Relativistic formulae for the energy level shifts due to electron self-energy corrections are derived within the external potential Bethe-Salpeter formalism. A rigorous treatment of QED corrections to the fine structure splittings of helium is carried out. Although some individual self-energy diagrams give contributions of order α 6mc 2, they are shown to sum to zero. In addition, the α 6mc 2 correction from vertex modifications in the presence of Coulomb photons does not contribute. Therefore, a rigorous treatment of all QED corrections of order α 6mc 2 to the fine structure splittings of helium now appears to be complete. © 1994 IOP Publishing Ltd

Drake, Gordon W. F.

Zhang, T.

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University of WindsorScholarship at UWindsorPhysics Publications Department of Physics1994A rigorous treatment of O(α 6mc 2) QEDcorrections to the fine structure splittings of heliumT. ZhangGordon W. F. DrakeUniversity of WindsorFollow this and additional works at: http://scholar.uwindsor.ca/physicspubPart of the Physics CommonsThis Article is brought to you for free and open access by the Department of Physics at Scholarship at UWindsor. It has been accepted for inclusion inPhysics Publications by an authorized administrator of Scholarship at UWindsor. For more information, please contact scholarship@uwindsor.ca.Recommended CitationZhang, T. and Drake, Gordon W. F.. (1994). A rigorous treatment of O(α 6mc 2) QED corrections to the fine structure splittings ofhelium. Journal of Physics B: Atomic, Molecular and Optical Physics, 27 (14), L311-L316.http://scholar.uwindsor.ca/physicspub/52A rigorous treatment of O( alpha 6mc2) QED corrections to the fine structure splittings ofheliumThis article has been downloaded from IOPscience. Please scroll down to see the full text article.1994 J. Phys. B: At. Mol. Opt. Phys. 27 L311(http://iopscience.iop.org/0953-4075/27/14/003)Download details:IP Address: 137.207.184.30The article was downloaded on 08/05/2013 at 19:36Please note that terms and conditions apply.View the table of contents for this issue, or go to the journal homepage for moreHome Search Collections Journals About Contact us My IOPscienceI. Phys. B At. Mol. Opt. phys. 27 (1994) L311-L316. Printed in the UK LETTER TO THE EDITOR A rigorous treatment of O(a6mc?) QED corrections to the fine structure splittings of helium Tao Zhang and G W F Drake Department of Physics, University of Wmdsor. Windsor. Ontario N9B 3P4. Canada Received 24 May 1994 Abshaet Relativistic formulae for Lhe energy level sh& due to elearan self-energy c o d o n s  are derived within the external potential &thesalpeter fonmli" A rigorous t"ent of QW corrections to the fine struclure spliuintings of helium is anfed out. Although some individual self-energy diagrams give mnhihutions of order a 6 d ,  Lhey are shown to sum to zero. In addition, the a 6 m s  correction from w t e x  modifidons in the presence of Coulomb photons does not contribute. TherefOR. a rigorous treatment of all QED corrections of order &c2 to the fine ~UucNre splittings of helium now appears to be complete. Fine structure splittings in the helium ls2p * state have recently atracted considerable attention because of dramatic improvements io experimental accuracy to f3 kHz (1 part in IO') [I]. Also, the lowest order Breit interaction terms of O(or4mc2) have now been calculated to better than 1 part in IO9 [21. However, a meaningful comparison of theay and experiment at the A3 ! d z  level of accuracy requires also terms of O(or6mc?) and perhaps higher. Operators for the spin-dependent terms of O(ry6mcZ) were obtained many years ago by Douglas and Kroll [31 (see also [41) but their derivation was in part phenomenological in their treatment of electron self-energy and vertex corrections. We have therefore re- examined their derivation. and introduced a rigorous relativistic formalism based on the Bethe-Salpeter equation for all radiative processes. This paper reports the final results of this work for spin-dependent terms up to O(or6mcz). It also lays the ground work for an extension to terms of O(a7 Inor mc2) and O(cr7mc2). In a previous paper [5], we presented a relativistic formula for the energy level shift due to vertex modifications to transverse photon exchange. A new or6mcz QED correction to the fine structure splittings of helium was found from the relativistic correction of the second order vertex modification to transverse photon exchange in the absence of Coulomb photons. In this paper, we present relativistic formulae for the energy level shifts due to electron self-energy corrections. Previously used phenomenological operators for electron self-energy corrections are verified. In addition, we show that the total spin-dependent contributions of order d m c 2  arising from electron self-energy corrections sum to zero. (We exclude thoughout spin-dependent contact terms which vanish for triplet states.) This result comes eom a non-trivial cancellation of various terms. Furthermore, we calculate the QED corrections due to the vertex modifications in the presence of Coulomb photons, which were not included in the previous calculation 151. The Feynman diagrams to be considered in this paper are shown in figure 1, in addition to the diagrams in figure 2 given previously [51. As before, we choose a mixed gauge in which the Feynman gauge is used for the radiative photon propagator, and the Coulomb 0953-4075/94/140311+06$19.50 0 1994 IOP Publishing Ltd L311 L312 Letter to the Editor Fwre 1. Feynman diagrams for elecvon selfenergy corrections. The curved wavy lines denote covariant photons, the straight wavy