The relativistic hydrogen-like atom : a theoretical laboratory for structure functions

1. Xavier Artru,Institut de Physique Nucléaire de Lyon, FranceTransversity 2005 Karima Benhizia,Mentouri University, Constantine, AlgeriaComo, 7- 10 sept. The relativistic hydrogen-like atom :a theoretical laboratory for structure functions i

2. Theoretical environment • pure QED • « atom » = hydrogen-like ion • (or ion with several e-, but neglecting e- - e- interactions) • Z ~ 102  Za ~ 1  relativistic bound state • Dirac equation  exact wave functions • neglect of nuclear recoil : MN >> m • neglect of nuclear spin (but can consider nuclear size) • neglect of Lamb shift : a (Za)4 << 1

3. What can we test in the atom • positivity constraints • sum rules • electric charge, • axial charge, • tensor charge, • magnetic moment of the atom = - e <b> • existence of electronic and positronic sea • T-even spin correlations : • with fixed kT : CNN , CLL , Cpp , CLp , CpL • with fixed b : CNN , CLL , Cpp , C0N , CN0 • T-odd correlations : • with fixed kT : C0N (Sivers) and CN0 (Boer-Mulders)

4. Deep-inelastic probes of the electron state Compton : g + e- (bound)  g + e- (free) at Moeller or Bhabha : eç + e- (bound)  eç + e- (free) s, t, u >> m2 annihilation : e+ + e- (bound) g + g i The experiment can be : • inclusive (single-arm detector)  measures k+ = k0 + kz • semi-inclusive (double-arm detector)  measures also kT • polarized or unpolarized

5. Scaling limit As scaling variable, we use : k+ = k0 + kz = - Q2 / Q- in the atom rest frame. ( We do not use xBj = k+/P+since it vanishes in the MN B limit ) There is no upper limit to |k+| . Typically,|k+- m| ~ (Za) m Like with quarks, we consider : q(k+) = unpolarized electron distribution Dq(k+) = helicity distribution dq(k+) = transversity distribution, as well as joint distributions in ( k+, kT) or ( k+, b)

6. Joint ( k+, kT ) distributions • Look at « infinite momentum frame », Pz >> M (replacing k+ by kz). • What is probed is the mechanical longitudinal momentum : • kzmec = kzcan – Az(x,y,z) ( kzcan = -iD = canonical ) • Trouble : kzmecdoes not commute with kx and ky (either canonical or mechanical) Speaking of a joint distribution in (kz ,kT) is heretical ! Nevertheless, in a gauge Az = 0 one can define joint distributions in kTcanANDkzcan(= kzmec ) • however : • kz and kT have not « equal rights » : Az is zero but not AT … • kTcan is not invariant under residual gauge freedom  FSI included or not.

7. « allowed » and gauge invariant joint distributions • Quantum mechanics allows joint distributions in : • ( z, b )  in the null-plane : ( X- , b ) • ( z, kTmec) ( at least in our atom, where [ kxmec, kymec ] = i e Bz = 0 ) • ( kzmec , b )  in the null-plane : ( k+(mec) , b ) • ( kzmec , Lz ) • Note : two-parton distributions r ( k+1 , k+2 , b12) involving a relative • impact parameter are used for double parton scattering. • In our case b is the relative electron – nucleus impact parameter.

8. Joint ( k+, b) distribution i i i i b q( k+, |b| ), and its spin correlations, can in principle be measured in double atom + atom collisions :

9. Spin-dependent distributions in ( k+, b) or ( k+, kT ) • without polarisation : q( k+, |b|) • - selecting an electron spin state | s > : q( k+, b, s ) • - with atom polarisation <S> : q( k+, b; <S> ) • with both polarisations : q( k+, b, s ; <S> ) • Everything can be expressed in terms of q( k+, |b| ) • and 7 correlation parameters : • C0N , CN0 , CNN , CLL , Cpp , CLp and CpL( k+, |b| ) • * Same with ( k+, kT ). p = direction of b or kT .

10. Positivity constraints & dictionnary with Amsterdam | CNN | < 1 , (1 ç CNN )2> (C0Nç CN0)2 + (CLLç Cpp)2 + (- CLpç CpL)2 They are most easily obtained in transversity (along N) basis. After removal of kinematical factors kT / MN or kT2 / (2MN2) : f1 C00 = f1 f1 C0N = f1Tperp (Sivers) f1 CN0 = - h1perp (Boer-Mulders) f1 CLL = g1 f1 CNN = h1- h1Tperp f1 Cpp = h1+ h1Tperp ( Kotzinian – Mulders – Tangerman) f1 CLp = g1T f1 CpL = h1Lperp

11. Basic formula electron wave function in the atom rest frame : (depends on the atom spin direction S) two-component spinor, after projection with 1+az : null-plane wave function : z0(b) arbitrary function gauge link : electron density in ( k+, b) : spin density : For ( k+, kT ) distributions, just take the Fourier transform of F( k+, b)

12. Results for impact parameter CNN = 1 CLL = Cpp = (w2 - v2) / (w2 + v2) C0N = CN0 = 2 w u / (w2 + v2) CLp = CpL = 0 w, v : real functions of k+ and b. After integration over b : C0N and CN0 disappear CNN - Cpp disappear ; ( CNN + Cpp ) /2  CTT CTT = ( 1 + CLL ) /2 ; saturates the Soffer inequality.

