The high-Z hydrogen-like atom: a model for polarized structure functions

The Dirac equation offers a precise analytical description of relativistic two-particle bound states, when one of the constituent is very heavy and radiative corrections are neglected. Looking at the high-Z hydrogen-like atom in the infinite momentum frame and treating the electron as a "parton...

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Datum:	2007
Hauptverfasser:	Artru, X., Benhizia, K.
Format:	Artikel
Sprache:	English
Veröffentlicht:	Національний науковий центр «Харківський фізико-технічний інститут» НАН України 2007
Schriftenreihe:	Вопросы атомной науки и техники
Schlagworte:	Elementary particle theory
Online Zugang:	http://dspace.nbuv.gov.ua/handle/123456789/110942
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spelling	irk-123456789-1109422017-01-08T03:02:26Z The high-Z hydrogen-like atom: a model for polarized structure functions Artru, X. Benhizia, K. Elementary particle theory The Dirac equation offers a precise analytical description of relativistic two-particle bound states, when one of the constituent is very heavy and radiative corrections are neglected. Looking at the high-Z hydrogen-like atom in the infinite momentum frame and treating the electron as a "parton", various properties usually attributed to the quark distributions in the nucleon are tested, in particular: Bjørken scaling; charge, helicity, transversity and momentum sum rules; existence of the parton sea; Soffer inequality; correlation between spin and transverse momentum (Sivers and Boer-Mulders effects); transverse displacement of the center-of-charge and its connection with the magnetic moment. Deep inelastic experiments with photon or positron beams at MeV energies, analogous to DIS or Drell-Yan reactions, are considered. Рівняння Дірака дає точний аналітичний опис релятивістських зв'язаних станів із двома частинками, якщо одна з них дуже важка, а радіаційними поправками можливо знехтувати. Розглядаючи воднеподібний атом з великим Z у системі нескінченного імпульсу та трактуючи електрон як "партон", перевірено різні властивості, звичайно приписувані розподілам кварків у нуклоні, зокрема: Б’йоркенівський скейлінг; правила сум для заряду, спіральності, поперечної поляризації, імпульсу; існування партонів “моря”; нерівність Соффера; кореляція між спином і поперечним імпульсом (ефекти Сіверса та Бура-Малдерса); поперечне зміщення центру заряду та його зв'язок з магнітним моментом. Розглянуто експерименти з глибоконепружного розсіювання фотонних або позитронних пучків мегаелектронвольтних енергій, подібні до глибоконепружного розсіяння або процесу Дрела-Яна. Уравнение Дирака дает точное аналитическое описание релятивистских связанных состояний двух частиц, если одна из них очень тяжелая, а радиационными поправками можно пренебречь. Рассматривая водородоподобный атом с большим Z в системе бесконечного импульса и трактуя электрон как "партон", проверены различные свойства, обычно приписываемые распределениям кварков в нуклоне, в частности, Бьёркеновский скейлинг; правила сумм для заряда, спиральности, поперечной поляризации, импульса; существование “моря” партонов; неравенство Соффера; корреляция между спином и поперечным импульсом (эффекты Сиверса и Бура-Малдерса); поперечное смещение центра заряда и его связь с магнитным моментом. Рассмотрены эксперименты по глубоконеупругим процессам на фотонных или позитронных пучках мегаэлектронвольтных энергий, аналогичные глубоко неупругому рассеянию, либо процессу Дрелла-Яна. 2007 Article The high-Z hydrogen-like atom: a model for polarized structure functions / X. Artru, K. Benhizia // Вопросы атомной науки и техники. — 2007. — № 3. — С. 98-103. — Бібліогр.: 14 назв. — англ. 1562-6016 PACS: 03.65.Pm, 11.55.Hx, 13.60.-r http://dspace.nbuv.gov.ua/handle/123456789/110942 en Вопросы атомной науки и техники Національний науковий центр «Харківський фізико-технічний інститут» НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
topic	Elementary particle theory Elementary particle theory
spellingShingle	Elementary particle theory Elementary particle theory Artru, X. Benhizia, K. The high-Z hydrogen-like atom: a model for polarized structure functions Вопросы атомной науки и техники
description	The Dirac equation offers a precise analytical description of relativistic two-particle bound states, when one of the constituent is very heavy and radiative corrections are neglected. Looking at the high-Z hydrogen-like atom in the infinite momentum frame and treating the electron as a "parton", various properties usually attributed to the quark distributions in the nucleon are tested, in particular: Bjørken scaling; charge, helicity, transversity and momentum sum rules; existence of the parton sea; Soffer inequality; correlation between spin and transverse momentum (Sivers and Boer-Mulders effects); transverse displacement of the center-of-charge and its connection with the magnetic moment. Deep inelastic experiments with photon or positron beams at MeV energies, analogous to DIS or Drell-Yan reactions, are considered.
