1. 高能物理学会第八届全国会员代表大会暨学术年会
Azimuthal Asymmetry in Semi-Inclusive Deep-Inelastic Lepton-Nucleon Scattering
梁作堂 (Liang Zuo-tang)
山东大学物理学院
(School of Physics, Shandong University)
南昌, 2010年4月18日
山东大学(Shandong University)
2010年4月　高能物理学会学术年会 南昌

2. 山东大学(Shandong University)
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
山东大学(Shandong University)
2010年4月　高能物理学会学术年会 南昌

3. 山东大学(Shandong University)
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
山东大学(Shandong University)
2010年4月　高能物理学会学术年会 南昌

4. 山东大学(Shandong University)
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 PDF’s
azimuthal asymmetries are examples of the sensitive measurable quantities.
For SIDIS
山东大学(Shandong University)
2010年4月　高能物理学会学术年会 南昌

5. Azimuthal asymmetry in unpolarized SIDIS
1977: Georgi & Politzer, “Clean test to pQCD”, Phys. Rev. Lett. 40,3 (77).
lepton plane
山东大学(Shandong University)
2010年4月　高能物理学会学术年会 南昌

6. 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:
(higher twist, power suppressed)
山东大学(Shandong University)
2010年4月　高能物理学会学术年会 南昌

7. Azimuthal asymmetry in polarized SIDIS
lepton plane
defined w.r.t. the lepton plane
defined w.r.t. the polarization direction
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: 1993’s proof is wrong because forgot gauge link.
山东大学(Shandong University)
2010年4月　高能物理学会学术年会 南昌

8. 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?
山东大学(Shandong University)
2010年4月　高能物理学会学术年会 南昌

9. 高能物理学会第八届全国会员代表大会暨学术年会
Gauge link in p.d.f. and
Collinear expansion in DIS and SIDIS
山东大学(Shandong University)
2010年4月　高能物理学会学术年会 南昌

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.
山东大学(Shandong University)
2010年4月　高能物理学会学术年会 南昌

11. Inclusive DIS with QCD interaction…
Parton distribution/correlation:
Not gauge invariant!
山东大学(Shandong University)
2010年4月　高能物理学会学术年会 南昌

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.
山东大学(Shandong University)
2010年4月　高能物理学会学术年会 南昌

13. Inclusive DIS with QCD interaction
Contain QCD interactions.
(Color) gauge invariant !
山东大学(Shandong University)
2010年4月　高能物理学会学术年会 南昌

14. 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).
山东大学(Shandong University)
2010年4月　高能物理学会学术年会 南昌

15. 高能物理学会第八届全国会员代表大会暨学术年会
Some of the consequences:
(1) SIDIS e+pe+q+X 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:  [hep-ph] .
(4) Twist-4 contributions and cos2
Song, Gao, Liang and Wang, in preparation.
山东大学(Shandong University)
2010年4月　高能物理学会学术年会 南昌

16. SIDIS with (leading twist): Direct consequence I
Consider the contribution from
Cross section for e+pe+q+X
Parton distributions =
Cross section for eqeq without 
(without transverse momentum) 
(with transverse momentum)
山东大学(Shandong University)
2010年4月　高能物理学会学术年会 南昌

17. SIDIS with (twist 3): Direct consequence II
A simplification of
left cut
independent of x2!
right cut
independent of x1!
山东大学(Shandong University)
2010年4月　高能物理学会学术年会 南昌

18. SIDIS with (twist 3): Direct consequence II
This leads to
Only
is relevant in semi-inclusive deep-inelastic lepton-nucleon scattering.
山东大学(Shandong University)
2010年4月　高能物理学会学术年会 南昌

19. 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
山东大学(Shandong University)
2010年4月　高能物理学会学术年会 南昌

20. SIDIS with (twist 3): differential cross-section to 1/Q
Unpolarized SIDIS:
Differential cross section:
Azimuthal asymmetry:
which reduces to Cahn’s result if we set g=0 so that
Transverse momentum dependent (TMD) quark distribution:
A practical way to study the parton correlation function
山东大学(Shandong University)
2010年4月　高能物理学会学术年会 南昌

21. 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.
山东大学(Shandong University)
2010年4月　高能物理学会学术年会 南昌

22. 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
山东大学(Shandong University)
2010年4月　高能物理学会学术年会 南昌

23. Nuclear dependence of the azimuthal asymmetry
Azimuthal asymmetry in
Azimuthal asymmetry in
Nuclear dependence of TMD parton distribution/correlation functions
山东大学(Shandong University)
2010年4月　高能物理学会学术年会 南昌

24. Nuclear dependence of the azimuthal asymmetry
An example: take a Gaussian for the transverse momentum dependence
for the case that
suppressed!
Integrated over :
山东大学(Shandong University)
2010年4月　高能物理学会学术年会 南昌

25. 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:
山东大学(Shandong University)
2010年4月　高能物理学会学术年会 南昌

26. 山东大学(Shandong University)
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 e+pe+q+X.
Naïve extension of TMD parton distributions convoluting with e+q  e+q 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.
山东大学(Shandong University)
2010年4月　高能物理学会学术年会 南昌

27. Thank you for your attention!
Outlook
Liang Zuo-tang et al (Shandong University)
Ma Bo-qiang et al (Peking University)
Chen Xiang-song et al (Hua-zhong University of Science and Technology)
Joint efforts in
(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!
山东大学(Shandong University)
2010年4月　高能物理学会学术年会 南昌