Title: Single Transverse Spin at RHIC II
1Single Transverse Spin at RHIC II
- Feng Yuan
- RBRC , Brookhaven National Laboratory
Collaborators Werner Vogelsang, Jianwei Qiu
2Outline
- 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
3Ultimate goal spin sum rule
Gluon spin
Orbital Ang. Momen
Quark spin
Transverse Spin
Longitudinal Spin
DVCS, FF,
4What 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
5Examples SSAs at RHIC
STAR
PHENIX
Central rapidity!!
BRAHMS preliminary
See also previous fixed target
experiments HERMES and COMPASS
DIS experiments
6Naïve 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)
7Why 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
8Why 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
9Novel 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
10SSAs from TMDs
- Semi-inclusive DIS
- Drell-Yan process at RHIC
- Jet-correlation at RHIC
11SSAs in DIS
- Semi-inclusive DIS (Vogelsang, Yuan, 05)
- See also Efremov, et al., 04, 05
Anselmino, et al., 05
12 13TMDs 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.
14SSA for Drell-Yan
15Asym. Jet correlation probe Gluon Sivers function
at RHIC
Boer, Vogelsang, PRD69094025,2004
16Di-jet Correlation
?2
?1
Jet2 P2?
Jet1 P1?
qT(1)(x)
qT(1/2)(x)
17cos? Asymmetry
Gluon Sivers
18Twist-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
-
19Whats 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
20Single 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
21P? 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
22Twist-3 diagrams for DY
Gluonic Pole
23Cross sections
- Unpolarized cross section
- Polarized cross section, e.g., the derivative
term
24- Expanding at q? Q,
-
- xQ/sqrtS ey
- Which should be reproduced by the Sivers function
at the same kinematical limit, by the
factorization
25Sivers function calculated from twist-three
- The derivative term for the Sivers function,
26Final Results
- P? dependence
- Which is valid for all P? range
Sivers function at low P?
Qiu-Sterman Twist-three
27- 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)
28Transition from Perturbative region to
Nonperturbative region?
- Compare different region of P?
Nonperturbative TMD
Perturbative region
29Summary
- 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