Yu-kun Song (USTC)2013.7.29 Weihai

YKS, Jian-hua Gao, Zuo-tang Liang, Xin-Nian Wang, Phys.Rev.D83:054010,2011

YKS, Jian-hua Gao, Zuo-tang Liang, Xin-Nian Wang, to be submitted

Higher twist effects in semi-inclusive DIS

Outline

Introduction to higher twist effects

Collinear expansion extended to SIDIS

Azimuthal asymmetries at twist-3 level

Nuclear effects and higher twist

Conclusions

Partonic picture of nucleon

Nucleon is the eigenstate of → Poincare invariance of induce

momentum/ angular momentum sum rules

→Test of QCD in strong coupling regime

QCDH

• 3 confined quarks• m_q ~ 200-300 MeV• static property• P, J shared by q

• a bunch of free partons• m_q ~ several MeV• hard scattering• P, J shared by q,qbar,g

Quark model(1960s) Parton model(1970s)

QCDL

1 , 1ˆ ˆ ˆ ˆ 2

. q g q gP P J J

Semi-inclusive DIS: a nice probe of nucleon

X

2 2ˆˆ O

Q

O

Sterman-Libby power counting

Leading twist

2

3 43 4ˆ ˆˆ ˆO O

Q Q Q

O

X

Higher twist (1/Q power corrections)

QCD radiative correction→“A clean test of QCD”[Georgi, Politzer, 1978]

Intrinsic [cahn,1978]

→

Power suppressed, higher twist(HT)!Magnitude of higher twist terms

~300 MeV , ~several GeV , ~10%

Not negligible for most SIDIS experiments.

k

2

2

2

2

2(2 ) 1cos

2 2

2 2cos 2

|

2

|

2

y y

y y

y

y y

k

Q

k

Q

k

Q

| |k

Q

Semi-inclusive DIS: a nice probe of nucleon

cos , cos 2 0

Collinear expansion:

Systematic way of calculating higher twist in DIS [Ellis, Furmanski, Petronzio, 1982, 1983; Qiu, 1990]

Extension to SIDIS [Liang, Wang, 2006] QCD multiple gluon scattering

→ gauge link + Higher twist terms

→ nuclear broadening [Liang, Wang, Zhou,2008]

nuclear modification of azimuthal asymmetries

[Liang, Wang, Zhou, 2008]

twist-4 corrections to unpolarized SIDIS

[YKS, Gao, Liang,Wang, 2010]

twist-3 corrections to doubly polarized SIDIS

[YKS, Gao, Liang, Wang, to be submitted]

Higher twist and collinear expansion

(0; )yLk

Leading twist: Collinear approximationBasis of QCD factorization theorem: Sterman-

Libby Power counting [Collins, 2011]

→ Leading contributions ~ Collinear approximation

Example: DISi ik x p

, ] [ , ] ([ )i A G xp A OG kQ

Ap

Ap

…

~ ( ) ( , )

( , ) | (0) (0; ) ( ) |2 2

ixp y

W h x f x k

dyf x k e N L y y N

Gauge invariant parton distribution function

• Collinear approximation• Ward identity

Higher twist: Collinear expansionLeading twist:

Non-leading twist: expansion near collinear limit

Collinear expansion is the natural and systematic way to extract HT effects.

Notice: for a well-defined expansion

, ] [ , ]

[ ,

( )

( ) [ , ] ...[ , ] ]

[

k xpi

i A G xp A

G xp

Q

GG k A k xp G xp AA

G

Ak

k

O

[[ , ] , ]i A GG xpk A

( )n nO

Gauge-invariant,So that they can be measured in Exps.

Expansion parameter

[Ellis, Furmanski, Petronzio, 1982,1983 ;Qiu,1990]

Collinear Expansion:1. Taylor expand at , and decompose

2. Apply Ward Identities

3. Sum up and rearrange all terms,

(0) 2 (0)(0) (0) ' ' ' '

' ' ' '

