Hiroyuki Kawamura (RIKEN)

QT 　 resummation in transversely polariz

ed 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)

Introduction

“tDY” process

Spin projects at RHIC

pp collider experiment with longitudinal/transverse polarization

2001 ~ 2004 ~ RUN4 s = 200 GeV2005 RUN5

gluon polarization ( ) :G x ,p p X ,p p Q Q X

( ) :q x

,p p X

p p l l X

− helicity structure of the proton

− transverse structure

− spin dependent dynamics

transversity distribution

large single spin asymmetry ↔ T-odd FF 21 ( , ) :TH z k

,p p X

Hiroyuki Kawamura (RIKEN)

Transversity distribution function

− chiral-odd (not measured in DIS)

Ralston & Soper ‘79

− relativistic effect

− Soffer’s inequality 2 ( ) ( ) ( )q x q x q x Soffer ‘95

− DGLAP splitting functions

1-loop : Artru & Mukhfi ’902-loop : Hayashigaki et.al. ‘97, Kumano&Miyama ‘97, Vogelsang ‘98

− last unmeasured twist-2 pdf

Hiroyuki Kawamura (RIKEN)

Transversely Polarized DY process

— Transverse asymmetry cos(2φ)

1-loop corrections to tDY

→ observe φof the final lepton

Vogelsang ‘98

• 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.

We calculated Q_T distribution of DY pair directly in D-dimension.

— Only q-qbar initial state contributes.

Hiroyuki Kawamura (RIKEN)

1-loop calculations

Partonic Cross Section

Tree + Virtual corrections

• calculation in MS-bar scheme• naive anti-commuting

Hiroyuki Kawamura (RIKEN)

Real emission

lengthy but all O(ε) terms cancel in collinear limit

Hiroyuki Kawamura (RIKEN)

1-loop result

X: singular at qT =0, Y: finite at qT =0

Splitting function Artru & Mukhi ‘90

Hiroyuki Kawamura (RIKEN)

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)

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)

Finite terms

O(а) fixed order calculation NLO resummation

NNLL etc.

Hiroyuki Kawamura (RIKEN)

General formula Collins, Soper ’81Collins, Soper, Sterman ‘85

• Momentum conservation → Impact parameter space b

• General formula

— A, B, C are perturbtively calculable.

Hiroyuki Kawamura (RIKEN)

• NLL approximation

Kodaira & Trantadue ‘82

Together with Y terms at O(α), we obtained the first resultof NLL Q_T resummation formula of tDY.

Resummation at NLL

• From 1-loop result,

→

consistent with general relations

De Florian & Grazzini ‘00

Hiroyuki Kawamura (RIKEN)

Contour deformation method

Landau pole at in Sudakov factor

Contour deformation in b-integration : b

bL

bmax

→

positive frequency

negative frequency

Hankel like fn.

C1

C2

— introduced in “Joint resummation”. Laenen et al. ‘01Kulesza 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)

Numerical calculations

1. PDF a model given by Martin, Shäfer, Stratmann,Vogelsang (‘98). −

2.　 Small b :

1( ) [ ( ) ( )]

2q x q x q x

2 2 2 2 2 20 0ln / ln( / 1)L Q b b L Q b b Catani et al. ‘93

Bozzi et al. ’03

at initial scale(1)

(2) evolved to complex scale : b0/b numerically

→ expS(b,Q) = 1 at b=0 (correct overall normalization)

3. Non-perturbative effects

simplest form :

↔ IR renormalon ambiguity from Landau pole

intrinsic kT

Hiroyuki Kawamura (RIKEN)

s = 100 GeV, Q = 10 GeV, y=0

FNP(b) =exp(-0.5b2)

Hiroyuki Kawamura (RIKEN)

s = 100 GeV, Q = 10 GeV, Y=0

Hiroyuki Kawamura (RIKEN)

s = 200 GeV, Q = 20 GeV, Y=0

Hiroyuki Kawamura (RIKEN)

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

Hiroyuki Kawamura (RIKEN)

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?