format	Article
author	Artru, X. Benhizia, K.
author_facet	Artru, X. Benhizia, K.
author_sort	Artru, X.
title	The high-Z hydrogen-like atom: a model for polarized structure functions
title_short	The high-Z hydrogen-like atom: a model for polarized structure functions
title_full	The high-Z hydrogen-like atom: a model for polarized structure functions
title_fullStr	The high-Z hydrogen-like atom: a model for polarized structure functions
title_full_unstemmed	The high-Z hydrogen-like atom: a model for polarized structure functions
title_sort	high-z hydrogen-like atom: a model for polarized structure functions
publisher	Національний науковий центр «Харківський фізико-технічний інститут» НАН України
publishDate	2007
topic_facet	Elementary particle theory
url	http://dspace.nbuv.gov.ua/handle/123456789/110942
citation_txt	The high-Z hydrogen-like atom: a model for polarized structure functions / X. Artru, K. Benhizia // Вопросы атомной науки и техники. — 2007. — № 3. — С. 98-103. — Бібліогр.: 14 назв. — англ.
series	Вопросы атомной науки и техники
work_keys_str_mv	AT artrux thehighzhydrogenlikeatomamodelforpolarizedstructurefunctions AT benhiziak thehighzhydrogenlikeatomamodelforpolarizedstructurefunctions AT artrux highzhydrogenlikeatomamodelforpolarizedstructurefunctions AT benhiziak highzhydrogenlikeatomamodelforpolarizedstructurefunctions
first_indexed	2025-07-08T01:22:56Z
last_indexed	2025-07-08T01:22:56Z
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fulltext	THE HIGH-Z HYDROGEN-LIKE ATOM: A MODEL FOR POLARIZED STRUCTURE FUNCTIONS X. Artru1 and K. Benhizia2 1Institut de Physique Nucléaire de Lyon, CNRS and Université Lyon-I. Domaine Scientifique de la Doua. 4, rue Enrico Fermi, F-69622 Villeurbanne, France; e-mail: x.artru@ipnl.in2p3.fr; 2Laboratoire de Physique Mathématiques et Physique Subatomique, Université Mentouri, Constantine, Algeria; e-mail: Beni.Karima@laposte.net The Dirac equation offers a precise analytical description of relativistic two-particle bound states, when one of the constituent is very heavy and radiative corrections are neglected. Looking at the high-Z hydrogen-like atom in the infinite momentum frame and treating the electron as a "parton", various properties usually attributed to the quark distributions in the nucleon are tested, in particular: Bjørken scaling; charge, helicity, transversity and mo- mentum sum rules; existence of the parton sea; Soffer inequality; correlation between spin and transverse momen- tum (Sivers and Boer-Mulders effects); transverse displacement of the center-of-charge and its connection with the magnetic moment. Deep inelastic experiments with photon or positron beams at MeV energies, analogous to DIS or Drell-Yan reactions, are considered. PACS: 03.65.Pm, 11.55.Hx, 13.60.-r 1. THEORETICAL FRAME The Dirac equation enables us to study the relativis- tic aspects of an hydrogen-like atom of large ( 1 , where ). It takes all or- ders in into account but neglects (i) the nucleus recoil, (ii) the nuclear spin and (iii) radiative corrections like the Lamb shift. So it is accurate at least to zeroth order in and . Applying a Lorentz boost, we have an explicit model of “doubly relativistic” two-body bound state (relativistic for the internal and external motions). In particular, boosting the atom to the “infi- nite momentum frame” (or looking it on the null-plane ), one has a model for the structure functions which appear in deep inelastic scattering on hadrons. In fact, since it neglects nucleus recoil, this model is best suited to mesons with one heavy quark. However many properties can be generalized to hadrons made of light quarks. A Z ~αZ = 0t z+ ( ) 137/14/2 ≅= πα e /e Am m Zα α In analogy with the quark distributions, we introduce the unpolarized electron distributions q k , and where takes the place of the Bjørken scaling variable, is