The relativistic hydrogenlike atom : a theoretical laboratory for structure functions

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The relativistic hydrogen-like atom a
theoretical laboratory for structure functions
Xavier Artru, Institut de Physique Nucléaire de
Lyon, France Transversity
2005 Karima Benhizia, Mentouri University,
Constantine, Algeria Como, 7- 10
sept.
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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 gtgt m
• neglect of nuclear spin (but can consider
  nuclear size)
• neglect of Lamb shift a (Za)4 ltlt 1

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What can we test in the atom

• positivity constraints
• sum rules
• electric charge,
• axial charge,
• tensor charge,
• magnetic moment of the atom - e ltbgt
• 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)

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Deep-inelastic probes of the electron state
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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)
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Joint ( k, kT ) distributions

• Look at  infinite momentum frame , Pz gtgt M
  (replacing k by kz).
• What is probed is the mechanical longitudinal
  momentum
• kzmec kzcan Az (x,y,z)
  ( kzcan -iD canonical )
• Trouble kzmec does 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
• kTcan AND kzcan
  ( 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.

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 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 ( k1 , k2 ,
  b12 ) involving a relative
• impact parameter are used for double parton
  scattering.
• In our case b is the relative electron nucleus
  impact parameter.
• 
  

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Joint ( k, b ) distribution
q( k, b ), and its spin correlations, can in
principle be measured in double atom atom
collisions
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Spin-dependent distributions in ( k, b ) or (
k, kT )

• without polarisation
  q( k, b )
• - selecting an electron spin state s gt
  q( k, b , s )
• - with atom polarisation ltSgt
  q( k, b ltSgt )
• with both polarisations
  q( k, b , s ltSgt )
• 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 .

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Positivity constraints dictionnary with
Amsterdam
CNN lt 1 , (1 ç CNN )2 gt (C0N ç CN0)2 (CLL
ç Cpp)2 (- CLp ç 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
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Basic formula
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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.
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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.

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Charge sum rules
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Burkardt connection
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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

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Electron-positron sea (2)
Electron states are eigenstates of
H H v . p H0 HI
with H0 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 H0 plane waves. The
deformed Dirac sea all y - states of negative
energy are occupied W gt
Pmlt0 a(ym ) W0 gt
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Electron-positron sea (3)

• Atom in state N 1 A1 gt
  a(y1 ) W gt
• DIS measures the number of electron in the
  plane-wave state Fk gt ,
• N(k) ltA1 a(Fk ) a(Fk ) A1 gt ( Fk ,
  y1 ) 2 Smlt0 ( 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 gt
• Ne(k) ltA1 a(F-k ) a(F-k ) A1 gt
  Smgt1 ( F-k , ym ) 2
• If the nucleus is fully stripped, the sum is over
  all positive m.

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Electron-positron sea (4)
Results Ne Skgt 0 Ne(k) Ne- Skgt
0 Ne-(k) Skgt 0 ( Fk , y1 ) 2 Ne-
(atom) - Ne- (nucleus) lt 1 Sklt 0 ( Fk , y1
) 2 Ne (nucleus) - Ne (atom) lt 1
( Ne- - Ne )_atom ( Ne- - Ne )_nucleus
1 Second braket renormalisation of the
nucleus charge
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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.

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Inégalités de spin
2 d q(x) lt q(x) D q(x)

How to realise an intricate state in the
t-channel (proton antiquark ? X)