Wigner Functions; PT-Dependent Factorization in SIDIS

1
Wigner Functions PT-Dependent Factorization in
SIDIS

• Xiangdong Ji
• University of Maryland

COMPASS Workshop, Paris, March 1-3, 2004
2
Outline

• Quantum phase-space (Wigner) distributions.
• GPD and phase-space picture of the nucleon.
• Transverse-momentum-dependent (TMD) parton
  distributions.
• Factorization theorem in semi-inclusive DIS.
• Single spin asymmetry transversity and the
  connection between Collins, Sivers and
  Efremov-Teryaev-Sterman-Qiu.

3
Motivation

• Elastic form-factors provide static
  coordinate-space 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 Lrp !

4
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

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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

Short of measuring the wave function, the Wigner
function contains the most complete (one-body)
info about a quantum system.
6
Simple Harmonic Oscillator
N5
N0
Husimi distribution positive definite!
7
Measuring Wigner function of a quantum Light!
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Quarks in the Proton

• Wigner operator
• Wigner distribution density for quarks having
  position r and 4-momentum k? (off-shell)

a la Saches
Ji (2003)
7-dimensional distribtuion
No known experiment can measure this!
9
Custom-made for high-energy processes

• In high-energy processes, one cannot measure k?
  (k0kz) and therefore, one must integrate this
  out.
• The reduced Wigner distribution is a function of
  6 variables r,k(k k?).
• After integrating over r, one gets
  transverse-momentum dependent (TDM) parton
  distributions.
• Alternatively, after integrating over k?, one
  gets a spatial distribution of quarks with fixed
  Feynman momentum k(k0kz)xM.

f(r,x)
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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
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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

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Up-Quark Charge Density at x0.4
z
y
x
13
Up-Quark Charge Denstiy at x0.01
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Up-Quark Density At x0.7
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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.

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TMD Parton Distribution

• Appear in the process 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-),
• ?qL(x, k-), ?qT(x, k-),
• dq(x, k-), dLq(x, k-), dTq(x, k-), dTq(x, k-)

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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
  vA0 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)

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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 d4 before the
  transverse-momentum 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?

19
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.

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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.

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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)

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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)
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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,)
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A general leading region in non-singular gauges
Ph
Ph
J
H
H
s
J
P
P
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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

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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 transverse-momentum.
• Soft-radiation is calculable at large pT

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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

k
p
p
S
The cross section can have a term depending on
the azimuthal angle of k
which produce an asymmetry AN when S flips SSA
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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.

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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.

Beyond the naïve parton model in which quarks are
collinear
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Novel Way to Generate Phase
Coulomb gluon
Some propagators in the tree diagrams go on-shell
No loop is needed to generate the phase!
Efremov Teryaev 1982 1984 Qiu Sterman
1991 1999
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Single Target-Spin Asymmetry in SIDIS

• Observed in HERMES exp.
• At low-Pt, this can be generated from Sivers
  distribution function and Collins fragmentation
  function (twist-2).
• At large-Pt, this can be generated from
  Efremov-Taryaev-Qiu-Sterman (ETQS) effect
  (twist-3).
• Boer, Mulders, and Pijlman (2003) observed that
  the moments of Sivers function is related to the
  twist-3 matrix elements of ETQS.

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Low P- Factorization

• If P- is on the order of the intrinsic
  transverse-momentum of the quarks in the nucleon.
  Then the factorization theorem involved
  un-integrated transversity distribution,

• One can measure the un-integrated transversity
• To get integrated one, one can integrate out P-
• with p- weighted. (soft factor disappears)

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When P- is large

• Soft factor produces most of the
  transverse-momentum, 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.
• Likewise for Sivers effect.
• The SSA is then a twist-three observable.

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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.

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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
  semi-inclusive 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.