Many-electron QED corrections to the g factor of lithiumlike ions
A. V. Volotka, D. A. Glazov, V. M. Shabaev, I. I. Tupitsyn, G. Plunien
aa r X i v : . [ phy s i c s . a t o m - ph ] A p r Many-electron QED corrections to the g factor of lithiumlike ions
A. V. Volotka, , D. A. Glazov, V. M. Shabaev, I. I. Tupitsyn, and G. Plunien Institut f¨ur Theoretische Physik, Technische Universit¨at Dresden,Mommsenstraße 13, D-01062 Dresden, Germany Department of Physics,St. Petersburg State University, Oulianovskaya 1,Petrodvorets, 198504 St. Petersburg, Russia
A rigorous QED evaluation of the two-photon exchange corrections to the g factor of lithiumlike ions ispresented. The screened self-energy corrections are calculated for the intermediate- Z region and its accuracyfor the high- Z region is essentially improved in comparison with that of previous calculations. As a result,the theoretical accuracy of the g factor of lithiumlike ions is significantly increased. The theoretical predictionobtained for the g factor of Si g th = 2 .
000 889 892(8) is in an excellent agreement with the correspondingexperimental value g exp = 2 .
000 889 889 9(21) [A. Wagner et al. , Phys. Rev. Lett. , 033003 (2013)].
PACS numbers: 31.30.J-, 31.30.js, 31.15.ac
Highly charged ions provide not only a unique scenario forprobing QED effects in the strongest electromagnetic fieldsbut also give access to an accurate determination of funda-mental physical constants and nuclear parameters. In recentyears, amazing progress has been made in the experimentaland theoretical investigations of the bound-electron g factor.High-precision measurements of the ground state g factor ofH-like carbon [1] and oxygen [2] and the related theoreticalcalculations provided determination of the electron mass. Re-cently, due to the substantial progress in the experimental ac-curacy of the g factor of H-like carbon and silicon the mass ofthe electron is once again substantially increased [3]. So farH- and Li-like silicon ions represent the heaviest ions, wherethe g factor has been measured [4–6]. To date, these experi-ments provide the most stringent tests of the bound-state QED(BS-QED) corrections in the presence of a magnetic field. Ac-curate measurements of the g factor in few-electron ions, suchas Li-like calcium and B-like argon [7], are already antici-pated. The investigations of the few-electron ions unlike H-like ions provide also an access to the many-electron QEDcorrections, which are represented by a different facet of theQED diagrams.The theoretical contributions to the g factor of Li-like ionscan be separated into one-electron and many-electron parts.The one-electron terms are similar to the corresponding cor-rections to the g factor of H-like ions. The many-electroncontributions, which define the main difference between theg factors of H- and Li-ions, were investigated in Refs. [8–13].The many-electron contributions are mainly determined bythe screened radiative and the interelectronic-interaction cor-rections. For low- Z ions, the screened radiative correctionswere obtained employing the perturbation theory to the lead-ing orders in αZ [9, 10]. For intermediate- Z ions, the screen-ing effect was evaluated by introducing the effective screen-ing potential in the QED calculations to all orders in αZ [11].For high- Z ions, the most accurate results for the screenedradiative corrections were obtained rigorously within a sys-tematic QED approach [12, 13]. The one-photon exchange diagrams, which represent the interelectronic-interaction cor-rection of the first order in /Z , were evaluated in the frame-work of QED in Ref. [8]. The second and higher-orders con-tributions of the interelectronic interaction were calculatedby means of the large scale configuration-interaction Dirac-Fock-Sturm (CI-DFS) method in Ref. [10]. However, untilnow, for all values of Z the theoretical uncertainty was de-termined by the interelectronic-interaction corrections and forthe intermediate- Z region also by the screened self-energycorrections. In the present Letter we report on the completeevaluation of the two-photon exchange and the screened self-energy corrections in the framework of a rigorous QED ap-proach within an extended Furry picture.In the extended Furry picture, to zeroth order we solvethe Dirac equation with an effective spherically symmetricpotential treating the interaction with the external Coulombpotential of the nucleus and the local screening potential ex-act to all orders. This approach significantly accelerates theconvergence of the perturbation expansion. We use differenttypes of the screening potential. The simplest choice is thecore-Hartree (CH) potential, which is created by the chargedensity distribution of the two core electrons in the s state.Other choices are the x α potentials: Kohn-Sham (KS), Dirac-Hartree (DH), and Dirac-Slater (DS), which were success-fully employed in previous calculations of highly charged ions[11, 14–19]. Moreover, we have also employed the Perdew-Zunger (PZ) potential [20] and the local Dirac-Fock (LDF)potential derived by inversion of the radial Dirac equation[21].Let us now turn to the evaluation of the two-photon ex-change corrections to the g factor of Li-like ions. These cor-rections are defined by diagrams of third order in the QED per-turbation theory. The corresponding diagrams are presentedin Fig. 1. The electron propagators in the figure have to betreated in the effective potential (we indicate this diagrammat-ically via the triple electron line). In contrast to the case of theoriginal Furry picture, in the extended Furry picture the addi-tional counterterm diagrams appear. These diagrams are de- × × × × × ×× ×× FIG. 1: Feynman diagrams representing the second-order interelectronic-interaction corrections to the g factor in local effective potentials.The wavy line indicates the photon propagator and the triple lines describe the electron propagators in the effective potential. The dashed lineterminated with the triangle denotes the interaction with the magnetic field. The counterterm diagrams are depicted in the second line. Thesymbol ⊗ represents the extra interaction term associated with the screening potential counterterm. picted in the second line in Fig. 1. They are associated with anextra interaction term represented graphically by the symbol ⊗ . Taking into account all possible permutations of the one-electron states, in total, we have to evaluate 36 three-electron,36 two-electron, and 2 one-electron diagrams, respectively.All together these diagrams form the complete gauge invari-ant set of the two-photon exchange contributions. The mostdifficult ones are the 16 two-electron diagrams depicted in thefirst line in Fig. 1. Each of this diagram contains a three-foldsummation over the complete Dirac spectrum and an integra-tion over the loop energy. Formal expressions for the diagramsin the first line are similar to those derived for the correspond-ing calculation of the hyperfine splitting and can be found inRef. [22]. The formulas derived there can be taken over but,instead of the hyperfine-interaction potential, we employ herethe interaction with a constant magnetic field and keep in mindthat the Dirac spectrum is now generated by solving the Diracequation with the effective potential. The derivation of the for-mal expressions for the diagrams of the second line is straightforward and will be presented elsewhere. More details aboutthe scheme of the numerical implementations can be foundin Ref. [22]. However, unlike the hyperfine splitting, in thecase of g factor the calculations are more involved due to thelarge cancellations of various terms and poor convergence ofthe partial-wave expansion. Nevertheless, we have substan-tially increased the accuracy of the all numerical integrationsand extended the partial-wave summation up to κ max = 15 .For a consistency check, we performed calculations both inFeynman and Coulomb gauges, and the results are found tobe gauge invariant with a very high accuracy.In Table I the interelectronic-interaction corrections to theg factor of Li-like silicon are given. The results are ob-tained with four different starting potentials: Coulomb, core-Hartree, Perdew-Zunger, and local Dirac-Fock potentials. Inthe extended Furry picture, the interelectronic interaction con-tributes already in the zeroth order, due to the presence ofthe screening potential in the Dirac equation. The one-photon TABLE I: Interelectronic-interaction corrections to the ground-stateg factor of Li-like Si ion in various starting potentials in units − . Coulomb CH PZ LDFZeroth order 348.267 321.632 349.636First order 321.592 − − − − − − − − (first order) and two-photon (second order) exchange correc-tions have been evaluated to all orders in αZ in the frame-work of rigorous QED approach. The higher-order correc-tions have been extracted from the calculations performedby means of the large-scale configuration-interaction Dirac-Fock-Sturm method described in Refs. [10, 23]. As it wasexpected, the employment of the extended Furry picture in-creases the convergence of the perturbation expansion. Thisallows us to reduce the absolute uncertainty of the higher or-der interelectronic-interaction corrections. Finally, the rigor-ous evaluation of the two-photon exchange corrections and theimproved calculations of the higher-order terms allow us tosignificantly increase the total accuracy of the interelectronic-interaction terms for all ions under consideration. E.g., in thecase of Si ion the previous result was .
000 314 903(74) [10], while the present calculation yields .
000 314 809(6) ,and in the case of Pb ion instead of previous value .
002 140 7(27) [10] we now receive .