the transverse momentum of the electron and the impact parameter is the variable conjugate to . We will also define the corre- sponding polarized distributions like where and S are the polarization vectors of the atom and the electron. We will particularly study: ( )+ = (b ( ,q k ( , )Tq k +k , )x y , ; ),e A+b S S ( , ),q k +b AS k + Tk Tk e • the differences between , the helicity distribu- tion and the transversity distribution ; ( )q k + ( )q k +∆ ( )q kδ + • the sum rules for the vector, axial and tensor charges and for the longitudinal momentum; • the correlations between , and or k , like the Sivers effect; AS eS b T • the existence of a non-zero for transverse S and its connection to the atom magnetic moment; 〈 〉b A • the positivity constraints; • the existence of an electron - positron sea and its role in the sum rules. As scaling variable we take the null-plane momen- tum of the electron measured in the atom rest frame, ( ) ( )0 ,rest frame inf. mom. frame = = / = . z A z A z A Bj k k k M k P M x + + (1) We prefer it to the Bjørken variable which is very small and depends on the nucleus mass. The kin- ematical limit for \| is but typical values are . Bjx \|k + atomM \| \|~e ek m Z mα+ − We hope in this study to get a better insight of rela- tivistic and spin effects in hadronic physics. The infinite momentum or null-plane description can also be inter- esting in atomic physics itself, since “deep inelastic” experiments can also be made with atoms, in particular: • Compton profile measurements: ( ) bound ( ) free ( )K e K Q eγ γ− − ′+ → + + k − , • Moeller or Bhabha scattering: ( ) bound ( ) free ( )e K e e K Q e k± − ± ′+ → + + • annihilation: ( ) bound ( ) ( )e K e K Q kγ γ+ −+ → + + . The Mandelstam variables , t Q and are supposed to be large compared to . A ten MeV beam is sufficient for that. We take the axis opposite to the beam direction. In the laboratory frame the final particles are ultrarelativistic nearly in the 2= ( )s K Q k ′+ + 2= 2= ( )u K k ′− 2 em ẑ PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2007, N3 (1), p.98-103. 98 ˆ−z k + Tk + −∇ q k direction. The components k and k of the elec- tron momentum just before the collision are given by + .T ), , ]k { i ) 1 1 , ' , ) T + T = i− ∇k − ∇ , )i .z b A+ ( , ( , k ( , k ) = Co . +Φ ,+ 2. ) b q ,+ b E E ,k Q+ + (2) ' (nucleus) = 'T T−k P k QT A e } Φb r   b   (3) can be measured with one detector, k and need two detectors. The definition (3) of k is ambigu- ous due to the final state Coulomb interaction. + Tk 2. JOINT DISTRIBUTION ( , )k +b Being observable quantities, the operators and should be defined in the gauge independent way k + 0= ( ) ( ( )z z T T Tk i V i A∂ − − ∂ − −r r r . (4) They do not commute: [ , where is the transverse part of the Coulomb force. Therefore one cannot define a joint distribution in an unambiguous way. Leaving this prob- lem for the next section, we can at least define the joint distribution in the impact parameter represen- tation. This quantity plays a role in double H + H colli- sions in which both nuclei and both electrons collide. = ( )T Ti V+ k r ( )TV r ( , T + k ) ( , )q k + b From the known the Dirac wave function of the hy- drogen atom (see, for instance [1-3]) ( , ) = ( ) ,iEtt −Ψ Ψr r (5) we can define the two-component null-plane wave func- tion in b and [4], Tk ( , ) = exp ( ( );k dz ik z Ez z +∞+ + −∞ Φ − + − χ∫b (6) 1 3 2 4 ( ) ( ) = ( ) = ; ( ) ( ) Ψ +Ψ Φ Ψ −Ψ  r r r r r (7) ( ) ( ( ) ( ) 0 0 , ' , sinh / sinh / z z z dz V x y z Z z b− − χ = = − α − ∫b (8) The "gauge link" exp{ ( }i z− χ transforms in the Coulomb gauge to in the null plane gauge ( = in the infinite momentum frame). The choice of corresponds to a residual gauge freedom. The quantity Ψ Ψ = 0 zA 0z 0 † 2) = ) ), /(2 ) edN q k d dk + +≡ Φ Φ π b b b (9) + +b will be temporarily interpreted as the