ˆ ˆ( ) 1 ( )ˆ ˆ( ) ( ) ... 2

H x H x AH k H x k k k A p A

k k k p

Collinear expansion in DIS

( , )ˆ ( )n ciH k i ik x p

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

1 2, 2 1

ˆ ˆ( ) ( )ˆ ˆ( , ), ( , )c L

c L R

H x H xH x x p H x x

k x x i

(0) (0) (1, ) ' (1) (2, ) ' ' (2)1 2 ' 1 2 1 2 ' ' 1 2

, , ,

1 ˆ ˆ ˆˆ ˆ ˆ( ) ( ) ( , ) ( , ) ( , , ) ( , , )2

c c

c L R c L M R

W H x x H x x x x H x x x x x x

A

(0)H

(1, )ˆ cH

(2, )ˆ cH

(0) (0) ˆˆW H (1) (1, )

,

ˆˆ c

c L R

W H

(2) (2, )

, ,

ˆˆ c

c L R M

W H

e N e X

In the low region, we consider the case when final state is a quark(jet)

Compared to DIS, the only difference is the kinematical factor

Collinear expansion is naturally extended to SIDISParton distribution/correlation functions are -

dependent

Collinear expansion in SIDIS e N e q X

3 32 (2 ) ( )k cK E k k q

(0)H

(1, )ˆ cH

(2, )ˆ cH

(0) (0) ˆˆ KW H (1) (1, )

,

ˆˆ c

c L R

W H K

(2) (2, )

, ,

ˆˆ c

c L R M

W H K

k

[Liang, Wang, 2007]

k

Form of hadronic tensor after collinear approximation

: color gauge invariant

Hadronic tensor for SIDIS

(0)

(

(0) (1, ) (2, )

2 2 2 2, , ,

2 (0)(0)

2

2 (11,

, )(1)

2

2 (2,)

)

)(1

2 2

ˆ ( , )

ˆ ( ,

...

1 ˆTr ,2

1 ˆTr ,4

1 ˆTr(2 )

)

[ ]

[ ]

[

c c

c L R c L R M

L

L

NB

L NB

x

dW dW dW d

k

x

W

d k d k d k d k

d Wh

d k

d Wh

d k q p

d Wh

d k q p

k

(2)

2 (2, )(2)

(2, ) (

2 2

2, )

(2, )

ˆTr ,

1 ˆTr

ˆ ˆ( , ) ( , )

ˆ ( ) .(2 )

,

] [ ]

[ ]

L N L NB B

M NB

M

N

d W

x k x k

x khd k q p

2,3,4

3,4,

4,

(0) (0; ) ( ) , (0) (0; ) ( ) ( ) ,

(0) (0) (0) (0; ) ( ) ,

y y y D y y

D D y y

L L

L

: Projection operator

k k xp

(0) (1, ) (2, ) (2, )ˆ , , ,L L M

Structure of correlation matricesExpand in spinor space

Constraints from parity invariance

(0)5

(0) (0)

(2 (0)

( 0) (00) (0)52

)

ˆ ,

1 ˆ ˆTr Tr2

[ ] [ ]d Wh h

d k

(0) 4

4

4

4

( , , ) , (0) ( ) ,

ˆ ˆ ˆ ˆ , (0) ( ) ,

, (0) ( ) ,

, (0) ( )

ik y

ik y

ik y

ik y

p k s d ye p s y p s

d ye p s PP y PP p s

d ye p s y p s

d ye p s y p

(0)

(0) (0)

,

( , , )

( , , ) ( , , )

s

p k s

p k s p k s

Structure of correlation matricesTime reversal invariance relate and

Lorentz covariance + Parity invariance,

† †( ,0 ;0 ,0 ) ( ,0 ;0 ,0 ) ( , ; , )( , ; , ) TL L y y y L L y y y

(0)1 1

(0)1 1

...

·...( ) ( )

T T T L

L T

ks ks i ii i

ksT T

iLi

f p k M s k

k sp k Ms k

M

f f f f f

g g g g g g

1 1 1 1, , ,

, ,

: twist-2 parton distribution functions

: twist-3 parton correlation funct, i, , ons, ,

T L T

T T L T T L

f f g g

f f f f g g g g

y

SIDISDY

SIDISf DYf

TMD PDF and correlation functionsTwist-2 TMD parton distribution functions

Twist-3 TMD parton correlation functions

2·

3

2·

3

1

31

·51

2

| (0) (0; ) ( ) | ,2(2 )

, | (0) (0; ) ( ) | , ,2(2 )

, | (0) (0;2(2 )

( , )

( , )

