X. Ji, PRL91, 062001 (2003)
A. Belitsky, X.Ji, F. Yuan, hep-ph/0307383
Outline
 A brief story of the proton
 The elastic form factors and charge distributions
in space
 The Feynman quark distributions
 Quantum phase-space (Wigner) distribution
 Wigner distributions of the quarks in the proton
 Quantum Phase-space tomography
 Conclusions
A Brief Story of the Proton
Protons, protons, everywhere
 The Proton is one of the most abundant particles
around us!
– The sun ☼ is almost entirely made of protons...
– And all other stars…
– And all atomic nuclei…
 The profile:
– Spin 1/2, making MRI (NMR) possible
– Mass 938.3 MeV/c2, making up ½ of our body weight
– Charge +1, making a H-atom by attracting an electron
What’s in A Proton? (Four Nobel Prizes)
 It was thought as a point-like particle, like electron
 In 1933, O. Stern measured the magnetic moment
of the proton, finding 2.8N, first evidence that the
proton is not point-like (Nobel prize, 1943)
 In 1955, R. Hofstadter measured the charge radius
of the proton, about 0.8fm.
(1fm = 10-13 cm, Nobel prize, 1961)
 In 1964, M. Gell-Mann and G. Zweig postulated
that there are three quarks in the proton: two ups
and one down (Nobel prize, 1969)
 In 1969, Friedman, Kendall, & Taylor find quarks
in the proton (Nobel prize, 1990)
QCD and Strong-Interactions
 Building blocks
– Quarks (u,d,s…, spin-1/2, mq ~ small, 3 colors)
– Gluons (spin-1, massless, 32 −1 colors)
 Interactions
1  a
L   (i    mq )  F F a  g s  A
4
In the low-energy region, it represents an extremely
relativistic, strongly coupled, quantum many-body
problem—one of the daunting challenges in theoretical physics
Clay Math. Inst., Cambridge, MA
$1M prize to solve QCD! (E. Witten)
The Proton in QCD
 We know a lot and we know little
2 up quarks (e = 2/3) + 1 down quark (e = −1/3)
+ any number of quark-antiquark pairs
+ any number of gluons
 Fundamental questions (from quarks to cosmos…)
– Origin of mass?
~ 90% comes from the motion of quarks & gluons
~ l0% from Higgs interactions (Tevertron, LHC)
– Proton spin budget?
– How are Elements formed?
the protons & neutrons interact to form atomic nuclei
Understanding the Proton
 Solving QCD
– Numerically simulation, like 4D stat. mech. systems
 Feynman path integral  Wick rotation
 Spacetime discretization  Monte Carlo simulation
– Effective field theories (large Nc, chiral physics,…)
 Experimental probes
– Study the quark and gluon structure through low and
high-energy scattering
– Require clean reaction mechanism
• Photon, electron & perturbative QCD
Elastic Form Factors & Charge
Distributions in Space
Form Factors & Microscopic Structure
 In studying the microscopic structure of matter,
the form factor (structure factor) F(q2) is one of the
most fundamental observables
– The Fourier Transformation (FT) of the form factor is
related to the spatial charge (matter) distributions !
 Examples
– The charge distribution in an atom/molecule
– The structure of crystals
– …
The Proton Elastic Form Factors
 First measured by Hofstadter et al in the mid
1950’s
Elastic electron scattering
k’
q
k
P’
P
 
 



q

2

2 i
p' j p  U  p' F1 q   F2 q
U  p

2M 

What does F1,2 tell us about the structure of the nucleon?
Sachs Interpretation of Form Factors
 According to Sachs, the FT of GE=F1−τF2 and
GM=F1+F2 are related to charge and magnetization
distributions.
 This is obtained by first constructing a wave
packet of the proton (a spatially-fixed proton)
d 3 p iRP
| R  
e  ( p) | p
3
(2 )
then measure the charge density relative to the
center  (r )   R  0 | j0 (r ) | R  0
Sachs Interpretation (Continued)

