Q T resummation in transversely polarized Drell-Yan process

1. QTresummation in transversely polarized Drell-Yan process Hiroyuki Kawamura (RIKEN) June 16, 2005 InternationalConference on QCD and Hadronic Physics work in common with Jiro Kodaira (KEK) Kazuhiro Tanaka (Juntendo) Hirotaka Shimizu (KEK)

2. Introduction  Spin projects at RHIC pp collider with longitudinal/transeverse polarization 2001 ~ 2004 RUN1~RUN4 s =200 GeV 2005 RUN5 s = 410 GeV − helicity structure of the proton gluon polarization − single transverse spin asymmetry T-odd FF − transverse structure tDY process transversity distribution

3. Transversity distribution Ralston & Soper ‘79 − last unmeasured twist-2 pdf − chiral-odd (not measured in DIS) − relativistic effect − Soffer’s inequality Soffer ‘95 − DGLAP splitting functions 1-loop : Artru & Mukhfi ’90 2-loop : Hayashigaki et.al. ‘97, Kumano&Miyama ‘97, Vogelsang ‘98

4. Transversely Polarized DY process  1-loop corrections − works done so far (1) QT integrated cross section • Massive gluon scheme : Vogelsang & Weber ‘93 • Dimensional reduction : Coutogouris,Kamal,& Merebashvilli ‘94 • Relation between Dim. Reg. & Dim. Red. : Kamal ’96 • → result in Dim. Reg. : Kamal ’98 (2) QT unintegrated cross section • Massive gluon scheme : Vogelsang & Weber ‘93 Direct calculation in D-dim. Transverse double spin asymmetry  cos(2φ) − phase space integral keeping azimuthal angle dependence No direct calculation so far → We calculated (1) & (2) in D-dim. directly and matched (2) to the resummation formula (formulated in MS-bar scheme).

5. Transversly polarized DY (2)  Momentum, Spin & Invariants Only q-qbar initial states contribute. azimuthal angle

6. 1-loop calculations Spin Dependent cross section  Tree + Virtual corrections

7. Real emission Very lengthy but all O(ε) terms cancel in collinear limit

8. − Phase space integration : difficult in general However at O(а), we only need to calculate for : and viable!! → We reproduced the former result for the total cross section and obtained the QT distribution (new result). • QT distribution at 1 loop level singular terms : → resummation non-singular terms : finite at qT =0 Unpol. DY : Altarelli, Ellis, Greco,martinelli (’84) − kinematic variables

9. 1-loop result X: singular at qT =0, Y: finite at qT =0

10. 1-loop result (cont’d)

11. QT resummation  QT distribution of final state particles → recoil logs ; QT << Q ;Soft gluon emission become important → resummation needed. Leading Logs (LL) Next to Leading Logs (NLL) Finite terms O(а) LO calculation  NLO resummation

12. Collins, Soper ’81 Collins, Soper, Sterman ‘85 General formula • Momentum conservation → Impact parameter space b • General formula

13. NLL resummation • NLL approximation Dokshitzer ’78 etc. Kodaira & Trantadue ‘82 Davies, Stirling,Webber ‘85 • Together with Y terms at O(а), we obtained NLL result for this process.

14. Numerical calculations • PDF − model saturating Soffer’s inequality at the initial scale • (Vogelsang et.al. ‘98) 2.Small b : Bozzi et.al. ’03 Catani et.al. ‘93 3. Large b : Landau singularity b b integration in complex plane bmax Introduced in “Joint resummation” Kulesza, Sterman, Vogelsang ’02 bL − no need to introduce bmax − purely perturbative definition at QT  0 cf. Minimal prescription in threshold resum.

15. s = 100 GeV, Q = 10 GeV, y=0 without FNP(b2)

16. s = 100 GeV, Q = 10 GeV, Y=0, with

17. Summary • Chiral-odd distribution can be measured in transversely polarized Drell-Yan process/semi-inclusive processes etc.. • We calculated O(а) corrections to qT-integrated/unintegrated • cross sections and matched them to the NLL resummation formula. • → obtained dσ/dqT reliable both at qT ~ Q,qT « Q . • We calculated numerically the soft gluon effects using a newly developed method. − b-integral defined by contour deformation • Different (s, Q2) , different FNP ,

18. b* prescription : bmax=0.5 GeV-1 s = 100 GeV, Q = 10 GeV, Y=0, with