QED radiative corrections to the $^2$P$_{1/2}$-$^2$P$_{3/2}$ fine-structure in fluorinelike ions

TL;DR

This paper presents precise, nonperturbative QED calculations of fine-structure energy corrections in fluorinelike ions, resolving previous discrepancies between theory and experiment and highlighting significant differences from approximate methods.

Contribution

It provides the first nonperturbative ab initio QED calculations of fine-structure in F-like ions, including all relevant effects, improving accuracy over prior approximate methods.

Findings

Discrepancies with approximate QED calculations are significant, especially for uranium.

The new calculations resolve previous theory-experiment discrepancies when combined with existing fine-structure data.

Self-energy contributions can differ in sign from earlier estimates, notably in heavy ions.

Abstract

Ab initio calculations of QED radiative corrections to the $^2P_{1/2}$ - $^2P_{3/2}$ fine-structure transition energy are performed for selected F-like ions. These calculations are nonperturbative in $\alpha Z$ and include all first-order and many-electron second-order effects in $\alpha$. When compared to approximate QED computations, a notable discrepancy is found especially for F-like uranium for which the predicted self-energy contributions even differ in sign. Moreover, all deviations between theory and experiment for the $^2P_{1/2}$ - $^2P_{3/2}$ fine-structure energies of F-like ions, reported recently by Li et al., Phys. Rev. A 98, 020502(R) (2018), are resolved if their highly accurate, non-QED fine-structure values are combined with the QED corrections ab initially evaluated here.

Tables

Table 1. Table 1: QED ( Δ ​ E tot ( 1 ) Δ subscript superscript 𝐸 1 tot \Delta E^{(1)}_{\rm tot} ) and screened QED ( Δ ​ E tot ( 2 ) Δ subscript superscript 𝐸 2 tot \Delta E^{(2)}_{\rm tot} ) radiative corrections to the P 3 / 2 2 − 2 P 1 / 2 superscript 2 superscript subscript 𝑃 3 2 2 subscript 𝑃 1 2 {}^{2}P_{3/2}-^{2}P_{1/2} transition energy in F-like Mo 33+ ( Z = 42 𝑍 42 Z=42 ) and U 83+ ( Z = 92 𝑍 92 Z=92 ) ions. The self-energy and vacuum polarization contributions are presented separately. Results are shown for three different starting potentials: Coulomb, CH1, and CH2. Our final values (Final) are compared with the results of the approximate treatments ( grasp2k , Welton, MQED) evaluated in Ref.  Li et al. ( 2018 ) . All values are in eV.

	$Z=42$				$Z=92$
Contr.	Coulomb	CH1	CH2		Coulomb	CH1	CH2
	Self-energy
$\Delta E_{\mathrm{tot}}^{\left(1\right)}$	0.2714	0.2272	0.2059		$-$0.658	$-$0.396	$-$0.423
$\Delta E_{\mathrm{tot}}^{\left(2\right)}$	$-$0.0588	$-$0.0112	0.0120		1.271	0.906	0.920
Final	0.2126	0.2160	0.2179		0.613	0.510	0.497
grasp2k	0.1987				$-$0.821
Welton	0.1967				$-$0.668
MQED	0.2248				$-$0.361
	Vacuum polarization
$\Delta E_{\mathrm{tot}}^{\left(1\right)}$	0.0079	0.0062	0.0055		2.596	2.290	2.190
$\Delta E_{\mathrm{tot}}^{\left(2\right)}$	$-$0.0043	$-$0.0021	$-$0.0013		$-$0.680	$-$0.330	$-$0.221
Final	0.0036	0.0041	0.0042		1.916	1.960	1.969
grasp2k	0.0046				2.148

Table 2. Table 2: Comparison of the QED radiative corrections evaluated here (This work) and in Ref.  Li et al. ( 2018 ) ( grasp2k , Welton, MQED) with the QED corrections deduced from the experiments (Exp). The latter values are obtained as the difference between the experimental results (Experiment, 2nd column) and theoretical values without QED effects (Non-QED, 4th column) calculated in Ref.  Li et al. ( 2018 ) . All values are in eV.

