New Approach to Many-Body-QED Calculations: Merging Quantum-Electro- Dynamics with Many-Body Perturbation

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1 1 New Appoch to Mny-Body-QED Clcultions: Meging Quntum-Electo- Dynmics with Mny-Body Petubtion Ingv Lindgen, Sten Slomonson, nd Dniel Hedendhl 1. Intoduction Abstct: A new method fo bound-stte QED clcultions on mny-electon systems is pesented tht is combintion of the non-qed mny-body technique fo qusidegenete systems nd the newly developed covint-evolution-opeto technique fo QED clcultions. The ltte technique hs been successfully pplied to the fine stuctue of excited sttes of medium-hevy heliumlike ions, nd it is expected tht the new method should be pplicble lso to light elements, hopefully down to neutl helium. PACS Nos.: Jv, Md, Jf nd Pw Résumé : Fench vesion of bstct (supplied by CJP) [Tduit p l édction] Thee exist essentilly two ppoches to bound-stte QED clcultions on mny-electon systems, which we cn efe to s the low-z nd the high-z ppoches, espectively. In the low-field ppoch the effects of eltivity nd QED e expnded in powes of α nd Zα, nomlly stting fom the Bethe-Slpete eqution [1]. This ws pioneeed in the 1950 s by Aki [2] nd Suche [3], who evluted ll contibutions to the helium enegy levels to ode α 3 H(tees). Following minly the ppoch of Suche, Dougls nd Koll [4] evluted in 1974 ll contibutions to the helium fine stuctue of ode α 4 H. In the 1990 s lso ll contibutions of ode α 5 H nd α 5 log α H hve been evluted by Zhng nd Dke [5, 6, 7] s well s by Pchucki nd Spistein [8, 9, 10]. Numeicl evlution of these effects hs been done pticully by Dke nd cowokes [11], using ccute Hylles-type of wve functions. With the contibutions of ode α 5 H the theoy hs eched the level of ccucy of the expeimentl esults fo the 1s2p stte of helium, lthough thee e still seious discepncies between theoy nd expeiment. In the high-z ppoch the stting point is Dic obitls nd electon popgtos (Geen s functions), geneted in the field of the nucleus, which in tems of fee electons coespond to expnsions with nucle intections, s illustted in Fig. 1. With these obitls/popgtos the covint multi- (one-, two-,...) photon exchnge, illustted in Fig. 2, is evluted numeiclly. This hs been the ppoch of vious goups nd successfully pplied to highly chged ions (see Refs [12, 13, 14], fo futhe efeences). Fo n isoelectonic sequence, s the heliumlike systems, the effect of multi-photon exchnge deceses oughly by fcto of Z fo ech dditionl photon. This implies tht the high-z ppoch conveges pidly fo hevy elements but quite slowly fo light elements. Fo pcticl esons, it is t pesent not fesible to evlute the exchnge of moe thn two covint photons in this wy, which, Ingv Lindgen, Sten Slomonson, nd Dniel Hedendhl. Deptment of Physics, Chlmes Univesity of Technology nd the Götebog Univesity, Götebog, Sweden Cn. J. Phys. : 1 9 ()

