Can We Learn Quark Orbital Motion from SSAs?

Feb. 24-26, 2006 Workshop on Parton Orbital Angular Momentum

1

Can We Learn Quark Orbital Motion from

SSAs?Feng Yuan

RIKEN/BNL Research CenterBrookhaven National

Laboratory

Feb. 24-26, 2006 2Workshop on Parton Orbital Angular Momentum

Outline Why naïve parton model fails

for SSAs Two mechanisms: Sivers and

twist-3 Unifying these two What we learn from SSA? Summary


Statistics: Big SSA! SSA! Systematics

AN is significant in the fragmentation region of the polarized beam: Valence feature

AANN and its sign show a strong and its sign show a strong dependence on the type of polarized dependence on the type of polarized beam and produced particles: beam and produced particles: Flavor Flavor dependencedependence

Why Does SSA Exist?Why Does SSA Exist? Single Spin Asymmetry is proportional to

Im (MN * MF)

where MN is the normal helicity amplitude

and MF is a spin flip amplitude

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) formally violate naïve time-reversal

invariance

Naïve Parton Model FailsNaïve 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) The only way to generate the hadron helicity-flip is

through quark helicity flip, which is proportional to current quark mass mq

To generate a phase difference, one has to have pQCD loop diagrams, proportional to αS

Therefore a generic pQCD prediction goes like AN ~ αS mq/Q

Every factor suppresses the SSA!


Beyond the Naïve Parton Beyond the Naïve Parton ModelModel

Transverse Momentum Dependent Parton Transverse Momentum Dependent Parton DistributionsDistributions Sivers function, Sivers 90Sivers function, Sivers 90 Collins function, Collins 93Collins function, Collins 93 Brodsky, Hwang, Schmidt 02Brodsky, Hwang, Schmidt 02

Collins 02 Collins 02

Belitsky, Ji, Yuan 02Belitsky, Ji, Yuan 02 Twist-three CorrelationsTwist-three Correlations

Efremov-Teryaev, 82, 84Efremov-Teryaev, 82, 84 Qiu-Sterman, 91,98Qiu-Sterman, 91,98


Parton Orbital Angular Parton Orbital Angular Momentum Momentum

and Gluon Spinand Gluon Spin The hadron helicity flip can be

generated by other mechanism in QCD Quark orbital angular momentum

(OAM): Therefore, the hadron helicity flip can occur without requiring the quark helicity flip.

1/2 −1/2

1/2 1/2−1

Beyond the naïve parton model in which quarks are collinear


Parton OAM and Gluons Parton OAM and Gluons (cont.)(cont.)

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

1/2 −1/2

1/2 1/2

Quark-gluon quark correlation function!

-1

Efremov & Teryaev: 1982 & 1984Qiu & Sterman: 1991 & 1999


Where are the PhasesWhere are the Phases

TMD: the TMD: the factorizable final factorizable final state interactions state interactions --- the gauge link in --- the gauge link in the definition of the definition of the TMDsthe TMDs

Twist-three quark-Twist-three quark-gluon correlation: gluon correlation: poles from the poles from the hard scattering hard scattering amplitudesamplitudes

Brodsky, Hwang, Schmidt, 02Collins, 02Ji, Belitsky, Yuan, 02

Efremov & Teryaev: 1982 & 1984Qiu & Sterman: 1991 & 1999


Unifying the Two Unifying the Two Mechanisms Mechanisms

(P(P?? dependence of DY) dependence of DY) At low PAt low P??, the non-perturbative TMD Sivers , the non-perturbative TMD Sivers

function will be responsible for its SSAfunction will be responsible for its SSA When PWhen P??»» Q, purely twist-3 contributions Q, purely twist-3 contributions

For intermediate PFor intermediate P??, , QCDQCD¿¿ P P??¿¿ Q, we should Q, we should see the transition between these twosee the transition between these two

An important issue, at PAn important issue, at P??¿¿ Q, these two Q, these two should emerge, showing consistence of the should emerge, showing consistence of the theorytheory

