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2006年9月 DESY Sino-German Workshop 梁作堂(Liang Zuo-tang) 山东大学(Shandong University)
1
Azimuthal Asymmetries InAzimuthal Asymmetries In Semi-inclusive Deep-Inelastic ScatteringSemi-inclusive Deep-Inelastic Scattering
Sino-German Workshop on Frontiers in QCD
Liang Zuo-tang, Shandong UniversityLiang Zuo-tang, Shandong University山东大学 梁作堂山东大学 梁作堂
Contents Introduction Collinear expansion in Semi-Inclusive DIS Results and discussions
Collaborator: Wang Xin-Nian (LBNL)
2006年9月 DESY Sino-German Workshop 梁作堂(Liang Zuo-tang) 山东大学(Shandong University)
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Introduction:
Sino-German Workshop on Frontiers in QCD
Intrinsic transverse momentum Gauge link
2006年9月 DESY Sino-German Workshop 梁作堂(Liang Zuo-tang) 山东大学(Shandong University)
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1977: Georgi & Politzer, “Clean test to pQCD”, Phys. Rev. Lett. 40,3 (77).1977: Georgi & Politzer, “Clean test to pQCD”, Phys. Rev. Lett. 40,3 (77).
Azimuthal asymmetry in Azimuthal asymmetry in unpolarizedunpolarized SiDIS SiDIS
, largefor )1()(
)(
2~cos 2/1 zz
yA
yB
,)1(1)( 2yyA ,)1)(2(2)( 2/1yyyB
hp
*
e
e lepton plan
hp
Xhepe
2006年9月 DESY Sino-German Workshop 梁作堂(Liang Zuo-tang) 山东大学(Shandong University)
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1978: Cahn, Intrinsic momentum effects. Phys. Lett. B78B, 269 (1978).1978: Cahn, Intrinsic momentum effects. Phys. Lett. B78B, 269 (1978).
Azimuthal asymmetry in Azimuthal asymmetry in unpolarizedunpolarized SiDIS SiDIS
Generalized parton model to include an intrinsic transverse momentum but no gluon radiation in the final state:
“Realistic studies”: Monte-Carlo, experiments etc.
In DIS at intermediate energies: intrinsic transverse momentum k effects have to be taken into account.
)(
)(||cos
yA
yB
Q
k
k
small ph, moderate Q, intrinsic k contribution dominates; large ph , large Q, Georgi-Politzer mechanism dominates.
2006年9月 DESY Sino-German Workshop 梁作堂(Liang Zuo-tang) 山东大学(Shandong University)
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Azimuthal asymmetry inAzimuthal asymmetry in polarized polarized SiDISSiDIS
Motivation: Single-spin asymmetry in pp collisionsMotivation: Single-spin asymmetry in pp collisions azimuthal asymmetry in singly polarized pp collisionsazimuthal asymmetry in singly polarized pp collisions
Xπpp )(
RL
RLN NN
NNA
up
forward
left
right
spin
2006年9月 DESY Sino-German Workshop 梁作堂(Liang Zuo-tang) 山东大学(Shandong University)
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History: theoretical studies on single-spin asymmetry in p(History: theoretical studies on single-spin asymmetry in p()+p)+p+X+X
1978: Kane, Pumplin, Repko, pQCD aN[q()+qq+q]=0.
Azimuthal asymmetry in Azimuthal asymmetry in polarized polarized SiDISSiDIS
Factorization )..(ˆ.)..( )()( funfragdfdpd XqqqXπpp
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”
2006年9月 DESY Sino-German Workshop 梁作堂(Liang Zuo-tang) 山东大学(Shandong University)
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Conclusion:Conclusion:
Azimuthal asymmetry in Azimuthal asymmetry in polarized polarized SiDISSiDIS
both Sivers and Collins effects can exist when gauge link is taken into account.
intrinsic transverse momentum and gauge link are important in studying azimuthal asymmetries in SiDIS
2006年9月 DESY Sino-German Workshop 梁作堂(Liang Zuo-tang) 山东大学(Shandong University)
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Semi-inclusive DIS with : Semi-inclusive DIS with : A BASIC QUESTION
The general framework to write the cross section
Cross section for epehX
Parton distributions
Cross section for eqeq without
)(xq )(zD Xhq
),( kxq
kxpkeqeqd |)( ),(
FXhq kzD
= k
(with transverse momentum, with gauge link)
(without transverse momentum: collinear approximation)
xpkeqeqd |)(
Frag. function for qh+X
)( Xheped
?
k
2006年9月 DESY Sino-German Workshop 梁作堂(Liang Zuo-tang) 山东大学(Shandong University)
9
Gauge link and
Collinear expansion in DIS and SiDIS.
