Soft gluon resummation in transversely polarized

Drell-Yan process at small transverse momentum

NPB777 (’07) 203 and more

― 　 double-spin asymmetries and novel asymptotic formula 　 ―

Kazuhiro Tanaka (Juntendo U)

H. Kawamura (RIKEN)

J. Kodaira (KEK)

T

TT

dd dA

dd d

— Transversity

• Chiral-odd : unmeasurable in inclusive DIS• First global fit (LO): SIDIS at HERMES, COMPASS + Collins function from BELLE Anselmino’s talk• Lattice study by QCDSF/UKQCD

• tDY data can provide an direct access to QCD corrections

Transversely Polarized Drell-Yan （ｔ DY) process

RHIC,J-PARC,GSI

p p X

0

( )

5( ) (0) ( )4

Pig d

i xi

n

i

t t

i

n Adf x e PS nn Se PSll g

ld y y l

p ^^ ^

×ò/ =/=ò

0

( )

( ) (0) ( )P4

ig di x

tn

i i

A n

i

tdf x e PS ne PSnlll

y y lp

×

/= =ò

ò

( ,0, ), (0, , )P P n nm + m -^ ^=0 0;

( )if xd

21 2

21 2

LO

( ) ( )cos(2 )

2 ( ) ( )

i ii

TTi i

i

i

i

e f x f xA

e f x f x

• bulk of dileptons produced at small QT • large perturbative corrections from gluon radiations make QT the relevant scale

NLO double-spin asymmetries

NLO NLO

2NLO

2T

TT

d dA

dQ dyd dQ dyd

Martin, Schafer, Stratmann, Vogelsang(’99)

TQ

2 , ,Q y

21 2

2

2 2

21 2

2< a few @RHIC

( , ) ( , )cos(2 )

2 ( , ) (

%, )

i ii

i ii

i

i

e f x f x

e f x f

Q

Q x

Q

Q

“sea-quark region” is probed at the scale2 210 GeVQ >

%　

double-spin asymmetries at a measured

2 2( )TT T

TTT

d ddQ dyd dQ

QdQ d dd Qy

A

21 2

NLO2

1

2

22

2

2

( , ) ( , )cos(2 )

> 2 ( , ) ( , )

i i ii

i i

T T

T ii

TT

T

e f x f x

e

Q

Qf f xA

Q

x Q

TQ Q

tDY at ap p measured TQX

2

2

fix

( )

d

,

e

T

QA B S

A BS

Q

P

x

Q

P

x

L

= +

=

?

, Q Qy yA BS Sx e x e-= = transversity

1 12 2

2 cos(2 ) ,( , , ; )( , ) ( ) ,A B A B

A BA B

T

A B

Tiiii x

i A Bi

xT

x xd dddQ dyd dQ

Hf Q fQx x

fx x x x

D sf

d x m d x mmå ò ò:2( , )TQ QÄG%

KLN cancellation

( )2 2 2ln Ts Q Qa

( )2 2ln Ts Q Qa

T

( )2TQd:

4 2D= - e

( )22

2

1 34

2 2Ts Qæ öp ÷ç ÷- - - +ç ÷ç ÷ç e eè ø

a d:

( )

( )( )2 2

2

2 2

2

ln1 3

2

2

1 3

T T

T

T

s

Q QQ Q

Q

++

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üïïï+ ýïïï÷ç

þç ø

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֏

a d

L

:

}

1 1

2

2

2

2

2

2

2

,1

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( , )

, ,

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1, ,

2 (2 )3

( ) ( )

( ) ( )

) (

cos

A B

AA B

A

B

B A B

T

T

i i x x

T

A B

A B

c

i ii ii

T

d x xdS b Qd b

Q

b

Q

b

x x

b b

N SQ

f C C

Y

fe

ddQ dyd dQ

e× x x

x x x xx x

a= f

ìïï´ íïïî

+

d d

D sf

å ò òò b Q

Next-to-leading logarithmic (NLL) resummation formula

( ) ( )2

2

2

2

2

2

2 2

1

log( , ) ( ) ( )Q

s s

b

QdS b Q A B

m

m

m

ì üæ öï ïï ï÷ç=- a m + a m÷í ýç ÷ç ÷ï ïè øï ïî þò

( )

( )

2 21 67 5

2 18 6 9

3

2

s ss F F G f

ss F

A C C C N

B C

p-

ì üæ ö æ öï ïa aï ï÷ ÷ç ça = + -÷ ÷í ýç ç÷ ÷÷ ççï ï è øè øp pï ïî þa

a =-p

Process-indep.: Kodaira, Trentadue (’82)

Davies, Stirling,Webber (’85)

Collins, Soper, Sterman ( ‘85)

