Q T resummation in transversely polarized Drell-Yan process

1. QTresummation in transversely polarized Drell-Yan process Hiroyuki Kawamura (RIKEN) Oct. 6, 2005 RADCOR2005, Shonan Village work in common with J. Kodaira (KEK) H. Shimizu (KEK) K. Tanaka (Juntendo U)

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

3. Transversity distribution function 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  — Only q-qbar initial state contributes. — Transverse asymmetry  cos(2φ) → observe φof the final lepton  1-loop corrections to tDY • No direct calculation in D-dim. • — D-dim. calculation keepingφ: cumbersome compared with unpol. case. • 1-loop, MS-bar for Q_T integrated cross section using scheme tr. Vogelsang ‘98 We calculated Q_T distribution of DY pair directly in D-dimension.

5. 1-loop calculations Partonic Cross Section • calculation in MS-bar scheme • naive anti-commuting  Tree + Virtual corrections

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

7. 1-loop result X: singular at qT =0, Y: finite at qT =0 Splitting function Artru & Mukhi ‘90

8. 1-loop result (cont’d) — All terms are finite as Q_T → 0 — By integrating X+Y w.r.t. Q_T, we reproduced the known result.

9. QT resummation  QT distribution of DY-pair → recoil logs ; QT << Q ;Soft gluon emission become important → resummation needed. Leading Logs (LL) Next to Leading Logs (NLL) NNLL etc. Finite terms O(а) fixed order calculation  NLO resummation

10. Collins, Soper ’81 Collins, Soper, Sterman ‘85 General formula • Momentum conservation → Impact parameter space b • General formula — A, B, C are perturbtively calculable.

11. Resummation at NLL • NLL approximation • From 1-loop result, consistent with general relations De Florian & Grazzini ‘00 Kodaira & Trantadue ‘82 → Together with Y terms at O(α), we obtained the first result of NLL Q_T resummation formula of tDY.

12. Contour deformation method Landau pole at in Sudakov factor b  Contour deformation in b-integration : C1 bmax bL → C2 Hankel like fn. positive frequency negative frequency — introduced in “Joint resummation”. Laenen et al. ‘01 Kulesza et al. ‘02 — no need to introduce bmax as in b* formalism. —reproduce perturbative results order by order. cf. Minimal prescription in threshold resummation

13. Numerical calculations • PDF − a model given by Martin, Shäfer, Stratmann,Vogelsang (‘98). (1) at initial scale (2) evolved to complex scale : b0/b numerically 2.Small b : Catani et al. ‘93 Bozzi et al. ’03 → expS(b,Q) = 1 at b=0 (correct overall normalization) 3. Non-perturbative effects ↔ IR renormalon ambiguity from Landau pole intrinsic kT simplest form :

14. s = 100 GeV, Q = 10 GeV, y=0 FNP(b)=exp(-0.5b2)

15. s = 100 GeV, Q = 10 GeV, Y=0

16. s = 200 GeV, Q = 20 GeV, Y=0

17. Double Spin Asymmetry : s = 100 GeV, Q = 10 GeV, Y=0

18. Summary • Chiral-odd distribution can be measured in transversely polarized Drell-Yan process by measuring φdependence of the cross section. • We calculated O(α) corrections to QT-distribution of DY pair • in MS-bar scheme. • The soft gluon effects are included all order resummation at NLL accuracy. — b-integral defined by contour deformation • Spin asymmetry  10 % at (S, Q, y) = (100GeV, 10GeV, 0) at most. — difficult to measure at RHIC — fixed target experiment at GSI?