						QED
$Z$	Experiment		Ref.	Non-QED Li et al. (2018)		Exp	This work	grasp2k	Welton	MQED
18	2	.240084(1)	Draganić et al. (2003)	2	.2348	0.0053	0.0055(7)	0.0049	0.0049	0.0063
22	5	.85434(19)	Peacock et al. (1984)	5	.8408	0.0135(2)	0.0139(10)	0.0125	0.0125	0.0154
26	12	.71818(25)	Peacock et al. (1984)	12	.6899	0.0282(3)	0.0292(16)	0.0266	0.0265	0.0318
28	17	.8486(7)	Peacock et al. (1984)	17	.8093	0.0393(7)	0.0404(16)	0.0368	0.0366	0.0435
36	55	.352(7)	Denne et al. (1989)	55	.238	0.114(7)	0.118(3)	0.108	0.107	0.123
39	79	.071(6)	Silwal et al. (2017)	78	.906	0.165(6)	0.164(3)	0.150	0.149	0.171
40	88	.465(25)	Myrnäs et al. (1994)	88	.306	0.159(25)	0.182(3)	0.167	0.165	0.189
42	109	.888(20)	Myrnäs et al. (1994)	109	.668	0.220(20)	0.222(4)	0.203	0.201	0.229
74	1388	.87(60)	Clementson et al. (2011)	1387	.60	1.27(60)	1.78(1)	1.50	1.48	1.67
92	3913	.54(16)	Beiersdorfer et al. (1998)	3911	.29	2.25(16)	2.47(2)	1.33	1.48	1.79

Equations

$$V_{\mathrm{scr}}(r)=\alpha {\int }_0^{\infty }\mathrm{d}{r}^{\prime }\frac{1}{r_{>}}\rho (r^{\prime })$$

$${\rho }_{CH1}(r)=\sum _{c=1s^2,2s^2}\left[G_c^2(r)+F_c^2(r)\right],$$

$${\rho }_{CH2}(r)=\sum _{c=1s^2,2s^2,2p_{1/2},2p_{3/2}^3}\left[G_c^2(r)+F_c^2(r)\right],$$

$$\Delta E_{\mathrm{tot}}^{(1)}=\Delta E_{2p_{3/2}}^{(1)}-\Delta E_{2p_{1/2}}^{(1)},$$

$$\Delta E_{\mathrm{tot}}^{(2)}$$

Full text

Since quantum electrodynamics (QED) has first been worked out at the end of 1940s, this theory has been found ubiquitous for studying all phenomena associated with electrically charged particles and has meanwhile reached a high level of accuracy. The great successes of QED have also lead to its cornerstone place in the description of other interactions in the standard model. In view of this decisive role, it is of a crucial importance to verify QED under different conditions and to set new limits for its validity. In these studies, highly charged ions provide a unique scenario for probing QED effects in the strongest electromagnetic fields accessible at present for experimental study. In recent years it was particularly shown that highly charged ions are a spectacular tool for high precision measurements. For example, it is worth to mention the experiments on binding and transition energies in few-electron heavy ions Brandau et al. (2003); Gumberidze et al. (2004, 2005); Beiersdorfer et al. (2005); Bruhns et al. (2007); Thorn et al. (2009); Mäckel et al. (2011); Amaro et al. (2012); Chantler et al. (2012); Gassner et al. (2018), which provided diverse tests of the bound-state QED. In the future, the advent of new accelerator facilities, e.g., FAIR in Germany and HIAF in China, as well as storage rings and ion traps will force the limits of accuracy to ever higher and higher levels.

