QCD Map of the Proton
Xiangdong Ji
University of Maryland
Outline
 An Alternative Formulation of Quantum
Mechanics
 Wigner parton distributions (WPD)
– mother of all distributions!
 Transverse-momentum dependent parton
distributions and pQCD factorization
 GPD & quantum phase-space tomography
 Summary
Alternative Formulations of
Quantum Mechanics
 Quantum mechanical wave functions are not
directly measurable in experiment. But is it
possible to formula quantum mechanics in terms of
observables?
– Heisenberg’s matrix mechanics (1925)
– Wigner’s phase-space distributions (1932)
– Feynman path integrals (1948)
– …
QM with phase-space distribution
 Phase-space formulation is based on the statistical
nature of quantum mechanics.
– 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 Boltzmann equation.
 Many identical copies of a quantum system can be
described by a similar phase-space distribution.
Wigner function
 Define as
 When integrated over p, one gets the coordinate
space density ρ(x)=|ψ(x)|2
– Measurable in elastic scattering
 When integrated over x, one gets the coordinate
space density n(p)=|ψ(p)|2
– Measurable in knock-out scattering
 Uncertainty principle Not positive definite in
general. But it is in the classical limit!
Wigner Distribution
 Wigner distributions are physical observables
– Real (hermitian)
– Super-observable!
O ( x, p )   dxdpO ( x, p )W ( x, p )
 Many applications
– heavy-ion collisions,
– quantum molecular dynamics,
– signal analysis,
– quantum information,
– optics,
– image processing…
Simple Harmonic Oscillator
N=0
N=5
•Phase-space distribution gives a vivid “classical” picture.
•Non-positive definiteness is the key for quantum interference
Phase-space tomography
 Phase-space distribution (a map) can be
constructed from slices with fixed momentum.
– For small p, the oscillator is at the turning point of the
oscillator potential.
– For large p, the oscillator is at the middle of the
potential
– For every p, we have a topographic picture of the
system which give a much detailed map of the system.
This information cannot be obtained from the densities in
space or momentum alone!
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
Wigner distributions for quarks in
proton
 Wigner operator (X. Ji,PRL91:062001,2003)
 Wigner distribution: “density” for quarks having
position r and 4-momentum k (off-shell)
a la Saches
7-dimensional distributions
No known experiment can measure this!
Custom-made for high-energy processes (I)
 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 6
variables [r,k=(k+ k)].
Mother of all SP distributions!
 Integrating over z, resulting a phase-space distribution
q(x, r k) through which parton saturation at small x is
easy to see.
dxq( x, r , k ) Nc
 4
2
2
d r d k
2
Custom-made for high-energy processes (II)
 Integrating over r, resulting transverse-momentum
dependent (TMD) parton distributions!
q(x, k)
Measurable in semi-inclusive DIS & Drell-Yan &..
A major subject of this kmeeting…

 Integrating over k, resulting a reduced Wigner
distribution
q(x,r)
The above are not related by Fourier transformation!
Wigner parton distributions &
offsprings
Mother Dis. W(r,p)
q(x, r, k)
Reduced
wigner dis
q(x,r)
TMDPD q (x, k)
PDF q(x)
Density ρ(r)
TMD Parton Distribution
 Appear in the processes in which hadron
transverse-momentum is measured, often together
with TMD fragmentation functions.
 The leading-twist ones are classified by Boer,
Mulders, and Tangerman (1996,1998)
– There are 8 of them
q(x, k┴), qT(x, k┴),
ΔqL(x, k┴), ΔqT(x, k┴),
δq(x, k┴), δLq(x, k┴), δTq(x, k┴), δT’q(x, k┴)
Factorization for SIDIS with P┴
 For traditional high-energy process with one hard
scale, inclusive DIS, Drell-Yan, jet
production,…soft divergences typically cancel,
except at the edges of phase-space.
 At present, we have two scales, Q and P┴ (could be
soft). Therefore, besides the collinear
divergences which can be factorized into TMD
parton distributions (not entirely as shown by the
energy-dependence), there are also soft
divergences which can be taken into account by
the soft factor.
X. Ji, F. Yuan, and J. P. Ma, PRD71:034005,2005
Example I
 Vertex corrections
q
p′
k
p
Four possible regions of gluon momentum k:
1) k is collinear to p (parton dis)
2) k is collinear to p′ (fragmentation)
3) k is soft (wilson line)
4) k is hard (pQCD correction)
A general reduced diagram
Factorization theorem
 For semi-inclusive DIS with small pT
~
• Hadron transverse-momentum is generated from
multiple sources.
• The soft factor is universal matrix elements of Wilson
lines and spin-independent.
• One-loop corrections to the hard-factor has been
calculated
Spin-Dependent processes
 Ji, Ma, Yuan, PLB597, 299 (2004);
PRD70:074021(2004)
Reduced Wigner Distributions and
GPDs
 The 4D reduced Wigner distribution f(r,x) is
related to Generalized parton distributions (GPD)
H and E through simple FT,
t= – q2
 ~ qz
H,E depend only on 3 variables. There is a rotational
symmetry in the transverse plane..
What is a GPD?
 A proton matrix element which is a hybrid of
elastic form factor and Feynman distribution
 Distributions depending on
x: fraction of the longitudinal momentum carried
by parton
t=q2: t-channel momentum transfer squared
ξ: skewness parameter (a new variable coming
from selection of a light-cone direction)
Review: M. Diehl, Phys. Rep. 388, 41 (2003)
X. Ji, Ann. Rev. Nucl. Part. Sci. 54, 413 (2004)
Charge and Current Distributions
in Phase-space
 Quark charge distributions at fixed x
 Quark current at fixed x in a spinning nucleon
A GPD or W-Parton Distribution Model

A parametrization which satisfies the following
Boundary Conditions: (A. Belitsky, X. Ji, and F. Yuan,
PRD 69,074014,2004)
– Reproduce measured Feynman distribution
– Reproduce measured form factors
– Polynomiality condition
– Positivity

Refinement
– Lattice QCD
– Experimental data
Imaging quarks at fixed Feynman-x
 For every choice of x, one can use the Wigner
distributions to picture the nucleon in 3-space;
quantum phase-space tomography!
z
by
bx
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.
Impact parameter space distribution
 Obtained by integrating over z, (Soper, Burkardt)
f ( x, b )   dz f ( x, b   r  , z )
 x and b are in different directions and therefore,
there is no quantum mechanical constraint.
– It is a true density
– Momentum density in the z-direction
– Coordinate density in the transverse plane.
QCD-Map: how to obtain it?
 Data
 Parametrizations
 Lattice QCD
Mass distribution
 Gravity plays an important role in cosmos and at
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
Spin sum rule
 One can calculate the total quark (orbital + spin)
contribution to the spin of the proton
 Amount of proton angular momentum carried by
quarks is
Jq 
1
1
Aq  0,  2   Bq  0,  2  
2
2
1
  dxx[ H ( x,  , 0,  2 )  E ( x,  , 0,  2 )
2
Summary
 One of the central goals for 12 GeV upgrade is to
obtain a QCD map of the proton: DNA sequencing
in biology
 TMD parton distributions: semi-inclusive
processes
 Quantum phase-space tomography
– Mass and spin of the proton