8 th circum-pan-pacific symposium on high energy spin physics
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page12011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)
page28th 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
page38th 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
48th Circum Pan-Pacific Spin Symposium
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
58th Circum Pan-Pacific Spin Symposium
2011年 6月 Cairns, Austrilia
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.
62011年 6月 Cairns, Austrilia
8th Circum Pan-Pacific Spin Symposium
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.
78th Circum Pan-Pacific Spin Symposium
2011年 6月 Cairns, Austrilia
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?
page88th 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
Look at the inclusive DIS first.
98th Circum Pan-Pacific Spin Symposium
2011年 6月 Cairns, Austrilia
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
108th Circum Pan-Pacific Spin Symposium
2011年 6月 Cairns, Austrilia
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
Inclusive DIS with QCD interaction
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
138th Circum Pan-Pacific Spin Symposium
2011年 6月 Cairns, Austrilia
Inclusive DIS with QCD interaction
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
148th Circum Pan-Pacific Spin Symposium
2011年 6月 Cairns, Austrilia
Inclusive DIS with QCD interaction
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).
158th Circum Pan-Pacific Spin Symposium
2011年 6月 Cairns, Austrilia
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.:
…
168th Circum Pan-Pacific Spin Symposium
2011年 6月 Cairns, Austrilia
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
178th Circum Pan-Pacific Spin Symposium
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
Semi-Inclusive DIS with QCD interaction
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
2011年 6月 Cairns, Austrilia
188th Circum Pan-Pacific Spin Symposium
(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
2011年 6月 Cairns, Austrilia
Semi-Inclusive DIS with QCD interaction
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
page198th 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
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).
208th Circum Pan-Pacific Spin Symposium
2011年 6月 Cairns, Austrilia
Semi-Inclusive DIS with QCD interaction
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
218th Circum Pan-Pacific Spin Symposium
2011年 6月 Cairns, Austrilia
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
228th Circum Pan-Pacific Spin Symposium
2011年 6月 Cairns, Austrilia
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
238th Circum Pan-Pacific Spin Symposium
2011年 6月 Cairns, Austrilia
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
248th Circum Pan-Pacific Spin Symposium
2011年 6月 Cairns, Austrilia
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
258th Circum Pan-Pacific Spin Symposium
2011年 6月 Cairns, Austrilia
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.
262011年 6月 Cairns, Austrilia
8th Circum Pan-Pacific Spin Symposium
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).
278th Circum Pan-Pacific Spin Symposium
Nuclear dependence of the azimuthal asymmetries
2011年 6月 Cairns, Austrilia
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
page288th 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
e N e q X
292011年 6月 Cairns, Austrilia
8th Circum Pan-Pacific Spin Symposium
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
308th Circum Pan-Pacific Spin Symposium
2011年 6月 Cairns, Austrilia
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!
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