Bound-state QED

Bound-state QED provides a proper and practicable description of few-electron systems. Both QED-radiative corrections and electron-electron interactions may be treated perturbatively with respect to the coupling a — e2, counting the number of virtual photons involved, while the interaction with the external nuclear fields is included to all orders in Za. 

The QED radiative effects are treated perturbatively by the inclusion of Hrad-Accordingly, the unperturbed Hamiltonian Ho reduces to the external-field problem (normal ordering is indicated by ) 

Even-even nuclei may be described by a spherically symmetric Coulomb potential Vext of an extended nucleus with charge number Z. Pure QED effects due to the interaction with the free radiation field are carried by the interaction Hamiltonian 

The field operator xj/ is now expanded into the set of solutions pi of the corresponding one-particle Dirac equation htn f i = E/0, according to 

The subscript i labels the principle quantum number and angular momentum quantum numbers (njlm). Here a,- and bj denote electron-annihilation and positron-creation operators, respectively, defined via diagonaiization of the unperturbed Hamiltonian (1.11) 

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Perhaps a problem more important for applications is to eliminate the nuclear effects and to test the bound state QED precisely or use the bound state QED for the determination of some fundamental physical constants. There are a few ways to manage this problem [11] and to expand the accuracy of the tests of bound state QED beyond a level of our knowledge of the nuclear structure effects. 

Both pure leptonic atoms, muonium and positronium, are not stable the muon lives 2.2 /is and the positronium atom annihilates into photons in a much shorter time. However, they are compatible with the hydrogen atom as an object to test bound state QED. 

In the case of the positronium spectrum the accuracy is on the MHz-level for most of the studied transitions (Is hyperfine splitting, Is — 2s interval, fine structure) [13] and the theory is slightly better than the experiment. The decay of positronium occurs as a result of the annihilation of the electron and the positron and its rate strongly depends on the properties of positronium as an atomic system and it also provides us with precise tests of bound state QED. Since the nuclear mass (of positronium) is the positron mass and me+ = me-, such tests with the positronium spectrum and decay rates allow one to check a specific sector of bound state QED which is not available with any other atomic systems. A few years ago the theoretical uncertainties were high with respect to the experimental ones, but after attempts of several groups [17,18,19,20] the theory became more accurate than the experiment. It seems that the challenge has been undertaken on the experimental side [13]. 

A number of data are available for Is and 2s hfs intervals in hydrogen, deuterium and the helium-3 ion. The potential of this difference for the hfs intervals in the helium-3 ion [21] with respect to testing bound state QED is compatible with the ground state hfs in muonium both values are sensitive to fourth-order perturbative contributions. The difference of the Lamb shift plays an important role in the evaluation of optical data on the hydrogen and deuterium spectrum [22]-... 

The most straightforward way refers to the Bethe-Salpeter equation, i.e. an equation for the two-body Green function. It may be solved for the Coulomb potential and a two-body perturbative theory can be developed starting from this solution. This method was rarely used in the bound state QED calculations, being very complicated. 

The fine structure constant a can be determined with the help of several methods. The most accurate test of QED involves the anomalous magnetic moment of the electron [40] and provides the most accurate way to determine a value for the fine structure constant. Recent progress in calculations of the helium fine structure has allowed one to expect that the comparison of experiment [23,24] and ongoing theoretical prediction [23] will provide us with a precise value of a. Since the values of the fundamental constants and, in particular, of the fine structure constant, can be reached in a number of different ways it is necessary to compare them. Some experiments can be correlated and the comparison is not trivial. A procedure to find the most precise value is called the adjustment of fundamental constants [39]. A more important target of the adjustment is to check the consistency of different precision experiments and to check if e.g. the bound state QED agrees with the electrical standards and solid state physics. 

Fundamental physical constants are universal and their values are needed for different problems of physics and metrology, far beyond the study of simple atoms. That makes the precision physics of simple atoms a subject of a general physical interest. The determination of constants is a necessary and important part of most of the so-called precision test of the QED and bound state QED and that makes the precision physics of simple atoms an important field of a general interest. 

Lamb shift measurements on muonic hydrogen, as now pursued with a novel intense source of slow muons at the Paul Scherrer Institute [86,87] promise to yield an accurate rms charge radius of the proton, so that bound state QED can be tested to new levels of scrutiny. 