22 1

2

2 2

21

22 3

22 2

2 2

LL

NLL

ln

ln

ln

n ns

n T

n

n

T

nn n

T TT T

Q

Q

Q Q

Q Q

Q

Q Q

-¥

- -

=

æ ö÷ç ÷ç ÷ç ÷çè ø

æ ö æ ö÷ ÷ç ç÷ ÷ç ç÷ ÷ç

ìï wïa íïïî

üïh z ï+ + ýïïþç÷ ÷ç çè ø è ø

å144444424444443

1444444444444442444444444444443

1 1

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(1 )( , ) (1

(( ) ( )

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)A BA B

A B A BA B

A A

A A

ii

sT A B

i i x

BT

x

A

s

i

i i F

i

d x xd

x x

Q QQ

Q Q x xY f R

bC

Q

C b e

f Qx x

x x x x

x x

a

px x

æ öa ÷ç ÷= d - + p -ç ÷ç ÷

= d d

ç pè ø

å ò ò

2,

valid up to corrections down by

, , is for

the resummation formula

: exact phase space integrals inMS scheme

0

un

(

first complete N

4 2

regular

iv

L

)

L

A B

A B

ii T

s

T

x xQ Q QR

D

x x·

·

a

· = - e

®

·

( )consistent with LO result in the massive gl

trans. spin case

no chi

is realized for ,

but

for : ( ) factorization formula

gluon distribution

a la CSS & CDG

L

ral-o

ersal

dds

str

u t

O

c ure

T sQ Q·

Ü

L O a: ?

uon scheme by Vogelsang, Weber ('93)

Kawamura,Kodaira,Shimizu,Tanaka, PTP 115 (’06) 667

{22

1

1

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T A B A B A A A

A AA A B B B B

A A A A Ae

yA A BA A B B B

A A B A A B B

A B

B

x

Aii T

x x x x xe C e

Q x x x x

d

x x

xx x x

x x

d x xx x e

x x x x

Q QR

t

xq x t d x

x x x

xd x d x d x

x x x

xd x d x q x t d

x x

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Q Qx x

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x x x x xe

x x x x

x x x x xe

x x x x

t

xq x t d x

x x x

xq x t d x

x x x

+

+

+ + +

+-

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= + +

üìï ï ï -ï ï ï - -í ýíï ï ï - -ïï ï îî þùüï- ï ú+ - - ýúï- - úïþû

2 2 2 2

, y yT T

A B

Q Q Q Qx e x e

S S

+ + -+ += =

2

00

( , ) ( , )1. -inte 2grati ( )o Jn: TiT

S b Q S b Qd b dbeb bQbe e¥

× = pò òb Q

1442443L L

All-order resummation IR Landau poleÞ

(1) (2)0 0H ( ) H ( )

2T TbQ bQ+

P

20

1

2 ( )1s Q

Qe b a Contour Contour

deformationdeformation 　　

NP( , ) ( , )2. ( )S b Q S b Q F be e® 14424432 2 3 3

2 31 b b+r L +r L= +L

2NPNP ( ) g bF b e-=

°2

NPNP 4 NP

NP

1( ) ,

4 2

Tk

gT T

gF k e k

g

- p= =

p

Laenen, Sterman, Vogelsang (’00) Kulesza, Sterman, Vogelsang (’02)

Landau pole

cut-off at bmax ： Collins,Soper,Sterman (’82)2 2

max1 /

bb

bb

b

intrinsic Tk

Elaboration of resummation formalism

systematic reorganization of the resummed series in the b-space2

0 ( )s Q L 2 2ln 1L Q b 1, 1

f 1/ /or 1 Tb

L

QQ

for 1, 0 /L b Q

(0)20

( ) ln(1 )FCh

2

(1) 2130 0

2 20

67 5

18 6 9

31 ln(1 )( ) ln (1 ) ln(1 )

2 1 2

ln(1 )2 1

F F

FG f

C Ch

CC N

(0) (1)2

1( , ) ( ) ( ) ( )

( ) ss

S b Q h h OQ

LL NLL NNLL

2,

0,

,2( ) ( )1/ exp ln(1 )2T q

N iNq

Ni

Pbf Qf

( , ) 1S b Qe

● 　 all-order resummation is accomplished at the ``partonic level’’

Bozzi, Catani, de Florian, Glazzini (’06)

● NNLL corrections are down by αs●

uniform accuracy classified by αs !

