sino-german workshop on frontiers in qcd

2006年9月 DESY Sino-German Workshop 梁作堂(Liang Zuo-tang) 山东大学(Shandong University)

1

Azimuthal Asymmetries InAzimuthal Asymmetries In Semi-inclusive Deep-Inelastic ScatteringSemi-inclusive Deep-Inelastic Scattering

Sino-German Workshop on Frontiers in QCD

Liang Zuo-tang, Shandong UniversityLiang Zuo-tang, Shandong University山东大学　梁作堂山东大学　梁作堂

Contents Introduction Collinear expansion in Semi-Inclusive DIS Results and discussions

Collaborator: Wang Xin-Nian (LBNL)


2

Introduction:


Intrinsic transverse momentum Gauge link


3

1977: Georgi & Politzer, “Clean test to pQCD”, Phys. Rev. Lett. 40,3 (77).1977: Georgi & Politzer, “Clean test to pQCD”, Phys. Rev. Lett. 40,3 (77).

Azimuthal asymmetry in Azimuthal asymmetry in unpolarizedunpolarized SiDIS SiDIS

, largefor )1()(

)(

2~cos 2/1 zz

yA

yB

,)1(1)( 2yyA ,)1)(2(2)( 2/1yyyB

hp

*

e

e lepton plan

hp

Xhepe


4

1978: Cahn, Intrinsic momentum effects. Phys. Lett. B78B, 269 (1978).1978: Cahn, Intrinsic momentum effects. Phys. Lett. B78B, 269 (1978).

Azimuthal asymmetry in Azimuthal asymmetry in unpolarizedunpolarized SiDIS SiDIS

Generalized parton model to include an intrinsic transverse momentum but no gluon radiation in the final state:

“Realistic studies”: Monte-Carlo, experiments etc.

In DIS at intermediate energies: intrinsic transverse momentum k effects have to be taken into account.

)(

)(||cos

yA

yB

Q

k

k

small ph, moderate Q, intrinsic k contribution dominates; large ph , large Q, Georgi-Politzer mechanism dominates.


5

Azimuthal asymmetry inAzimuthal asymmetry in polarized polarized SiDISSiDIS

Motivation: Single-spin asymmetry in pp collisionsMotivation: Single-spin asymmetry in pp collisions azimuthal asymmetry in singly polarized pp collisionsazimuthal asymmetry in singly polarized pp collisions

Xπpp )(

RL

RLN NN

NNA

up

forward

left

right

spin


6

History: theoretical studies on single-spin asymmetry in p(History: theoretical studies on single-spin asymmetry in p()+p)+p+X+X

1978: Kane, Pumplin, Repko, pQCD aN[q()+qq+q]=0.

Azimuthal asymmetry in Azimuthal asymmetry in polarized polarized SiDISSiDIS

Factorization )..(ˆ.)..( )()( funfragdfdpd XqqqXπpp

1991: Sivers, asymmetric quark distribution (Sivers effect)

Collins, Proof of non-existence of Sivers effect, asymmetric fragmentation function (Collins effect)

2002: Brodsky, Hwang, Schmidt, quark orbital angular momentum & “final state interaction”.

Ji, Yuan, final state interaction = gauge link

Collins, 1993’s proof is wrong because forgot gauge link,

1993: Boros, Liang, Meng, quark orbital angular momentum & “surface effect”


7

Conclusion:Conclusion:

Azimuthal asymmetry in Azimuthal asymmetry in polarized polarized SiDISSiDIS

both Sivers and Collins effects can exist when gauge link is taken into account.

intrinsic transverse momentum and gauge link are important in studying azimuthal asymmetries in SiDIS


8

Semi-inclusive DIS with : Semi-inclusive DIS with : A BASIC QUESTION

The general framework to write the cross section

Cross section for epehX

Parton distributions

Cross section for eqeq without

)(xq )(zD Xhq

),( kxq

kxpkeqeqd |)( ),(

FXhq kzD

= k

(with transverse momentum, with gauge link)

(without transverse momentum: collinear approximation)

xpkeqeqd |)(

Frag. function for qh+X

)( Xheped

？

k


9

Gauge link and

Collinear expansion in DIS and SiDIS.