From a theoretical viewpoint, all tests of QED for many-electron atoms and ions are restricted by the uncertainty of the electron-electron correlation effects. Since these correlation effects rapidly increase with the number of electrons, rigorous QED calculations up to the second order in $\alpha$ have been so far performed only for ions with one, two, three, four, five, or eleven electrons (see, e.g., Refs. Soff and Mohr (1988); Mohr (1992); Persson et al. (1996); Yerokhin et al. (1999); Labzowsky et al. (1999); Yerokhin et al. (2003); Artemyev et al. (2005, 2007); Kozhedub et al. (2010); Sapirstein and Cheng (2011); Artemyev et al. (2013); Malyshev et al. (2014, 2015); Yerokhin and Shabaev (2015); Sapirstein and Cheng (2015); Malyshev et al. (2017); Blundell (1993) and references therein). For ions with more electrons, in contrast, ab initio QED calculations have been performed only in first order in $\alpha$ Blundell (1993); Sapirstein and Cheng (2002, 2003); Chen et al. (2006) or various QED model potentials were applied to include the QED contributions approximately Welton (1948); Indelicato and Desclaux (1990); Pyykkö and Zhao (2003); Flambaum and Ginges (2005); Lowe et al. (2013); Shabaev et al. (2013). In Refs. Ginges and Berengut (2016); Tupitsyn et al. (2016) it has been shown that the rigorous first-order in $\alpha$ and QED model potential calculations give similar predictions. Moreover, the QED corrections evaluated by the approximate methods unambiguously bring in agreement theoretical predictions with experimental results, e.g., in the case of boronlike argon Draganić et al. (2003) or neutral gold Pašteka et al. (2017). However, despite these examples, it is still not clear how accurate the QED contributions obtained by such approaches are. In particular, in recent papers of Li et al. Li et al. (2018) and Si et al. Si et al. (2018) accurate theoretical calculations have been performed for the ground-state transition energies in F-like and Co-like ions and have shown that the correlation effects are strongly reduced by so-called Layzer quenching effect Layzer (1959); Layzer and Bahcall (1962). Thus, the accurate account for the correlations becomes possible for these transitions and this allows one to test the QED corrections. However, the theoretical results for some F-like ions are substantially different from the experimental values, as it has been reported in Ref. Li et al. (2018). Moreover, a careful analysis of this discrepancy showed that it does not arise from missing correlations contributions. A particular large discrepancy arises for F-like uranium for which different QED model computations all give similar results but which all deviate by 3-6 $\sigma$ from the experiment Beiersdorfer et al. (1998).

Here, we present ab initio calculations of the QED radiative corrections for the fine-structure $1s^{2}2s^{2}2p^{5}\,{}^{2}P_{3/2}-^{2}P_{1/2}$ transition energy in F-like ions. The first-order and many-electron second-order in $\alpha$ diagrams are rigorously evaluated. In particular, we show that for F-like U83+ ion the QED corrections obtained in our work strongly disagrees with the model-potential values. Moreover, when we employ our QED values instead of the QED results obtained in Ref. Li et al. (2018) we find excellent agreement between theoretical and experimental values.

For highly charged ions, ab initio QED calculations should be relativistic from the very beginning. For these ions, in fact, the parameter $\alpha Z$ ($Z$ is the nuclear charge number) cannot be utilized as an expansion parameter and, hence, the calculations must be performed to all orders in $\alpha Z$. As usual, let us start from the Dirac equation with the Coulomb or effective local potential, an approach that is known also as the original or extended Furry picture. The many-electron wave functions are then constructed from the one-electron Dirac solution as the Slater determinants. In the extended Furry representation, the effective potential is the sum of the Coulomb and screening potentials. In the present study, we employ two core-Hartree-type of potentials

[TABLE]

as screening potentials. Here, $\rho$ is the density of the core electrons. First potential (CH1) is constructed from the density of $1s^{2}2s^{2}$ shells electrons:

[TABLE]

while for the second one (CH2) we refer $1s^{2}2s^{2}2p_{1/2}2p_{3/2}^{3}$ electrons as core electrons:

[TABLE]

where $G_{c}$ and $F_{c}$ are the upper and lower radial components of the one-electron Dirac wave function, respectively. The normalization integral of density $\rho_{\rm CH1}$ gives four, while the integral of $\rho_{\rm CH2}$ gives eight, which corresponds to the number of core electrons in both cases.

Further, one has to develop the QED perturbation expansion. In the first order in $\alpha$, the radiative corrections are representing by the diagrams depicted in Fig. 1.

Their contributions to the $1s^{2}2s^{2}2p^{5}\,{}^{2}P_{3/2}-^{2}P_{1/2}$ transition energy $\Delta E^{(1)}_{\rm tot}$ can be expressed as follows

[TABLE]

where $\Delta E^{(1)}_{2p_{3/2}}$ and $\Delta E^{(1)}_{2p_{1/2}}$ are the one-electron QED corrections to the $2p_{3/2}$ and $2p_{1/2}$ states, respectively. In the next (second) order in $\alpha$, the radiative corrections are given by the one-electron two-loop diagrams and many-electron (screened) one-loop diagrams, the latter are presented in Fig. 2.