° ° ° }

1

2

2

2

, ,2

12

22

,

1

2

( , )

1 1( , ,) ,, ,,

22

3

( ) ( ) ) ( )(

B

A B

A B

B

B

T

A x

T A Bij q q

Ag

i Bi j

i A

c Ax

j

T

x x

dS b Q

b bfb

b

b

d

Q Q x x

d

C C Yf

eN SQ

ddQ dyd dQ

eb Q

=

×

x x

x

x

x x

ì xa ïï= íï xïî

´ +

sf ò

å

ò ò

NLL resummation formula for spin-independent cross section

from Altarelli, Ellis, Greco, Martinelli (’84);

° ° ° ° 2, , , , , ,( ), ( ), ( ), ( )A A A

A A A

T A Bgq qx x x

b b b Q Q x xC C C Yx x x

Collins, Soper, Sterman (’85)

( , )S b Qe is universal !( )2

NPNPNP( , ) ( , ) ( ) ( ) = g bS b Q S b Q F b eb Fe e -®

same contour deformation for the b-integration

same elaborations in terms of 20 ( )s Q L 2 2ln 1L Q b

Gluon distribution ( ,1/ )gf b participates !

Transversity distribution 0

2( )

5,( ) ( ( )P0)4

ig dtn A tx

j j j

nid

f x e PS n PSn Se^ ^^

×

= /ò

/ò llld m y y lg

pμ-dep.: NLO DGLAP evolution

Hayashigaki, Kanazawa, Koike (’97)

Kumano, Miyama (’97) 　 Vogelsang(’98)

0

0

,

,

( ) : GRV98

( ) : GRSV00

j

j

f x

f x

m

D m

, ,,

( ) ( )( )

2j j

j

f x f xf x

æ öm+D m÷ç ÷d m £ç ÷ç ÷çè øSoffer (’95)

x-dep.: 0 0

0 0, ,,

( ) ( ) at ( ) 0.6 Ge

2Vj j

j

f x f xf x

m +D mm = md ;

Vogelsang et al (’98)

2NP 0.9 GeV 0.5 GeV from unpolarized DY da taTg k

Nonperturbative inputs

Gaussian smearing factor:

2NPNP ( ) = g bF b e-

Dilepton QT spectrum in tDY Kawamura, Kodaira, Shimizu, Tanaka (’06)

pp collision @ RHIC s = 200 GeV, Q = 5GeV, y=2, φ=0

NLL+LO: total (“NLL+Y’’) prediction

2 2( )TT T

TTT

d ddQ dyd dQ

QdQ d dd Qy

A

2T

T

d

dQ dyd dQ

2

T

d

dQ dyd dQ

NLO LL4.0% ( )TT TT TQA A

( )TT TQA

2T

T

d

dQ dyd dQ

2

T

d

dQ dyd dQ

〔 Kawamura, Kodaira, Tanaka (’07) 〕

flat behavior: universality of soft gluon effectsnonuniversal contribution at NLL enhances

NLL+LO NLL( ) ( )T TTTTT Q QA A

NLL ( )TT TQA

NLL+LO ( )TT TQA @ RHIC (y=2, φ=0)

200 GeVS 500 GeVS

"deepersmaller '' sea- ( ) for quark regionTTT QA and/or smaller larger Q S

NLL NLO+LO ( ) is always larger than by about 20-30%. T TT T TQ AA

( )TT TQA

pp collision @ J-PARC s = 10 GeV, y=0, φ=0

NLL+LO ( )TT TQA

2 GeVQ

flat behavir due to universal soft gluon effects

enhancement mechanism from nonuniversal NLL effects

NLO LL12.8% ( )TT TT TQA A

larger asymmetries than the RHIC case moderate-x region

Novel asymptotoic formulaNLL+LO NLL NLL( ) ( ) ( 0)T TTT TT TTTQ Q Q A A A

NLL

2NP

1 1

2

2

2 2

22

1

2,

( , ) ,1 1

, ,

2 (2

( ) ( )( ) ( )

)3

cos

A B

A B

A B

A B

A B

T

T

i iA

x x

Bi ii b

i

c

i

T

xd xdS b Q gdb b

bb b

N SQ

e f fC C

ddQ dyd dQ

eb Q× x x

xx x x

- x x

a= f

ì üï ïï ïï ï´ í ýï ïï ïïd

ïîd

þ

D sf

å ò òò

(1)( ) ( )2 2

21 2

0

2 21/ 11 2

4 ( )( , (

(, )/)

)SP SP

ih

s SP ispi spie

Qe f x fb bx

Q- z l + l

æ öp ÷ç ÷ç ÷ç ÷¢¢ ÷ç b a z lèd

ødå

0TQ

(0) (1)2

1( , ) ( ) ( )

( )s

S b Q h hQ

20

(0)( )NP

2 2 20

( )( )

( ) ( )s Q

s s

ghe

Q Q Q

( ) 0SP 2

02 ( )1SP

s QSPb e

Q

Saddle-point formula at the NLL level ( ) (NNLL for ) 0TsO O Q

“degree 0 approximation’’ suggested by CSS (’85)

