Stringent tests of QED using highly charged ions

TL;DR

This paper reviews recent experimental and theoretical tests of quantum electrodynamics (QED) using highly charged ions, focusing on Lamb shift, hyperfine splitting, and g-factor measurements, and explores future prospects at supercritical fields.

Contribution

It provides a comprehensive review of current QED tests with highly charged ions and discusses new experimental approaches and future opportunities at supercritical fields.

Findings

Agreement between theory and experiment for Lamb shift and transition energies

Recent progress in hyperfine splitting and g-factor isotope shift measurements

Potential for testing QED in supercritical fields in low-energy heavy-ion collisions

Abstract

The present status of tests of QED with highly charged ions is reviewed. The theoretical predictions for the Lamb shift and the transition energies are compared with available experimental data. Recent achievements in studies of the hyperfine splitting and the $g$-factor isotope shift with highly charged ions are reported. Special attention is paid to tests of QED within and beyond the Furry picture at strong-coupling regime. Prospects for tests of QED at supercritical fields that can be created in low-energy heavy-ion collisions are discussed as well.

Tables

Table 1. Table 1: The ground-state Lamb shift in 238 U 91+ , in eV.

Contribution	Value
Furry picture QED	265.19(33)
Finite nuclear size	198.54(19)
Nuclear recoil	0.46
Nuclear polarization	0.20(10)
Total theory	463.99(39)
Experiment gum05	460.2(4.6)

Table 2. Table 2: The 1 ​ s ​ 2 ​ p 1 ​ P 1 − 1 ​ s 2 ​ S 0 1 1 𝑠 2 superscript 𝑝 1 subscript 𝑃 1 1 superscript 𝑠 2 superscript subscript 𝑆 0 1 1s2p\,^{1}P_{1}-1s^{2}\,{}^{1}S_{0} transition energy in He-like titanium, in eV.

Contribution	Value
Breit approximation	4751.706
QED	$-$2.059
Total theory (this work)	4749.647(2)
Total theory art05	4749.644(1)
Experiment cha12	4749.85(7)

Table 3. Table 3: Isotope shift of the g 𝑔 g factor of Li-like calcium, Ca 17 + 40 − 48 Ca 17 + superscript 48 superscript superscript Ca limit-from 17 40 superscript Ca limit-from 17 {}^{40}{\rm Ca}^{17+}-\;^{48}{\rm Ca}^{17+} .

Contribution	Value
Nuclear recoil: one-electron non-QED	0.000 000 012 246
Nuclear recoil: interelectronic int.	$-$0.000 000 001 302
Nuclear recoil: QED	0.000 000 000 114(12)
Finite nuclear size	0.000 000 000 004(9)
Total theory sha17	0.000 000 011 056(16)
Experiment koe16	0.000 000 011 7(14)

Equations

$$(-i\vec{\alpha }\cdot \vec{\nabla }+m\beta +V_{\mathrm{C}}(r))\psi (\vec{r})=E\psi (\vec{r}).$$

$${\Delta }^{\prime }E=\Delta E^{(2s)}-\xi \Delta E^{(1s)},$$

Full text

††journal: Hyperfine Interactions

1 Introduction

For almost four decades since the creation of quantum electrodynamics (QED) in its present form, tests of QED were mainly restricted by light atomic systems: hydrogen, helium, positronium, muonium, etc. The calculations of these systems are generally based on expansions in two small parameters: $\alpha$ and $\alpha Z$, where $\alpha$ is the fine structure constant and $Z$ is the nuclear charge number. So, the studies of these systems provided tests of QED in few lowest orders in $\alpha$ and $\alpha Z$ only. A possibility to extend this rather small region of the QED tests appeared in the late 1980s when first high-precision measurements with heavy few-electron ions were performed. In contrast to light atoms, the parameter $\alpha Z$ in highly charged ions is no longer small. Therefore, the calculations of these ions must be performed to all orders in $\alpha Z$. From one side, it provides us with serious technical and, in some cases, even conceptual problems but, from the other side, it gives us a unique opportunity to test QED in a new region: nonperturbative in $\alpha Z$ regime.