2 2 Cn. J. Phys. Vol., = Bound el. Fee el. Nucle intections... Fig. 1. The bound electon obitl, geneted in the field of the tomic nucleus, cn be epesented by fee electon with nucle potentil intections, ech coesponding to fcto of Zα in the nlyticl expnsion. howeve, leds to good ccucy fo highly chged ions. Fo light elements, on the othe hnd, whee the electon coeltion is eltively stong, this ppoch is not fesible.... Fig. 2. Exmples of covint multi-photon exchnge digms. Thee is pesently get inteest in the fine stuctue of the lowest 3 P stte of neutl helium, both expeimentlly nd theoeticlly, since compison between theoy nd expeiment cn yield n ccute vlue of the fine-stuctue constnt α. Vey ccute mesuement of the seption 3 P 1 3 P 0 hs been pefomed [15] nd even moe ccute expeiments e unde wy s epoted t the ICAP 2004 confeence in Rio de Jneio. Accute QED clcultions cn only be pefomed with the low-z ppoch, but thee is some confusion bout the theoeticl esults [10, 16, 11, 17]. The esults disgee, nd the most complete clcultion of Dke [11] diffes by 4-5 stndd devitions fom expeiments fo the 3 P 1 3 P 0 tnsition (using the ccepted vlue of α) nd even moe fo the 3 P 2 3 P 1 tnsition. The eson fo this poblem is not cle. It is obvious tht the nlytic ppoch becomes dsticlly moe complicted nd moe eo pone fo ech ode of petubtion, nd it is questionble if it will eve be possible to go beyond the ode pesently eched. In the pesent ppe we shll discuss n ltentive ppoch tht my mke it possible to extend the numeicl ppoch to low Z, possibly s f s down to neutl helium. This is bsed upon the covint-evolution-opeto ppoch, which we hve successfully used in evluting the finestuctue septions of heliumlike ions down to Z = 9 [18, 14]. In the poposed extension we shll bse the evlution on coelted pi functions, insted of hydogenic Dic obitls, in this wy including the dominting effect of electon coeltion into the QED clcultions. In pinciple, we will be ble to evlute ll effects due to so-clled Coulomb-ldde digms s well s those involving single tnsvese photon with bity numbe of Coulomb cossings. This will constitute possibility to test the nlyticl ppoch. The contibutions due to two tnsvese photons e of two kinds, in the ppes quoted bove denoted by E T T nd E T T, epesenting educible nd ieducible effects, espectively. It hs been shown by Zhng [6] tht the exchnge of thee tnsvese photons does not ffect the fine stuctue

3 Lindgen et l. 3 to ode α 5 H. We will be ble to evlute the educible two-photon effect by iteting single-photon exchnge, nd theefoe the only effect tht we will not be ble to tet numeicl on this level in the ne futue is the ieducible two-photon effect. Tht hs then to be included by mens of n nlyticl ppoximtion. In ddition, of couse, ou ppoch will utomticlly include mny effects of highe odes, but not in complete mnne. To wht extent we will be ble to evlute the ditive effects numeiclly is pesently unde investigtion. To begin with, we shll use the phenomenologicl ppoch, using the nomlous mgnetic moment of the electon, which will be coect to ode α 4. As bckgound to the desciption of ou new technique, we shll fist biefly summize the non-qed mny-body technique fo qusi-degenete systems, followed by summy of the high-z ppoches, the S-mtix method nd the ecently intoduced covint-evolution-opeto method [18, 14]. 2. Mny-Body Theoy In the stndd mny-body petubtion theoy (MBPT) we wnt to solve the Schödinge eqution fo numbe of tget sttes H Ψ α = E α Ψ α (α = 1, 2 d) (1) The Hmiltonin is ptitioned into n unpetubed model Hmiltonin nd petubtion H = H 0 H (2) Fo ech tget stte we hve model stte, Ψ 0 = P Ψ, which is n eigenfunction of H 0 nd confined to model spce with the pojection opeto P. The wve opeto tnsfoms the model sttes to the tget sttes Ψ α = Ω Ψ α 0 (α = 1, 2 d) nd stisfies the genelized Bloch eqution [19] [Ω, H 0 ] P = ( H Ω Ω P H Ω ) P (3) ssuming intemedite nomliztion P ΩP = P. Pojecting the Schödinge eqution (1) on the model spce yields P HΩ Ψ α 0 = E α Ψ α 0 (4) which shows tht the model functions e eigenfunctions of the effective Hmiltonin H eff = P HΩP with the eigenvlues equl to the exct eigenvlues of the tget sttes. Using second quntiztion, the wve opeto cn be septed into one-, two-,..body pts [19] nd insetion into the Bloch eqution (3) leds to the equtions Ω = 1 Ω 1 Ω 2 (5) [Ω n, H 0 ] P = ( H Ω Ω P H Ω ) n P (6) If we conside heliumlike systems, stting with hydogenic obitls geneted in the field of the nucleus, the wve opeto hs only two-body component, Ω = 1 Ω 2, stisfying the pi eqution [Ω 2, H 0 ] P = ( H H Ω 2 )2 P Ω 2H eff,2 (7) whee H is the Coulomb intection 1/ 12 nd H eff = P H ΩP is the petubtive pt of the effective Hmiltonin (4), H eff = P H 0 P H eff. The lst tem in the pi eqution (7) epesents so-clled folded digms. This yields the mtix elements of the wve opeto s Ω 2 b = s 1/ 12 b s 1/ 12 tu tu Ω 2 b ε ε b ε ε s (8)