10

InclusiveInclusive DIS: DIS: where does the gauge link come from?where does the gauge link come from?

DIS without QCD interactionDIS without QCD interaction

)],(ˆ)(ˆTr[)2(

),,( 4

4

qkHk,p,Sφkd

SpqW

SpzψψSpezdk,p,Sφ ikz ,|)()0(|,)(ˆ 4

))(()2()(),(ˆ 2qkqkqkH

No QCD interactions.Not (color) gauge invariant.

contracted with the leptonic tensor)e(e -- qqd

parton distributions

The hard part:

The matrix element:

)()2(,|)0(||)0(|,),,( 44X

X

pqpSpJXXJSpSpqW


11

Inclusive DIS with QCD interactionInclusive DIS with QCD interaction

SpzψψSpezdk,p,S ikz ,|)()0(|,)(ˆ 4)0(

)],(ˆ)(ˆTr[)2(

),,( )0()0(4

4)0( qkHk,p,S

kdSpqW

...),,(),,(),,(),,( )2()1()0( SpqWSpqWSpqWSpqW

SpzψygAψSpyedzdp,S,kk ρyzikyik

ρ ,|)()()0(|,),(ˆ )(4421

)1( 21

)],,(ˆ),(ˆTr[)2()2(

),,( 21)1(

21)1(

42

4

41

4)1( qkkHp,S,kk

kdkdSpqW ρ

μνρμν

… ...

Not gauge invariant!

parton correlation function


12


Collinear expansion:

......),(ˆ

),(ˆ),(ˆ'1

'

1

21(1)

21(1)(1)

kk

pxpxHpxpxHqkH σ

ρμνρ

μνρ

μν

......)(ˆ

)(ˆ),(ˆ'

')0(

)0()0(

ρρ

μνμνμν k

k

xpHxpHqkH

Decomposition of the gluon field:

)()()( yAωpn

pyAnyA ρ'

ρ'ρ

ρρ

ρ'ρ

ρ'ρ

ρ'ρ nngω

Using the identities such as,

),,(ˆ)(ˆ(1)

(0)

xpxpHk

xpH ρμνρ

μν

pkx /

Ellis, Furmanski, Petronzio, (82) Qiu, Sterman (90,91)

Expanding the hard parts around k =xp:

to replace the derivatives etc.

ixx

pxHpxpxHp μνρ

μν

12

1(0)

21L)(1, )(ˆ

),(ˆ

)(' xpkkω ρ'ρ

)(2

130 kkk

)0,1,0(

n

)0,0,1(

n


13


SpzψzψSpezdk,p,S ikz ,|)(),0()0(|,)(ˆ 4)0( L

)](ˆ)(ˆTr[)2(

),,(~ )0()0(

4

4)0( xpHk,p,S

kdSpqW μνμν

...),,(~

),,(~

),,(~

),,( )2()1()0( SpqWSpqWSpqWSpqW μνμνμνμν

SpzψzyyDyψSpyedzdp,S,kk ρyzikyik

ρ ,|)(),()(),0()0(|,),(ˆ )(4421

)1( 21 LL

]),(ˆ),(ˆTr[)2()2(

),,(~

21)1(

21)1(

42

4

41

4)1( ρ'

ρρ

μνρ'μν ωpxpxHp,S,kkkdkd

SpqW

Contain QCD interactions.(Color) gauge invariant !