Sino-German Workshop on Frontiers in QCD
2006年9月 DESY Sino-German Workshop 梁作堂(Liang Zuo-tang) 山东大学(Shandong University)
10
InclusiveInclusive DIS: DIS: where does the gauge link come from?where does the gauge link come from?
DIS without QCD interactionDIS without QCD interaction
)],(ˆ)(ˆTr[)2(
),,( 4
4
qkHk,p,Sφkd
SpqW
SpzψψSpezdk,p,Sφ ikz ,|)()0(|,)(ˆ 4
))(()2()(),(ˆ 2qkqkqkH
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
2006年9月 DESY Sino-German Workshop 梁作堂(Liang Zuo-tang) 山东大学(Shandong University)
11
Inclusive DIS with QCD interactionInclusive 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 correlation function
2006年9月 DESY Sino-German Workshop 梁作堂(Liang Zuo-tang) 山东大学(Shandong University)
12
Inclusive DIS with QCD interactionInclusive DIS with QCD interaction
Collinear expansion:
......),(ˆ
),(ˆ),(ˆ'1
'
1
21(1)
21(1)(1)
kk
pxpxHpxpxHqkH σ
ρμνρ
μνρ
μν
......)(ˆ
)(ˆ),(ˆ'
')0(
)0()0(
ρρ
μνμνμν k
k
xpHxpHqkH
Decomposition of the gluon field:
)()()( yAωpn
pyAnyA ρ'
ρ'ρ
ρρ
ρ'ρ
ρ'ρ
ρ'ρ nngω
Using the identities such as,
),,(ˆ)(ˆ(1)
(0)
xpxpHk
xpH ρμνρ
μν
pkx /
Ellis, Furmanski, Petronzio, (82) Qiu, Sterman (90,91)
Expanding the hard parts around k =xp:
to replace the derivatives etc.
ixx
pxHpxpxHp μνρ
μν
12
1(0)
21L)(1, )(ˆ
),(ˆ
)(' xpkkω ρ'ρ
)(2
130 kkk
)0,1,0(
n
)0,0,1(
n
2006年9月 DESY Sino-German Workshop 梁作堂(Liang Zuo-tang) 山东大学(Shandong University)
13
Inclusive DIS with QCD interactionInclusive DIS with QCD interaction
SpzψzψSpezdk,p,S ikz ,|)(),0()0(|,)(ˆ 4)0( L
)](ˆ)(ˆTr[)2(
),,(~ )0()0(
4
4)0( xpHk,p,S
kdSpqW μνμν
...),,(~
),,(~
),,(~
),,( )2()1()0( SpqWSpqWSpqWSpqW μνμνμνμν
SpzψzyyDyψSpyedzdp,S,kk ρyzikyik
ρ ,|)(),()(),0()0(|,),(ˆ )(4421
)1( 21 LL
]),(ˆ),(ˆTr[)2()2(
),,(~
21)1(
21)1(
42
4
41
4)1( ρ'
ρρ
μνρ'μν ωpxpxHp,S,kkkdkd
SpqW
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
2006年9月 DESY Sino-German Workshop 梁作堂(Liang Zuo-tang) 山东大学(Shandong University)
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Inclusive DIS with QCD interactionInclusive DIS with QCD interaction
...),,(~
),,(~
),,(~
),,( )2()1()0( SpqWSpqWSpqWSpqW μνμνμνμν
)](ˆ)(ˆTr[)](ˆ)(ˆTr[)2(
),,(~ )0()0()0()0(
4
4)0( xpHx,p,SdxxpHk,p,S
kdSpqW μνμνμν
)(ˆ)()2(
)(ˆ )0(4
4)0( k,p,S
p
kx
kdx,p,S
SpzψzψSpedz zixp ,|)0,,0(),0()0(|,2
L
...)/2()(ˆ )0()0( x,p,Sx,p,S ...)()( 1
)0( xfpx,p,S
Spzψp
zψSpedz
xf zixp ,|)0,,0(2
),0()0(|,2
)(1
L
Definition of parton distributions
2006年9月 DESY Sino-German Workshop 梁作堂(Liang Zuo-tang) 山东大学(Shandong University)
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Inclusive DIS with QCD interactionInclusive DIS with QCD interaction
Conclusion: Conclusion:
Collinear expansion is the necessary procedure Collinear expansion is the necessary procedure to obtain the gauge invariant parton to obtain the gauge invariant parton distributions.distributions.
