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Xavier Artru, Institut de Physique Nucléaire de Lyon, France Transversity 2005 Karima Benhizia, Mentouri University, Constantine, Algeria Como, 7- 10 sept. The relativistic hydrogen-like atom : a theoretical laboratory for structure functions. i. Theoretical environment. pure QED - PowerPoint PPT Presentation
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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 2005Karima Benhizia, Mentouri University, Constantine, Algeria Como, 7- 10 sept.
Theoretical environment
• pure QED• « atom » = hydrogen-like ion
(or ion with several e- , but neglecting e- - e- interactions)
• Z ~ 102 Z ~ 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 : (Z)4 << 1
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)
Deep-inelastic probes of the electron state
The experiment can be :
• inclusive (single-arm detector) measures k+ = k0 + kz • semi-inclusive (double-arm detector) measures also kT
• polarized or unpolarized
Compton : + e- (bound) + e- (free) at Moeller or Bhabha : eç + e- (bound) eç + e- (free) s, t, u >> m2
annihilation : e+ + e- (bound) +
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| ~ (Z) m
Like with quarks, we consider :
q(k+) = unpolarized electron distribution
q(k+) = helicity distribution
q(k+) = transversity distribution,
as well as joint distributions in ( k+, kT) or ( k+, b)
Joint ( k+, kT ) distributions
Look at « infinite momentum frame », Pz >> M (replacing k+ by kz).
What is probed is the mechanical longitudinal momentum :
kzmec = kz
can – 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 kz
can (= 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.
« 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 , ky
mec ] = i e Bz = 0 )
• ( kzmec , b ) in the null-plane : ( k+(mec) , b )
• ( kzmec , Lz )
Note : two-parton distributions ( 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.
Joint ( k+, b ) distribution
q( k+, |b| ), and its spin correlations, can in principle be measured in double atom + atom collisions :
b
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 .
Positivity constraints & dictionnary with Amsterdam
| CNN | < 1 ,
(1 ç CNN )2 > (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 kT
2 / (2MN2) :
f1 C00 = f1
f1 C0N = f1Tperp (Sivers)
f1 CN0 = - h1perp (Boer-Mulders)
f1 CLL = g1
f1 CNN = h1 - h1T
perp
f1 Cpp = h1 + h1T
perp ( Kotzinian – Mulders – Tangerman)
f1 CLp = g1T
f1 CpL = h1Lperp
Basic formula
42
31
)(r
)( )( i - i i- exp )( r b, b,
zzEzkdzk
(z/b) sinh Z - )z'y,(x, )( z
)(z
1-
)( 00
bb
b,
z
z
V'dzz
),( ),( ) ; ,( kkkq b bS b
)( )( ) ; ( s k,k,k,q b bSb
4321
0
)t( r,
gauge link :
electron density in ( k+, b ) :
spin density :
electron wave function in the atom rest frame :(depends on the atom spin direction S)
two-component spinor, after projection with 1+z :
null-plane wave function :
z0(b) arbitrary function
For ( k+, kT ) distributions, just take the Fourier transform of ( k+, b )
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.
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.
Charge sum rules
Integration over k+ and b yields the electric, axial and tensor charges
) 2q q (
1
31 1
q , 1
31 - 1
q q
2
2
2
2
1
,
)(Z - 1 mE , 1
Z 2
case Z = 1 : q = 1/3 (= « spin crisis »!...) , q = 2/3
with
Burkardt connection
Classically, for a particle of any spin J perpendicular to the figure:
G = centre of energy C = centre of charge
(1) d . e = normal magnetic moment 0 = (e J) / M
(2) d’ . e = anomalous magnetic moment a
(3) (d + d’) . e = total magnetic moment = 0 a
(2) or (3) OK for hydrogen-like atom. a = = -e (1+2) / (6m)
at rest after ultra-relativistic boost
.C
.G.C,G
d’d
Electron-positron seaRecall : 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
),( ),(
kk b b
Electron-positron sea (2)
Electron states are eigenstates of
H’ = H – v . p = H’0 + HI
with H’0 = . p + m – v . p , (v = atom velocity)
HI = - . A + A0 ( A(x,y,z,t) = moving field of the nucleus)
The term – v . p takes into account the recoil of the nucleus.
« m - state » = eigenstate of H’. « k - state » = eigenstate of H’0 = plane waves.
The deformed Dirac sea : all - states of negative energy’ are occupied :
| > = Pm<0 a*(m ) | 0 >
Electron-positron sea (3)
Atom in state N° 1 : |A1 > = a*(1 ) | >
DIS measures the number of electron in the plane-wave state | k > ,
N(k) = <A1 | a*(k ) a(k ) | A1 > = | ( k , 1 ) |2 + Sm<0 | ( k , m ) |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 | -k > :
Ne+(k) = <A1 | a(-k ) a*(-k ) | A1 > = Sm>1 | ( -k , m ) |2
If the nucleus is fully stripped, the sum is over all positive m.
Electron-positron sea (4)
Results :
Ne+= Sk> 0 Ne+(k) ; Ne- = Sk> 0 Ne-(k)
Sk> 0 | ( k , 1 ) |2 = Ne- (atom) - Ne- (nucleus) < 1
Sk< 0 | ( k , 1 ) |2 = Ne+ (nucleus) - Ne+ (atom) < 1
( Ne- - Ne+ )_atom – ( Ne- - Ne+ )_nucleus = 1
Second braket = renormalisation of the nucleus charge
• 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.
Conclusions
Inégalités de spin
2 | q(x) | < q(x) + q(x)
How to realise an intricate state in the t-channel (proton + antiquark X) :
p q
X
a c
(spin 0) (spin 0)
b (spin 1/2)
structure function