8 th Circum-Pan-Pacific Symposium on High Energy Spin Physics

年 6月　 Cairns, Austrilia

8th Circum Pan-Pacific Spin Symposium

8th Circum-Pan-Pacific Symposium on High Energy Spin Physics

Higher Twist Contributionsto the Azimuthal Asymmetries in Semi-Inclusive DIS

梁作堂 (Liang Zuo-tang)

Cairns, Austrlia, 2011 年 6 月 22日

山东大学物理学院(School of Physics, Shandong University)

th Circum Pan-Pacific Spin Symposium 2011年 6月　 Cairns, Austrilia

Contents

Introduction: Why higher twist? How should we take higher twist contributions into

account systematically? Azimuthal asymmetries in the un-polarized semi-

inclusive DIS process up to twist-4 Summary & Conclusions

XqeNe


2011年 6月　 Cairns, Austrilia

Azimuthal asymmetry in unpolarized SIDIS

Definition

lepton plane

in c.m. frame, z: proton’s momentum directionxz-plane = lepton plane, x-component of lepton momentum is

positive.

* p

hp

*

e

e

hp

zx

Xhepe



Azimuthal asymmetry in unpolarized SIDIS

Georgi & Politzer, PRL(1977): “Clean test to pQCD”.

1/2( )cos ~ (1 ) 2 ( )

for large ,

B yz

A yz

,)1(1)( 2yyA

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

Cahn, PLB(1978): Intrinsic momentum effects. (Cahn’s effect)Generalized parton model to include an intrinsic

2)1(11)2(2||cosy

yyQk

k

22

2

)1(1)1(2||2cos

yy

Qk

(twist 3) (twist 4)

In current experiments, Q ~ several GeV; ~ 0.3 0.7GeVk 3cos | ~ 0.1twist 4cos 2 | ~ 0.01twist the same order of the exp. results.



Azimuthal asymmetry in polarized SIDISTriggered by the striking single-spin asymmetry observed in p()+p+X

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

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”

Since 2001, experimental results from HERMES and COMPASS.



Azimuthal asymmetry in polarized SIDIS

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

intrinsic transverse momentum induced higher twist effects and gauge link are important in studying azimuthal asymmetries in SIDIS.

Questions:Where does the gauge come from?How should we take intrinsic transverse momentum induced higher twist effects into account consistently?


Contents




XqeNe

Look at the inclusive DIS first.



Inclusive DIS: where does the gauge link come from?

DIS without QCD interaction

)],(ˆ)(ˆTr[)2(

),,( 4

4

qkHk,p,Sφkd

SpqW

SpzψψSpezdk,p,Sφ ikz ,|)()0(|,)(ˆ 4

2)()2()(),(ˆ qkqkqkH

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



Inclusive 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 distribution/correlation:

…multiple gluon scattering

; )()2()(),(ˆ 2)0( qkqkqkH 212

2

21);,1( )()2()(

)()(),(ˆ qkiqk

qkqkqkH siL

112011年 6月　 Cairns, Austrilia8th Circum Pan-Pacific Spin Symposium

Inclusive DIS with QCD interactionCollinear approximation:

(1) (1) (1)1 2 1 1 2 2 1 2

ˆ ˆ ˆ( , , ) ( , , ) ( , )ρ ρ ρμν μν μνH k k q H k x p k x p q H x x

(0) (0) (0)ˆ ˆ ˆ( , ) ( , ) ( )μν μν μνH k q H k xp q H x

Keep only the longitudinal component of the gluon field:( ) ( ) ρ

ρ

pA y n A y

n p

pkx /

Approximate the hard parts by those at k =xp:

Using the Ward identities such as, ixxxH

xxHp μνρμν

12

1(0)

21L)(1, )(ˆ

),(ˆ

)(2

130 kkk

)0,1,0(

n

)0,0,1(

n

4(0) (0) (0) (0)

4ˆ ˆˆ ˆ( , , ) Tr ( ) ( ) Tr (x ) ( )

(2 )μνd k

W q p S k, p,S H x dx , p,S H x

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

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

2

0

)0,,0(0

yzzyAdyig

yAdyyAdyigyAdyigPez

z

L

4(0) (0)