In the present investigation, we do not consider the two-loop diagrams. Their contributions estimated by the one-electron values given in Ref. Yerokhin and Shabaev (2015) are found to be smaller than our final uncertainty. Note that for the extended Furry picture one needs to account for the so-called counterterm diagrams in addition to the diagrams depicted in Fig. 2. In these diagrams, the photon which mediates the interelectronic interaction is replaced by the interaction with the screening potential. Finally, the second-order contributions to the 2P1/2 - 2P3/2 fine-structure transition energy can be combined into:

[TABLE]

Here, $\Delta E^{(2)}_{(1s)2p_{3/2}}$, $\Delta E^{(2)}_{(1s)2p_{1/2}}$ and $\Delta E^{(2)}_{(2s)2p_{3/2}}$, $\Delta E^{(2)}_{(2s)2p_{1/2}}$ represent the screening effects to the radiative corrections to the $2p_{3/2}$ and $2p_{1/2}$ states due the interaction with $1s^{2}$ and $2s^{2}$ shells, respectively; $\Delta E^{(2)}_{(2p_{3/2})}$ and $\Delta E^{(2)}_{(2p_{1/2})}$ correspond to the screening corrections due the filled $2p_{3/2}^{4}$ and $2p_{1/2}^{2}$ shells, respectively; while $\Delta E^{(2)}_{(2p_{1/2})2p_{3/2}}$ is the $2p_{1/2}^{2}$ screening effect to the radiative correction of the $2p_{3/2}$ state. Last line of Eq. (5) displays the counterterms, which arise in the extended Furry representation. The detailed formulas for all the contributions listed in Eqs. (4) and (5) are presented, e.g., in Refs. Yerokhin et al. (1999); Kozhedub et al. (2010); Malyshev et al. (2017), and need not to be given here, nor do we need to discuss the renormalization of these diagrams, cf. Refs. Soff and Mohr (1988); Mohr (1992); Persson et al. (1996); Yerokhin et al. (1999); Labzowsky et al. (1999); Yerokhin et al. (2003); Artemyev et al. (2005, 2007); Kozhedub et al. (2010); Sapirstein and Cheng (2011); Artemyev et al. (2013); Malyshev et al. (2014, 2015); Yerokhin and Shabaev (2015); Sapirstein and Cheng (2015); Malyshev et al. (2017); Blundell (1993) and references therein.

All diagrams are evaluated numerically by making use of the dual-kinetic-balance finite basis set method Shabaev et al. (2004) with basis functions constructed from $B$ splines Sapirstein and Johnson (1996). These calculations involve an infinite summation over the angular-momentum quantum number $\kappa$ of intermediate states. This summation has been extended up to $|\kappa_{\rm max}|=12$ and the remaining part of the sum is estimated by a least-square inverse-polynomial fitting.

The QED and screened QED radiative corrections given by Eqs. (4) and (5) have been calculated for ${}^{2}P_{3/2}-^{2}P_{1/2}$ fine-structure splitting of F-like ions. For Mo33+ and U83+ ions, Table 1 presents the results for the self-energy and vacuum polarization corrections separately for three different cases: Coulomb potential and two core-Hartree potentials described above.

As seen from this table, the employment of the screening potential decreases (screens) the first-order contributions. This reduction happened due to partially account for the next order (e.g., second order) diagrams, accelerates the convergence of the higher-order screening corrections. Moreover, it also illustrates the main idea of approximate methods to screen the hydrogenic QED results in a way that the next order many-electron screened corrections, which are absent in the approximate treatments, become negligible. In Table 1 we also compare our final results for the self-energy and vacuum polarization corrections with values obtained within the approximate treatments in Ref. Li et al. (2018). In particular, there are several approaches: “grasp2k” - the standard approach in grasp2k Jönsson et al. (2013), “Welton” - the Welton interpretation Welton (1948) of the self-energy implemented by Lowe et al. Lowe et al. (2013), and “MQED” - a model QED operator approach Shabaev et al. (2013). The comparison reveals a reasonable agreement for the case of Mo33+. For F-like U83+ ion, however, the vacuum polarization corrections are close, while the self-energy values, evaluated with any approximate methods, differ even in sign. The reason for this deviation is the partial cancellation of the two (first-order) terms $\Delta E^{(1)}_{2p_{3/2}}$ and $\Delta E^{(1)}_{2p_{1/2}}$. This cancellation leads to a situation when the many-electron screening effects $\Delta E^{(2)}_{\rm tot}$ absent in the approximate treatments start to be dominant. In order to estimate an accuracy, the spread of the final results through starting potentials is employed. In addition, we use the scaling of the higher-order screened diagrams $(\alpha/8\pi)(\alpha Z)^{4}/Z^{2}$ multiplied by a factor 5. The final uncertainty is taken as a maximum of those two. We note here, that the calculations have been also performed with the Kohn-Sham local potential Kohn and Sham (1965). However, due to the marginal difference with the CH2 results, we do not tabulate those values.

