April 21, 2006 DIS2006, KEK 1

Unifying the Mechanisms for SSAs

in Hard ProcessFeng Yuan

RIKEN/BNL Research CenterBrookhaven National

Laboratory

April 21, 2006 2DIS2006, KEK

Outline Naïve parton model fails for

large SSAs Two mechanisms: Sivers and

twist-3 Unifying these two Summary

April 21, 2006 3DIS2006, KEK

Naïve Parton Model Fails to Naïve Parton Model Fails to

Explain Large SSAsExplain 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

April 21, 2006 4DIS2006, KEK

Two Mechanisms in QCDTwo Mechanisms in QCD

Transverse Momentum Dependent Parton Transverse Momentum Dependent Parton DistributionsDistributions Sivers function, Sivers 90Sivers function, Sivers 90 Collins function, Collins 93Collins function, Collins 93 Gauge invariant definition of the TMDs: Brodsky, Gauge invariant definition of the TMDs: Brodsky,

Hwang, Schmidt 02; Collins 02 ; Belitsky, Ji, Yuan Hwang, Schmidt 02; Collins 02 ; Belitsky, Ji, Yuan 02; Boer, Mulders, Pijman, 0302; Boer, Mulders, Pijman, 03

The QCD factorization: Ji, Ma, Yuan, 04; Collins, The QCD factorization: Ji, Ma, Yuan, 04; Collins, Metz, 04Metz, 04

Twist-three Correlations (collinear Twist-three Correlations (collinear factorization)factorization) Efremov-Teryaev, 82, 84Efremov-Teryaev, 82, 84 Qiu-Sterman, 91,98Qiu-Sterman, 91,98

April 21, 2006 5DIS2006, KEK

Previous Attempts to Connect Previous Attempts to Connect these Twothese Two

Relations at the matrix elements levelRelations at the matrix elements level

TTFF(x,x)(x,x)==ss d d22k |k|k |k|22 qqTT(x,k)(x,k)

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

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

Qiu-Sterman Sivers

April 21, 2006 6DIS2006, KEK

Unifying the Two Unifying the Two Mechanisms Mechanisms

(P(P?? dependence of SSAs) dependence of SSAs) 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 merge, showing consistence of the should merge, showing consistence of the theorytheory

(Ji, Qiu, Vogelsang, Yuan, hep-ph/0602239, 4023: Drell-Yan hep-ph/0604128: SIDIS)

April 21, 2006 7DIS2006, KEK

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

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

Fragmentation function:\hat q(x’)

Collinear Factorization:

Qiu,Sterman, 91

April 21, 2006 8DIS2006, KEK

The TMD FactorizationThe TMD Factorization

April 21, 2006 9DIS2006, KEK

Factorization guidelinesFactorization guidelines

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

April 21, 2006 10DIS2006, KEK

Detailed Calculations:Detailed Calculations:Soft and Hard PolesSoft and Hard Poles

Soft: xSoft: xgg=0=0

Hard: xHard: xgg= 0 = 0

April 21, 2006 11DIS2006, KEK

Cross sectionsCross sections

Summing all diagrams, Summing all diagrams,

For PFor P??»» Q, many other terms Q, many other terms contribute, see Eguchi, Koike, Tanaka, contribute, see Eguchi, Koike, Tanaka, hep-ph/0604003hep-ph/0604003

April 21, 2006 12DIS2006, KEK

PPhh??¿¿Q limitQ limit

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

Hard-pole contributions play very Hard-pole contributions play very important role to get the above results, important role to get the above results, which should be reproduced by the Sivers which should be reproduced by the Sivers function at the same kinematical limit, function at the same kinematical limit, from the TMD factorizationfrom the TMD factorization

April 21, 2006 13DIS2006, KEK

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

April 21, 2006 14DIS2006, KEK

Fragmentation functionFragmentation function at at pp??ÀÀQCDQCD

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

April 21, 2006 15DIS2006, KEK

Soft FacotorSoft Facotor

April 21, 2006 16DIS2006, KEK

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

soft polessoft poles

April 21, 2006 17DIS2006, KEK

Hard Poles for Sivers Hard Poles for Sivers FunctionFunction

April 21, 2006 18DIS2006, KEK

Sivers Function at Large Sivers Function at Large kk??

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

Drell-Yan Sivers function has opposite signDrell-Yan Sivers function has opposite sign Plugging this into the factorization Plugging this into the factorization

formula, we indeed reproduce the formula, we indeed reproduce the polarized cross section calculated from polarized cross section calculated from twist-3 correlationtwist-3 correlation

April 21, 2006 19DIS2006, KEK

Concluding RemarksConcluding Remarks

Two mechanisms generating the large SSAs Two mechanisms generating the large SSAs are unified for a physical process and are unified for a physical process and observableobservable

This unification will impose a rigorous This unification will impose a rigorous constraint for phenomenological studies of constraint for phenomenological studies of SSAsSSAs

Extension to other processes, and other Extension to other processes, and other observables are also interested to followobservables are also interested to follow

PPtt scan of SSAs will show the transition from scan of SSAs will show the transition from perturbative region to nonperturbative regionperturbative region to nonperturbative region

April 21, 2006 20DIS2006, KEK

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