The basic idea of the theoretical approach to the QED calculations of highly charged few-electron ions can be formulated as follows. Since the number of electrons in these ions is much smaller than the nuclear charge number, to zeroth order we can take into account only the interaction of the electrons with the Coulomb field of the nucleus, $V_{\rm C}(r)$, and neglect the interaction of the electrons with each other. This means that to zeroth order the electrons obey the Dirac equation ($\hbar=c=1$),

[TABLE]

Then, the interelectronic-interaction and QED effects are accounted for by perturbation theory in the parameters $1/Z$ and $\alpha$, respectively. Due to smallness of these parameters, precise QED calculations of these systems are definitely possible. Moreover, for very heavy ions the parameter $1/Z$ becomes comparable with $\alpha$ and, therefore, all contributions can be classified by the parameter $\alpha$. This perturbation theory can be further improved by incorporating into the Dirac equation a screening potential, $V_{\rm scr}(r)$, which partly accounts for the electron-electron interaction effects. Then, in the higher orders, to avoid double counting of some electron-electron interaction effects, one should add the interaction with the potential $-V_{\rm scr}(r)$. These two schemes of the QED perturbation approach are known as the Furry and the extended Furry picture, respectively.

2 Lamb shift and transition energies

In Table 1 we present the theoretical contributions to the $1s$ state Lamb shift in H-like uranium gla11 . The Lamb shift is defined as the difference between the exact total energy and the energy which is derived from the Dirac equation for the point-charge nucleus. The finite nuclear size effect was evaluated for the Fermi model of the nuclear charge distribution, including the nuclear deformation effect koz08 . The Furry picture of QED incorporates the contributions of the first order in $\alpha$ (one-loop QED) yer15 and the second-order in $\alpha$ (two-loop QED) yer03 . The calculation of the nuclear recoil effect sha98 requires using QED beyond the Furry picture at strong-coupling regime. Finally, one should account for the nuclear-polarization effect, which in terms of the Feynman diagrams is described by two-photon exchange between the electron and the nucleus with excited intermediate nuclear states plu95 ; nef96 . As one can see from Table 1, the total theoretical Lamb shift value agrees well with the experimental one gum05 but has ten times better accuracy. This provides a test of QED at strong field on a 2% level.

A higher accuracy was achieved in experiments with heavy Li-like ions sch91 ; bra03 ; bei05 . The comparison of the most precise experiment on the $2p_{1/2}-2s$ transition energy in Li-like uranium bei05 with the related theory gla11 ; koz08 ; yer07 ; sap11 provides a test of strong-field QED on a 0.2% level.

In Ref. cha12 , on the grounds of new measurements with heliumlike titanium ($Z=22$) and a statistical analysis of all the data available in literature, it was claimed that there exists a systematic discrepancy between theory and experiment, which scales approximately as $Z^{3}$. Although the experimental results of a number of works (see Refs. tra09 ; ama12 ; kub14 ; bei15 ; epp15 ; mac18 ) for the transition energies in He-like ions agree with the most elaborated QED calculations art05 , independent theoretical predictions would be very desirable. In Table 2 we present the results of our calculation of the $1s2p\,^{1}P_{1}-1s^{2}\,{}^{1}S_{0}$ transition energy in heliumlike titanium and compare them with the theoretical result of Ref. art05 and the experiment cha12 . As one can see from the table, our new result is rather close to the previous theoretical prediction and disagrees with the experimental result of Ref. cha12 .