We can also compare our values with experiments and other approximate results. In Table 2 the measured fine-structure transition energies are presented in the second column.

The fourth column of this table gives the results of the grasp2k calculations Li et al. (2018), which includes all the corrections except of QED. We can also consider a difference between the experimental results and the non-QED theoretical values. If we assume that in the theoretical calculations Li et al. (2018) all the non-QED corrections are accurately taken into account, we can interpret the difference experiment - theory as the experimentally given QED contribution and keep only the experimental uncertainty in the difference. These experimental QED contributions are given in the fifth column of the table. Here, we should note, that these values contain not only the radiative but also the many-electron QED corrections as well as the relativistic nuclear recoil contributions, which go beyond those implemented in the grasp2k. The relativistic nuclear recoil corrections suppressed by $(\alpha Z)^{4}m/M$ (with $M$ the mass of the nucleus) are quite small, while the many-electron QED entering at order $(\alpha Z)^{3}/Z^{2}$ might be important. In the sixth column the QED corrections evaluated starting with the CH2 potential (see above) are presented as our final results. Its uncertainty is estimated as a difference between the values obtained in the CH1 and CH2 potentials. Comparing the experimentally deduced QED contributions with those obtained in this work one finds an excellent agreement. From this comparison, we can conclude: (i) the correlation effects are indeed strongly diminished for the case under consideration as it has been demonstrated in Ref. Li et al. (2018); (ii) the Breit and, in particular, the frequency-dependent Breit interaction contributions are very well taken under control in the grasp2k calculations; (iii) the many-electron QED effects appear to be small similar as previously revealed for He- and Li-like ions Mohr and Sapirstein (2000); Yerokhin et al. (2001); (iv) the disagreement between theory and experiment discovered for some ions is now resolved. This is summarized in Fig. 3 which shows the difference between various fine-structure energies and accurate non-QED calculations given also in 5th - 9th columns of Table 2 relatively to the non-QED results. Experimental results for this ratio are compared with our QED calculations as well as with the approximate results of grasp2k, Welton, and MQED computations.

Let us also compare the QED radiative corrections obtained here with the results of approximate approaches. As approximate results presented in the last three columns, grasp2k, Welton, and MQED, we use the data reported in Ref. Li et al. (2018). The comparison with the rigorous calculations reveals that the grasp2k and Welton approaches differ by about 10% for all ions except U83+. The difference with the MQED results starts from 15% for Ar9+ and rapidly improves to 3% for $Z=42$. In the case of uranium ion the approximate treatments underestimate the QED correction on a level of 46% (grasp2k), 40% (Welton), and 28% (MQED). The reason for this is the cancellation of the first-order corrections for the $2p_{3/2}$ and $2p_{1/2}$ states discussed above. Moreover, we expect similar findings for the ground-state transition in B-, C-, N-, and O-like uranium ions.

To summarize, rigorous QED calculations have been performed for the $1s^{2}2s^{2}2p^{5}\,{}^{2}P_{3/2}-^{2}P_{1/2}$ fine-structure transition energy of selected F-like ions. The radiative corrections have been evaluated in particular up to the second order of the perturbation theory in the framework of the original and extended Furry pictures. If our QED computations are combined with highly accurate but non-QED fine-structure data by Li et al. Li et al. (2018), the discrepancy with the experimental data is resolved and significant improvement of the agreement between theoretical and experimental data has been achieved.

We are grateful R. Hutton for useful discussions. This work was supported by DFG (Grant No. VO1707/1-3). A.V.V. acknowledges the support of the CAS President International Fellowship Initiative (PIFI). X.M. was supported by NSFC (Grant No. 11320101003).