20 ( )s Q L

° °

NLL

2NP

1

2

, ,2 2

2

2

1

2

1,

2

( , ) 1 1( , ) ,,

2

( ) ( ) (

2

3

)B

AB

A BB

BT

A

i A

x

Bij q q g

c

i A

A

j

x

bji

T

x xdS b Q gd b bfb b

b

N

C

SQ

dfe C

ddQ dyd dQ

eb Q

=

× x

x x x

- x x

a=

ì üï ïxï ïï ï´ í ýï ïxï ïï ïî þ

sf

åòò ò

0TQ

(1)( ) ( )2

222 1

2

0

21/ 1( , )1 2

( ,( )

/ )4 ( )

SP SPs

h

s SPi i

ip spie

Qe f x f

Qbxb- z l + l

æ öp ÷ç ÷ç ÷ç ÷¢¢ ÷ç b a z lè øå

“very large prefactor” is universal !

gluon distribution decouples ! ( ) (NNLL for ) 0TsO O Q

extension of LL-level saddle point formula by Parisi, Petronzio (’79) to NLL level

20 ( )s Q L

NLL+LO NLL( 0) ( )T TT TTT Q Q A A2 2

2

21 2

21 2

2

1( , )cos(2

( , )

( ,

/ 1/

1/ 1/) )2 (

)

,

SP SP

SP

i i ii

i i i SPi

e f x f x

e

b b

xbf x f b

Most of soft gluon resummation effets cancel out LOreminiscent of TTA

unconventional scale associated with saddle point, instead of 2

NP22 (dependence on 1 G , is weak)1 V/ eSP Qb gQ

2Q

NNLL corrections would grow at small-x region probed by RHIC.

Useful for direct comparison with experiment

Clearly demonstrates the enhancement mechanism observed in numerical study

a part of NLL-level contribution survives

pp col in liY tD sion p p X

ˆ q qd d

NLOTTA Barone, Cafarella, Coriano, Guzzi, Ratcliffe (‘05)

HOTTA

Shimizu, Sterman, Yokoya, Vogelsang (’05)

directly probe the valence-quark transversity at GSI

For QT unobserved:

Large asymmetries ～ 20-40%

QCD corrections are rather small 10%

We consider the QCD corrections for QT observed case

“threshold resummation’’ at the NNL level

( )TT TQA@ GSI s = 14.5 GeV, y=0, φ=0

NLL+LO ( )TT TQA

4 GeVQ

NLO LL24% ( )TTT TTA Q A

larger asymmetries than RHIC and J-PARC case

collpp ision

flat behavir due to universal soft gluon effects“valence region”

LO is rather robustTTAenhancement mechanism is not significant

NLL+LO NLL( 0) ( )T TT TTT Q Q A A

,

,

2 22

21 2

2

1

2

2 1/ 1/

1/

( , ) ( , )

( , ) ( 1/,

(2 )

2 )

cos iSP SP

SP

qi i i

SPq

iq

i i

q

i

e f x f x

e

b b

b bf x f x

unconventional scale 2 221 GeV1/ SP Qb

NNLL corrections are small

Useful for direct comparison with experiment

does not cause siginificant

modification in the valence region probed by GSI

Summary:

the first complete formula for

well-defined, finite cross section and asymmetrie

MS scheme

all-order r

NLL+LO transverse spin

esummation recoil

channel

logsof due

s

to soft gluon radiat

( )

T TT QA

·

·

( ) ( ) ( )2 1 2 2 2 2 2 2 2 3 2 2ln , ln ln

universal structur

for

ˆ

( ) with satisfying Soffer

transvers inequa

0

ity

i

,

l

o

ity,

e

n

( )

n n n n

T T T

n n

s s s

T

TT T

T

Q Q Q Q Q Q

S f C

Q

C

Q

f

f

d Y

x

eA

- - -a a a

D s Ä Ä Ä sÄ Ä +d

m·

d

d

®

:

flat in the small region

nonuniversal contributions:

universal soft gluo

( ) at

RHIC & J-PAR

C

( ) at

n effects

GSI T

T

TT TT

TTT TT

T

A

AQ

Q

QA

A

A

>

·

:2 2

2 2

21 2

NLL+LO NLL2

1 2

22 2

( , ) ( , )0

( , ) ( , )

1 GeV

useful to extr

cos(2 )( ) ( )

2

soft gluon effe

1

c

/ 1/

1/ 1/

1ts at NL

a

L can be absorbed entirely into

ct tran v

/

s

i i ii

i i ii

SP SP

SP SP

SP

TT TTT

e f x f xb b

bQ Q

e f x f

Qb

x bA

　

» =f

=d då

å: =

ersity directly from experimental data

Transversely Polarized DY at small QT in QCD