lines transverse photons, and the broken lines instantaneous Coulomb photons Figure 2. Feynman diagrams for vertex corrections to transverse photon exchange. The curved wavy lines denote covariant photons, the straight wavy lines transverse photons. and the broken lines instantaneous Coulomb photons. gauge is used for the photon propagator between electrons. We adopt the same notation as used in reference [5]. The energy level shift from figure l ( n )  is is the eigenfunction in the Coulomb ladder approximation defined in [6] and 131. L+ is defined by Letter to the Editor L313 Evaluating the corrections in equation (1). we reproduce the u5mcz operators and A&) obtained by Douglas and Rol l  131, arising from electron self-energy modifications to the external potential and electron-electron Coulomb interaction. Together with the vertex correction of order d m c Z  in [5], we obtain the well known anomalous magnetic moment fine structure contributions [3] expressed in terms of Breit operators as 4 H 5  = - 2H& + -H:,, + 2H4 2n "II 3 As expected, no a6mc2 QED correction to the fine structure splittings is found. electron self-energy modification to transverse photon exchange The diagrams shown in figures I(bHe) give the following energy correction due to L314 Letter to the Editor 1 E - & @ ]  - k ) - & & + k )  XU; YPYPYP 1 1 X E - 90 - H @ ~  - IC - q) - +IC) - L:~+(P, +k)iC.c:z+fpz +k) + is XI+C@l - k ,PZ+k) ) .  (3) The last two terms are associated with figures I(d) and (e). They can be shown to cancel with another two terms arising from second order Brillouin-Wigner perturbation theory. To illustrated this cancellation, we start from the Brillouin-Wigner perturbation formula A E  = E - EC = ($cIHA 11 - U E ,  &)HA]-' l&c) where V E .  &) = ( E  - H d - ' [ l  - I&)(&ell is the Green function. The perturbative Hamiltonian H A  is defined by H A  = L++Ic(l - L++) - L--Ic 4- -D.FgA(F - gA)-'Ic J -Zi where 3 = ( E  / 2  + E  - H i ) ( E / 2  - E - Hz) D = E - Hi - Hz and ga is the interaction integral operator. The energy level shift is expanded as A E  = AE") + A E @ )  + . . . where AE") = (@NAI&) AEO = (@clHarHdh) etc. Equation (3) is derived from AECI). A term arising from A E @ )  is where gr and gs are transverse photon and electron self-energy operators. Equation (8) expands into the corresponding second-order RayleighSchrodinger perturbation term, which is of nominal order a7mcz, plus two additional terms which cancel with the last two terms in equation (3). However, the first two terms in equation (3) do contribute to the fine splittings of order a%". Without the presence of Coulomb photons, the first term is given by Letter to the Editor L315 where p = ( E  - k  - E@i - k )  - E @ z ) , O )  - P I  and E@) = -, The renormalized elecvon self-energy modification function is The second term in equation (3) is then where 4rr m To order a6mc2, the two corrections given by equations (9) and (11) cancel with each other. In the presence of Coulomb photons, the QED corrections from the first two terms in equation (3) contribute to the fine structure splittings only to O(a7mc2) and higher. A similar analysis applies to figures l(f)-(i). Here the energy correction is L316 Letter to the Editor 1 1 X E + o  - E @ ] )  -&@i + k) - L t + @ ~ , p z +  W L + + @ I , P I  + k )  + isff' xl4c@1 - k , ~ i l + k ) ) .  (13) As before, the last two terms cancel with two terms derived from second order Brillouin- Wigner perturbation theory, and the spin-dependent parts of the first two terms cancel with each other to order u6mcz. Therefore, the total spin-dependent contribution from electron self-energy modifications of order a6mG is zero. In our previous paper [SI, we considered the additional diagrams shown in figure 2. In the approximation that Coulomb photons are neglected, we found a previously unsuspected contribution of (14) ff4 12x2 AE,  = -2- (MvI, T ~ ) I Q ~  . (5 x vl/i) IM-~. V 2 ) )  to the fine sbllcture splittings of helium. We have now verified that this result is correct to order a6mc2 even if additional Coulomb photons are included. in summary, we have presented fully relativistic formulae for the energy level shift due to electron self-energy modifications. Using these formulae, the previously known QED corrections from electron self-energy are put on a rigorous footing. No new spin-dependent contribution of order a6m$ is found beyond that recently reported [SI. We believe that the now known terms from Douglas and Kroll [3] and Zhang and Drake [SI are likely to be the total QED corrections of order g4mcZ to the fine structure splittings of helium. The relativistic formulae presented here and in [SI are necessary for an extension of this work to terms of order a7mc2. One of the authors ('E) would like to thank Dr Lixin Xiao for numerous discussions. The Natural Sciences and Engineering Research Council of Canada is acknowledged for financial support. References [I] Shiner D, Dixson R and Zhao P 1994 Phys. RCK Lett. 72 1802 121 Drake 0 W F and Yan Z C  1992 Phys. Rev. A 46 2378 131 Douglas M H and Kroll N M 1974 Ann. Phys., NY82 89 Douglas M H 1971 P M  tkesk University of California (unpublished) 141 Lewis M L and and Serafino P H 1978 Phys. Rev. A 18 867 [SI Zhang T and Drake G W F 1994 Phys. Rev. Lett. in press 1.51 Sucher J 1958 PhD fhesir Colombia University (unpublished) 

A rigorous treatment of O(α 6mc 2) QED corrections to the fine structure splittings of helium

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A rigorous treatment of O(α 6mc 2) QED corrections to the fine structure splittings of helium

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