Azimuthal Asymmetry in Semi-Inclusive Deep-Inelastic Lepton-Nucleon Scattering

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• Azimuthal Asymmetry in Semi-Inclusive
  Deep-Inelastic Lepton-Nucleon Scattering

??? (Liang Zuo-tang)
???????? (School of Physics, Shandong University)
??, 2010?4?18?
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Contents

• Introduction
• Collinear expansion in Semi-inclusive deeply
  inelastic eN scattering (SIDIS)
• Azimuthal asymmetry in SIDIS
• Nuclear dependence of the azimuthal asymmetry in
  SIDIS
• Summary and outlook

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Introduction

• High energy reactions

Structure of Nucleon
Inclusive Deep-Inelastic Scattering (DIS)
Parton Distribution Functions
Semi-Inclusive Deep-Inelastic Scattering (SIDIS)
Parton Correlation Functions e.g. Sivers function
... ... ... ...
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Introduction

• Tasks

(1) Establish the relation between the measurable
quantities and the parton distribution and/or
correlation functions (2) Phenomenology of the
parton distribution and correlation
functions (3) Physical interpretation of the
parton distribution and correlation functions.
For inclusive DIS
(1)
(2) Global fit
PDF parametrizations
(3) Simple probability interpretaion of the PDFs
azimuthal asymmetries are examples of the
sensitive measurable quantities.
For SIDIS
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Azimuthal asymmetry in unpolarized SIDIS

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

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Azimuthal asymmetry in unpolarized SIDIS

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

(higher twist, power suppressed)
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Azimuthal asymmetry in polarized SIDIS

• History triggered by single-spin asymmetry in
  p(?)p??X

1993, Collins Proof of non-existence using the
definition of p.d.f.
2002, Brodsky, Hwang, Schmidt an explicit
example of the existence of the asymmetry by
taking quark orbital angular momentum
final state interaction (multi-parton
scattering) into account.
Collins 1993s proof is wrong because forgot
gauge link.
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Azimuthal asymmetry in SIDIS

• Lessons

intrinsic transverse momentum (higher twist) and
gauge link are important in studying azimuthal
asymmetries in SIDIS.
Questions
What is gauge link? where does it come from? What
is the relation between the asymmetries and
higher twist parton correlation functions?
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• Gauge link in p.d.f. and
• Collinear expansion in DIS and SIDIS

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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
Parton distribution/correlation
Not gauge invariant!
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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
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Inclusive DIS with QCD interaction
Contain QCD interactions. (Color) gauge invariant
!
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Semi-Inclusive DIS with QCD interaction

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

We successfully showed that, the same collinear
expansion technique applies, and it leads to
Liang Wang, PRD (2007).
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Some of the consequences (1) SIDIS
ep?eqX upto twist-3
Liang and Wang, Phys. Rev. D
(2007) (2) Transverse momentum broadening
in nucleus
Liang, Wang Zhou, Phys. Rev. D (2008)
(3) Nuclear dependence of the azimuthal
asymmetry.
Gao, Liang and Wang, arXiv 1001.3146 hep-ph
. (4) Twist-4 contributions and
?cos2??
Song, Gao, Liang and Wang, in
preparation.
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SIDIS with (leading twist) 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 (twist 3) Direct consequence II

• A simplification of

left cut
independent of x2!
independent of x1!
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SIDIS with (twist 3) Direct consequence II

• This leads to

Only
is relevant in semi-inclusive deep-inelastic
lepton-nucleon scattering.
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SIDIS with (twist 3) differential
cross-section to 1/Q

• Unpolarized SIDIS

? Expanding the matrices ?s in terms of ?s
? The general Lorentz-structure of the
coefficient functions are
un-integrated parton distriution/correlation
functions
? Equation of motion
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SIDIS with (twist 3) differential
cross-section to 1/Q

• Unpolarized SIDIS

Differential cross section
Azimuthal asymmetry
Transverse momentum dependent (TMD) quark
distribution
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Transverse momentum broadening in nucleus
Transverse momentum dependent (TMD) quark
distribution
Gauge link comes from
Replace N by A, the gluons can connect to
different nucleons in A.
Nuclear enhancement or transverse momentum
broadening
Transverse momentum broadening should be obtained
from the gauge link.
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Transverse momentum dependent PDF in nucleus
With maximal two gluon approximation,
i.e., only maximal two gluons are connected
to one nucleon
transverse momentum broadening squared
nucleon density in nucleus A
gluon distribution function in nucleon N
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Nuclear dependence of the azimuthal asymmetry
Azimuthal asymmetry in
Azimuthal asymmetry in
Nuclear dependence of TMD parton
distribution/correlation functions
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Nuclear dependence of the azimuthal asymmetry
An example take a Gaussian for the transverse
momentum dependence
for the case that
suppressed!
Integrated over
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SIDIS with (twist 4) differential
cross-section to 1/Q2

• Unpolarized SIDIS

Differential cross section (up to twist 4)
The asymmetry
Another parton correlation function
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Summary

• Progresses have been made in finding the correct
  relation between measurable quantities in SIDIS
  and parton correlation functions.

• Collinear expansion has been extended to SIDIS
  ep?eqX.
• Naïve extension of TMD parton distributions
  convoluting with eq ? eq cross section to
  include intrinsic transverse momentum is
  incorrect.
• The existence of the projection operator
  simplifies the higher twist calculations very
  much. Calculations up to twist 4 have been
  carried out.
• In maximal two gluon approximation,
• Azimuthal asymmetry in eA is suppressed, e.g.

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Outlook

• (1) A complete calculation including
  fragmentation, both in unpolarized and polarized
  case, also to other reactions
• (2) Phenomenology of the related parton
  correlation functions
• (3) Physical interpretation of the related parton
  correlation functions.

Thank you for your attention!