1. QT resummation 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)
Hiroyuki Kawamura (RIKEN)

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

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
Hiroyuki Kawamura (RIKEN)

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.
Hiroyuki Kawamura (RIKEN)

5. 1-loop calculations Partonic Cross Section
calculation in MS-bar scheme
naive anti-commuting
 Tree + Virtual corrections
Hiroyuki Kawamura (RIKEN)

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

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

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.
Hiroyuki Kawamura (RIKEN)

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
Hiroyuki Kawamura (RIKEN)

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

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.
Hiroyuki Kawamura (RIKEN)

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
Hiroyuki Kawamura (RIKEN)

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
simplest form :
intrinsic kT
Hiroyuki Kawamura (RIKEN)

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

15. s = 100 GeV, Q = 10 GeV, Y=0
Hiroyuki Kawamura (RIKEN)

16. s = 200 GeV, Q = 20 GeV, Y=0
Hiroyuki Kawamura (RIKEN)

17. Double Spin Asymmetry : s = 100 GeV, Q = 10 GeV, Y=0
Hiroyuki Kawamura (RIKEN)

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?
Hiroyuki Kawamura (RIKEN)