13. Saturation of the inequalities • The spin inequalities come from the positivity of the density matrix • of the (e+ atom) system in the t-channel. • When the all other commuting degrees of freedom are fixed (orbital momentum, spin of spectators, radiation field…) the (e+ atom) system is a pure state (in our mind) • the density matrix is of rank one • a maximal set of inequalities is saturated. • Then one predicts, without any calculation : • CNN = +1 CNN = -1 • (A) CLL = + Cpp OR :(B)CLL = - Cpp • C0N = + CN0 C0N = - CN0 • CLp = - CpL CLp = + CpL • We have (A) at fixed b and k+ , (B) at fixed kT and k+ . • If the atom is in the negative parity state P1/2, (A) and (B) are interchanged.

14. Charge sum rules Integration over k+ and b yields the electric, axial and tensor charges with case Za = 1 : Dq = 1/3 (= « spin crisis »!...) , dq = 2/3

15. Burkardt connection Classically, for a particle of any spin J perpendicular to the figure: at rest after ultra-relativistic boost .C,G .G d .C d’ G = centre of energy C = centre of charge (1) d . e = normal magnetic moment m0 = (e J) / M (2) d’ . e = anomalous magnetic moment ma (3) (d + d’) . e = total magnetic moment m = m0 +ma (2) or (3) OK for hydrogen-like atom. ma = m = -e (1+2g) / (6m)

16. Electron-positron sea • Recall : electron density in a polarized atom = • q( k+, b ; S) = • It is positive everywhere and non-vanishing for both signs of k+ . • On the other hand, its integral is 1. • What is the meaning of q( k+) for negative k+ ? • Why is the integral of q( k+) on positive k+ less than unity ? Next : Interpretation in terms of deformed Dirac sea and parton-like sea

17. Electron-positron sea (2) Electron states are eigenstates of H’ = H – v . p = H’0 + HI with H’0 = a . p + b m – v . p , (v = atom velocity) HI = - a . A + A0 ( Am(x,y,z,t) = moving field of the nucleus) The term – v . p takes into account the recoil of the nucleus. « ym - state » = eigenstate of H’. « Fk - state » = eigenstate of H’0 = plane waves. The deformed Dirac sea : all y - states of negative energy’ are occupied : | W > = Pm<0 a*(ym ) | W0 >

18. Electron-positron sea (3) Atom in state N° 1 : |A1 > = a*(y1 ) | W > DIS measures the number of electron in the plane-wave state | Fk > , • N(k) = <A1 | a*(Fk ) a(Fk ) | A1 > = | (Fk , y1 ) |2 + Sm<0| (Fk , ym ) |2 ; The first term is the one considered up to now. The second term exists even for a fully stripped nucleus. It represents the virtual electron cloud which may become by scattering with the probe. DIS can also pick-up positrons in states | F-k > : Ne+(k) = <A1 | a(F-k ) a*(F-k ) | A1 > = Sm>1| (F-k , ym ) |2 If the nucleus is fully stripped, the sum is over all positive m.

19. Electron-positron sea (4) Results : Ne+= Sk> 0 Ne+(k) ; Ne- = Sk> 0 Ne-(k) Sk> 0 | (Fk , y1 ) |2 = Ne- (atom) - Ne- (nucleus) < 1 Sk< 0 | (Fk , y1 ) |2 = Ne+ (nucleus) - Ne+ (atom) < 1 ( Ne- - Ne+ )_atom – ( Ne- - Ne+ )_nucleus = 1 Second braket = renormalisation of the nucleus charge

20. Conclusions • The hydrogen-like atom at high Z has many expected, calculable • and fashionable DIS properties. • The connection between magnetic moment and average impact • parameter is transparent there. • We have not yet studied the joint ( k+, kT ) distributions with final state • interaction. We only took z0(b) = 0 for the origin of the gauge link. • Positivity constaints, when saturated due to lack of spectator entropy, • have a very predictive behaviour • The coulomb field generates an electron positron sea. Due to that and • to charge renormalisation, neither the number of electron, nor the • difference electrons – positrons is equal to one.

21. Inégalités de spin b (spin 1/2) c a p q (spin 0) (spin 0) structure function X 2 | d q(x) | < q(x) + D q(x) How to realise an intricate state in the t-channel (proton + antiquark  X) :