RSC Meeting, RIKEN

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Unique Description for Single Transverse Spin Asymmetries

Feng Yuan

RBRC, Brookhaven National Laboratory

Collaborators: Kouvaris, Ji, Qiu, Vogelsang

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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

z

x

y

S?

P P’

K?

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Why Does SSA Exist?

Single Spin Asymmetry requiresHelicity 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

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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 αS mq/Q

We have to go beyond the naïve parton model to understand the large SSAs observed in hadronic reactions

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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, Pijman, 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, 99-02

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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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Connecting these Two

Relations at the matrix elements level

TF(x,x)=s d2k |k|2 qT(x,k)

Boer, Mulders, Pijman 03 Attempts to connect these two in some

processes, no definite conclusion:Ma, Wang, 03Bacchetta, 05

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, PLB638,178;PRD73,094017; PRL97, 082002, 2006)

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Recall the TMD Factorization

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SIDIS: at Large P?

When q?ÀQCD, 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

Twist-3 quark-gluon Correlation: TF(x1,x2)

Fragmentation function:\hat q(x’)

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 Vogelsang’s talk,

Kouvaris-Qiu-Vogelsang-

Yuan, hep-ph/0609238

not corrected for acceptance and smearing

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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 Nonzero leading order in a model-

independent way, for example

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At this order, the asymmetry can be related to that in DIS

qTDIS--- Sivers function from DIS

q?--- imbalance of the dijet

Hsivers depends on subprocess (see Bomhof’s talk)

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Predictions for dijet-correlation AN

VY05: Vogelsang-Yuan, Phys.Rev.D72:054028,2005 Sivers functions fit to the HERMES data

Model I: uT(1/2)/u=-0.81x(1-x),dT

(1/2)/u=1.86x(1-x) Model II: uT

(1/2)/u=-0.75x(1-x),dT(1/2)/d=2.76x(1-x)

Caution: Sudakov effects may reduce the asymmetry, Boer-Vogelsang 04

VY05

Modified Initial/final

u-quark

d-quark

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Global Picture for SSAs

TMD

Quark-gluon correlation

SIDIS

Drell-Yan

Gluon SiversCharmonium

Dijet-Corr.

Single Inclusiveat RHIC

Dijet at RHIC

Large q?

Drell-Yan

Large P?

SIDIS

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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