002 139 34(4) .Let us now turn to the screened self-energy corrections tothe g factor of Li-like ions. In Refs. [12, 13] these correc-tions have been rigorously evaluated only for the Pb and U ions. The reasons for this are twofold. First isthe large numerical cancellations which occur in the point-by-point difference. Second is the poor convergence of thepartial-wave expansion. In order to overcome these problems,we have performed the calculations in the extended Furry pic-ture and employed a special treatment of the many-potentialterms. The Feynman diagrams in the extended Furry pic-ture corresponding to the screened self-energy corrections tothe g factor are presented in Fig. 2. The corresponding ex-pressions derived in Refs. [12, 13] remain formally the samebut keeping in mind that the Dirac spectrum is now gener-ated by solving the Dirac equation with the effective poten-tial. In the second line of Fig. 2 the additional countertermdiagrams are depicted. The derivation of the formal expres-sions for them is relatively simple and will be presented else-where. The employment of the extended Furry picture al-lows us to substantially reduce the numerical cancellationsof different terms as well as to improve convergence of thepartial-wave expansion. However, in order to improve theconvergence even further, we have employed a specific treat-ment of some many-potential terms. The standard way totreat the vertex and reducible corrections is to separate terms(zero-potential contributions) in which bound-electron propa-gators are replaced by free propagators. The remaining many-potential terms being ultraviolet finite are generally calculateddirectly in coordinate space [24]. However, for gaining bet-ter control over the partial-wave summation we separate alsothe so-called one-potential contributions. In this way the one-potential terms are treated in the momentum space. Suchtreatment of the one-potential term was applied in previouscalculations of the one-electron self-energy corrections to theg factor in Refs. [25–29] and to the magnetic-dipole transi-tion amplitude in Ref. [30]. Here, we extend this procedureto the evaluation of the screened self-energy corrections to theg factor. Performing the analysis of the convergence of thepartial-wave expansion for different terms we have found, thatsuch treatment should be applied to the terms (C1) Eq. (32),(H3) Eq. (38), (I1) Eq. (47), and (I3) Eq. (49) in Ref. [13]. Thecorresponding one-potential contributions are given by the ex-pressions: ∆ E SE(C1)(1)SQED = − πiα X P ( − P Z d p d p ′ d q d k (2 π ) k × ¯ ψ a ( p ) γ µ S F ( p − k ) γ h V eff ( q ) S F ( p − k − q ) × T ( p − p ′ − q ) + T ( q ) S F ( p − k − q ) V eff ( p − p ′ − q ) i × γ S F ( p ′ − k ) γ µ ψ ζ b | PaPb ( p ′ ) + ( a ↔ b ) , (1) ∆ E SE(H3)(1)SQED = − πiα Z d p d p ′ d k (2 π ) k ¯ ψ a ( p ) × ∂∂ε a h γ µ S F ( p − k ) γ V eff ( p − p ′ ) S F ( p ′ − k ) γ µ i × ψ η a ( p ′ ) + ( a ↔ b ) , (2) ∆ E SE(I1)(1)SQED = − πiα Z d p d p ′ d k (2 π ) k ¯ ψ a ( p ) × ∂∂ε a h γ µ S F ( p − k ) γ T ( p − p ′ ) S F ( p ′ − k ) γ µ i ψ a ( p ′ ) × X P ( − P h ab | I (∆) | P aP b i + ( a ↔ b ) , (3) ∆ E SE(I3)(1)SQED = − πiα Z d p d k (2 π ) k × ¯ ψ a ( p ) ∂ ∂ε a h γ µ S F ( p − k ) γ µ i ψ a ( p ′ ) h a | T | a i× X P ( − P h ab | I (∆) | P aP b i + ( a ↔ b ) , (4)where p = ( ε a , p ) , p ′ = ( ε a , p ′ ) , q = ( ε a , q ) , ∆ = ε a − ε P a ,and the notation ( a ↔ b ) stands for the contribution with in-terchanged labels a and b ; γ µ = ( γ , γ ) are the Dirac matri-ces, S F ( p ) = ( γ · p − m ) − is the free-electron propagator,the interelectronic-interaction operator I ( ε ) and its derivativesare defined in a similar way as in Ref. [13], and V eff is the ef-fective potential being the sum of the nuclear and screeningpotentials. T is the operator of interaction with a constantmagnetic field, which reads in the momentum space: T ( p ) = iµ (2 π ) [ α × ∇ p δ ( p )] · H , (5)where µ = | e | / is the Bohr magneton and H is the magneticfield directed along the z axis. The wave function | η a i is givenby the expression: | η a i = X P ( − P ( | a i h h ζ b | P aP b | T | a i + h ζ a | P bP a | T | b i + h ab | I ′ (∆) | P aP b i (cid:16) h a | T | a i − h b | T | b i (cid:17)i + | ξ a ih ab | I (∆) | P aP b i + | ζ b | P aP b ih a | T | a i ) , (6)and the wave functions | ξ i and | ζ i are defined similar as inRef. [12].The ultraviolet-finite one-potential terms given by Eqs. (1)-(4) have been evaluated in the momentum space. The corre-sponding expressions in the coordinate space have been sub-tracted from the related many-potential terms by means ofpoint-by-point difference. The partial-wave expansion for themany-potential terms was terminated at κ max = 15 , and theremainder of the sum was estimated by a least-square polyno-mial fitting and by the ǫ -algorithm of the Pad´e approximation.As a result, we have significantly increased the accuracy ofthe screened self-energy correction. In the case of Si ionthe previous result was − .