3 Hyperfine splitting

High-precision measurements of the hyperfine splitting (HFS) in heavy H-like ions were performed in Refs. kla94 ; cre96 ; cre98 ; see98 ; bei01 ; ull15 . The main goal of these experiments was to test QED in a unique combination of the strongest electric and magnetic fields. For instance, the average magnetic field experienced by the electron in H-like bismuth amounts to about 30000 T, which is 1000 times stronger than the field obtained with the strongest superconducting magnet. Despite a very high precision of these measurements, from $10^{-4}$ to $10^{-5}$, their comparison with theory could not provide tests of QED because of a large theoretical uncertainty of the nuclear magnetization distribution correction (so-called Bohr-Weisskopf effect) sha97 ; sen02 . To overcome this problem, in Ref. sha01 it was proposed to study a specific difference of the HFS values for Li- and H-like ions of the same heavy isotope,

[TABLE]

where the parameter $\xi$ must be chosen to cancel the Bohr-Weisskopf effect. It was shown sha01 that both the parameter $\xi$ and the specific difference $\Delta^{\prime}E$ are very stable with respect to possible variations of microscopic nuclear models and, therefore, can be calculated to a very high accuracy. In case of 209Bi, one finds $\xi=0.16886$. The most elaborated theoretical calculations for 209Bi employing the nuclear magnetic moment $\mu/\mu_{N}=4.1106(2)$ rag89 lead to the specific difference $\Delta^{\prime}E=-61.320(6)$ meV vol12 . This theoretical value includes the QED contribution of 0.229(2) meV. It means that tests of QED effects on the HFS are possible, provided the HFS measurements for H- and Li-like bismuth are carried out to the required accuracy.

The recent experiments for 209Bi ull17 resulted in the value $\Delta^{\prime}E=-61.012(5)(21)$ meV, which was in strong disagreement with the theoretical prediction. This disagreement was resolved in Ref. skr18 , where new calculations of the magnetic shielding constants in 209Bi(NO3)3 and 209BiF${}_{6}^{-}$ clearly showed that the widely used value for the nuclear magnetic moment, $\mu/\mu_{N}=4.1106(2)$ rag89 , is incorrect. Moreover, these calculations combined with new NMR measurements of the nuclear magnetic moment in 209Bi(NO3)3 and 209BiF${}_{6}^{-}$ skr18 lead to a value $\mu/\mu_{N}=4.092(2)$. With this magnetic moment, the theoretical value of the specific difference amounts to $\Delta^{\prime}E=−61.043(5)(30)$ meV, where the first uncertainty is due to uncalculated corrections and remaining nuclear effects, while the second one is due to the uncertainty of the new nuclear magnetic moment value. This theoretical value agrees well with the experiment.

The most precise to-date value for the nuclear magnetic moment of 209Bi can be obtained via equating the theoretical and experimental results on the specific difference. This leads to $\mu/\mu_{N}=4.0900(15)$ skr18 . However, new measurements of the nuclear magnetic moment as well as the HFS in H- and Li-like 209Bi are needed to provide stringent QED tests.

4 Bound-electron $g$ factor

High-precision measurements of the $g$ factor of highly charged ions were first performed for H-like carbon haf00 . The uncertainty of the experimental value obtained in that work was mainly defined by the uncertainty of the electron mass which was accepted that time. Combined with the related advance in theory of the $g$ factor, which included evaluations of the higher-order relativistic recoil and QED corrections sha02 ; yer02 , this resulted in four-times improvement of the accuracy of the electron mass. Later, the $g$ factor measurements and the related theoretical predictions were improved in accuracy and extended to other ions (see, e.g., Refs. pac05 ; stu13 ; wag13 ; stu14 ; sha15 ; cza18 and references therein). As a result, in Refs. stu14 ; zat17 the precision of the atomic mass of the electron was further improved by a factor of 13.

Recently, the isotope shift of the $g$ factor of Li-like calcium, ACa17+ with $A=40$ and $A=48$, was measured koe16 . From the theoretical side, the value of the isotope shift in calcium is mainly defined by the nuclear recoil effect (mass shift). The theoretical result presented in Ref. koe16 included the QED calculation of the one-electron recoil effect and the evaluation of the two-electron recoil effect using an extrapolation of the results obtained in Refs. yan01 ; yan02 in the framework of a two-component Breit-approximation approach heg75 . This theoretical value was in agreement with the experimental one but at the edge of the experimental uncertainty. In Ref. sha17 the two-electron recoil contribution was recalculated within the Breit approximation using a four-component approach sha01b . It was found that the obtained result strongly disagrees with the corresponding result based on the two-component approach yan01 ; yan02 . A detailed analysis sha17 showed that the disagreement was caused by omitting some important terms in the calculation scheme formulated within the two-component approach in Ref. heg75 .