electron distribu- tion in the atom. One has indeed 2( ) = ( , 1 2 2 dk dkq k d b k + ++∞ +∞+ + −∞ −∞∫ ∫ ∫π π (10) q k However a significant re-interpretation will be given in Section 5. The gauge link makes invariant under a gauge transformation, for instance , . Such a shift of the potential is practically realized when elec- trons are added in far outer shells. Intuitively, this addi- tion does not change the momentum distribution of the deeply bound electrons. ( )q k ( )r r( ) ( ) ConstV V→ +r r ( ) nst→ + 3. JOINT DISTRIBUTION ( , )T k +k Notwithstanding the non-commutativity mentioned earlier, one can make a transversal Fourier transform of (6), 2( , ) = ( , )i TT k d e k−+ ⋅Φ ∫ k bk b b (11) and define a longitudinal-transverse momentum distri- bution †( , ) = ( , ) ( , )T T Tq k k k+ +Φ Φk k k (12) normalized to 2( ) = ( , ) /(2 )T Tq k q k d π+ +∫ k k (13) ( ,Tq k+k 0 =z −∞ 0z depends on , which is a remnant of the ambiguity. However, an appropriate choice of turns this apparent disease into an advantage [5–7]. Taking for the Compton reaction, the factor e in (6) just takes care of the final state interaction: it de- scribes the distortion of the scattered electron wave function by the Coulomb potential, in the eikonal ap- proximation. Similarly, taking for the annihila- tion reaction, it describes the distortion of the initial positron wave function. Thus , which depends on , has no precise intrinsic character. One can just consider a “most intrinsic” definition with . 0z 0z i− χ = 0 ( , )zb 0 =z +∞ ( ,Tq k +k ) 0z 4. SPIN DEPENDENCE OF THE ELECTRON DENSITY In formulas (5-12) the angular momentum state of the atom was not specified. We assume that the electron is in the fundamental , state and the nu- cleus is spinless. Let = 1n = 2AS = 1/2j 〈 〉j and denote the atom and electron polarization vectors. The unpolar- ized electron density in ( , space in a fully polar- ized ( \| ) atom is = 2e 〈 〉S s 1 A+ . ⋅n )+kb \|=AS †( , ; ) = ( , ; ) ( , ; )A Aq k k k+ +Φ Φb S b S b S (14) and the electron polarization is given by † ( , ; ) ( , ; ) = ( , ; ) ( , ; ) e A A A A S k q k k k + + + +Φ Φ b S b S b S σ b S (15) Taking into account parity and angular momentum conservations, the density of electrons with polarization in a polarized atom can be written as eS ( ) 0 0ˆ ˆ( , , ; ) = ( , ) / 2 [1e A A e n nq k C C+ + + ⋅ +b S S b S n S ˆ ˆ ˆ( )( ) ( ) ˆ ˆ ˆ( ) ( )( )], e A e A e A nn ll z z l z e A e A l z C C S S C S C S C π π ππ + ⋅ ⋅ + + ⋅ + ⋅ + ⋅ ⋅ S n S n S π S π S π S π (16) where and n z . The C 's also are func- tions of b and . A similar equation can be written in the representation. Integrating (16) over leaves the following spin correlations: ˆ = /bπ b ˆ ˆ ˆ= × π ,i j k + Tk b 99 ( , ; ) = 12 ( ) ( ) ( ) , e A e A e A z z T T q k q k q k S S q kδ + + + + + ∆ + ⋅  S S S S (17) where and are the helicity and trans- versity distributions. ( )q k +∆ (q kδ + ) .   4.1. FORMULAS FOR THE POLARIZED DENSITIES IN AND ( ,( , )k +b )T k +k For the state, Φ of Eq.(11) can be writ- ten as = 1/2zj + ˆ( , ; = ) = ; ˆ( , ; = ) = A i A T i w k ive w k ve + φ + φ   Φ +  −   Φ +  −  b S z k S z (18) =z For the state, = 1/2zj − ˆ( , ; = ) = ; ˆ( , ; = ) = . i A i A T ive k S w ve k S w − φ + φ +   Φ −      Φ −     b z k z (19) For other orientations of , one takes linear com- binations of (18) and (19). The distribution de- pends of only in an over-all phase. Choosing , and are real and given by AS ( , )k +b 0z 0 = 0z ( , )v k +b ( , )w k +b ( , )= ikv b z iEz i zdz e f r r w r i z +∞ − −∞ ξ    + − χ    + ξ    ∫ b (20) ( )/ , 3 †= ( ; ) ( ; )A Aq d Ψ Ψ∫ r r S r S (26) where = /(1 )Zξ α + γ , 2= / = 1 ( )eE m Zγ − α and 1/2 1/2 11( ) = (2 ) exp( ) 8 (1 2 ) e ef r m Z r m Z rγ+ γ− + γ α − πΓ + γ  α (21) is the 1S radial wave function. Then, 2 2 2 2 0 0 2 2 2 2 ( , ) = ; ( , ) = 1; ( , ) = ( , ) = 2 /( ); ( , ) = ( , ) = ( )/( ); ( , ) = ( , ) = 0. nn n n ll l l q b k w v C b k C b k C b k wu w v C b k C b k w v w v C b k C b k ππ π π + + + + + + + + + − + − + (22) Note that , which gives an asymmet- rical impact parameter profile for a transversely polar- ized atom. 0 ( , ) 0nC b k + ≠ The distribution depends on . Taking makes (8) divergent. In practice we will as- sume that \| \| is large but finite, accounting for a screening of the Coulomb potential. It gives ( , )T k +k 0z 0z 0 =z ∞∓ ( ) ( ) ( ) ( )[ ]bzzbzZz /2ln/sinh, 00 1 ε−α−=χ −b , (23) with and , the upper sign corre- sponding to Compton scattering and the lower sign to annihilation. Modulo an overall phase, (0) = 0ε ( ) = 1ε ∞∓ ∓   = 1; ( 00 0 1 ( ) ( , ))= 2 ( ) ( , ) iZ z T T w J k b w b k b db b v J k b v b k + ∞ αε +    π      ∫ . (24) The analogue of (22) is 2 2 * 2 2 0 0 2 2 2 2 * 2 2 ( , ) = \| \| \| \| ; ( , ) = 1; ( , ) = ( , ) = 2 ( )/(\| \| \| \| ); ( , ) = ( , ) = (\| \| \| \| )/(\| \| \| \| ). ( , ) = ( , ) = 2 ( )/(\| \| \| \| ). T nn T n T n T ll T T l T l T q k k w v C k k C k k C k k v w w v C k k C k k w v w v C k k C k k v w w v + + + + + + ππ + + π π + ℑ + − + − ℜ + (25) These coefficients are related to the structure func- tions listed in Ref.[8]. For the "most intrinsic" gauge , and are real so that C k (no Sivers effects). This is in accordance with time reversal invariance [7]. For the “Compton” and "annihilation" gauges ( ), and v are complex numbers, so that Sivers [9-10] effect ( and the Boer- Mulders [8] effect ( ) take place. 0 = 0z w 0 v 1 0 ( , ) = 0n T k + ) 0k ≠ ∓ w 0 k + ( ,n TC k + ( , ) 0n T ≠0C k In the Compton case the factor b behaves like a converging cylindrical wave. Multiplying , it op- erates as a boost toward the axis, interpreted as the "focusing" of the final particle by the Coulomb field [10]. This focusing converts the asymmetry in b for a transversely polarized atom into the Sivers asymmetry in . The opposite effect (defocusing of the positron) takes place in the annihilation case. iZ− α ( )Φ r z Tk 4.2. SUM RULES Integrating (17) over , one obtains the vector, ax- ial and tensor charges k + 2 3 † 2 1 /3= ( ; ) ( ; ) = 1 A A Aq d − ξ ∆ ⋅ Ψ Ψ + ; ξ∫S r r S Σ r S (27) 2 3 † 2 1 /3= ( ; ) ( ; ) = 1 A A Aq d + ξ δ ⋅ Ψ β Ψ + . ξ∫S r r S Σ r S (28) Note the big "helicity crisis", instead of 1 as naively expected, for . = 1/3q∆ = 1Zα 4.2.1. SUM RULE FOR THE ATOM MAGNETIC MOMENT Consider a classical object of mass , charge , spin and time-averaged magnetic moment M Q J µ in its rest frame. In this frame, the centre of energy r and the average center of charge 〈 coincide, say at . Upon a boost of velocity , and 〈 un- dergo the lateral displacements G Cr C 〉r v= 0r Gr 〉 = / , = /G CM× 〈 〉 ×b v J b v .Qµ (29) Gb and coincide if the gyromagnetic ratio has the Dirac value . In our case, is negligible due to the large nucleus mass, therefore the magnetic moment is almost totally anomalous. In the infinite momentum or null-plane frame ( ) one observes an electric dipole moment [11] C〈 〉b /Q M Gb ˆv z atom ˆ= Ae− 〈 〉 µ ×b z ,S x (30) which we can calculate from C b . Weighting (14) with b for S 0 ( , )n k + x ≡ ˆ=A y one obtains 100 = (1 2 )/(6 )ex〈 〉 − + mγ (31) n which is in accordance with the relativistic result for the atomic magnetic moment = (1 2 )/(6 )A ee mµ − + γ (ig- noring the anomalous magnetic moment of the electron itself). 