4 4 Cn. J. Phys. Vol., s b = s b t s u b /b H eff s b/ b = s b s b s b folded Fig. 3. The pi function fo two-electon system is equivlent to n infinite sequence of ldde digms (including the folded digms). summed ove the intemedite sttes tu, whee = b H eff,2 b (incl. exchnge) is due to the folded digms. The stte b lies in the model spce nd the sttes s nd tu outside this spce (ε etc. e the obitl enegy eigenvlues). The pi eqution (7) cn be solved non-petubtively by ecusion, which coesponds to petubtive expnsion to ll odes nd epesents the exct solution of the Schödinge eqution fo this system. This cn be epesented gphiclly s in Fig. 3. Fo the numeicl solution we use the method of discetiztion, developed by Slomonson nd Öste [20, 21]. The pi functions e used to evlute the elements of the effective Hmiltonin (4) digonl s well s non-digonl nd the digonliztion then yields the model functions nd the enegies with coesponding ccucy. This ppoch is pticully effective in the cse of qusi-degenecy, whee unpetubed sttes cn be stongly mixed. By including such sttes in the model spce, thei mixing will be ccounted fo to ll odes of petubtion theoy. A good illusttion of the powe of the method is the ppliction to the fine stuctue of the 1s2p 3 P stte of heliumlike ions, whee it hs been shown to convege in ll cses down to neutl helium [22]. The conventionl ppoch with single model function, on the othe hnd, fils fo light elements, nd even model functions tht e eigenfunctions of the fist-ode effective Hmiltonin does not led to convegence fo Z 6 [23]. 3. S-mtix fomultion The stndd ppoch fo bound-stte QED clcultions is the S-mtix fomultion [12]. This is bsed upon the time-evolution opeto, defined by Ψ(t) = U(t, t 0 ) Ψ(t 0 ). This cn be expnded s U(t, t 0 ) = 1 ( i) n n=1 n! t t 0 d 4 x n... t t 0 d 4 x 1 T D [ H I (x n )... H I(x 1 ) ] (9)

5 Lindgen et l. 5 whee T D is the Wick time-odeing opeto nd H I(x) = e (x) α µ A µ (x)(x) (10) is the petubtion in the fom of the electon-field intection in the intection pictue [24]. e is hee the bsolute vlue of the electonic chge (positive numbe), (x)/ (x) epesent the electon-field opetos nd A µ (x) the electomgnetic field. A dmping fcto H I (x) H I (x) e γ t is intoduced, so tht the Hmiltonin ppoches the unpetubed Hmiltonin, when t 0 ±. The evolution opeto fom the unpetubed stte fo the exchnge of single photon between two electons is given by (using eltivistic units, m = c = = ɛ 0 = 1) SU (2) (t, ) = e2 2 t d 4 x 1 d 4 x 2 (x 1) (x 2) α µ 1 id Fµν(x 1 x 2 ) α ν 2 (x 1 ) (x 2 ) e γ( t 1 t 2 ) nd illustted in Fig. 4. D Fµν (x 1 x 2 ) is hee the photon popgto, epesented by the contction of the electomgnetic-field opetos of the petubtions H I (x 1) nd H I (x 2). The S mtix is defined by S = U(, ). Pefoming the time integtions yields fo the single-photon exchnge nd the coesponding enegy shift (11) s S (2) b = 2πi δ(ε ε b ε ε s ) s V eq b (12) E = s H (1) eff b = δ(ε ε b ε ε s ) s V eq b (13) Hee, V eq = α µ 1 id Fµν(x 1 x 2 ) α ν 2 is the effective intection nd in the Feynmn guge given by whee nd q = ε ε = ε s ε b. V eq (q) = 0 e2 dk f(k) 2k q 2 k 2 iη f(k) = 4π 2 (1 α 1 α 2 ) sin(k 12 ) 12 (14) t = t s 1 2 b Fig. 4. Gphicl epesenttion of the stndd evolution opeto fo single-photon exchnge. The outgoing lines epesent hee positive-enegy electon sttes. The delt fcto bove shows tht enegy must be conseved in the S mtix. Theefoe, off-digonl elements of the effective Hmiltonin between sttes with diffeent unpetubed enegies cnnot be evluted, which is seious limittion, implying tht the method cnnot be pplied to the cse of qusi-degenecy, s descibed in the MBPT section. An dditionl shotcoming is tht no infomtion of the wve function is obtined.

