Single (transverse) Spin Asymmetry

1
Single (transverse) Spin Asymmetry QCD
Factorization

• Xiangdong Ji
• University of Maryland

Workshop on SSA, BNL, June 1, 2005
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Outline

1. General Remarks
2. DIS/Drell Yan processes
3. p?p ? pX friends
4. Summary

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Single (transverse) Spin Asymmetry

• SSA is a general phenomenon in physics, and it
  exists so long as there are
• A single transverse spin
• A mechanism for helicity flip
• Initial and/or state interactions
• Ok, SSA is an interesting phenomenon, but what
  do you learn about QCD from it? or Why do we
  have to spend time to measure it?
• We have some models that fit the data (who cares
  about models? we have QCD)
• We learn something about the nucleon spin
  structure (what exactly do you learn? And why
  that is interesting? can you check it in lattice
  QCD? What is it missing if we dont measure it?)
• .

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Pertubative Nonperturbative Mechanisms

• In general, however, the physics mechanism for
  SSA in strong interactions can be either be
  perturbative non-perturbative,
• pp? to pp at low energy non-perturbative
• What one would like to understand is the SSA in
  perturbative regiongt we hope to learn something
  simple, maybe!
• There must be some hard momentum
• Perturbative description of the cross section
  must be valid ?
• Factorization
• A good description of spin-averaged cross
  sections

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SSA processes
p?p -gt pX friends
DIS Drell-Yan
Hard scale
Q2
PT
QCD factorization In TMDs
Small PT?QCD
Non-perturbative
QCD factorization In TMDs ? Twist-3 effects
QCD factorization In TMDs Twist-3 effects
Q2,s PT ?QCD
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SIDIS at low pT

• Single-jet production
• If the target is transversely polarized, the
  current jet with a transverse momentum kT has a
  SSA which allows a QCD factorization theorem even
  when kT is on the order ?QCD
• The SSA is of order 1 in the scaling limit, i.e.
  a twist-2 effect!

q
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Factorization for SIDIS with P-

• Must consider generic Feynman diagrams with
  partons having transverse momentum, and gluon
  loops.
• We have two observable scales, Q and P- (soft).
  We consider leading order effects in P- /Q.
• The gluons can be hard, soft and collinear. Can
  one absorb these contributions into different
  factors in the cross sections.
• X. Ji, F. Yuan, and J. P. Ma, PRD71034005,2005

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Example at one-loop

• 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 lines) 4) k is hard (pQCD correction)
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A general reduced diagram

• Leading contribution in p- /Q.

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Factorization

• Factoring into parton distribution,
  fragmentation function, and soft factor

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TMD parton distributions

• The unintegrated parton distributions is defined
  as
• where the light-cone gauge link is
• the usual parton distribution may be regarded
  as

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Classification

• The leading-twist ones are classified by Boer,
  Mulders, and Tangerman (1996,1998)
• There are 8 of them, corresponding to the number
  of quark-quark scattering amplitudes without
  T-constraint
• q(x, k-), qT(x, k-) (sivers),
• ?qL(x, k-), ?qT(x, k-),
• dq(x, k-), dLq(x, k-), dTq(x, k-), dTq(x, k-)
• Similarly, one can define fragmentation functions

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Sivers Function

• A transverse-momentum-dependent parton
  distribution which builds in the physics of SSA!

k
P
S
The distribution of the parton transverse
momentum is not symmetric in azimuth, it has a
distribution in S (p k). Since kT is small,
the distribution comes from non-perturbative
structure physics.
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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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Light-Cone Gauge Pitfalls

• It seems that if one choose the light-cone gauge,
  the gauge link effect disappears.
• FSI can be shifted ENTIRELY to the initial state
  (advanced boundary condition). Hence the FSI
  effects must come from the LC wave functions.
• LCWF components are not real, they have
  nontrivial phase factors!
• A complete gauge-independent TMD PD contains a
  additional FSI gauge link at ? 8 which does
  not vanish in the light-cone gauge
• Conjectured by Ji Yuan (2002)
• Proved by Belitsky, Ji Yuan (2002)

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The extra FSI gauge link

• Through an explicit calculation, one can show
  that the standard definition of TMD PD is
  modified by an additional gauge link
• Gauge link arises from the eikonal phase
  accumulation of final state particle traveling in
  its trajectory. Although the dominant phase
  accumulation is in the light-cone direction,
  however, the phase accumulation happens also in
  the transverse direction.

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SSA in A Simple Model

• A proton consists of a scalar diquark and a
  quark, interacting through U(1) gauge boson
  (Brodsky, Hwang, and Schmidt, PLB, 2002).
• The parton distribution asymmetry can be obtained
  from calculating Sivers function in light-cone
  gauge (Ji Yuan)

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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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Spin-Dependent processes

• Ji, Ma, Yuan, PLB597, 299 (2004)
  PRD70074021(2004)

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Additional Structure Functions
Sivers effect
Collins effect
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As PT becomes large

• If PT become hard (PT ?QCD), so long as Q PT
  the above factorization formula still works!
• On the other hand, in this region one can
  calculate the PT dependence perturbatively,
• The pT dependence in the soft factor is easily to
  calculate..
• Expanding in parton momentum, one leads to the
  following

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As PT becomes large

• The pT dependence in the TMDs can also be
  calculated through one-gluon exchange
• The soft matrix element is the twist-3 matrix
  elements TD

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Putting all together

• One should obtain a SSA calculated in Qiu-Sterman
  approach (H. Eguchi Y. Koike)

Therefore, SSA becomes twist-3, JI, Ma, Yuan (to
be published)
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Relation between TMDs Twist-3?

• The TMD approach for DIS/DY works for both small
  and perturbative, but moderate PT.
• At small PT, it is a twist-two effect
• At moderate PT, it is a twist-three effect.
• The TMD approach is more general, but not
  necessary at moderate PT
• The twist-3 approach works only at large PT, but
  is the most economical there!

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SSA processes
p?p -gt pX friends
DIS Drell-Yan
Hard scale
Q2
PT
QCD factorization In TMDs
Small PT?QCD
Non-perturbative
QCD factorization In TMDs ? Twist-3 effects
QCD factorization In TMDs Twist-3 effects
Q2,s PT ?QCD
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p?p ? pX friends

• PT must be large so that perturbative QCD works.
• In this region, it is not need to use the TMD
  formalism. The twist-3 approach is sufficient.
• Phases are generated perturbatively.

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Perturbative 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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A possible exception

• Is it possible that at moderate pT, the
  intrinsic transverse-momentum effect is so large
  that it cannot be expanded?
• Soft function is still perturbative...
• One could include the Sudakov form factors
• I dont know yet an argument to rule this out.
  However, I dont know an example where this is
  true.
• Difficulty
• No proof of factorization (may be it will work!)
• The gauge links on the TMDs might be very
  complicated (both initial and final state
  interactions are present).

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Conclusion

• For SIDIS/DY with small and moderate transverse
  momentum, there is a QCD factorization theorems
  involving TMDs.
• At moderate P-, one recovers the twist-3
  mechanism (ETQS).
• For pp-gtpX at perturbative P-, twist-3 mechanism
  seems to be complete.
• One has yet to find a TMD type of factorization
  for pp-gtpX at perturbative P- and the TMD
  distributions might not be related to those in
  SIDIS/DY.