Single Transverse Spin at RHIC II

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Single Transverse Spin at RHIC II

• Feng Yuan
• RBRC , Brookhaven National Laboratory

Collaborators Werner Vogelsang, Jianwei Qiu
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Outline

• Introduction
• More RBRC Workshop on SSA (June 1-3)
  http//quark.phy.bnl.gov/fyuan/workshop/summer05_
  program.htm
• SSAs at RHIC TMDs
• Twist-three mechanism (Qiu-Sterman)
• Emergence of these two mechanisms
• Summary

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Ultimate goal spin sum rule
Gluon spin
Orbital Ang. Momen
Quark spin
Transverse Spin
Longitudinal Spin
DVCS, FF,
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What is Single Spin Asymmetry?

• Consider scattering of a transversely-polarized
  spin-1/2 hadron (S, p) with another hadron,
  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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Examples SSAs at RHIC
STAR
PHENIX
Central rapidity!!
BRAHMS preliminary
See also previous fixed target
experiments HERMES and COMPASS
DIS experiments
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Naïve Parton Model Fails

• 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
• Two ways to generate sizable SSAs
• Transverse momentum dependent (TMD) parton
  distributions
• Twist-three mechanism (Qiu-Sterman)

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Why Transverse Momentum is Relevant

• Because we want to have nonzero quark orbital
  angular momentum (OAM) to flip the hadron
  helicity
• Which means the quark not only moves along the
  longitudinal direction, but also the transverse
  direction
• -- We have to consider the nonzero transverse
  momentum of the quark in nucleon for these
  processes

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Why Twist-Three?

• A collinear gluon carries one unit of angular
  momentum because of its spin. Therefore, one can
  have a coherent gluon interaction

-1
1/2
1/2
-1/2
1/2
Quark-gluon quark correlation function! Qiu-Sterma
n Mechanism
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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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SSAs from TMDs

• Semi-inclusive DIS
• Drell-Yan process at RHIC
• Jet-correlation at RHIC

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SSAs in DIS

• Semi-inclusive DIS (Vogelsang, Yuan, 05)
• See also Efremov, et al., 04, 05
  Anselmino, et al., 05

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• COMPASS

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TMDs at RHIC

• Drell-Yan
• SSA for Drell-Yan Sivers function has
  opposite sign, qTDY-qTDIS, because of the gauge
  link changing direction.
• Di-Jet Correlation
• There is no factorization proof yet. It is
  likely factorizable in terms of TMDs. However,
  the universality of Sivers function for this case
  is not clear yet. We assume they are the same as
  DY.

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SSA for Drell-Yan
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Asym. Jet correlation probe Gluon Sivers function
at RHIC
Boer, Vogelsang, PRD69094025,2004
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Di-jet Correlation
?2
?1
Jet2 P2?
Jet1 P1?
qT(1)(x)
qT(1/2)(x)
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cos? Asymmetry
Gluon Sivers
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Twist-Three SSAs at RHIC

• We are working on several projects, which will
  bring the twist-three studies from a model
  perspective to a QCD theory for SSAs at hadron
  colliders
• Including plenty processes
• Drell-Yan
• Inclusive hadron/jet (adding the nonderivative
  terms
• Di-jet/di-hadron

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Whats the difference between inclusive jet and
di-jet

• The unpolarized scattering amplitudes are the
  same for these two cases, e.g., qq?qq
• However, the single transverse polarized
  amplitudes are not the same because the detailed
  calculations show that they depend on the other
  jet is observed or not
• Which means that these two observables will
  provide cross check for the theory, especially at
  dedicated RHIC II program

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Single inclusive lepton and lepton pair

• Will also be different observables
• A next-to-leading order QCD correction is in
  progress for inclusive lepton process for single
  transverse polarized scattering
• We will be able to, at the first time, study the
  evolution of the Qiu-Sterman matrix element, and
  the scaling violation for SSA
• Eventually, all these observables can be studied
  rigorously in QCD at NLO, which will provide the
  same solid foundation for transverse spin as that
  for longitudinal spin

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P? dependence of DY(Emergence of the two
mechanisms)

• At low P?, the non-perturbative TMD Sivers
  function will be responsible for its SSA
• At large P? Q, twist-three mechanism will be
  dominant, while the resummation should also be
  considered
• When P? Q, purely twist-3 contributions
• An important issue, at P? Q, these two should
  emerge, showing consistence of the theory

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Twist-3 diagrams for DY
Gluonic Pole
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Cross sections

• Unpolarized cross section
• Polarized cross section, e.g., the derivative
  term

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• Expanding at q? Q,
• xQ/sqrtS ey
• Which should be reproduced by the Sivers function
  at the same kinematical limit, by the
  factorization

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Sivers function calculated from twist-three

• The derivative term for the Sivers function,

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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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• On top of the above formula, a QCD resummation
  can also be performed for the polarized cross
  section
• Using spin-dependent Collins-Soper equation
  (Idilbi et al., PRD 2003)

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Transition from Perturbative region to
Nonperturbative region?

• Compare different region of P?

Nonperturbative TMD
Perturbative region
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Summary

• RHIC II is a very dedicated place to study
  transverse spin physics. Many channels can be
  measured and investigated in much details
• From these studies, we can better understand the
  QCD dynamics and the nucleon structure
• One example (DY) demonstrated the emergence of
  the two mechanisms generating the SSAs