000 000 218(46) [10], while thepresent calculation yields − .
000 000 242(5) , and in the caseof Pb ion instead of previous value − .
000 003 3(2) [12, 13] we now receive − .
000 003 44(2) .In Table II, the individual contributions and the total valuesof the g factor for Li-like silicon Si , calcium Ca , × × × × × × FIG. 2: Feynman diagrams representing the screened self-energy corrections to the g factor in local effective potentials. The wavy lineindicates the photon propagator and the triple lines describe the electron propagators in the effective potential. The dashed line terminatedwith the triangle denotes the interaction with the magnetic field. The counterterm diagrams are depicted in the second line. The symbol ⊗ represents the extra interaction term associated with the screening potential counterterm. lead Pb and uranium U are presented togetherwith the previously reported theoretical results and the exper-imental value for the case of silicon. The screened self-energyand interelectronic-interaction corrections calculated in thisLetter allow to substantially increase the theoretical accuracyfor all ions under consideration. The other contributions to theg factor presented in Table II were considered in detail in ourprevious studies [10–13]. Comparison with the experimentalvalue for Li-like silicon ion provides tests of relativistic in-terelectronic interaction on a level of − , the one-electronBS-QED on a level of 0.7%, and the screened BS-QED on alevel of 3%. Thus, the current studies provide the most ac-curate test of the many-electron QED effects in the case of gfactor. The further improvement of the g factor theory for Li-like ions requires at first the rigorous evaluation of the three-photon exchange diagrams and the subsequent betterment ofthe screened self-energy contribution for the intermediate- Z region, and the one-electron two-loop and nuclear recoil cor-rections for the high- Z region.The techniques and numerical methods developed can alsobe extended for the g factor of B-like ions, where the corre-sponding studies can also lead to an independent determina-tion of the fine-structure constant [31].The work reported in this paper was supported by DFG(Grant No. VO 1707/1-2), RFBR (Grants No. 13-02-00630 and 14-02-31316), and Saint Petersburg State Univer-sity (Grants No. 11.0.15.2010 and 11.38.269.2014). D.A.G.acknowledges financial support by the FAIR – Russia Re-search Center and by the “Dynasty” foundation. [1] H. H¨affner, T. Beier, N. Hermanspahn, H.-J. Kluge, W. Quint,S. Stahl, J. Verd´u, and G. Werth, Phys. Rev. Lett. , 5308(2000).[2] J. Verd´u, S. Djeki´c, S. Stahl, T. Valenzuela, M. Vogel, G. Werth, T. Beier, H.-J. Kluge, and W. Quint, Phys. Rev. Lett. , 093002(2004).[3] S. Sturm, F. K¨ohler, J. Zatorski, A. Wagner, Z. Harman,G. Werth, W. Quint, C. H. Keitel, and K. Blaum, Nature ,467 (2014).[4] S. Sturm, A. Wagner, B. Schabinger, J. Zatorski, Z. Harman,W. Quint, G. Werth, C. H. Keitel, and K. Blaum, Phys. Rev.Lett. , 023002 (2011).[5] S. Sturm, A. Wagner, M. Kretzschmar, W. Quint, G. Werth, andK. Blaum, Phys. Rev. A , 030501(R) (2013).