4ˆ ˆ(x ) ( ) (x )

(2 )d k k

, p, S x , p,Sp

only leading twist contribution can be obtained

122011年 6月　 Cairns, Austrilia8th Circum Pan-Pacific Spin Symposium


Collinear expansion:

......),(ˆ

),(ˆ),(ˆ'1

'

1

21(1)

21(1)(1)

kk

xxHxxHqkH σ

ρμνρ

μνρ

μν

......)(ˆ

)(ˆ),(ˆ'

')0(

)0()0(

ρρ

μνμνμν k

kxH

xHqkH

Decomposition of the gluon field:

)()()( yAωpn

pyAnyA ρ'

ρ'ρ

ρρ

ρ'ρ

ρ'ρ

ρ'ρ nngω

pkx /

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

Expanding the hard parts around k =xp:

Using the Ward identities such as,

to replace the derivatives etc.

(0)1(1,L)

1 22 1

ˆ ( )ˆ ( , ) μνρμν

H xp H x x

x x i

)(' xpkkω ρ'ρ

)(2

130 kkk

)0,1,0(

n

)0,0,1(

n

),,(ˆ)(ˆ(1)

(0)

xxHk

xH ρμνρ

μν




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

)(ˆTr)2(

),,(~ )0(4

4)0( k,p,S

kdSpqWμν

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

SpzψzyyDyψSpyedzdp,S,kk ρyzikyik

ρ ,|)(),()(),0()0(|,),(ˆ )(4421

)1( 21 LL

),(ˆTr)2()2(

),,(~21

)1(4

24

41

4)1( p,S,kk

kdkdSpqW ρ'μν

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

)(ˆ )0( xH μν )(ˆ )0( xH μν

),(ˆ21

)1( xxH ρμν

ρ'ρω ρ'ρω),(ˆ

21)1( xxH ρ

μν

twist-2 and higher

twist-3 and higher




Conclusion:

Question: How about semi-inclusive DIS?

Gauge link comes from the multiple gluon scattering and collinear expansion is the necessary procedure to obtain the correct form of gauge invariant parton distributions.

For simplicity and clarity, consider first e +p e + q + X, i.e., NO fragmentation.

ZTL & X.N. Wang, PRD (2007).



Semi-Inclusive DIS with QCD interactionSIDIS e +p e + q (jet) + X:

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

X

si pqpSpJXkXkJSpSkpqW

)],(ˆ)(ˆTr[)2(

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

4),0( qkHk,p,S

kdkSpqW sisi

...)',,,()',,,()',,,( ),2(),1(),0( kSpqWkSpqWkSpqW sisisi

)'()2()(),(ˆ 44),0( qkkqkqkH si

)()2()(),(ˆ 2)0( qkqkqkH C.f.:

…



Semi-Inclusive DIS with QCD interaction

Using the mathematical identity:

)],(ˆ)(ˆTr[)2(

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

4),0( qkHk,p,S

kdkSpqW si

)'()2)(2( 33

' qkkEk

),,()0( SpqW

)'()2()2( )()2()'()2( 3'

3244 qkkEqkqkk k

)'()2)(2)(,(ˆ),(ˆ 33'

)0(),0( qkkEqkHqkH ksi

)'()2)(2)(,,(ˆ),,(ˆ 33'21

),,1(21

),,,1( qkkEqkkHqkkH ckcsic

We obtain:

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

42

4

41

4),1(

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

kdkd

kSpqW siμν )],,(ˆ),(ˆTr[ 21

),1(21

)1(

RL,cqkkHp,S,kk ρc

μνρ

),,()1( SpqW


4( ) (0)

4

2 (

(0)

(

3 3'

2'

0 0))

ˆ( , , , ') Tr[ ( ) ](2 )

ˆ

(2 )(

T

2 ) ( ' )

(2 ) ( 'r[ ( , ) ]

ˆ )

ˆ ( ) )

(μν

μ

μ k

ν

si

k

νd k

W q p S k k, p,S

dxd k x k ,

EH x

H

k k q

Ep, x kS k


Under collinear approximation, i.e.,

( ) ( ) ρρ

pA y n A y

n p

(0,)3333''

