1. Single (transverse) Spin Asymmetry & QCD Factorization Single (transverse) Spin Asymmetry & QCD Factorization Xiangdong Ji University of Maryland — Workshop on SSA, BNL, June 1, 2005 —

2. Outline 1.General Remarks 2.DIS/Drell Yan processes 3. p  p  πX & friends 4.Summary

3. Single (transverse) Spin Asymmetry  SSA is a general phenomenon in physics, and it exists so long as there are –A single transverse spin –A mechanism for helicity flip –Initial and/or state interactions  “Ok, SSA is an interesting phenomenon, but what do you learn about QCD from it?” or “Why do we have to spend $ & time to measure it?” We have some models that fit the data (who cares about models? we have QCD)We have some models that fit the data (who cares about models? we have QCD) We learn something about the nucleon spin structure (what exactly do you learn? And why that is interesting? can you check it in lattice QCD? What is it missing if we don’t measure it?)We learn something about the nucleon spin structure (what exactly do you learn? And why that is interesting? can you check it in lattice QCD? What is it missing if we don’t measure it?) ….….

4. Pertubative & Nonperturbative Mechanisms  In general, however, the physics mechanism for SSA in strong interactions can be either be perturbative & non-perturbative, –pp  to pp at low energy: non-perturbative –What one would like to understand is the SSA in perturbative region=> we hope to learn something simple, maybe! There must be some hard momentum: Perturbative description of the cross section must be valid  FactorizationFactorization A good description of spin-averaged cross sectionsA good description of spin-averaged cross sections

5. SSA & processes DIS & Drell-Yan  p  p -> πX & friends Hard scale Q2Q2 PTPT Small P T ~Λ QCD QCD factorization In TMD’s Non-perturbative Q 2,s» P T » Λ QCD QCD factorization In TMD’s Twist-3 effects QCD factorization In TMD’s ? Twist-3 effects

6. SIDIS at low p T  Single-jet production  If the target is transversely polarized, the current jet with a transverse momentum k T has a SSA which allows a QCD factorization theorem even when k T is on the order Λ QCD  The SSA is of order 1 in the scaling limit, i.e. a twist-2 effect! q k’ PP k X

7. Factorization for SIDIS with P ┴  Must consider generic Feynman diagrams with partons having transverse momentum, and gluon loops.  We have two observable scales, Q and P ┴ (soft). We consider leading order effects in P ┴ /Q.  The gluons can be hard, soft and collinear. Can one absorb these contributions into different factors in the cross sections. –X. Ji, F. Yuan, and J. P. Ma, PRD71:034005,2005

8. Example at one-loop  Vertex corrections Four possible regions of gluon momentum k: 1) k is collinear to p (parton dis) 2) k is collinear to p′ (fragmentation) 3) k is soft (wilson lines) 4) k is hard (pQCD correction) p p′p′ q k

9. A general reduced diagram  Leading contribution in p ┴ /Q.

10. Factorization  Factoring into parton distribution, fragmentation function, and soft factor:

11. TMD parton distributions  The unintegrated parton distributions is defined as where the “light-cone” gauge link is the usual parton distribution may be regarded as the usual parton distribution may be regarded as

12. Classification  The leading-twist ones are classified by Boer, Mulders, and Tangerman (1996,1998) –There are 8 of them, corresponding to the number of quark-quark scattering amplitudes without T-constraint q(x, k ┴ ), q T (x, k ┴ ) (sivers), q(x, k ┴ ), q T (x, k ┴ ) (sivers), Δq L (x, k ┴ ), Δq T (x, k ┴ ), δq(x, k ┴ ), δ L q(x, k ┴ ), δ T q(x, k ┴ ), δ T’ q(x, k ┴ ) –Similarly, one can define fragmentation functions

13. Sivers’ Function  A transverse-momentum-dependent parton distribution which builds in the physics of SSA! S k P The distribution of the parton transverse momentum is not symmetric in azimuth, it has a distribution in S ·(p × k). Since k T is small, the distribution comes from non-perturbative structure physics.

14. Physics of a Sivers Function  Hadron helicity flip –This can be accomplished through non-perturbative mechanics (chiral symmetric breaking) in hadron structure. –The quarks can be in both s and p waves in relativistic quark models (MIT bag).  FSI (phase) –The hadron structure has no FSI phase, therefore Sivers function vanish by time-reversal (Collins, 1993) –FSI can arise from the scattering of jet with background gluon field in the nucleon (collins, 2002) –The resulting gauge link is part of the parton dis.