...)0,',0(')0,,0()()0,,0(1),0(00

2

0

)0,,0(0

yzzyAdyig

yAdyyAdyigyAdyigPez

z

L

)()( ygAiyD ρρρ


14


...),,(~

),,(~

),,(~

),,( )2()1()0( SpqWSpqWSpqWSpqW μνμνμνμν

)](ˆ)(ˆTr[)](ˆ)(ˆTr[)2(

),,(~ )0()0()0()0(

4

4)0( xpHx,p,SdxxpHk,p,S

kdSpqW μνμνμν

)(ˆ)()2(

)(ˆ )0(4

4)0( k,p,S

p

kx

kdx,p,S

SpzψzψSpedz zixp ,|)0,,0(),0()0(|,2

L

...)/2()(ˆ )0()0( x,p,Sx,p,S ...)()( 1

)0( xfpx,p,S

Spzψp

zψSpedz

xf zixp ,|)0,,0(2

),0()0(|,2

)(1

L

Definition of parton distributions


15


Conclusion: Conclusion:

Collinear expansion is the necessary procedure Collinear expansion is the necessary procedure to obtain the gauge invariant parton to obtain the gauge invariant parton distributions.distributions.

Question: How about semi-inclusive DIS?


16

Semi-InclusiveSemi-Inclusive DIS with QCD interaction DIS with QCD interactionConsider first e +pe + q + X, i.e., no fragmentation

)()2(,|)0(|,','|)0(|,),',,( 44)(X

X

si pqpSpJXkXkJSpSkpqW

)],(ˆ)(ˆTr[)2(

)',,,( )0()0(4

4),0( qkHk,p,S

kdkSpqW si

...)',,,()',,,()',,,()',,,( ),2(),1(),0()( kSpqWkSpqWkSpqWkSpqW sisisisi

42

4

41

4),1(

)2()2()',,,(

kdkd

kSpqW siμν

)'()2)(2( 33' qkkEk

)],,(ˆ),(ˆTr[ 21),1(

21)1(

RL,c

qkkHp,S,kk ρcμνρ

)'()2)(2( 33

' qkkE ck

… ...

),,()0( SpqW ),,()1( SpqW


17

Semi-Inclusive DIS with QCD interactionSemi-Inclusive DIS with QCD interaction

The same collinear expansion technique leads toThe same collinear expansion technique leads to

)'()2)(2( 33' qkkEk

...)',,,(~

)',,,(~

)',,,(~

)',,,( ),2(),1()0( kSpqWkSpqWkSpqWkSpqW siμν

siμν

,siμν

(si)μν

] ),(ˆ ),(ˆ[Tr)2()2(

)',,,(~

21),1(

21)1(

RL,c4

24

41

4),1( ρ'

ρρc

μνρ'si

μν ωpxpxHp,S,kkkdkd

kSpqW

)](ˆ)(ˆTr[)2(

)',,,(~ )0()0(

4

4),0( xpHk,p,S

kdkSpqW μν

siμν

)'()2)(2( 33' qkkE ck

Gauge invariant parton distributions/correlations (with gauge link)


18

Consequences and discussions



19

SiDIS with : SiDIS with : Direct consequence IDirect consequence I

Consider the contribution from Consider the contribution from

Cross section for epeqX

Parton distributions

Cross section for eqeq without

)(xq

),( kxq

xpkeqeqd |)(

=k

(with transverse momentum)

(without transverse momentum)

)( Xqeped

xpkeqeqd |)(

)',,,(~ ),0( kSpqW si

k


20

SiDIS with : SiDIS with : Direct consequence IIDirect consequence II

A great simplification of : A great simplification of : '

21),1( ),(ˆ ρ

ρρL ωpxpxH

'21

),1( ),(ˆ ρρ

ρR ωpxpxH

k

)',,,(~ ),1( kSpqW si

μν

left cut

right cut

nnωxxδ

pqρ ρ

ρB

'1 )(

2

)(ˆ1

')1( pxH ρ

nnωxxδ

pqρ ρ

ρB

'2 )(

2

' 02

)1(0 )(ˆ

pxH ρ


21

SiDIS with : SiDIS with : Direct consequence IIDirect consequence II

This leads toThis leads to

)'()2)(2)]((ˆ)(ˆTr[)2(

Re2 33'

)1()1(4

4

qkkExpHk,p,Skd

kρ

μνρ

SpzψzDzψSpezdp,Sk,kkd

p,Sk ρikz

ρρ ,|)()(),0()0(|,),(ˆ)2(

),(ˆ 42

)1(4

24

)1( L

Only

k

] ),(ˆ ),(ˆ[Tr)2()2(

)',,,(~

21),1(

21)1(

RL,c4

24

41

4),1( ρ'

ρρc

μνρ'si

μν ωpxpxHp,S,kkkdkd

kSpqW

)'()2)(2( 33' qkkE ck

is relevant in semi-inclusive deep-inelastic lepton-nucleon scattering.


