Unique Description for SSAs in DIS and Hadronic Collisions

1
Unique Description for SSAs in DIS and Hadronic
Collisions

• Feng Yuan
• RBRC, Brookhaven National Laboratory

Collaborators Kouvaris, Ji, Qiu, Vogelsang
2
What is Single Spin Asymmetry?

• Scattering a transverse spin polarized proton on
  unpolarized target (another hadron or a photon)
• the cross section contains a term

K?
S?
P
P
3
Why Does SSA Exist?

• Single Spin Asymmetry requires
• Helicity flip one must have a reaction mechanism
  for the hadron to change its helicity (in a cut
  diagram)
• Final State Interactions (FSI) to generate a
  phase difference between two amplitudes
• The phase difference is needed because the
  structure S (p k) violate the naïve
  time-reversal invariance

4
Naïve Parton Model Fails to Explain Large SSAs

• If the underlying scattering mechanism is hard,
  the naïve parton model generates a very small
  SSA (G. Kane et al, PRL41, 1978)
• It is in general suppressed by aS mq/Q
• We have to go beyond the naïve parton model to
  understand the large SSAs observed in hadronic
  reactions

5
Two Mechanisms in QCD

• Transverse Momentum Dependent (TMD) Parton
  Distributions and Fragmentations
• Sivers function, Sivers 90
• Collins function, Collins 93
• Gauge invariant definition of the TMDs Brodsky,
  Hwang, Schmidt 02 Collins 02 Belitsky, Ji,
  Yuan 02 Boer, Mulders, Pijlman, 03
• The QCD factorization Ji, Ma, Yuan, 04 Collins,
  Metz, 04
• Twist-three Correlations (collinear
  factorization)
• Efremov-Teryaev, 82, 84
• Qiu-Sterman, 91,98
• Kanazawa-Koike, 00-04

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How Do They Contribute?

• TMD the quark orbital angular momentum leads to
  hadron helicity flip
• The factorizable final state interactions --- the
  gauge link provides the phase

• Twist-three the gluon carries spin, flipping
  hadron helicity
• The phase comes from the poles in the hard
  scattering amplitudes

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Global Picture for SSAs
SIDIS
Large P? SIDIS
Drell-Yan
TMD
Gluon Sivers Charmonium
Large q? Drell-Yan
Quark-gluon correlation
Dijet-Corr.
Dijet at RHIC
Single Inclusive at RHIC
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Territories and unification

• Twist-three the single inclusive hadron
  production in pp, require large P?
• TMD low P?, require additional hard scale
  like Q2 in DIS and Drell-Yan, P? Q
• Connecting these two, at the matrix elements
  level
• TF(x,x)s d2k k2 qT(x,k)
• Boer, Mulders, Pijlman 03

Qiu-Sterman
Sivers
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Unifying the Two Mechanisms (P? dependence of
SSAs)

• At low P?, the non-perturbative TMD Sivers
  function will be responsible for its SSA
• When P? Q, purely twist-3 contributions
• For intermediate P?, ?QCD P? Q, we should see
  the transition between these two
• An important issue, at P? Q, these two should
  merge, showing consistence of the theory

(Ji, Qiu, Vogelsang, Yuan, PRL97,
082002PRD73,094017

PLB638,178, 2006)
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Recall the TMD Factorization
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SIDIS at Large P?

• When q?À?QCD, the Pt dependence of the TMD parton
  distribution and fragmentation functions can be
  calculated from pQCD, because of hard gluon
  radiation
• Single Spin Asymmetry at large P? is not
  suppressed by 1/Q, but by 1/P?

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SSA in the Twist-3 approach
Fragmentation function \hat q(x)
Twist-3 quark-gluon Correlation TF(x1,x2)
Collinear Factorization
Qiu,Sterman, 91
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Factorization guidelines
Reduced diagrams for different regions of the
gluon momentum along P direction, P, and soft
Collins-Soper 81
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Final Results

• P? dependence
• Which is valid for all P? range

Sivers function at low P?
Qiu-Sterman Twist-three
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Transition from Perturbative region to
Nonperturbative region?

• Compare different region of P?

Nonperturbative TMD
Perturbative region
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More over

• At low P? Q, the second term vanishes, use the
  Sivers function only
• P?-moment of the asymmetry can be calculated from
  twist-three matrix element
• In SIDIS, for the Sivers asymmetry

Boer-Mulders-Tangelmann, 96,98
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Compare to the HERMES data

• TF from the fit to single inclusive hadron
  asymmetry data (fixed target RHIC)
• See Vogelsangs talk, Kouvaris-Qiu-Vogelsang-Yuan,
  hep-ph/0609238
• Indicate the consistency of SSAs in DIS and
  hadron collisions

not corrected for acceptance and smearing
See also, Efremov, et al., PLB612,233 (20005)
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Dijet-correlation at RHIC

• Proposed by Boer-Vogelsang
• Initial state and/or final state interactions?
• Bacchetta-Bomhof-Mulders-Pijlman,04-06
• We can illustrate the initial and final state
  interactions in a model-independent way, at
  nonzero leading order, for example

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At this order, the asymmetry can be related to
that in DIS, in leading power of q?/P?

• qTDIS--- Sivers function from DIS
• q?--- imbalance of the dijet
• Hsivers depends on subprocess (see Bomhofs
  talk)

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Predictions for dijet-correlation AN
d-quark
Modified Initial/final
VY05
u-quark

• VY05 Vogelsang-Yuan, Phys.Rev.D72054028,2005
• Sivers functions fit to the HERMES data
• Model II uT(1/2)/u-0.75x(1-x),dT(1/2)/d2.76x(1-
  x)
• Cautions Scale difference, HERMES ltQ2gt2GeV2
• Sudakov effects may affect the asymmetry,
  Boer-Vogelsang 04

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Global Picture for SSAs
SIDIS
Large P? SIDIS
Drell-Yan
TMD
Gluon Sivers Charmonium
Large q? Drell-Yan
Quark-gluon correlation
Dijet-Corr.
Dijet at RHIC
Single Inclusive at RHIC
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Summary

• We are in the early stages of a very exciting era
  of transverse spin physics studies
• Many data are coming out, and new experiments are
  proposed to provide a detailed understanding of
  the spin degrees of freedom, especially for the
  quark orbital motion
• We will learn more about nucleon structure from
  these studies