6 6 Cn. J. Phys. Vol., 4. Covint evolution-opeto method In ode to emedy the shotcomings of the S-mtix method fo bound-stte clcultions, we hve intoduced modifiction, efeed to s the covint evolution-opeto method [18, 14]. We then etun to the evolution opeto (11), nd in ode to mke this covint we llow the outgoing obitls to epesent positive- s well s negtive-enegy sttes. This leds to the expession U (2) Cov (t, ) = e2 d 3 x 2 1d 3 x 2 (x 1) (x 2) d 4 x 1 d 4 x 2 is F (x 1, x 1 ) is F (x 2, x 2 ) α µ 1 id Fµν(x 2 x 1 ) α2 ν (x 2 ) (x 1 ) e γ( t 1 t 2 ) (15) illustted by the time-odeed digms in Fig. 5. S F (x, x) is the electon popgto o singleelecton Geen s function, epesented by the intenl lines of the ight-most digm. t = t s 1 2 b 1 2 s b 1 2 s b = ± t = t S F s ± S F 1 2 b Fig. 5. Gphicl epesenttion of the covint evolution opeto fo single-photon exchnge. The outgoing lines cn hee epesent positive- s well s negtive-enegy sttes. Evluting the single-photon exchnge (15) yields the fist-ode effective Hmiltonin H (1) eff = V eq(q, q ) = 0 [ 1 dk f(k) q (k iγ) 1 ] q (k iγ) (16) whee q = ε ε, q = ε b ε s nd epesents hole/pticle sttes. In contst to the coesponding S-mtix expession (12), enegy need not hee be conseved. This mens tht lso non-digonl elements cn be evluted, which is needed in ode to pply the method to qusi-degenecy s descibed bove. The covint evolution opeto lso yields infomtion bout the wve function. The mtix element of the fist-ode wve opeto is given by s Ω b = s V eq(q, q ) b q q (17) We hve pplied the covint-evolution-opeto method to evlute the fine stuctue of the 1s2s 3 P stte of some heliumlike ions [18, 14], nd the esults e exhibited in Tble 1, whee compison is mde with the expeimentl esults s well s with clcultions of Dke [25, 11] nd Plnte et l. [23] (We efe to Refs [18, 14] fo moe detiled efeences nd discussions). Ou clcultion epesents the fist numeicl evlution of QED effects of the fine stuctue of heliumlike ions, including the stongly mixed 3 P 1 stte. Pevious clcultions, using the S-mtix method, hs been esticted to the pue sttes 3 P 2 nd 3 P 0 [26]. An ltentive ppoch fo QED clcultions on qusi-degenete sttes is the two-times Geen sfunction method, developed by Shbev nd cowokes [13]. At lest in its pesent fom, this method does not yield ny infomtion bout the wve function (wve opeto) nd cn theefoe not be used s bsis fo the combined QED-MBPT ppoch to be pesented in the next section.