( , )

ixp y ik y

ixp y ik yks

ixp yL

i

T

k y

dy d ye p y y p

dy d

f x k

f x k

g

y Me p s y y p s

dy d yx k e p y

L

L

L

2· 5

31

) ( ) | , ,

, | (0) (0; ) ( ) | , ,2(2 )

( , ) ixp y ikT

y

y p

Mg x

dy d ye p s y y p s

k sk

L

Unpolarized PDF

Sivers

Helicity distribution

Worm-gear

2·

3 2

2·

3

2·

3

( , )

(

| (0) (0; ) ( ) | ,2(2 )

, | (0) (0; ) ( ) | , ,2(2 )

,2(2 )

, )

( , )

ixp y ik y

ijj iixp y ik y

ixp y ik y

T

T

p dy d y ke p y y p

k

kdy d yp

f x k

f x e p s y y p sM k s

p dy d ye p

k

sf x k

L

L

2·

3 2

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

, | (0) (0; ) ( ) | , ,2( )

( , )2

ks

ijj iixp y ik y

L

sy y p s

k s

kp dy d ye p y y p

kf x k

L

L

color gauge invariant !

Structure of correlation matricesSimilar for

QCD equation of motion, ,induce relations

(1, ) (1, ) (1, )

(1, )

(1, )

5

...

ˆ

...

( )

( )

ks i ii

L L L

LT T L

LT T

i

Lks i

i

p k M s k

ip k Ms k

0i D

Re ,

Re ,

Re ,

Re ,

( )( )( )( )

T T

L L

T T

T

L

T

x

x

x

x

f

f

f

f

Im ,

Im ,

Im ,

Im .

( )( )( )( )

T T

L L

T T

T

L

T

g

g

g

x

x

x

xg

(1, ) (1, ),L R

Relations from QCD EOMSum up and , one has (up to twist-3)

Explicit color gauge invariance for and .Explicit EM gauge invariance

(0)W (1, ) (1, ),L RW W

2

{ }2

{ } { }

[

1

]

1

1 1

1( 2 )

·

( 2 ) ( 2 )· ·

· ( 2 )

·

( ) ( )

( ) ( )

ks ksT B

i iB i B i

ks

T

T L

L T TB

f f f f

f f

d Wd k q x p

d k p q

Mq x p s q x p k

p q p q

k s ii k q x p

Mg g

qg

pg

[ ] [ ] ( 2 ) ( 2 )· ·B i B iT

i iLg

iM iq x p s q x p k

p q qg

p

(0)

20

dW

dq

k

if ig

Consistency to DISIntegration over , one has

where

because of Time-reversal invariance.For DIS at twist-3 only contribute.

2d k

1 { }

1 [ ]

( ) ( 2 ) ( )·

( ) ( 2 ) ( )2 ·

iB i T

iL B i T

MW d f x q x p s f x

p q

iMi g x q x p s g x

p q

22

2

22

2

( ) ( , ) ( , ) ,2

( ) ( , ) ( , ) .2

[ ]

[ ]

T T T

T T T

kf x d k f x k f x k

M

kg x d k g x k g x k

M

( ) 0Tf x

( )Tg x

Azimuthal asymmetries at twist-3 levelCross section for

Twist-3 parton correlation function

QCD equation of motion implies

e e XN q

2 2 2 2

4 4

2

2 2 | |{ ( ) ( ) cos

( , )

(

..

co, )

2(

.

2s

}(2 )

) 1

2 2B

e

B

m q BB

B

e dx dQ p dk d k kd A y B y x

Qf f

x f x k

f x

y

Q

y

y Q k

k

y

2

3 2(0) (0;( , ) ) ( )

(2 ) 2ix y

Bp ik yp dy d y k

e N L y y Nf kk

x

(1)

(1

2

3 2

2

53 2)

( , )

( , )

( )(0) (0; ) ( )

(2 ) 2

( )(0) (0; ) ( )

(2 ) 2

ixp y ik y

iji jixp y ik

B

By

dy d y k D ye N L y y N

k

k D ydy d ye N L y y N

k

x k

x k

[Liang,Wang,2007]

(1) (1)Re( )Bx f

Azimuthal asymmetries at twist-4 level

Cross section for

Twist-4 parton correlation functions

(1) (1)2 2

(

2 2em 2

2

1) (1)

2

2

2

2 22 2

2

2

2 | |[1 (1 ) ] 4(2 ) 1 cos

|

( ,

|4(1 ) [ ]cos 2

| | 28(1 )

