Higher-order QED effects

The agremnent between the experiments results and the theoretical results of Artemyev et al. and Plante et al. is in most cases quite good. Nevertheless, Chantler et al. have in a series of papers claimed that there are significant discrepancies between theory and experiments in a number of cases [17,18]— up to the order of 100 meV. We have found in our calculations that the effects beyond second order for the ground states of medium-heavy ions are only of the order of a few meV (the effect on the excited state should be even smaller), thus considerably smaller than the effects that Chantler et al. claim to have found. Therefore, if these discrepancies are real, they must have other causes than higher-order QED effects. 

This paper gives a brief survey of the variational and asymptotic expansion methods used to solve the nonrelativistic problem, and the principal effects that must be taken into account in order to estimate the higher-order QED corrections. The lowest order QED shift can now be calculated to high precision, and there has been much recent progress on the next-to-lowest order terms. 

Thus for a determination of a from a g factor measurement it would be desirable to choose an ion where Z is sufficiently high to get a small uncertainty in a but the influence of higher order QED contributions is not too large. Ca19+ seems to be a good choice. If we assume the same experimental accuracy on that ion as presently obtained in C5+ we would obtain a fractional uncertainty in a of 8 10-8. This is comparable to other present determinations of a from Quantum Hall or Josephson effect. The envisaged improvement in the experimental g factor by one order of magnitude would make the a determination competitive with that extracted from the g factor of the free electron. 

Abstract. The usefulness of study of hyperfine splitting in the hydrogen atom is limited on a level of 10 ppm by our knowledge of the proton structure. One way to go beyond 10 ppm is to study a specific difference of the hyperfine structure intervals 8Au2 — Avi. Nuclear effects for axe not important this difference and it is of use to study higher-order QED corrections. 

A very important point is the overall uncertainty of our calculations and we consider here particularly the contribution of the QED part to the uncertainty. The second order vacuum polarization effects (<5VP2) are of order (rx2/tt2)Ef (cf. relativistic corrections are of order (Za)2Ep))- Another source of uncertainty arises from higher order recoil effects (<5rec2) which can be estimated as <5rec2 = m/M ATec. ... 

Accurate calculations for the Lamb shift and hfs of hydrogen-like atoms are limited by their nuclear structure and higher-order QED corrections. In the case of low-Z Lamb shift, the finite-nuclear-size effects can be taken into account easily if we know the nuclear charge radius. 

Above we have proposed to measure the the small difference frequency df = f(1S-2S) - 3 f(2S-nS). This frequency depends critically on the Lamb shifts of the participating levels, and can provide a stringent test of QED. For n=100, the theoretical uncertainty of df is dominated by the nuclear size effect ( 70 kHz) and by approximations in the computation of electron structure corrections and uncalculated higher order QED corrections (65 kHz). The contribution of the Rydberg constant (1 kHz), the electron mass (0.05 kHz), and the fine structure constant (4 kHz) are negligible by comparison. The QED computations can be improved, and if theory is correct, a precision measurement of df can provide accurate new values for the charge radii of the proton and deuteron. 

It is true that in this procedure we miss some second-order QED effects which can be evaluated by standard QED methods. To mix these in a general way with electron correlation is beyond reach for the moment. We have found, however, that higher-order correlation is considerably more important than second-order QED effects for medium-heavy elements. Therefore, this procedure does include the most important effects of many-body QED in the cases studied. 

DCB is correct to second order in the fine-structure constant a, and is expected to be highly accurate for all neutral and weakly-ionized atoms [8]. Higher quantum electrodynamic (QED) terms are required for strongly-ionized species these are outside the scope of this chapter. A comprehensive discussion of higher QED effects and other aspects of relativistic atomic physics may be found in the proceedings of the 1988 Santa Barbara program [9]. 

The uncertainty of theoretical calculations including the estimation of missing or uncalculated terms has been receiving increasing scrutiny as techniques have advanced. One of the most recent two-electron Lamb shift calculations by Persson et al. [9] estimates missing correlation effects in QED contributions at 0.1 eV for all elements or 20 ppm of transition energies in medium Z ions. In earlier work, Drake [4] claimed uncertainty for Z = 23 was < 0.005 eV or 1 ppm of helium-like resonance lines due to uncalculated higher order terms. Some of the latest theoretical calculations for the w transition in medium Z ions are summarized in Table 3. 

Persson et al. states that the missing correlation effects in their two-electron QED calculations is estimated to be of the order of 0.1 eV for all elements. Formally this should only be only be applied to the range of elements 32 < Z < 92. The associated uncertainties for Z < 32 are unknown, but could be expected to increase in this regime. In the calculations of Drake, the uncertainty due to relativistic correlation effects in QED scales as a4Z4. The sources of the uncertainty are quite different in the calculations of Drake, and of Persson et al.. The lowest order Lamb shift is of order a3 ZA, and so the leading two-electron correction is of order a3Z3, i.e. smaller by a factor of 1 /Z. Higher order correlation... 

One will probably have to use some hybrid approaches, in which one treats relativistic one-particle contributions to higher order than two-particle contributions, in particular, where one neglects the coupling between relativity and electron correlation, and where one adds QED effects in an ad-hoc way. 

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