[6] A. Wagner, S. Sturm, F. K¨ohler, D. A. Glazov, A. V. Volotka,G. Plunien, W. Quint, G. Werth, V. M. Shabaev, and K. Blaum,Phys. Rev. Lett. , 033003 (2013).[7] D. von Lindenfels, M. Wiesel, D. A. Glazov, A. V. Volotka,M. M. Sokolov, V. M. Shabaev, G. Plunien, W. Quint, G. Birkl,A. Martin, et al., Phys. Rev. A , 023412 (2013).[8] V. M. Shabaev, D. A. Glazov, M. B. Shabaeva, V. A. Yerokhin,G. Plunien, and G. Soff, Phys. Rev. A , 062104 (2002).[9] Z.-C. Yan, J. Phys. B , 1885 (2002).[10] D. A. Glazov, V. M. Shabaev, I. I. Tupitsyn, A. V. Volotka, V. A.Yerokhin, G. Plunien, and G. Soff, Phys. Rev. A , 062104(2004).[11] D. A. Glazov, A. V. Volotka, V. M. Shabaev, I. I. Tupitsyn, andG. Plunien, Phys. Lett. A , 330 (2006).[12] A. V. Volotka, D. A. Glazov, V. M. Shabaev, I. I. Tupitsyn, andG. Plunien, Phys. Rev. Lett. , 033005 (2009).[13] D. A. Glazov, A. V. Volotka, V. M. Shabaev, I. I. Tupitsyn, andG. Plunien, Phys. Rev. A , 062112 (2010).[14] J. Sapirstein and K. T. Cheng, Phys. Rev. A , 032506 (2001).[15] J. Sapirstein and K. T. Cheng, Phys. Rev. A , 022502 (2001).[16] J. Sapirstein and K. T. Cheng, Phys. Rev. A , 042501 (2002).[17] A. V. Volotka, D. A. Glazov, I. I. Tupitsyn, N. S. Oreshk-ina, G. Plunien, and V. M. Shabaev, Phys. Rev. A , 062507(2008).[18] Y. S. Kozhedub, A. V. Volotka, A. N. Artemyev, D. A. Glazov,G. Plunien, V. M. Shabaev, I. I. Tupitsyn, and T. St¨ohlker, Phys.Rev. A , 042513 (2010).[19] J. Sapirstein and K. T. Cheng, Phys. Rev. A , 012504 (2011).[20] J. P. Perdew and A. Zunger, Phys. Rev. B , 5048 (1981).[21] V. M. Shabaev, I. I. Tupitsyn, K. Pachucki, G. Plunien, and V. A.Yerokhin, Phys. Rev. A , 062105 (2005).[22] A. V. Volotka, D. A. Glazov, O. V. Andreev, V. M. Shabaev, I. I. TABLE II: Individual contributions to the ground-state g factor of Li-like ions and comparison with the previously reported theoretical valuesas well as with the experimental result for the Si ion. Si
11+ 40 Ca
17+ 208 Pb
79+ 238 U Dirac value (point nucleus) 1.998 254 751 1.996 426 011 1.932 002 904 1.910 722 624Finite nuclear size 0.000 000 003 0.000 000 014 0.000 078 57(14) 0.000 241 62(36)QED, ∼ α ∼ α − − − − − − − − − − a b c c b Experiment 2.000 889 889 9(21) aa Wagner et al. [6]; b Glazov et al. [10]; c Glazov et al. [13].Tupitsyn, and G. Plunien, Phys. Rev. Lett. , 073001 (2012).[23] I. I. Tupitsyn, A. V. Volotka, D. A. Glazov, V. M. Shabaev,G. Plunien, J. R. Crespo L´opez-Urrutia, A. Lapierre, and J. Ull-rich, Phys. Rev. A , 062503 (2005).[24] S. A. Blundell, K. T. Cheng, and J. Sapirstein, Phys. Rev. A ,1857 (1997).[25] H. Persson, S. Salomonson, P. Sunnergren, and I. Lindgren,Phys. Rev. A , R2499 (1997).[26] T. Beier, I. Lindgren, H. Persson, S. Salomonson, P. Sun-nergren, H. H¨affner, and N. Hermanspahn, Phys. Rev. A ,032510 (2000).[27] V. A. Yerokhin, P. Indelicato, and V. M. Shabaev, Phys. Rev. Lett. , 143001 (2002).[28] V. A. Yerokhin, P. Indelicato, and V. M. Shabaev, Phys. Rev. A , 052503 (2004).[29] V. A. Yerokhin and U. D. Jentschura, Phys. Rev. A , 012502(2010).[30] A. V. Volotka, D. A. Glazov, G. Plunien, V. M. Shabaev, andI. I. Tupitsyn, Eur. Phys. J. D , 293 (2006).[31] V. M. Shabaev, D. A. Glazov, N. S. Oreshkina, A. V. Volotka,G. Plunien, H.-J. Kluge, and W. Quint, Phys. Rev. Lett.96