalous magnetic moment of a free electron, and in particular... 

These coefficients can be studied at Z = 10 — 20 or calculated. With Z higher than 20 it is necessary to take into consideration a2(Za)6 terms which can contain temrs up to the cube of the low-energy logarithm (ln(Za)) and we have a problem of higher-order two-loop corrections. That is now one of the most important theoretical problems in bound-state QED. In particular, it essentially limits computational accuracy for... 

In the case of the bound g factor for the first time there is an opportunity to study the higher-order two-loop corrections experimentally in detail with varying value of Z. 

Ivanov and Karshenboim [11] have determined a theoretical value for the 25 i/2 — 2P3/2 transition in 14jV6+ of 834.928(7) cm-1, where the error is dominated by the uncertainty in the two-loop binding corrections. Other QED corrections, and the nuclear size correction (based on the stated error of the nuclear rms charge radius measurements), contribute only about 0.001 cm-1. Hence an experimental precision of 0.002 cm-1 or better would provide an interesting test of the two-loop corrections. This corresponds to subdivision of the 8 cm-1 natural resonance FWHM by a factor of 4000. 

One-particle spectral properties. The two-loop corrections are also of interest for the quasi-particle weight z (cf. Eq. 7) at the Fermi level which is governed by the scaling equation... 

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