22

SiDIS with : differential cross-section to 1/QSiDIS with : differential cross-section to 1/Q

Further calculations in the standard way: Further calculations in the standard way:

,2/)()(ˆ )0()0( k,p,Sk,p,S 2/])(~)([)(ˆ

5)1()1()1(

k,p,Sk,p,Sk,p,S

MfSkpfkfpk,p,S T /)( 1''

1)0(

.../)( )1(2

2)1(1

)1()1( MSgkkkpSMppkk,p,S ss

.../~)(~~)(~ )1(2

2)1(1

)1()1( MSgkkkipSiMpkipk,p,S ss

Expanding the matrices ’s in terms of ’s:

The general Lorentz-structure of the coefficient functions are:

k

un-integrated parton distribution Sivers’ function

Equation of motion)1(1

)1(112

~ssTf

M

kxp

,~ )1()1(

xf


23

SiDIS with : differential cross-section to 1/QSiDIS with : differential cross-section to 1/Q

To the order of 1/Q:

)sin(||

)( )2(

2114

22

4

22

sTBqem f

M

kfyA

kddkpdQdx

Q

ed

k

sssM

k sin)~3(2

)1(2

)1(22

2

should be used to extract Sivers function and so on.

cos)~(||2

)( )1(

fx

M

k

Q

MyB B

)2sin()~(2

)1(2

)1(22

2

sssM

k

where is the azimuthal angle of s S


24

SiDIS with : SiDIS with : An observationAn observationk

By integrating overBy integrating over

sssQM

kyB sin)~3(

2)( )1(

2)1(2

2

the terms vanish if integrated over k- from -∞ to +∞,

no single-spin asymmetry for inclusive DIS, but … …

14

22

4

22

)( )2(

fyAkddkpdQdx

Q

ed Bqem

ssin

a single-spin asymmetry at fixed k-


25

Single-spin asymmetry for the “triggered” inclusive DIS

Consider Consider

sss

q

qem

B QM

kyBfyAkd

dkp

Q

e

dQdx

d

sin)~3(

2)()(

)2()1(2

)1(2

2

1

0

234

22

2

Xhepe trigger )(

a large momentum hadron in photon direction

2/32/

2/32/)(

||

||

ss

ss

dd

dd

dd

ddA trig

N

1

0

2

0

)1(2

)1(22

22

)(

)~3(2

)(

fkddkyA

Mk

kddkQ

MyB

q

q

ss

Xrealqepe )()(

a jet of hadrons

It corresponds to a “triggered” inclusive DIS, i.e.,


26

Single-spin asymmetry for the “triggered” inclusive DIS

Xhepe trigger )(

q

sssB M

kkddkGx

0

)1(2

)1(22

22 )~3(

At the hadron level, it corresponds to

),()]2()2([2

1),,(

2

1 2),( QxGSpxqpxqqp

qSpW BsBBtrigs

1

)( )(F

G

Q

MxyB

dd

ddA sBtrig

N

Measurements by e.g. HERMES at HERA?


27

Conclusions and Outlook

Collinear expansion can be extended to SiDIS;Collinear expansion can be extended to SiDIS;

Naïve extension of parton distribution convoluting Naïve extension of parton distribution convoluting with with eq eq eq eq cross section to include intrinsic cross section to include intrinsic transverse momentum is incorrect. transverse momentum is incorrect.

The existence of the projection operator The existence of the projection operator simplifies very much.simplifies very much.

A new single-spin asymmetry is predicted to 1/Q.A new single-spin asymmetry is predicted to 1/Q.

Extending to include fragmentation is underway.Extending to include fragmentation is underway.

' ρρω

)',,,(~ ),1( kSpqW si

μν

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