4.3. POSITIVITY CONSTRAINTS The spin correlations between the electron and the atom can be encoded in a positive-definite "grand den- sity matrix" [12], R = t A eR C µ ν µν  σ ⊗ σ  . (32) Here µ and run from 0 to 3, summation is under- stood over repeated indices, and C . can be seen as the density matrix of the final state in the crossed reaction . Be- sides the trivial conditions \| the positivity of gives ν 0 = Iσ atom(→ \| 1ij ≤ 00 = 1 ( )A e++ −S S R R nucleus ) e C 2 2 2 0 0(1 ) ( ) ( ) ( ) .nn n n ll l lC C C C C C Cππ π π± ≥ ± + ± + ∓ 2 + + (33) These two inequalities agree with those of Ref. [13]. Together with \| they are saturated by (22) or (25). This maximal strength of the spin correlation means that the information contained in the atom polari- zation is fully transferred to the electron, once the other degrees of freedom ( and b or ) have been fixed. If we integrate over , for instance, some information is lost and some positivity conditions get non-saturated. The same happens if there are “spectators” electrons which keep part of the information for themselves. \| 1llC ≤ k + k + Tk After integration over b or k , we are left with the Soffer inequality [14], T 2 \| ( ) \| ( ) ( ),q k q k q kδ + +≤ + ∆ (34) which are saturated by (26-28). Note that a complete anti-correlation between the atom and the electron spins, and , leading to = = = 1ll nnC C Cππ − = 0i jC ≠ ( , , ; ) = ( , ) (1 )/2,e A e Aq k q k+ + − ⋅b S S b S S violates the positivity conditions, although the last ex- pression is positive for any S and . In fact such a correlation would make e ,A e + 〉 AS ,A e\| \|R+ A e 〈 〉 + 〉 〉 negative for some entangled states \| , in the crossed channel, in particular the spin-singlet state [12]. 5. THE ELECTRON-POSITRON SEA The charge rule (26) receives positive contributions from both positive and negative values of k . So the contribution of the positive k domain is less than unity. On the other hand, physical electrons have posi- tive . It seems therefore that there is less than one physical electron in the atom. This paradox is solved by the second quantification and the introduction of the electron-positron sea. + + k + Let us denote by \| an electron state in the Cou- lomb field. Quantizing the states in a box, we take n to be integer. Negative 's are assigned to negative energy states. Positive 's up to label the bound states ( ) and the remaining ones from to label the positive energy scattering states, . Let \| be the plane wave with four- momentum and spin , solution of the free Dirac equation. The destruction and creation operators in the interacting and free bases are related by n〉 n em , k s a− − = \| n′ Bn k s Dir atom〈 〉 < <e nm E− +∞ n eE m≥ + k = ,k sα † †\| =n nH a 0k k + atom <0 ( , ) \| , e n N s k s − ′ + 〈∑ k † na 1Bn + . ,k s 1 2 〉 \| n a〈 〉∑ †a † , 2\| . k s 〉 s , † \|a−∞ k ,k s atom , e e k k N N − − ion , \| (k s n〈 〉 → Φ k 0>0 ,T = \| s ∑ ∑ ) ,k s (q k + k + k + s a e tom ( ,N k + atom= k s− − 〉α α ion atom e e+ + = <0k+ q− ∫ k + k + ( at =e eQ N atom ( ) ( ) ( = = ion q k + q k + )− + + = 1e om e − ion eQ ion e eN N − − N ion atomN− † † , = \| ,n n n n a n k sα 〈∑ (35) In the Dirac hole theory, the hydrogen-like atom is in the Fock state ac-bare nucleus .〉 〉 (36) “Dirac-bare” means that all Dirac states, including the negative energy ones, are empty. The number of electrons of momentum and spin (with positive and ) in the atom is s 2=\| , \| \|k s nα α 〈 (37) A stripped ion is a "Dirac-dressed" nucleus, all nega- tive energy states being occupied. For the ion the factor of (36) is missing and the first term of (37) is absent. By difference, 2\| \| = >0 2k dkn + +− 〈 〉 π∫ (38) ), the last expression being for the continuum limit . Positrons are holes in the Dirac sea. The number of positrons of momentum k (with positive 0k and ) and spin in the atom is † 2 , , 0< ) = \| , \| \|k s n n s k s n− − ′≠ ′〈 〉 〈− −∑ . (39) For the ion, the condition is relaxed. By dif- ference, n n′ ≠ 2 dkN N k + + π (40) ( ), where we have made the change of variable = −k k . The sum rule (26) can therefore be interpreted in the following way: • for , ( distrib. in atom) - ( distrib. in ion) > 0 e− e− • for , ( distrib. in ion) - ( distrib. in atom). < 0 e+ e+ Thus ) = 1,e eN + + (41) where each bracket ∈ . Introducing , this