(0)(0) ˆ(,)(2)(2)(')(2)(2)( ˆ()(' ,)) ˆ sikk HkqEk HkkqEk qxkq H

(1, , , ) 3 31

(1, , )1 2

(1, , )1

2 '

3 3'2

ˆ ( , , ) (2 )(2 ) ( ' )

ˆ ( , , )ˆ ( , ) (2 )(2 ) ( ' )

c sik c

k c

c

c

H k k H k k q

H x

q E k k q

k qx E k

(0) (0)2

ˆ ˆ( , ) ( - ) ( )(2 )

dk dk kx k , p, S x k, p,S

p

we obtain



(4

(0, ) 3 3'

0 ))4

0( (2 )(2 ) (ˆ( , , , ') Tr[ ( ) ])

) '(

ˆ (2

)ksi

ν μνμd k

W q p S k k, p, HS E qx k k



The same collinear expansion leads to

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

siμν

,siμν

(si)μν

4 4(1, ) (1)1 2

1 24 4c L,

(1, )1 2

R

ˆ( , , , ') Tr[ ( , ) ](2 ) (

ˆ2 )

( , ) s c ρμν

ρ'ρ

iμν ρ'

d k d kW q p S k k ,k p, S x ωH x

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

Gauge invariant parton distributions/correlations (with gauge link)

4 4 4(2, ) (2)1 2

' 1 24 4 4c L,C,R

(2, )1 2

ˆ( , , , ') Tr[ ( , , ) ](2 ) (2

ˆ ( ,)

,(2 )

)siμ

ρ'ν ρ ρ

c ''

ρμν

d k d k d kW q p S k k ,k k p x ω ωH xS x,

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


Contents




e N e q X

ZTL & X.N. Wang, PRD (2007); J. Gao, ZTL, & X.N. Wang, PRC (2009); Y.K. Song, J.H. Gao, ZTL & X.N. Wang, PRD (2010).




For

)'()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 ωxxHp,S,kkkdkd

kSpqW

)](ˆ)(ˆTr[)2(

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

4),0( xHk,p,S

kdkSpqW μν

siμν

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

twist-2 and higher

] ),,(ˆ ),,(ˆ[Tr)2()2()2(

)',,,(~21

),2(21

)2('

RC,L,c4

4

42

4

41

4),2( 'ρ'

ρρc

μνρ'si

μν ωωxxxHp,Sk,kkkdkdkd

kSpqW

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

e N e q X

twist-3 and highertwist-4 and higher



SIDIS with : Direct consequence I

Consider the contribution from

Cross section for epeqX

Parton distributions

Cross section for eqeq without

)( xq

),( kxq ˆ ( ) |k xpd eq eq

=k

(take intrinsic transverse momentum into account)

(neglect the intrinsic transverse momentum )

)( Xqeped

xpkeqeqd |)(

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

k

k

( ')k

TMD Parton distributions



SIDIS with : Direct consequence II

A simplification of

k

)',,,(~ ),1( kSpqW siμν

right cut

left cut independent of x2!

independent of x1!

'21

),1( ),(ˆ ρρ

ρR ωxxH

nnωxxδpq

ρ ρρB

'

2 )(2

02')1(

0 )(ˆ xH ρ

'21

),1( ),(ˆ ρρ

ρL ωxxH

nnωxxδpq

ρ ρρB

'

1 )(2

)(ˆ1

')1( xH ρ



SIDIS with : Direct consequence II

This leads to

SpzψzDzψSpezd

Sp,kk,kd

p,Sk

ρikz

ρρ

,|)()(),0()0(|,

) , (ˆ)2(

),(ˆ

4

2)1(

42

4)1(

L

Only

k

)'()2)(2)]((ˆ)(ˆTr[)2(

Re2 33'

)1()1(4

4

qkkExpHk,p,Skd

kρ

μνρ

] ),(ˆ ),(ˆ[Tr)2()2(

)',,,(~21

),1(21

)1(

RL,c4

24

41

4),1( ρ'

ρρc

μνρ'si

μν ωxxHp,S,kkkdkd

kSpqW

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

contributes in semi-inclusive deep-inelastic lepton-nucleon scattering.

