Yu-kun Song (USTC)2013.10.28 Jinan

YKS, Jian-hua Gao, Zuo-tang Liang, Xin-Nian Wang, arXiv:1308.1159

Collinear expansion and SIDIS at twist-3

Outline

Introduction

Collinear expansion extended to SIDIS

Azimuthal asymmetries & nuclear effects

Discussions and outlook

A large scale (Q) and a small scale (k)Large scale → factorizationSmall scale → structure information (intrinsic k )

An ideal probe of nucleon/nuclear structure !TMD factorization works at leading twistHigher twist

Gauge invariant parton correlation functionsFactorization proof/argumentsNLO calculations

Experiments : Compass, Jlab, EIC,…

Semi-inclusive DIS

X

2 2ˆˆ O

Q

O

Sterman-Libby power counting

Leading twist

Gauge invariant parton distribution functionsFinite, perturbatively calculable partonic cross

section

Leading twist: collinear approximation

Higher twist: collinear expansion

Gauge invariant parton correlation functionsFinite, perturbatively calculable partonic cross

section

2

3 43 4ˆ ˆˆ ˆO O

Q Q Q

O

X

Higher twist (1/Q power corrections)

Systematic applications to SIDIS

Collinear expansion in DIS Ellis, Furmanski, Petronzio, 1982, 1983 Qiu, 1990

Collinear expansion applied to SIDIS e+N →e+q+X Liang, Wang, 2006

Nuclear medium effects of azimuthal asymmetriesGauge link → nuclear modification of PDFs Liang, Wang, Zhou, 2008

SIDIS e+N →e+q+X at twist-4 YKS, Gao, Liang, Wang, 2011

Doublely polarized SIDIS e+N →e+q+X at twist-3 YKS, Gao, Liang, Wang, Arxiv: 1308.1159

[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 SIDIS

Parton 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, 2006]

k

: color gauge invariant

Hadronic tensor for SIDIS(0) (1, ) (2, )

2 2 2 2, , ,

2 (0)(0)

2

2 (1, )(1)

2

2 (2, )(1)

2

(0)

(1, )

2

...

ˆ ( , )

ˆ ( ,

1 ˆTr ,2

1 ˆTr ,4

)

1 ˆ{Tr(2 )

[ ]

[ ]

[

c c

c L R c L R M

L

L

NB

L NB

x

dW dW dW dW

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

x k

(2, )

(2, )(

(2, )

2)

2 (2, )(2)

2 2

ˆ ( , )

ˆ ( , )

ˆ ( ,

ˆ Tr },

1 ˆTr .(2

))

]

[ ]

[ ]

L NB

L NB

MM

NB

N

x k

d Wh

d k

x k

x kq p

2,3,4

3,4,

4,

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

(0) (0; ) (

)

(0) (0 )) ,

D y

D

y y y y

yD y

L L

L

: Projection operator

k k xp

(0) (1, ) (2, ) (2, )ˆ , , ,L L M

Structure of correlation matrices

Time 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

Structure of correlation matrices

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, )

(1, ) (

(0)

1 )(0) ,

Re Im

Re Im

L L

L L

nx

p

nx

p

Relations from QCD EOM

Sum 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

Unpolarized SIDIS at twist-4 levelCross 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

13

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

Doubly polarized e+N →e+q+X at twist-3

[YKS, Gao, Liang, Wang, 2013]( ) ),(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

kT - broadening of PDF in a nucleus

Two facts about the gauge linkGenerated by QCD multiple gluon scattering

between struck quark and mediumIt induce physical effects, cannot be removed by

a wise choice of gauge

→ Different interaction induce different PDFs: fNq

& fA

q

~ | (0) (0; ) ( ) |

~ | (0) (0; ) (

,

|) )

( )

( ,

N

A

q

q

f x k

f

N y y N

x yk A y A

L

L

More FSI !

kT - broadening of PDF in a nucleus

Relations between nucleon and nuclear PDFs simplify under “maximal two-gluon exchange ” approximation.

It is just Gaussian broadening.2 /

( , ) ( )NqN

k

q

ef x k f x

2

2( )

2

/

, ((

( ) ))

FNq

F

kAq

ef x k Af x

2 ˆ( )F d q

2F

22( ) /2

2

( , ) ( , )

Fk lqA N

qF

Af x k d l e f x l

More FSI diffuse the scattered parton!

Nuclear modification of ˂cosφ>˂cos2φ>

Nuclear twist-3/4 parton correlation function

Gaussian ansatz for distributionTake identical Gaussian parameter for parton

distribution/correlation functions

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

˂cosφ>˂cos2φ> are Suppressed!

Nuclear modification for

depend on dependence

Nuclear modification of ˂sinφ>LU

sin LU

2F

2 22

2 2 2

sin 1 1 1 1exp .

sin( ) [( ) ]

eALU FeNLU F F F

k

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

kT - dependence

Nuclear modification of ˂sinφ>LU

Sensitive to the ratio of γ/α !

Discussions and outlook

Collinear expansion is the systematic and essential method

to calculate higher twist effects to SIDIS.

Gauge invariance of correlation functions are automatically

fulfilled as a result of collinear expansion.

Azimuthal asymmetries for doubly polarized e+N →e+q+X are

obtained up to twist-3, and for unpolarized case up to twist-4.

Much more abundant azimuthal asymmetries at high twist,

and their gauge invariant expressions are obtained.

Extension to hadronic production process are interesting and

are underway.Thanks for your attention!