) ( , )

( , ) ( , )

( , ( , )])[

qBB

BB

B

B B

B

B

e kdy y y x

dxdyd k Q y Q

ky x

Q

d

dxdyd k

d

dxdyd

f x k f x k

x k x k

x k xy kk x

xQk

( )

(2, )2

2

22

2 2

2

2

| |2[1 (1

( , )

( , )) ]

B

B

B

LB

f x k

d

dxd

M

Q

ky

yd k Qxx k

(1) (1)2

22

2

2

( , ) ( , )cos

| | [ ]2(12

(( ,) )

)

1 1B B

B

Bk xy x k x k

f xy kQ

(1)2

(1)2

2 2

4 3

2{ } 5

4 3

2, (0) (0( , )

( , )

) (0; ) ( ) ,(2 ) 2

, (0) (0) (0; ) ( ) ,(2 ) 2

ixp y ik y

ix

B

kB

p y i y

k k k g dy d ye p s D L y y p s

k

ik k dy d ye p s D L yx k y s

k

x k

p

19

[YKS, Gao, Liang, Wang,2011] e N e q X

Doubly polarized at twist-3 e N e q X

[YKS, Gao, Liang, Wang, to be published]( ) ),(le N e q Xs

2 2em

1

2

2

1

2

2 2

2

2,

2 | |( ) ( ) ,

| | 2( ) ( ) ,

2 2

2 | |( )

cos

sin sin sin(2 )

sin ,

2

[( ) ]

UU l LU UT UL l LL l LT

UU

UT

q

B

B

B

B

T T Ts s

B

sT

L

L

LU

U

ed

dx dyd k Q y

x kA y B y

Q

k x M k kA y B y

M

F F s F F F s F

F f f

F

F

F

f f f f

f

Q M M

x kB y

Q

x

2 2

1

21 2

| |( ) ,

2 | |( ) ( ) ,

| | 2( ) ( )

sin

cos

cos2 2

cos cos 2[( ) ]

L LB

LL

LT s sT TB

T T s

kD y

Q

x kC y D y

Q

k x M k kC y

g

g g

g D yM Q M

g g gM

F

F

Leading twist

Twist-3 asymmetries

broadening of PDF in a nucleus

QCD multiple scattering cause broadending.

The form of broadening is simplified whenLocal color confinementA>>1Weak correlation between nucleons

If nucleon PDF take Gaussian form,

( , )qAf x k

22( ) /2

2

( , ) ( , )Fk l

F

Nq qA A

f x k d l e f x l

[Liang, Wang, Zhou, PRD2008]

2 /

( , ) ( )NqN

k

q

ef x k f x

2

2( )

2

/

, ((

( ) ))

FN

qF

kA

q

ef x k Af x

Tk

Tk

2 ˆ( )F d q

2F Broadening!

Nuclear modification of Nuclear twist-3/4 parton correlation function

Gaussian ansatz for distributionTake identical Gaussian parameter for parton

distribution/correlation functions

cc oos s, 2

22

22

( ) /2

2

( ) /(1)

2

22 (1)

2

2

222

( , ) ( , )

(ˆ2( )

, ) ( , )

F

F

k lA Nq q

l

F

A

F

k N

Af x k d l e f x l

Ax k d l e

k l

k

kl

l l

kx

2

2 2

cos cos 2,

cos cos 2eA eA

N F e Fe N

Tk

Suppressed!

Nuclear modification for

depend on dependence

Nuclear modification of sin LU

2F

2 22

2 2 2

sin 1 1 1 1exp .

sin( ) [( ) ]

eALU FeNLU F F F

k

sin LU

2(twist-2), (twist-3), F

dependence

Tk

Nuclear modification of sin LU

Sensitive to the ratio of ! /

Conclusions & outlooks Collinear expansion is naturally extended to SIDIS. Cross

section and azimuthal asymmetries for doubly polarized

are obtained up to twist-3, and unpolarized SIDIS up

to twist-4.

Much more abundant azimuthal asymmetries at high twist,

and their gauge invariant expressions are obtained.

Azimuthal asymetries act as a good probe of nuclear

properties. They are sensitive to Gaussian parameters of

HT correlation fuctions.

Numeric study of HT correlation functions, HT effects in

fragmentation functions, ,…, are underway.

ee q XN

Thanks for your attention!

e XN he