can be rewritten as [0,1] e− − Q Q (42) is the electronic charge renormalization of the ion on the null plane. It is more likely positive, maybe infi- 101 nite for a point-like nucleus. The renormalization of the atom is equal to it. It may be interesting to relate with the result of covariant QED. atom 1eQ − ato edN − ( ) γ + γ + (q k+ ion matter NP+ NE field = e ionQ m/dk seγ + → γ → ) P M + ν ( )r { } dx dx ∫ ∫ 00T 0 0z zT T atom eE E M 2 +E B = = P M eB ≡ − + , , and are separately measured in the deep inelastic reactions listed in Section 1 and their generalizations to the elec- tron-positron sea, for instance atom/edN dk + + ion /edN dk − + ions / edN dk + + a ( ) free ( ).K e K Q e k± ′→ γ + + ± e− N ) ) d xdydz (43) P P A “sea” electron can be equally understood in the sense given by Feyman in the parton model or by Dirac in the hole theory. It gives the second term of (37) and the whole process is slow fastA A e+′ + + + . (44) P k A “sea” positron is understood in the Feynman sense only. In the hole theory, an electron of large negative energy is lifted to a bound or slow free state \| . It gives the right-hand side of (39). The whole process is n′〉 , fast .n n nH H e− + ′ ′+ + γ (45) Of course, one can permute the roles of the electron and the positron in the Dirac theory; then Feynman and Dirac sea positrons become equivalent. 6. MOMENTUM SUM RULE obeys a momentum sum rule which, like the charge sum rule, applies to the difference between the atom and the ion. The null-plane momentum of the ion (=nucleus) can be decomposed into a matter part and a Coulomb field part: { }matter field= =N NP P+ + ++ E . (46) includes the momentum of the electron cloud which renormalize the ion charge. is the flux of the component of the energy- momentum tensor T of the nucleus Coulomb field across the null plane: { }field NP+ E , 0= zT T Tν ν+ µν { }NE { } { }( ( ( ) 0 00 0 0 2 2 = = . z N N N z z zz x y P T T dydz T T T T dydz E E + ν ν ν+ σ + + + + ∫E E E (47) We have used , = (1,0,0,1)d dνσ ( )2= 2 x yE B − , , . Similarly, for the atom, we have 00 2 2=zz z zT T E B− − y xE B { }matter bare field= = ;A N e N eP P P+ + + ++ + +E E Be .t ) (48) the electron magnetic field being included. Here , and take only into account the difference be- tween the atomic and ionic electron clouds. Subtracting (46) from (48), bare eP+ =A N e inM P P+ ++ (49) intP+ results from the crossed terms in and or of T . Its value is NE eE eB µν ( ) (3 3 = 2 4 4= = . 3 3 int Nx ex ey Ny ey ex N e P d E E B E E B d E E V +  + + −  ⋅ 〈 〉 ∫ ∫ r r (50) V〈 〉 is the average potential energy. The terms in have disappeared upon angular integration. N eE B { }bare field=e e P+ + ++ E = zk E i+ − ∂ is the mean value of the null- plane mechanical momentum k of the physical elec- tron (more precisely the “atom - minus – ion” part of it). Inserting in Eqs.(9), one obtains + e 4= ( ) = 2 3e dk q k E ++∞+ + + −∞ − 〈 〉∫ π (51) V / with 3 † 2= ( ) ( ) ( ) = ( )eV d V r m Z〈 〉 Ψ Ψ − α .γ∫ r r r (52) Eqs.(49-51) constitute the momentum sum rule. 7. CONCLUSION This study has shown the rich spin and structure of the hydrogen-like atom at large when it is ob- served in the infinite momentum (or null-plane) frame. Without the complications of QCD, like gluon self- interaction and confinement, many properties attributed to the leading twist hadronic structure functions have been found and clearly interpreted here, in particular: the sum rules, the spin crisis, the connection between and the Sivers effect, the relation between and the magnetic moment, the role of spectators in the positivity constraints, the existence of a Feynman sea. With this "theoretical laboratory" one may also investi- gate non-leading twist structure functions, elastic form factor a la Isgur-Wise, etc. Our results are interesting also in pure QED. We have seen a connection