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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
Feb. 24-26, 2006 3Workshop on Parton Orbital Angular Momentum
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!
Feb. 24-26, 2006 6Workshop on Parton Orbital Angular Momentum
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
Feb. 24-26, 2006 7Workshop on Parton Orbital Angular Momentum
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
Feb. 24-26, 2006 8Workshop on Parton Orbital Angular Momentum
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
Feb. 24-26, 2006 9Workshop on Parton Orbital Angular Momentum
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
Feb. 24-26, 2006 10Workshop on Parton Orbital Angular Momentum
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)
Feb. 24-26, 2006 11Workshop on Parton Orbital Angular Momentum
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
Feb. 24-26, 2006 12Workshop on Parton Orbital Angular Momentum
Soft and Hard PolesSoft and Hard Poles
Soft: xSoft: xgg=0=0
Hard: xHard: xgg= 0 = 0
Feb. 24-26, 2006 13Workshop on Parton Orbital Angular Momentum
Diagrams from Soft Diagrams from Soft PolesPoles
Feb. 24-26, 2006 14Workshop on Parton Orbital Angular Momentum
Diagrams from Hard Diagrams from Hard PolesPoles
Feb. 24-26, 2006 15Workshop on Parton Orbital Angular Momentum
Cross sectionsCross sections
Unpolarized cross sectionUnpolarized cross section
Polarized cross section, Polarized cross section,
Feb. 24-26, 2006 16Workshop on Parton Orbital Angular Momentum
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
Feb. 24-26, 2006 17Workshop on Parton Orbital Angular Momentum
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
Feb. 24-26, 2006 18Workshop on Parton Orbital Angular Momentum
TMD AntiquarkTMD Antiquark at k at k??
ÀÀQCDQCD
See, e.g., Ji, Ma, Yuan, 04
Feb. 24-26, 2006 19Workshop on Parton Orbital Angular Momentum
Soft FacotorSoft Facotor
Feb. 24-26, 2006 20Workshop on Parton Orbital Angular Momentum
Sivers Function from twist-Sivers Function from twist-3: 3:
soft polessoft poles
Feb. 24-26, 2006 21Workshop on Parton Orbital Angular Momentum
Hard Poles for Sivers Hard Poles for Sivers FunctionFunction
Feb. 24-26, 2006 22Workshop on Parton Orbital Angular Momentum
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
Feb. 24-26, 2006 23Workshop on Parton Orbital Angular Momentum
Factorization ArgumentsFactorization Arguments
Reduced diagrams for different regions of the gluon momentum:along P direction, P’, and soft Collins-Soper 81
Feb. 24-26, 2006 24Workshop on Parton Orbital Angular Momentum
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
Feb. 24-26, 2006 25Workshop on Parton Orbital Angular Momentum
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
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What do we learn from What do we learn from SSA?SSA?
Feb. 24-26, 2006 27Workshop on Parton Orbital Angular Momentum
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)
Feb. 24-26, 2006 28Workshop on Parton Orbital Angular Momentum
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
Feb. 24-26, 2006 29Workshop on Parton Orbital Angular Momentum
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