collinear expansion and sidis at twist-3
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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 SIDISCollinear 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, 2011Doublely 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 Ak 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 dWd 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 yD
y y y yyD y
L LL
: Projection operatork k xp
(0) (1, ) (2, ) (2, )ˆ , , ,L L M
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 s p k Ms kM
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
nxp
nxp
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 qM q x p s q x p kp q p q
k s ii k q x pM
g gq
gp
g
[ ] [ ] ( 2 ) ( 2 )· ·B i B iT
i iLgiM iq x p s q x p k
p q qg
p
(0)
2 0dWd
qk
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 kd y y y xdxdyd k Q y Q
ky xQ
ddxdyd k
ddxdyd
f x k f x k
x k x k
x k xy kk xxQk
( )
(2, )2
2
22
2 2
2
2
| |2[1 (1
( , )
( , )) ]
B
B
B
LB
f x k
ddxd
MQ
kyyd k Q
xx k
(1) (1)2
22
2
2
( , ) ( , )cos | | [ ]2(12(( ,) )
)1 1
B B
B
Bk xy x k x kf 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 y e p s D L y y p sk
ik k dy d y e p s D L yx k y sk
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
eddx dyd k Q y
x kA y B yQ
k x M k kA y B yM
F F s F F F s F
F f f
F
F
F
f f f f
f
Q M M
x k B yQ
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
k D yQ
x kC y D yQ
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: fN
q
& fAq
~ | (0) (0; ) ( ) |
~ | (0) (0; ) (
,
|) )
( )
( ,
N
A
q
q
f x k
f
N y y N
x yk A y A
LL
More FSI !
kT - broadening of PDF in a nucleusRelations between nucleon and nuclear PDFs
simplify under “maximal two-gluon exchange ” approximation.
It is just Gaussian broadening.2 /
( , ) ( )NqN
k
qef x k f x
22( )
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 lk
k ll lk
x
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!
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