1. PQCD mechanisms for single (transverse) spin asymmetry in Drell-Yan production PQCD mechanisms for single (transverse) spin asymmetry in Drell-Yan production Xiangdong Ji University of Maryland — Workshop on hadron structure at J-PARC, KEK Nov. 30, 2005 —

2. Outline 1.Introduction to single spin asymmetry (SSA) and pQCD mechanisms 2.SSA at low-q T and transverse-momentum- dependent (TMD) parton distributions 3.SSA at large-q T and twist-3 mechanism 4.Conclusion

3. An Example of Single Spin Asymmetry An Example of Single Spin Asymmetry  Consider scattering of a transversely-polarized spin-1/2 hadron (S, p) with another hadron, observing a particle of momentum k S k p The cross section can have a term depending on the azimuthal angle of k which produce an asymmetry A N when S flips: SSA p’ z x y

4. Why Does SSA Exist?  SSA is proportional to Im (F N * F F ) Im (F N * F F ) where F N : the normal helicity amplitude and F F : a spin flip amplitude and F F : a spin flip amplitude  Thus its existence requires –Helicity flip : one must have a reaction mechanism for the hadron to change its helicity (in a cut diagram). –Final State Interactions (FSI or ISI): to general a phase difference between two amplitudes. The phase difference is needed because the structure The phase difference is needed because the structure S ·(p × k) formally violate time-reversal invariance. S ·(p × k) formally violate time-reversal invariance.

5. Pertubative & Nonperturbative Mechanisms for SSA  In general, the physics mechanism for SSA in strong interactions can be due to either perturbative or non-perturbative physics –pp  to pp at low energy: non-perturbative  What is interested here is the SSA in perturbative QCD region==> we hope to learn something simple---maybe! –There must be some hard momentums A QCD factorizationA QCD factorization A good description of spin-averaged cross sectionsA good description of spin-averaged cross sections

6. Naïve Parton Model Fails to get large SSA  However, the underlying scattering mechanism cannot be entirely perturbative. 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 m q. –To generate a phase difference, one has to have pQCD loop diagrams, proportional to α S. Therefore the model generates Therefore the model generates A N ~ α S m q /Q less than 0.1 per cent, A N ~ α S m q /Q less than 0.1 per cent, Every factor suppresses the SSA!

7. QCD factorization and large SSA  QCD factorization introduces non-perturbative hadron structure functions which help to enhance the SSA relative to that in parton model –Twist-3 matrix effects (Efremov-Teryaev-Qiu- Sterman) hadron spin-flip through gluons and hence the quark mass is replaced by Λ QCD. hadron spin-flip through gluons and hence the quark mass is replaced by Λ QCD. –Transverse-momentum-dependent (TMD) parton distribution (Sivers function) non-perturbative generation of ISI or FSI phases a twist-2 effect: no 1/Q suppression a twist-2 effect: no 1/Q suppression

8. 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 P T » Λ QCD Twist-3 effects

9. Drell-Yan at J-Parc  Drell-Yan is one of the simplest processes to test the SSA mechanisms because –The process is clean in theory and exp. –There no fragmentation function involved.  J-Parc can to do a good measurement –Look at lepton pairs of invariant mass 2-6 GeV, requiring large-x partons requiring large-x partons –Large number of events at small p T ~ few hundred MeV … soft-gluon radiation is suppressed … soft-gluon radiation is suppressed test transition between twist-2 and twist-3 SSA. test transition between twist-2 and twist-3 SSA.

10. Drell-Yan Cross Section at J-Parcs No. of events is large when q t < 2 GeV and invariant mass of the lepton pair < 2-3 GeV

11. Small q T Drell-Yan Pair

12. TMD Parton Distributions  When q t is small, parton transverse momentum in the proton must be considered. Introduce TMD parton distributions

13. Classification  The leading-twist TMDPD 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 ┴ ), δq L (x, k ┴ ), δq T (x, k ┴ ), δq T’ (x, k ┴ ) –Similarly, one can define fragmentation functions

14. 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.

15. Physics of 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 ISI or FSI phase, therefore Sivers function vanish by time-reversal (Collins, 1993) –FSI can arise from the scattering of parton with background gluon field in the nucleon (collins, 2002) –The resulting gauge link is part of the parton dis.

16. Sivers function 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 (Ji & Yuan)

17. Factorization for Drell-Yan  Must consider generic Feynman diagrams with partons having transverse momentum, and gluon loops.  The gluons can be hard, soft and collinear. Can one absorb these contributions into different factors in the cross sections

18. Drell-Yan Factorization – –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. Factorization for Drell-Yan  For Drell-Yan production

20. SSA for Drell-Yan at JPAC (integrated over P T ) Sivers function fit from Vogelsang, Yuan, Phys.Rev.D72:054028,2005

21. Large q T Drell-Yan pair

22. Large q T DY pair  Must be produced by a hard gluon radiation, which can be calculated in QCD perturbation theory.  Single spin asymmetry can be produced by propagation of partons of unpolarized proton in the spin-dependent gluon field of polarized proton.  The effects of the polarized (electric like) gluon field can be described by a twist-3 matrix element

23. Perturbative way to generate ISI phase at large q t 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

24. Twist-3 mechanism for Drell-Yan Ji, Qiu, Vogelsang, Yuan, to be published

25. Twist-3 Mechanisms for SSA Ji, Qiu, Vogelsang, Yuan, to be published

27. Relation between TMD factorization & twist-3 effect  There is a common kinematic region that both approaches work Λ QCD « q T « Q Λ QCD « q T « Q –Twist-3 approach should work because q T is large compared to Λ QCD –TMD QCD factorization should work because q T is much smaller than Q 2

28. Twist-three approach at q T « Q  The twist-3 approach works at large, perturbative q T, even when q T « Q

29. TMD factorization at Λ QCD «q T «Q  The TMD approach for DIS/DY works for both small and perturbative, but moderate q T. –At small q T, it is a twist-two effect –At moderate q T, SSA goes like 1/q T it is a twist-three effect! it is a twist-three effect!  How to generate a twist-3 effect? Go back to the factorization formula….

30. As q T becomes large…  One can calculate the q T dependence perturbatively, –The p T dependence in the soft factor is easily to calculate.. –Expanding in parton momentum, one leads to the following

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

32. Putting all together  One should obtain a SSA, same as the twist-3 approach…  So far the two results do not agree!  Possible solutions –Current approach of Qiu-Sterman type of calculation must be reconsidered. Break down of factorization at the pole (Glauber contribution) ? Break down of factorization at the pole (Glauber contribution) ? –Transverse-momentum going through hard scattering has been neglected. Is it necessary to pick it back to get full twist-3 effects? Will be resolved soon theoretically

33. Conclusion  Drell-Yan is the cleanest process to study pQCD SSA mechanisms, and J-PARC is an excellent facility to do it.  At small q T, one can learn about the new TMD parton distribution---the Sivers function--- correlation of quark momentum distribution with the proton polarization.  At large q T, one can learn about the twist-3 correlation---the polarization of the color electric field in the polarized nucleon.  In both cases, one can learn a great deal about the spin structure of the proton.