(Ji, Qiu, Vogelsang, Yuan, to appear)


A General Diagram in A General Diagram in Twist-3 Twist-3

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

Antiquark distribution:\bar q(x’)

Collinear Factorization:

Qiu,Sterman, 91


Soft and Hard PolesSoft and Hard Poles

Soft: xSoft: xgg=0=0

Hard: xHard: xgg= 0 = 0


Diagrams from Soft Diagrams from Soft PolesPoles


Diagrams from Hard Diagrams from Hard PolesPoles


Cross sectionsCross sections

Unpolarized cross sectionUnpolarized cross section

Polarized cross section, Polarized cross section,


Low qLow q?? limit limit

Keeping the leading order of qKeeping the leading order of q??/Q,/Q,

Which should be reproduced by the Sivers Which should be reproduced by the Sivers function at the same kinematical limit, by function at the same kinematical limit, by the factorizationthe factorization


TMD FactorizationTMD Factorization

When qWhen q??¿¿ Q, a TMD factorization holds, Q, a TMD factorization holds,

When qWhen q??ÀÀQCDQCD, all distributions and soft , all distributions and soft factor can be calculated from pQCD, by factor can be calculated from pQCD, by radiating a hard gluonradiating a hard gluon


TMD AntiquarkTMD Antiquark at k at k??

ÀÀQCDQCD

See, e.g., Ji, Ma, Yuan, 04


Soft FacotorSoft Facotor


Sivers Function from twist-Sivers Function from twist-3: 3:

soft polessoft poles


Hard Poles for Sivers Hard Poles for Sivers FunctionFunction


Sivers Function at Large Sivers Function at Large kk??

1/k1/k??4 4 follows a power countingfollows a power counting

Plugging this into the factorization Plugging this into the factorization formula, we indeed reproduce the formula, we indeed reproduce the polarized cross section calculated polarized cross section calculated from twist-3 correlationfrom twist-3 correlation


Factorization ArgumentsFactorization Arguments

Reduced diagrams for different regions of the gluon momentum:along P direction, P’, and soft Collins-Soper 81


Final ResultsFinal Results

PP?? dependence dependence

Which is valid for all PWhich is valid for all P?? range range

Sivers function at low P? Qiu-Sterman Twist-three


Transition from Transition from Perturbative region to Perturbative region to

Nonperturbative region?Nonperturbative region? Compare different region of PCompare different region of P??

Nonperturbative TMD Perturbative region

Feb. 24-26, 2006 Workshop on Parton Orbital Angular Momentum

26

What do we learn from What do we learn from SSA?SSA?


Nonzero Sivers function Nonzero Sivers function impliesimplies

Nonzero Quark Orbital Angular Momentum

e.g, Siver’s function ~ the wave function amplitude with orbital angular momentum! Vanishes if quarks only in s-state!

Friends: Pauli Form Factor F2(t) Spin-dependent structure function g2(x) Generalized Parton Distribution E(x, ξ, t)


LLzz≠0 Amplitude and ≠0 Amplitude and Sivers FunctionSivers Function

All distributions can be calculated using the wave function. The amplitudes are not real because of FSI. Siver’s function:

Similar expressions for F2(Q), g2(x) and E(x,t))

Ji, Ma, Yuan, Nucl. Phys. B (2003)Ji, Ma, Yuan, Nucl. Phys. B (2003)

Lz=0 Lz=1


Concluding RemarksConcluding Remarks

Nonzero Sivers function indeed Nonzero Sivers function indeed indicates the existence of the Quark indicates the existence of the Quark Orbital Angular MomentumOrbital Angular Momentum

However, there is no definite However, there is no definite relation between these two so farrelation between these two so far

We, as theorists, need to work hard We, as theorists, need to work hard for that goal, as asked by for that goal, as asked by experimentalistsexperimentalists

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Can We Learn Quark Orbital Motion from SSAs?

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