Similar for )',,,(~ ),2( kSpqW siμν



SIDIS with : differential cross-section to 1/Q2k

A complete twist-4 result for

pzψzψpezddz

kxf zkizixpq |)(

2),0()0(|

)2(),( 3

2

L

pzψkk

zψpezddz

kxf zkizixpq |)(

2),0()0(|

)2(),( 23

2

L

TMD quark distribution:

Quark correlation functions such as :

pzψzDzψpezddz

dkkkxk zkizixp |)()(2

),0()0(|)2(

)ˆˆ2(),( 3

2)1(

22

L

cos),(||1)2(4),()1(1

12 22

22

2 kxxfQk

yykxfyyQe

kdxdydd

qqqem

2cos),(~),(||)1(4 )1(2

)1(22

2

kxkxx

Qk

y

),(2),(~),(||)1(8 )(2

22)1(

2)1(

22

2

kxfQ

Mxkxkxx

Qk

y q

),(||)1(12 ),2(

22

22 kxx

Qk

y L

XqeNe



SIDIS with : Azimuthal asymmetriesk

Up to twist-4),(),(||

)1(11)2(2

cos 2

kxfkxxf

Qk

yyy

q

q

),()],(~),([||

)1(1)1(22cos

)1(2

)1(2

2

2

2

kxf

kxkxxQk

yy

q

If g=0, i.e., no multiple gluon scattering (results by Cahn):

2

2

20||

)1(1)1(2|2cos

Qk

yy

g

,||)1(1

1)2(2|cos 20 Q

ky

yyg

proportional to ,||

Qk

proportional to ,||2

2

Qk

A good place to study such correlation functions and effects of multiple gluon scattering.



Transverse momentum broadening in nucleus

Replace N by A, the gluons can connect to different nucleons in A.

Gauge link comes from:

+ +…+ … +…

Nuclear enhancementTransverse momentum broadeningTransverse momentum broadening should be contained in the gauge link

With “maximal two gluon approximation”, i.e. taking only the case that two gluons are connected to one nucleon into account,

),(

),( 22 /)(2

2

lxfeldΔA

kxf Nq

lk

F

Aq

F

ZTL, X.N. Wang & J. Zhou, PRD (2008).


Nuclear dependence of the azimuthal asymmetries


An example: take a Gaussian for the transverse momentum dependence

,)( 1),( /2

kNq

Nq exfkxf

/2

)( 1),( kN

qNq exfkxf

)/(

2

22

)( )(

),( FkNq

F

Aq exf

Akxf

)/(2

2

22

)( )(

),( FkNq

F

Aq exf

Akxf

2

22

222

2

22

))(()()-(exp

)()(

coscos

kFF

FF

F

F

eN

eA

2F

for the case that suppressed

! Integrated over :kFeN

eA

2coscos


Contents




e N e q X



SummaryComparison of the results already obtained

g=0, i.e., no multiple gluon scattering

gauge link 1L

Sivers function 0),(1 kxf T

no A-dependence

other effects ...........

g=0, i.e., with multiple gluon scattering

1 ( , ) 0Tf x k

2

2(2 ) 1 | |cos ,

1 (1 )y y k

y Q

2

2 2

| |2(1 )cos 21 (1 )

kyy Q

Sivers function

transverse momentum broadening in nuclei

other effects ...........

2 (1) (1)2 2

2 2

| | [ ( , ) ( , )]2(1 )cos 21 (1 ) ( , )q

k x x k x kyy Q f x k

2

( , )2(2 ) 1 | |cos

1 (1 ) ( , )q

q

xf x ky y ky Q f x k

gauge link 1L

Measurements of cos , 2cos and their A-dependence in

effect of multiple gluon scattering and structure of nucleon.

' ( )l N l q jet X

Cahn’s results

and compare the obtained results with those obtained in 1978 by Cahn



Summary Gauge link is result of multiple gluon scattering and

collinear expansion is a necessary procedure to obtain the correct form.

Collinear expansion can be extended to SIDIS e+pe+q+X. Naïve extension of TMD parton distributions convoluting

with eq eq cross section to include intrinsic transverse momentum is incorrect.

Many other consequences.

Thank you for your attention!

8 th Circum-Pan-Pacific Symposium on High Energy Spin Physics

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