between the nucleus charge renormalization and the unpolarized deep inelastic structure function of the cloud of a stripped ion target. Thus the charge renormalization can be analyzed experimentally in deep inelastic Compton, Moller or annihilation processes. The same reactions at the 10 MeV energy scale can test the relativistic correc- tions to the electronic wave functions of large atoms. Tk ± Z Z 0〈 〉 ≠b 〈 〉b e The PowerPoint document presented at QEDSP2006, which contains figures not presented here, can be obtained upon request to one author (X. A.). REFERENCES 1. L.D. Bjørken, S.D. Drell. Relativistic Quantum Me- chanics. New York: “McGraw-Hill”, 1964. 2. A.I. Akhiezer, V.B. Berestetskii. Quantum Electro- dynamics. New York : “Interscience”, 1965, 868 p. 3. A. Messiah. Mécanique Quantique, v. 2. “Dunod”, 1964. 4. X. Artru, K. Benhizia. The relativistic hydrogen atom: a theoretical laboratory for structure functions //Proc. Int. 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Rev. D. 1991, v. 43 p. 261-253. 11. M. Burkardt. Chromodynamic lensing and trans- verse single spin asymmetries //Nucl. Phys. A. 2004, v. 735, p. 185-199. 12. X. Artru, J.-M. Richard. General constraints on spin observables, applications to antiproton proton to antilambda lambda and to polarized quark distributions //Proc. of the Xth Advance Research Workshop on High Energy Spin Physics, NATO ARW DUBNA-SPIN-03, Dubna, 2003. P. 19-28, hep-ph/0401234, 2004, 10p. 13. A. Bacchetta, M. Boglione, A. Henneman, P.J. Mulders. Bounds on transverse momentum de- pendent distribution and fragmentation functions //Phys. Rev. Lett. 2000, v. 85, p. 712-715. 14. J. Soffer. Positivity constraints for spin-dependent parton distributions //Phys. Rev. Lett. 1995, v. 74, p. 1292-1294. ВОДОРОДОПОДОБНЫЙ АТОМ С БОЛЬШИМ Z: МОДЕЛЬ ДЛЯ ПОЛЯРИЗАЦИОННЫХ СТРУКТУРНЫХ ФУНКЦИЙ К. Артру, К. Бенхизия Уравнение Дирака дает точное аналитическое описание релятивистских связанных состояний двух час- тиц, если одна из них очень тяжелая, а радиационными поправками можно пренебречь. Рассматривая водо- родоподобный атом с большим Z в системе бесконечного импульса и трактуя электрон как "партон", прове- рены различные свойства, обычно приписываемые распределениям кварков в нуклоне, в частности, Бьёрке- новский скейлинг; правила сумм для заряда, спиральности, поперечной поляризации, импульса; существо- вание “моря” партонов; неравенство Соффера; корреляция между спином и поперечным импульсом (эффек- ты Сиверса и Бура-Малдерса); поперечное смещение центра заряда и его связь с магнитным моментом. Рас- смотрены эксперименты по глубоконеупругим процессам на фотонных или позитронных пучках мегаэлек- тронвольтных энергий, аналогичные глубоко неупругому рассеянию, либо процессу Дрелла-Яна. ВОДНЕПОДІБНИЙ АТОМ З ВЕЛИКИМ Z: МОДЕЛЬ ДЛЯ ПОЛЯРИЗАЦІЙНИХ СТРУКТУРНИХ ФУНКЦІЙ К. Артру, К. Бенхізія Рівняння Дірака дає точний аналітичний опис релятивістських зв'язаних станів із двома частинками, як- що одна з них дуже важка, а радіаційними поправками можливо знехтувати. Розглядаючи воднеподібний атом з великим Z у системі нескінченного імпульсу та трактуючи електрон як "партон", перевірено різні властивості, звичайно приписувані розподілам кварків у нуклоні, зокрема: Б’йоркенівський скейлінг; прави- ла сум для заряду, спіральності, поперечної поляризації, імпульсу; існування партонів “моря”; нерівність Соффера; кореляція між спином і поперечним імпульсом (ефекти Сіверса та Бура-Малдерса); поперечне зміщення центру заряду та його зв'язок з магнітним моментом. Розглянуто експерименти з глибоко- непружного розсіювання фотонних або позитронних пучків мегаелектронвольтних енергій, подібні до гли- боконепружного розсіяння або процесу Дрела-Яна. 103

The high-Z hydrogen-like atom: a model for polarized structure functions

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