In Table 3, the individual contributions to the isotope shift of the $g$ factor of Li-like calcium, ${}^{40}{\rm Ca}^{17+}-\;^{48}{\rm Ca}^{17+}$, are presented. It can be seen that the theoretical and experimental results are in good agreement with each other. This gives the first test of the relativistic theory of the recoil effect in highly charged ions in presence of magnetic field. As was shown in Ref. mal17 , the study of a specific difference of the $g$ factors of H- and Li-like lead can provide a test of the QED recoil effect on a few-percent level. This would give the first test of QED at strong-coupling regime beyond the Furry picture.

5 Supercritical fields

In Fig. 1 we display the energy levels of an electron in a Coulomb field as functions of the nuclear charge number $Z$. For point nuclei the $1s$ level exists only up to $Z\approx 137$. However, for extended nuclei the ground state level goes continuously down and at $Z\approx 173$ “dives”  into the negative-energy Dirac continuum (see, e.g., book gre85 and references therein). If this level was originally empty, its diving into the negative-energy Dirac continuum should result in so-called spontaneous emission of two positrons. Since there are no nuclei with so high $Z$, the only way to observe this fundamental effect is to study low-energy heavy-ion collisions.

If we consider the behavior of the low-lying energy levels of a quasimolecule formed by colliding uranium ions at the energy near the Coulomb barrier ($\sim$ 5.9 MeV/u) as functions of time, we find that the supercritical field exists for about $10^{-21}$ seconds only gre85 . Unfortunately, this time is by two orders of magnitude smaller than the time required for the spontaneous positron emission, and the positrons are mainly created due to the dynamical (induced) mechanism. This was one of the main reasons why the experiments performed at GSI more than 30 years ago could not prove or disprove the spontaneous positron creation gre85 . One may expect, however, that investigations of quantum dynamics of various processes which take place in low-energy collisions of bare nuclei or few-electron ions at both subcritical and supercritical regimes can prove or disprove the “diving”  scenario which should lead to the spontaneous pair creation. From the theoretical side, new methods should be developed to perform these investigations in all details. The calculations of the pair-creation probabilities performed many years ago by Frankfurt ’s group gre85 were limited by the monopole approximation. In this approximation the expansion of the two-center nuclear potential is restricted to the zero-order spherical harmonic term. The first calculations of the pair-creation probabilities beyond the monopole approximation have been performed in Refs. mal17a ; pop18 . These calculations at the energy near the Coulomb barrier showed that the difference between the exact and monopole-approximation results varies from about 6% at the zero impact parameter, $b=0$, to about 30% at $b=40$ fm.

6 Conclusion and outlook

In this paper we have reviewed the recent achievements in the study of QED with highly charged ions. To date, strong-field QED has been mainly tested in the Lamb-shift experiments with heavy ions. However, during the last years a great progress was made from both experimental and theoretical sides in investigations of the hyperfine splitting and the $g$ factor with highly charged ions. High-precision measurements of the $g$ factor of heavy few-electron ions are anticipated in the nearest future at the Max-Planck-Institut für Kernphysik in Heidelberg and at the HITRAP/FAIR facilities in Darmstadt. Extention of these measurements to ions with nonzero nuclear spin could also provide the most precise determination of the nuclear magnetic moments, which are urgently needed for the HFS studies. Further developments of the theoretical methods to describe quantum dynamics of electrons in low-energy heavy-ion collisions are also required. It is expected that investigations of these collisions can prove or disprove the “diving”  scenario which leads to the spontaneous pair creation.

Acknowledgements

This work was supported by the Russian Science Foundation (Grant No. 17-12-01097).