At present the good agreement within two standard deviations between the fine structure constant determined from muonium hyperfine structure and the one from the electron magnetic anomaly is generally considered the best test of internal consistency of QED, as one case involves bound state QED and the other one QED of free particles. 

Helium and helium-like ions are the prototypical many-electron system. All the bound-state QED physics of one-electron atoms is still present, of course, but with considerable added complication due to the electron-electron interaction. 

The g Factor of Hydrogenic Ions A Test of Bound State QED... 

We have performed an experiment to measure the g factor of the electron bound to a Carbon nucleus in a Hydrogen-like C5+ ion [9]. As shown below, the result of our measurement represents a significant test of bound state QED contributions and also accounts for the recoil correction from the finite mass of the carbon nucleus. The experiments are performed on single C5+ ions confined in a Penning ion trap at low temperatures, almost completely isolated from the environment. As outlined in the last paragraph the extension of our experiments to other highly charged systems opens a number of possibilities for future measurements of fundamental quantities such as the electrons mass or the fine structure constant. 

The g Factor of Hydrogenic Ions A Test of Bound State QED Table 1. Theoretical contributions to [Pg.207]

Conventionally, the evaluation of bound-state QED corrections is made tractable by including the vacuum fluctuations in several steps. The corrections thus calculated are called radiative corrections, and their evaluation can be made by making two expansions. The first is in powers of (a/7r) and denotes the number of photon propagator loops present. The second is in the number of photon exchanges with the nucleus and is in powers of (Zot), where Z is the nuclear charge. The expression in equation 1 shows the first two terms of the (o/tt) expansion for the case of hydrogenic S-states, i.e. up to two photon loops. In this expression the second expansion has not yet been made and the (Za) dependence is still contained within the functions F and H. 

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-... 

Contrary to this outstanding success, investigations on the bound-state QED modifications to gj were rather sparse until the mid 1990 s. Crotch and Hegstrom [12,13,14] as well as Faustov [15,16] and Close and Osborn [17] performed calculations on the first terms of a (Za) expansion for QED and recoil corrections to gj. Experiments were carried out on hydrogen and deuterium ([18,19,20] and references therein) as well as on He+ [21]. The existing theoretical calculations were then sufficient to describe the experimental results. 

Table 1. Known theoretical contributions to the gj factor of an electron bound in the ground state of 12C5+. All values axe given in units of 10 9. The error estimates are discussed in the text. If no error is given, it is less than 0.5 x 10-10. The errors for the total value axe due to the (Za) expansion fox the xecoil contxibution, the numerical uncertainties for the QED effects of order (a/ir), and the estimated error for the bound-state QED contribution of order (a/7r)2. In order not to underestimate any systematic effect, the numerical errors were linearly added...

Until now any precision test of bound state QED [2] was always realized in the way that only a final theoretical figure could be compared with some experimental result and no term of any theoretical expression could be tested separately. Any theoretical expression is a function of few parameters and one of them is the nuclear charge Z. However, there was no way to measure any function of Z. Only very few particular values of Z were available for precision experiments of the Lamb shift, fine and hyperfme structure. Now this has been changed dramatically. 

Recently a bound electron g factor in the hydrogen-like carbon ion was measured very accurately by the Mainz-GSI collaboration [1]. We consider here this new opportunity to test bound state QED and to determine precisely some fundamental constants from the study of the bound electron g factor. 

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. 

Now let us discuss possible precision tests of bound state QED. First we need to discuss what experimental results have been available up-to-now [2] ... 

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... 

Concluding our consideration we would like to underline, that the study of the g factor of a bound electron [1] offers a new opportunity for us to precisely test bound state QED theory and to determine two important fundamental constants the fine structure constant a and the electron-to-proton mass ratio m/mp. The experiment can be performed at any Z with about the same accuracy [1] and one can expect new data at medium Z which will allow to verify the present ability to estimate unknown higher-order corrections (i. e. theoretical uncertainty) in both low-Z and high-Z calculations. 

This function is interesting because it contains the energy levels predicted by Bound-State QED one can show [24] that... 

The current state-of-the-art in non-perturbative calculations (in Za) of atomic energy levels within Bound-State QED consists in the theoretical evaluation... 

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