Transcript Document

Wigner Functions;
PT-Dependent Factorization in SIDIS
Xiangdong Ji
University of Maryland
— COMPASS Workshop, Paris, March 1-3, 2004 —
Outline
1. Quantum phase-space (Wigner) distributions.
2. GPD and phase-space picture of the nucleon.
3. Transverse-momentum-dependent (TMD)
parton distributions.
4. Factorization theorem in semi-inclusive DIS.
5. Single spin asymmetry: transversity and the
connection between Collins, Sivers and EfremovTeryaev-Sterman-Qiu.
Motivation
 Elastic form-factors provide static coordinatespace charge and current distributions (in the
sense of Sachs, for example), but no information
on the dynamical motion.
 Feynman parton densities give momentum-space
distributions of constituents, but no information
of the spatial location of the partons.
 But sometimes, we need to know the position and
momentum of the constituents.
– For example, one need to know r and p to
calculate L=r×p !
Phase-space Distribution?
 The state of a classical particle is specified
completely 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).
 In quantum mechanics, because of the uncertainty
principle, the phase-space information is a
“luxury”, 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 a quantum Light!
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!
Ji (2003)
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
6 variables [r,k=(k+ k)].
1. After integrating over r, one gets
transverse-momentum dependent
(TDM) parton distributions.
2. 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 quarks; 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
A GPD or Wigner Function Model

A parametrization which satisfies the following
Boundary Conditions: (A. Belitsky, X. Ji, and F.
Yuan, hep-ph/0307383, to appear in PRD)
– 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
Up-Quark Charge Denstiy at x=0.01
Up-Quark Density At x=0.7
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 M. Burkardt (2000) and Soper.
TMD Parton Distribution
 Appear in the process in which hadron transversemomentum 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┴)
UV Scale-dependence
 The ultraviolet-scale dependence is very simple. It
obeys an evolution equation depending on the
anomalous dimension of the quark field in the
v·A=0 gauge.
 However, we know the integrated parton
distributions have a complicated scale-dependence
(DGLAP-evolution)
 Additional UV divergences are generated through
integration over transverse-momentum, which
implies that
∫µ d2k┴ q(x, k┴)  q(x,µ)
Consistency of UV Regularization
 Feynman parton distributions are available in the
scheme: dimensional regularization, minimal
subtraction.
 This cannot be implemented for TMD parton
distributions because d=4 before the transversemomentum is integrated.
 On the other hand, it is difficult to implement a
momentum cut-off scheme for gauge theories…
( I love to have one for many other reasons!)
 Therefore, it is highly nontrivial that
∫ d2k┴ q(x, k┴)  Fey. Dis. known from fits?
Gauge Invariance?
 Can be made gauge-independent by inserting a
gauge link going out to infinity in some direction v
(in non-singular gauges).
 In singular gauges, the issue is more complicated
– Ji & Yuan (2003) conjectured a link at infinity to
reproduce the SSA in a model by Brodsky et. al.
– Belitsky, Ji & Yuan (2003) derived the gauge link
– Boer, Mulders, and Pijlman (2003): implications for real
processes
 If the link is not along the light-cone (used by
Collins and Soper, and others…). The integration
over k┴ does not recover the usual parton
distribution.
Evolution In Gluon Rapidity
 The transverse momentum of the quarks can be
generated by soft gluon radiation. As the energy
of the nucleon becomes large, more gluon radiation
(larger gluon rapidity) contributes to generate a
fixed transverse-momentum.
 The evolution equation in energy or gluon rapidity
has been derived by Collins and Soper (1981), but
is non-perturbative if k┴, is small.
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 (to be published)
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)
Example II
 Gluon Radiation
q
p′
k
p
The dominating topology is the quark carrying most
of the energy and momentum
1) k is collinear to p (parton dis)
2) k is collinear to p′ (fragmentation)
3) k is soft (Wilson line)
The best-way to handle all these is the soft-collinear
effective field theory… (Bauer, Fleming, Steward,…)
A general leading region in non-singular gauges
Ph
Ph
J
H
H
s
J
P
P
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
Sudakov double logs and soft radiation
 Soft-radiation generates the so-called Sudakov
double logarithms ln2Q2/p2T and makes the
hadrons with small-pT exponentially suppressed.
 Soft-radiation tends to wash out the (transverse)
spin effects at very high-energy, de-coupling the
correlation between spin and transversemomentum.
 Soft-radiation is calculable at large pT
What is a Single Spin Asymmetry (SSA)?
 Consider scattering of a transversely-polarized
spin-1/2 hadron (S, p) with another hadron (or
photon), observing a particle of momentum k
x
k
p
p’
S
z
y
The cross section can have a term depending on the azimuthal
angle of k
  