Question: How about semi-inclusive DIS?
2006年9月 DESY Sino-German Workshop 梁作堂(Liang Zuo-tang) 山东大学(Shandong University)
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Semi-InclusiveSemi-Inclusive DIS with QCD interaction DIS with QCD interactionConsider first e +pe + q + X, i.e., no fragmentation
)()2(,|)0(|,','|)0(|,),',,( 44)(X
X
si pqpSpJXkXkJSpSkpqW
)],(ˆ)(ˆTr[)2(
)',,,( )0()0(4
4),0( qkHk,p,S
kdkSpqW si
...)',,,()',,,()',,,()',,,( ),2(),1(),0()( kSpqWkSpqWkSpqWkSpqW sisisisi
42
4
41
4),1(
)2()2()',,,(
kdkd
kSpqW siμν
)'()2)(2( 33' qkkEk
)],,(ˆ),(ˆTr[ 21),1(
21)1(
RL,c
qkkHp,S,kk ρcμνρ
)'()2)(2( 33
' qkkE ck
… ...
),,()0( SpqW ),,()1( SpqW
2006年9月 DESY Sino-German Workshop 梁作堂(Liang Zuo-tang) 山东大学(Shandong University)
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Semi-Inclusive DIS with QCD interactionSemi-Inclusive DIS with QCD interaction
The same collinear expansion technique leads toThe same collinear expansion technique leads to
)'()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 ωpxpxHp,S,kkkdkd
kSpqW
)](ˆ)(ˆTr[)2(
)',,,(~ )0()0(
4
4),0( xpHk,p,S
kdkSpqW μν
siμν
)'()2)(2( 33' qkkE ck
Gauge invariant parton distributions/correlations (with gauge link)
2006年9月 DESY Sino-German Workshop 梁作堂(Liang Zuo-tang) 山东大学(Shandong University)
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Consequences and discussions
Sino-German Workshop on Frontiers in QCD
2006年9月 DESY Sino-German Workshop 梁作堂(Liang Zuo-tang) 山东大学(Shandong University)
19
SiDIS with : SiDIS with : Direct consequence IDirect consequence I
Consider the contribution from Consider the contribution from
Cross section for epeqX
Parton distributions
Cross section for eqeq without
)(xq
),( kxq
xpkeqeqd |)(
=k
(with transverse momentum)
(without transverse momentum)
)( Xqeped
xpkeqeqd |)(
)',,,(~ ),0( kSpqW si
k
2006年9月 DESY Sino-German Workshop 梁作堂(Liang Zuo-tang) 山东大学(Shandong University)
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SiDIS with : SiDIS with : Direct consequence IIDirect consequence II
A great simplification of : A great simplification of : '
21),1( ),(ˆ ρ
ρρL ωpxpxH
'21
),1( ),(ˆ ρρ
ρR ωpxpxH
k
)',,,(~ ),1( kSpqW si
μν
left cut
right cut
nnωxxδ
pqρ ρ
ρB
'1 )(
2
)(ˆ1
')1( pxH ρ
nnωxxδ
pqρ ρ
ρB
'2 )(
2
' 02
)1(0 )(ˆ
pxH ρ
2006年9月 DESY Sino-German Workshop 梁作堂(Liang Zuo-tang) 山东大学(Shandong University)
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SiDIS with : SiDIS with : Direct consequence IIDirect consequence II
This leads toThis leads to
)'()2)(2)]((ˆ)(ˆTr[)2(
Re2 33'
)1()1(4
4
qkkExpHk,p,Skd
kρ
μνρ
SpzψzDzψSpezdp,Sk,kkd
p,Sk ρikz
ρρ ,|)()(),0()0(|,),(ˆ)2(
),(ˆ 42
)1(4
24
)1( L
Only
k
] ),(ˆ ),(ˆ[Tr)2()2(
)',,,(~
21),1(
21)1(
RL,c4
24
41
4),1( ρ'
ρρc
μνρ'si
μν ωpxpxHp,S,kkkdkd
kSpqW
)'()2)(2( 33' qkkE ck
is relevant in semi-inclusive deep-inelastic lepton-nucleon scattering.