7 Lindgen et l. 7 Tble 1. The 1s2p 3 P fine stuctue of He-like ions. Vlues fo Z=2,3 given in MHz nd the emining in µhtee. Z 3 P 1 3 P 0 3 P 2 3 P 0 3 P 2 3 P (9) (10) Expt l (18) (31) Dke (66) (66) Expt l (1,5) (5) Dke 9 701(10) 4364,517(6) Expt l (5) Dke Plnte Pesent wok (7) 8458(2) Expt l 1361(6) 8455(6) Dke Plnte Pesent wok (30) Expt l (60) Dke Plnte Pesent wok 5. Combined high- nd low-z ppoch In this section we shll descibe the new ppoch to mny-body-qed clcultions tht is unde development t ou lbotoy. This is combintion of the mny-body ppoch fo qusi-degenecy, descibed in section 2, nd the covint-evolution-opeto method fo bound-stte QED, descibed in the pevious section. Fig. 6. Gphicl epesenttion of pi function (Fig. 3) with uncontcted photon (folded digms left out). The bsic ide is to stt with the pi functions, illustted in Fig. 3, nd to dd the petubtion (10), epesenting the intection between the electomgnetic field nd one of the electons. This is illustted by the leftmost digm in Fig. 6. Futhe itetions of the pi eqution yields dditionl Coulomb intections, s epesented by the emining digms of the figue. (In pinciple, these intections cn lso be instntneous Beit intections.) This equies one pi function fo ech momentum k of the photon. The photon cn be closed by contcting with nothe petubtion (10), eithe on the othe electon, which yields covint photon intection between the electons o on the sme electon, which yields self-enegy intection (the ltte with the pope enomliztion). This yields digms of the type illustted in Fig. 7.

8 8 Cn. J. Phys. Vol., Fig. 7. The uncontcted photon in Fig. 6 cn be closed by intecting with the sme o the othe electon. Finlly, the pi function cn be iteted futhe with Coulomb intections, which yields digms of the type illustted in Fig. 8. The fist digm epesents complete covint photon between the electons, evluted with eltivistic coelted wve function. The next digm epesents the dominting pt of the cossed-photon digm, whee one of the photons is etded nd the othe is instntneous (Coulomb o Coulomb-Beit) nd so on. The sitution is nlogous fo the self-enegy digms, whee the dditionl intections yield vetex coections. Fo the time being it does not seem fesible to tet moe thn one covint photon in this mnne. It hs been shown by Zhng nd Dke [6, 27] tht one nd two tnsvese photons with bity numbe of Coulomb cossings ffect the helium fine stuctue to ode α 5 H, while the exchnge of thee tnsvese photons does not ente on this level. Fig. 8. Futhe itetions of the pi functions in Fig. 7 yields QED effects evluted with fully coelted eltivistic wve functions. 6. Summy nd discussion Hee we hve outlined method fo mny-body-qed clcultions tht is pesently unde development. It is bsiclly combintion of the mny-body technique fo qusi-degenecy nd the ecently developed covint-evolution-opeto technique fo bound-stte clcultions. We expect the method to be pplicble to ll nucle chges, but ou min inteest will be on light elements, whee the tditionl high-z ppoches fil. The use of eltivistic bound-stte obitls/popgtos implies tht ou numeicl esults will utomticlly coespond to mny tems in the expnsion in powes of Zα of the nlyticl low-z ppoch. In ddition we cn evlute the effect of single tnsvese photon with bity Coulomb cossings, s illustted in Fig. 8. The only non-ditive effect enteing in ode α 5 H tht we cnnot evlute numeiclly t pesent is the ieducible two-photon effect, which then hs to be included nlyticlly. To begin with, the ditive effects will be included using nlyticl expessions. Ou gol is to be ble to evlute the fine-stuctue seption of neutl helium ccutely, nd believe tht combintion of the numeicl nd nlyticl ppoches will hve the best pobbility fo success. It is still n open question, though, wht ccucy cn ctully be eched.

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