d ~ S  ( p  k )
which produce an asymmetry AN when S flips: SSA
Why Does SSA Exist?
 Single Spin Asymmetry is proportional to
Im (FN * FF)
where FN is the normal helicity amplitude
and FF is a spin flip amplitude
– Helicity flip: one must have a reaction
mechanism for the hadron to change its helicity
(in a cut diagram).
– Final State Interactions (FSI): to general a
phase difference between two amplitudes.
The phase difference is needed because the
structure
S ·(p × k) formally violate time-reversal
invariance.
Parton Orbital Angular Momentum
and Gluon Spin
 The hadron helicity flip can be generated by other
mechanism in QCD
– Quark orbital angular momentum (OAM): the
quarks have transverse momentum in hadrons.
Therefore, the hadron helicity flip can occur
without requiring the quark helicity flip.
1/2
1/2
1/2−1
−1/2
Beyond the naïve parton model in which quarks are collinear
Novel Way to Generate Phase
Coulomb
gluon
Some propagators in the tree diagrams go on-shell
1
1
2
2

P

i

(
k

m
)
2
2
2
2
k  m  i
k m
No loop is needed to generate the phase!
Efremov & Teryaev: 1982 & 1984
Qiu & Sterman: 1991 & 1999
Single Target-Spin Asymmetry in SIDIS
 Observed in HERMES exp.
k
k’
X
P
 At low-Pt, this can be generated from Siver’s distribution
function and Collins fragmentation function (twist-2).
 At large-Pt, this can be generated from Efremov-TaryaevQiu-Sterman (ETQS) effect (twist-3).
 Boer, Mulders, and Pijlman (2003) observed that the
moments of Siver’s function is related to the twist-3 matrix
elements of ETQS.
Low P┴ Factorization
 If P┴ is on the order of the intrinsic transversemomentum of the quarks in the nucleon. Then the
factorization theorem involved un-integrated
transversity distribution,
1) One can measure the un-integrated transversity
2) To get integrated one, one can integrate out P┴
with p┴ weighted. (soft factor disappears…)
When P┴ is large…
 Soft factor produces most of the transversemomentum, and it can be lumped to hard
contribution.
 The transverse-momentum in the parton
distribution can be integrated over, yielding the
transversity distribution. Or when the momentum
is large, it can be factorized in terms of the
transversity distribution.
 The transverse-momentum in the Collins
fragmentation function can also be integrated out,
yielding the ETSQ twist-three fragmentation
matrix elements. Or when the momentum is large,
it can be factorized in terms of the transversity
distribution.
Physics of a Sivers Function
 Hadron helicity flip
– This can be accomplished through non-perturbative
mechanics (chiral symmetric breaking) in hadron
structure.
– The quarks can be in both s and p waves in relativistic
quark models (MIT bag).
 FSI (phase)
– The hadron structure has no FSI phase, therefore
Sivers function vanish by time-reversal (Collins, 1993)
– FSI can arise from the scattering of jet with background
gluon field in the nucleon (collins, 2002)
– The resulting gauge link is part of the parton dis.
Conclusion
 GPDs are quantum phase-space distributions, and
can be used to visualize 3D quark distributions at
fixed Feynman momentum
 There is now a factorization theorem for semiinclusive hadron production at low pt, which
involves soft gluon effects, allowing study pQCD
corrections systematically.
 According to the theorem, what one learns from
SSA at low pt is unintegrated transversity
distribution.
 At large pt, SSA is a twist-three effects, the
factorization theorem reduces to the result of
Efremov-Teryaev-Qiu-Sterman.