PQCD mechanisms for single (transverse) spin asymmetry in Drell-Yan production

1. 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 • Introduction to single spin asymmetry (SSA) and pQCD mechanisms • SSA at low-qT and transverse-momentum-dependent (TMD) parton distributions • SSA at large-qT and twist-3 mechanism • Conclusion

3. x z y 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 k p p’ S The cross section can have a term depending on the azimuthal angle of k which produce an asymmetry AN when S flips: SSA

4. Why Does SSA Exist? • SSA is proportional to Im (FN * FF) where FN : the normal helicity amplitude and FF : 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 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 factorization • A 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 mq. • To generate a phase difference, one has to have pQCD loop diagrams, proportional to αS. Therefore the model generates AN ~ αS mq/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 ΛQCD. • Transverse-momentum-dependent (TMD) parton distribution (Sivers function) non-perturbative generation of ISI or FSI phases a twist-2 effect: no 1/Q suppression

8. SSA & Processes pp -> πX & friends DIS & Drell-Yan Hard scale Q2 PT QCD factorization In TMD’s Small PT~ΛQCD Non-perturbative Twist-3 effects Twist-3 effects PT»ΛQCD

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 • Large number of events at small pT ~ few hundred MeV … soft-gluon radiation is suppressed test transition between twist-2 and twist-3 SSA.

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

11. Small qT Drell-Yan Pair

12. TMD Parton Distributions • When qt 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┴), qT(x, k┴) (sivers), ΔqL(x, k┴), ΔqT(x, k┴), δq(x, k┴),δqL (x, k┴),δqT (x, k┴), δqT’(x, k┴) • Similarly, one can define fragmentation functions

14. Sivers function • A transverse-momentum-dependent parton distribution which builds in the physics of SSA! k P S The distribution of the parton transverse momentum is not symmetric in azimuth, it has a distribution in S ·(p × k). Since kT 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 PT) Sivers function fit from Vogelsang, Yuan, Phys.Rev.D72:054028,2005

21. Large qT Drell-Yan pair

22. Large qT 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 qt Coulomb gluon Some propagators in the tree diagrams go on-shell No loop is needed to generate the phase! 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 ΛQCD « qT « Q • Twist-3 approach should work because qT is large compared to ΛQCD • TMD QCD factorization should work because qT is much smaller than Q2

28. Twist-three approach at qT « Q • The twist-3 approach works at large, perturbative qT, even when qT « Q

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

30. As qT becomes large… • One can calculate the qT dependence perturbatively, • The pT dependence in the soft factor is easily to calculate.. • Expanding in parton momentum, one leads to the following

31. As qT becomes large… • The qT dependence in the TMDs can also be calculated through one-gluon exchange… • The soft matrix element is the twist-3 matrix elements TF

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) ? • 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 qT, one can learn about the new TMD parton distribution---the Sivers function---correlation of quark momentum distribution with the proton polarization. • At large qT, 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.