15. Light-Cone Gauge Pitfalls  It seems that if one choose the light-cone gauge, the gauge link effect disappears. –FSI can be shifted ENTIRELY to the initial state (advanced boundary condition). Hence the FSI effects must come from the LC wave functions. – LCWF components are not real, they have nontrivial phase factors!  A complete gauge-independent TMD PD contains a additional FSI gauge link at ξ ± = ∞ which does not vanish in the light-cone gauge –Conjectured by Ji & Yuan (2002) –Proved by Belitsky, Ji & Yuan (2002)

16. The extra FSI gauge link  Through an explicit calculation, one can show that the standard definition of TMD PD is modified by an additional gauge link – Gauge link arises from the eikonal phase accumulation of final state particle traveling in its trajectory. Although the dominant phase accumulation is in the light-cone direction, however, the phase accumulation happens also in the transverse direction.

17. SSA in A Simple Model  A proton consists of a scalar diquark and a quark, interacting through U(1) gauge boson (Brodsky, Hwang, and Schmidt, PLB, 2002).  The parton distribution asymmetry can be obtained from calculating Sivers’ function in light- cone gauge (Ji & Yuan)

18. Factorization theorem  For semi-inclusive DIS with small p T ~ Hadron transverse-momentum is generated from multiple sources. The soft factor is universal matrix elements of Wilson lines and spin-independent. One-loop corrections to the hard-factor has been calculated

19. Spin-Dependent processes  Ji, Ma, Yuan, PLB597, 299 (2004); PR  Ji, Ma, Yuan, PLB597, 299 (2004); PRD70:074021(2004)

20. Additional Structure Functions Sivers effect Collins effect

21. As P T becomes large…  If P T become hard (  If P T become hard (P T » Λ QCD ), so long as Q» P T the above factorization formula still works!  On the other hand, in this region one can calculate the P T dependence perturbatively, –The p T dependence in the soft factor is easily to calculate.. –Expanding in parton momentum, one leads to the following

22. As P T becomes large… –The p T dependence in the TMDs can also be calculated through one-gluon exchange…  The soft matrix element is the twist-3 matrix elements T D

23. Putting all together  One should obtain a SSA calculated in Qiu- Sterman approach (H. Eguchi & Y. Koike) Therefore, SSA becomes twist-3, JI, Ma, Yuan (to be published)

24. Relation between TMDs & Twist-3?  The TMD approach for DIS/DY works for both small and perturbative, but moderate P T. –At small P T, it is a twist-two effect –At moderate P T, it is a twist-three effect.  The TMD approach is more general, but not necessary at moderate P T  The twist-3 approach works only at large P T, but is the most economical there!

25. SSA & processes DIS & Drell-Yan  p  p -> πX & friends Hard scale Q2Q2 PTPT Small P T ~Λ QCD QCD factorization In TMD’s Non-perturbative Q 2,s» P T » Λ QCD QCD factorization In TMD’s Twist-3 effects QCD factorization In TMD’s ? Twist-3 effects

26. p  p  πX & friends  P T must be large so that perturbative QCD works.  In this region, it is not need to use the TMD formalism. The twist-3 approach is sufficient.  Phases are generated perturbatively.

27. Perturbative Way to Generate Phase Some propagators in the tree diagrams go on-shell No loop is needed to generate the phase! Coulomb gluon Efremov & Teryaev: 1982 & 1984 Qiu & Sterman: 1991 & 1999

28. A possible exception  Is it possible that at moderate p T, the intrinsic transverse-momentum effect is so large that it cannot be expanded? –Soft function is still perturbative... –One could include the Sudakov form factors  I don’t know yet an argument to rule this out. However, I don’t know an example where this is true.  Difficulty: – No proof of factorization (may be it will work!) – The gauge links on the TMDs might be very complicated (both initial and final state interactions are present).

29. Conclusion  For SIDIS/DY with small and moderate transverse momentum, there is a QCD factorization theorems involving TMDs.  At moderate P ┴, one recovers the twist-3 mechanism (ETQS).  For pp->πX at perturbative P ┴, twist-3 mechanism seems to be complete.  One has yet to find a TMD type of factorization for pp->πX at perturbative P ┴ ; and the TMD distributions might not be related to those in SIDIS/DY.