Azimuthal Asymmetries In

1
Sino-German Workshop on Frontiers in QCD

• Azimuthal Asymmetries In
• Semi-inclusive Deep-Inelastic Scattering

Liang Zuo-tang, Shandong University ???? ???
Collaborator Wang Xin-Nian (LBNL)

• Contents
• Introduction
• Collinear expansion in Semi-Inclusive DIS
• Results and discussions

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Sino-German Workshop on Frontiers in QCD

• Introduction

• Intrinsic transverse momentum
• Gauge link

3
Azimuthal asymmetry in unpolarized SiDIS

• 1977 Georgi Politzer, Clean test to pQCD,
  Phys. Rev. Lett. 40,3 (77).

lepton plan
4
Azimuthal asymmetry in unpolarized SiDIS

• 1978 Cahn, Intrinsic momentum effects. Phys.
  Lett. B78B, 269 (1978).

Generalized parton model to include an intrinsic
transverse momentum but no gluon
radiation in the final state
Realistic studies Monte-Carlo, experiments
etc.
small ph?, moderate Q, intrinsic k?
contribution dominates large ph ?, large Q,
Georgi-Politzer mechanism dominates.
In DIS at intermediate energies intrinsic
transverse momentum k? effects have to be taken
into account.
5
Azimuthal asymmetry in polarized SiDIS

• Motivation Single-spin asymmetry in pp
  collisions
• azimuthal
  asymmetry in singly polarized pp collisions

up
left
spin
forward
right
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Azimuthal asymmetry in polarized SiDIS

• History theoretical studies on single-spin
  asymmetry in p(?)p??X

Factorization
1978 Kane, Pumplin, Repko, pQCD
aNq(?)q?qq0.
1991 Sivers, asymmetric quark distribution
(Sivers effect)
1993 Boros, Liang, Meng, quark orbital angular
momentum
surface 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, 1993s proof is wrong because
forgot gauge link,
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Azimuthal asymmetry in polarized SiDIS

• Conclusion

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
8
Semi-inclusive DIS with A BASIC QUESTION

• The general framework to write the cross section

Cross section for ep?ehX
Parton distributions
Cross section for eq?eq without
Frag. function for q?hX

?
?
(without transverse momentum collinear
approximation)
?
(with transverse momentum, with gauge link)
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Sino-German Workshop on Frontiers in QCD

• Gauge link and
• Collinear expansion in DIS and SiDIS.

10
Inclusive DIS where does the gauge link come
from?

• DIS without QCD interaction

The hard part
contracted with the leptonic tensor
The matrix element
parton distributions
No QCD interactions. Not (color) gauge invariant.
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Inclusive DIS with QCD interaction
...
Not gauge invariant!
parton correlation function
12
Inclusive DIS with QCD interaction
Ellis, Furmanski, Petronzio, (82) Qiu, Sterman
(90,91)

• Collinear expansion

? Expanding the hard parts around k xp
? Decomposition of the gluon field
? Using the identities such as,
to replace the derivatives etc.
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Inclusive DIS with QCD interaction
Contain QCD interactions. (Color) gauge invariant
!
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Inclusive DIS with QCD interaction
Definition of parton distributions
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Inclusive DIS with QCD interaction

• Conclusion
• Collinear expansion is the necessary procedure to
  obtain the gauge invariant parton distributions.

Question How about semi-inclusive DIS?
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Semi-Inclusive DIS with QCD interaction

• Consider first e p?e q X, i.e., no
  fragmentation

...
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Semi-Inclusive DIS with QCD interaction

• The same collinear expansion technique leads to

Gauge invariant parton distributions/correlations
(with gauge link)
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Sino-German Workshop on Frontiers in QCD

• Consequences and discussions

19
SiDIS with Direct consequence I

• Consider the contribution from

Cross section for ep?eqX
Parton distributions
Cross section for eq?eq without

?
(without transverse momentum)
?
(with transverse momentum)
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SiDIS with Direct consequence II

• A great simplification of
  

left cut
right cut
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SiDIS with Direct consequence II

• This leads to

Only
is relevant in semi-inclusive deep-inelastic
lepton-nucleon scattering.
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SiDIS with differential cross-section to
1/Q

• Further calculations in the standard way

? Expanding the matrices ?s in terms of ?s
? The general Lorentz-structure of the
coefficient functions are
un-integrated parton distribution
Sivers function
? Equation of motion
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SiDIS with differential cross-section to
1/Q
To the order of 1/Q
should be used to extract Sivers function and so
on.
where is the azimuthal angle of
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SiDIS with An observation
By integrating over
a single-spin asymmetry at fixed k-
the terms vanish if integrated over k-
from -8 to 8, no single-spin
asymmetry for inclusive DIS, but
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Single-spin asymmetry for the triggered
inclusive DIS

• Consider

a jet of hadrons
It corresponds to a triggered inclusive DIS,
i.e.,
a large momentum hadron in photon direction
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Single-spin asymmetry for the triggered
inclusive DIS
At the hadron level, it corresponds to
Measurements by e.g. HERMES at HERA?
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Conclusions and Outlook

• Collinear expansion can be extended to SiDIS
• Naïve extension of parton distribution
  convoluting with eq ? eq cross section to include
  intrinsic transverse momentum is incorrect.
• The existence of the projection operator
  simplifies very much.
• A new single-spin asymmetry is predicted to 1/Q.
• Extending to include fragmentation is underway.