2006年9月 DESY Sino-German Workshop 梁作堂(Liang Zuo-tang) 山东大学(Shandong University)
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SiDIS with : differential cross-section to 1/QSiDIS with : differential cross-section to 1/Q
Further calculations in the standard way: Further calculations in the standard way:
,2/)()(ˆ )0()0( k,p,Sk,p,S 2/])(~)([)(ˆ
5)1()1()1(
k,p,Sk,p,Sk,p,S
MfSkpfkfpk,p,S T /)( 1''
1)0(
.../)( )1(2
2)1(1
)1()1( MSgkkkpSMppkk,p,S ss
.../~)(~~)(~ )1(2
2)1(1
)1()1( MSgkkkipSiMpkipk,p,S ss
Expanding the matrices ’s in terms of ’s:
The general Lorentz-structure of the coefficient functions are:
k
un-integrated parton distribution Sivers’ function
Equation of motion)1(1
)1(112
~ssTf
M
kxp
,~ )1()1(
xf
2006年9月 DESY Sino-German Workshop 梁作堂(Liang Zuo-tang) 山东大学(Shandong University)
23
SiDIS with : differential cross-section to 1/QSiDIS with : differential cross-section to 1/Q
To the order of 1/Q:
)sin(||
)( )2(
2114
22
4
22
sTBqem f
M
kfyA
kddkpdQdx
Q
ed
k
sssM
k sin)~3(2
)1(2
)1(22
2
should be used to extract Sivers function and so on.
cos)~(||2
)( )1(
fx
M
k
Q
MyB B
)2sin()~(2
)1(2
)1(22
2
sssM
k
where is the azimuthal angle of s S
2006年9月 DESY Sino-German Workshop 梁作堂(Liang Zuo-tang) 山东大学(Shandong University)
24
SiDIS with : SiDIS with : An observationAn observationk
By integrating overBy integrating over
sssQM
kyB sin)~3(
2)( )1(
2)1(2
2
the terms vanish if integrated over k- from -∞ to +∞,
no single-spin asymmetry for inclusive DIS, but … …
14
22
4
22
)( )2(
fyAkddkpdQdx
Q
ed Bqem
ssin
a single-spin asymmetry at fixed k-
2006年9月 DESY Sino-German Workshop 梁作堂(Liang Zuo-tang) 山东大学(Shandong University)
25
Single-spin asymmetry for the “triggered” inclusive DIS
Consider Consider
sss
q
qem
B QM
kyBfyAkd
dkp
Q
e
dQdx
d
sin)~3(
2)()(
)2()1(2
)1(2
2
1
0
234
22
2
Xhepe trigger )(
a large momentum hadron in photon direction
2/32/
2/32/)(
||
||
ss
ss
dd
dd
dd
ddA trig
N
1
0
2
0
)1(2
)1(22
22
)(
)~3(2
)(
fkddkyA
Mk
kddkQ
MyB
q
q
ss
Xrealqepe )()(
a jet of hadrons
It corresponds to a “triggered” inclusive DIS, i.e.,
2006年9月 DESY Sino-German Workshop 梁作堂(Liang Zuo-tang) 山东大学(Shandong University)
26
Single-spin asymmetry for the “triggered” inclusive DIS
Xhepe trigger )(
q
sssB M
kkddkGx
0
)1(2
)1(22
22 )~3(
At the hadron level, it corresponds to
),()]2()2([2
1),,(
2
1 2),( QxGSpxqpxqqp
qSpW BsBBtrigs
1
)( )(F
G
Q
MxyB
dd
ddA sBtrig
N
Measurements by e.g. HERMES at HERA?
2006年9月 DESY Sino-German Workshop 梁作堂(Liang Zuo-tang) 山东大学(Shandong University)
27
Conclusions and Outlook
Collinear expansion can be extended to SiDIS;Collinear expansion can be extended to SiDIS;
Naïve extension of parton distribution convoluting Naïve extension of parton distribution convoluting with with eq eq eq eq cross section to include intrinsic cross section to include intrinsic transverse momentum is incorrect. transverse momentum is incorrect.
The existence of the projection operator The existence of the projection operator simplifies very much.simplifies very much.
A new single-spin asymmetry is predicted to 1/Q.A new single-spin asymmetry is predicted to 1/Q.
Extending to include fragmentation is underway.Extending to include fragmentation is underway.
' ρρω
)',,,(~ ),1( kSpqW si
μν
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