Download Proton tomography with Wigner distributions

April 29th, 2005
DIS2005, Madison, WI
Quantum imaging of the
proton via Wigner distributions
Andrei Belitsky
Arizona State University
Based on A.B., Xiangdong Ji, Feng Yuan, hep-ph/0307383
For a review see A.B., Anatoly Radyushkin, hep-ph/0504030
Traditional probes of nucleon structure
• Elastic electron-proton scattering ep T e’p’:
e
  (q)
p1
p2  (0)  (0) p1
local operator
p2
• Inelastic electron-proton scattering ep T
e’X :
e
  (q)
k
p
p  (0)  ( z  ) p   dx eixp
light-cone operator
 
z
f ( x)
Physics of form factors (Breit frame)
 | r |~ 1 | p |
y
 ( p)
Localized proton as a wave packet:

R



 
d3p
ip  R
e

(2 ) 3
 p 
RN

p
Charge distribution in the wave packet:

 (r ) 

 
R  0 j0 ( r ) R  0
r
z
x
1||

•
Size of the wave-packet << system size:
 r  R N
•
Resolution scale >> size of the wave packet (one does not want to
see the wave packet):
 r  1 
•
Size of the wave packet >> Compton wave length (to be
insensitive to wave nature of the proton):


1 RN    p  M N



 r  1 M
N
 iq  r 


2


d
x
e

r

q
2
j
(
0
)

q
2

2
M
G
(
Q
)
0
N
E

3
Electric form factor in the Breit frame is (with reservations) a Fourier transform of the charge distribution.
Physics in the infinite momentum frame
momentum frame of
a fast moving proton
proton at rest
pz  
r
x1 pz

Lorentz boost
xi pz
x

z
no spatial extent
y
y
y
Parton distributions:
Form factors:
xp
r
Distribution of
quarks in transverse plane irrespective of their
longitudinal motion.
 z ~ 1 Q
x
z
p
 (r )
Density of partons
of a given longitudinal momentum
x = k||/p measured
with transverse resolution ~1/Q
x
z
p
f ( x)
No corrections!
0
x
0
r
x 1
r
Impact parameter parton distributions
Soper ’77
Burkardt ’00
Localized wave packet in transverse plane:

p , R
d 2 p ip   R 

e
  p  p  , p
2
(2 )
The probability of a parton to possess the
momentum fractions x at transverse position r
f ( x, r )   dz  eixp
 
z
y
 z ~ 1 Q
p  ,0  ( z  , r )  (0 , r ) p  ,0
xp
r
Generalized parton distributions
simultaneously carry information
on both longitudinal and transverse distribution of partons in a
fast moving nucleon
f ( x, r )   d 2   e
 
 ir  
Muller et al. ’94
Ji ’96
Radyushkin ’96
x
z
p
f ( x, r )
H ( x,  0, 2 )
GPDs carry information on the
angular momentum of partons.
0
1
Q: What is the physical significance of skewness
h?
r
Wigner function in quantum mechanics
W ( p, r )   dD e ipD    r  12 D  r  12 D 
Wigner ’32
Contains full information about the single particle wave function.
Properties:
•
•
•
Real
It can and most often does go negative: a hallmark of interference!
Projections lead to probabilities:
 dp W ( p, r )   dq e
•
iqr 
F (q),
 dr W ( p, r )  n( p)
Operator expectation values:
Weyl-ordered
A( p, r )   dpdr W ( p, r ) A( p, r )
The quantum-mechanical uncertainty principle restrict the amount of localization that a Wigner
distribution might have. This yields a “fuzzy” phase-space description of the system compared
to the “sharp” determination of its momentum and coordinates separately. Wigner distribution
provides an appealing opportunity to characterize a quantum state using the classical concept of
the phase space.
Wigner function of 1D harmonic oscillator
p 2
 2x
H ( p, x) 
 m
2m
2
n0
2
n2
n  10
lim   0 , n   W n ( p , x ) ~   H ( p , x )  E 
WKB Wigner distribution resides on classical trajectories in phase space:
  C ( x ) e iS

W ( p, x)  C
2


  p 
S 

x 
Measurement of QM Wigner distributions
Mach-Zender interferometry of quantum
state of light:
•
Quantum state tomography of dissociated
molecules:
quantum
state filter
laser
weak coherent state
SPCM
•
phase-diffused coherent state
Banaszek et al.’99
Skovsen et al.’03
Wigner distributions of the nucleon
Introduce the Wigner operator
 



i   , k    k  
2
W r , k  , k     d  d   e
 r   2   r   2 
Define quark quasi-probability distribution in the proton (in the Breit frame)

W [r , k  , k  ] 
•



d 3q 


q
2
W
r
,
k
,
k

q
2


 ( 2 ) 3
•
Generalized parton distributions

d 2k


f (r , k  )  
W
[
r
,k,k ]
( 2 ) 2

d 3q
 
F x ,  ,  2 
3
( 2 )
(skewness:
Unintegrated parton distributions

q ( x, k  ) 

  q z 2 E  q z 2 M N2  q 2 4 )



d 3r
W
[
r
,k ,k ]
2
( 2 )
Viewing nucleon through momentum filters
z
g*
Feynman momentum:
x
y
Moderate
Low
rz
rx
ry
rz
rx
distance in fm
High
Limitations on the interpretation
•
 | r |~ 1 | p |
y
Transverse dynamics:


1 RN    p  M N
 ( p )
r
Longitudinal dynamics:
•
•
RN
The longitudinal position of partons is set by
skewness:
r z ~ 1 z ~ 1 (E )
p
x
z
Typical longitudinal momentum in the wave
packet:
r z ~ 1 p z
•
z-
The nonlocality of the probe:
z  ~ 1 ( xE)
d
rz
Constraints:
1/MN
•
z  p z  M N
z   r z
z
rz~1/h
RN
-zx  
The classical interpretation of GPDs as Wigner quasiprobabilities is valid in deep DGLAP domain!
Measurement of nucleon Wigner distribution
Exclusive processes sensitive to GPDs:
•
Compton-induced processes:
eN  e' N '  , N  N ' l l  , eN  e' N ' l l  , N  lN ' 
•
Hard re-scattering processes:
eN  e' N ' M l , eN  e' N ' M h , N  N ' M l , N  N ' l  l 
•
Diffractive processes:
N  (2 jets ) N ' , N  (2 jets ) N ' ,   N  Ds N ' , lN  l N '
Leptoproduction of a real photon (cf. Mach-Zender interferometry):
scanned area of the surface as
a functions of lepton energy
reference beam
beam
QM
superposition
principle
detector

object beam
x
ep  e' p' l l 
Parton’s Wigner distributions determine 3D structure of hadrons and are measurable via GPDs.