Calculate the FT of the charge density, which now
depends on the wave-packet profile
q
q
q
q
F (q)  dP *( P  ) ( P  ) P  | j0 | P  
2
2
2
2

Additional assumptions
– The wave packet has no dependence on the relative
momentum q
– |φ(P)|2 ~ δ(P)
F (q)  q / 2 | j0 | q / 2
Matrix element
In the Breit frame
Up-Quark Charge Distribution
fm
fm
Effects of Relativity
 Relativistic effects
– The proton cannot be localized to a distance better than
1/M because of Zitterbewegung
– When the momentum transfer is large, the proton
recoils after scattering, generating Lorentz contraction
 The effects are weak if
1/(RM) « 1 (R is the radius)
For the proton, it is ~ 1/4.
For the hydrogen atom, it is ~ 10-5
Feynman Quark Distribution
Momentum Distributions
 While the form factors provide the static 3D
picture, but they do not yield info about the
dynamical motion of the constituents.
 To see this, we need to know the momentum space
distributions of the particles.
This can be measured through single-particle knock-out
experiments
 Well-known Examples:
– Nuclear system: quasi-elastic scattering
– Liquid helium & BEC: neutron scattering
Feynman Quark Distributions
 Measurable in deep-inelastic scattering
 Quark distribution as matrix element in QCD
0
1 d
f  x  
P  (0)  e
2 2

 ig
 d


A (  )
 (  ) P
– where ξ± = (ξ 0± ξ 3)/2 are light-cone coordinates.
Infinite Momentum Frame (IMF)
 The interpretation is the simplest when the proton
travels at the speed of light (momentum P∞).
The quantum configurations are frozen in time
because of the Lorentz dilation.
Density of quarks with longitudinal momentum xP (with
transverse momentum integrated over)
“Feynman momentum” x takes value from –1 to 1,
Negative x corresponds to antiquark.
Rest-Frame Interpretation
 Quark spectral function
S (k )   (2 ) 4  4 ( P  k  Pn ) |  n |   (k ) | P |2
n
– Probability of finding a quark in the proton with energy
E=k0, 3-momentum k, defined in the rest frame of the
nucleon
A concept well-known in many-body physics
 Relation to parton distributions
d 4k
3
f ( x)  

(
x

(
E

k
) / M ) S (k )
4
(2 )
– Feynman momentum is a linear combination of quark
energy and momentum projection in the rest frame.
Present status
 GRV, CTEQ, MRS distributions
CTEQ6: J. Pumplin et al
JHEP 0207, 012 (2002)
Quantum Phase-space
(Wigner) Distribution
Phase-space Distribution?
 The state of a classical particle is specified by its
coordinate and momentum (x,p): phase-space
– A state of classical identical particle system can be
described by a phase-space distribution f(x,p). Time
evolution of f(x,p) obeys the Boltzmann equation.
 In quantum mechanics, because of the uncertainty
principle, the phase-space distributions seem
useless, but…
 Wigner introduced the first phase-space
distribution in quantum mechanics (1932)
– Heavy-ion collisions, quantum molecular dynamics,
signal analysis, quantum info, optics, image
processing…
Wigner function
 Define as
– When integrated over x (p), one gets the momentum
(probability) density.
– Not positive definite in general, but is in classical limit.
– Any dynamical variable can be calculated as
O( x, p)   dxdpO( x, p)W ( x, p)
Short of measuring the wave function, the Wigner function
contains the most complete (one-body) info about a quantum system
Simple Harmonic Oscillator
N=0
Husimi distribution: positive definite!
N=5
Measuring Wigner function
of Quantum Light
Measuring Wigner function
of the Vibrational State in a Molecule
Quantum State Tomography of
Dissociateng molecules
Skovsen et al.
(Denmark) PRL91, 090604
Quantum Phase-Space
Distribution for Quarks
Quarks in the Proton
 Wigner operator
 Wigner distribution: “density” for quarks having
position r and 4-momentum k (off-shell)
a la Saches
7-dimensional distribtuion
No known experiment can measure this!
Custom-made for high-energy
processes
 In high-energy processes, one cannot measure k =
(k0–kz) and therefore, one must integrate this out.
 The reduced Wigner distribution is a function of six
variables [r,k=(k+ k)].
– After integrating over r, one gets transverse-momentum
dependent parton distributions
– Alternatively, after integrating over k, one gets a
spatial distribution of quarks with fixed Feynman
momentum k+=(k0+kz)=xM.
f(r,x)
Proton images at a fixed x
 For every choice of x, one can use the Wigner
distribution to picture the nucleon; This is
analogous to viewing the proton through the x
(momentum) filters!
 The distribution is related to Generalized parton
distributions (GPD) through
t= – q2
 ~ qz
What is a GPD?
 A proton matrix element which is a hybrid of
elastic form factor and Feynman distribution
 Depends on
x: fraction of the longitudinal momentum carried
by parton
t=q2: t-channel momentum transfer squared
ξ: skewness parameter
Charge Density and Current
in Phase-space
 Quark charge density at fixed x
 Quark current at fixed x in a spinning nucleon
Mass distribution
 Gravity plays important role in cosmos and Plank
scale. In the atomic world, the gravity is too weak
to be significant (old view).
 The phase-space quark distribution allows to
determine the mass distribution in the proton by
integrating over x-weighted density,
– Where A, B and C are gravitational form factors
Spin of the Proton
 Was thought to be carried by the spin of the three
valence quarks
 Polarized deep-inelastic scattering found that only
20-30% are in the spin of the quarks.
 Integrate over the x-weighted phase-space current,
one gets the momentum current
 One can calculate the total quark (orbital + spin)
contribution to the spin of the proton
How to measure the GPDs?
 Compton Scattering
k’
k
– Complicated in general
 In the Bjorken limit
• Single quark scattering
• Photon wind
• Non-invasive surgery
• Deeply virtual Compton scattering
First Evidence of DVCS
HERA ep Collider in
DESY, Hamburg
Zeus detector
Present and Future Experiments
 HERMES Coll. in DESY and CLAS Coll. in
Jefferson Lab has made further measurements of
DVCS and related processes.
 COMPASS at CERN, taking data
 Jefferson Lab 12 GeV upgrade
– DVCS and related processes & hadron spectrocopy
 Electron-ion collider (EIC)
– 2010? RHIC, JLab?
Quantum Phase-space
Tomography
A GPD or Wigner Function Model

A parametrization which satisfies the following
Boundary Conditions: (A. Belitsky, X. Ji, and F.
Yuan, hep-ph/0307383)
– Reproduce measured Feynman distribution
– Reproduce measured form factors
– Polynomiality condition
– Positivity

Refinement
– Lattice QCD
– Experimental data
Up-Quark Charge Density at x=0.4
z
y
x
Surface of constant charge denstiy
Up-Quark Charge Denstiy at x=0.01
Surface of Constant Charge Density
Up Quark Density at x=0.7
Up-Quark Density At x=0.7
Surface of Constant Charge Density
Charge Denstiy at Negative x
Charge Denstiy in the MIT Bag
Comments
 If one puts the pictures at all x together, one gets a
spherically round nucleon! (Wigner-Eckart theorem)
 If one integrates over the distribution along the z
direction, one gets the 2D impact parameter space
pictures of Burkardt and Soper.
Conclusions
 Form factors provide the spatial distribution,
Feynman distribution provide the momentumspace density. They do not provide any info on
space-momentum correlation.
 The quark and gluon Wigner distributions are the
correlated momentum & coordinate distributions,
allowing us to picture the proton at every Feynman
x, and are measurable!