1. ShanDong University Jian Zhou
TMDs at small x
Jian Zhou
ShanDong University
Parton Distributions in Modern Era, @TD Lee Institute, SJTU, 2016, Dec.

2. Outline: TMD factorization at small x Spin dependent TMDs at small x
1: unpolarized target
2: transversely polarized target
Elliptic gluon GTMD inside a large nucleus

3. Parton distributions at small x
Gluon radiation at small x

4. Derive the TMD factorization formula I
Starting from the full CGC result,
Balitsky 1996
Gelis, Jalilian-Marian 2003
Taylor expanding the impact factor (PT >>QS ),
Integrating out k1T,
F. Dominguez, B-W. Xiao, F. Yuan 2011
F. Dominguez, C. Marquet, B-W. Xiao, F. Yuan 2011

5. Derive the TMD factorization formula II
The cross section then reads,
One can identify,
F. Dominguez, C. Marquet, B-W. Xiao, F. Yuan 2011
CGC
TMD
Derivative of impact factor in kT
Hard part
Derivative of Wilson lines in xT
Gluon TMDs

6. Physical picture behind the effective TMD factorization
Low jet PT ≤ KT ( only CGC applicable )
four Wilson lines
High jet PT >> KT ( both CGC and TMD applicable )
Gluons can not resolve the internal
structure of the color dipole system.
Collapse to two semi-finite Wilson lines

7. An additional hard scale is essential to ensure a local probe NLO analysis

8. Gluon initiated Drell-Yan process
M2 >>pT2, TMD factorization, resummed by Collins-Soper equation
S>>M2, Kt factorization, resummed by BFKL equation
, Collins, Soper
1991, Catani, Ciafaloni and Hautmann
1991, Collins and R. K. Ellis
The overlap region
An explicit NLO cross section calculation shows that both the large logarithm appear.
2013, Mueller, Xiao, Yuan
Such joint resummation has been also disscussed in other literatures.
2015, Balitsky and Tarasov; Marzani

9. Does it accommodate both type large logarithms?
In collinear calculation absorbed into PDF.
One natural question:
Does it accommodate both type large logarithms?

10. The main strategy both large logarithms are absent at LO;
In a simple quark model, at tree level:
both large logarithms are absent at LO;
how is it dressed by quantum corrections at NLO?

11. Real graphs Sample diagrams: Fig.a is the only diagram contributing to
both the CS and the BFKL evolution kernel
Calculation formulated in the Ji-Ma-Yuan scheme.

12. One remark In the infinite momentum frame P+ --> ∞ BFKL
rapidity divergence
Collins-Soper
rapidity divergence

13. Virtual graphs
Fig.a & Fig.b BFKL kernel
Fig.c CS kernel

14. Collecting all results
In the leading logarithm approximations,
Small x gluon TMD at NLO reads,
2016, ZJ

15. BFKL equation: CS equation:
The resulting gluon TMD indeed simultaneously satisfies the both
BFKL equation:
CS equation:

16. Small x TMDs in CGC at NLO
Sample diagrams
Collinear divergence is absent
Match small x TMDs onto multiple point functions instead of PDFs.
Two step evolution: S M kT2
Xiao, Yuan and ZJ, in preparation.

17. NLO analysis further justifies the use of TMD factorization at small x.

18. Spin dependent TMDs at small x
1: Inside an unpolarized target

19. Spin dependent gluon distributions
Circular polarization
(helicity distribution)
Linear polarization
Unpolarized

20. Linearly polarized gluon TMD
Mulders, Rodrigues, 2001
nonperturabtive object
1: fit to experimental data
2: computed on lattice
3: or ...

21. Quasi-classical model--the MV model
At small x, a semi-hard scale(QS) emerges, gluon TMD in the MV model:
Kovchegov, 96
J. Marian, Kovner, Mclerran & Weigert, 97

22. Gluon TMDs in the MV model
The linearly polarized gluon TMDs in the MV model,
Metz & ZJ, 2011
Weizsäcker-Williams(WW) distribution:
Dipole distribution:
positivity bound saturated for any value of kt

23. CGC PCGC(polarized CGC)

24. 2: Inside a transversely polarized target

25. Three T-odd gluon TMDs Are the T-odd gluon TMDs relevant at small x?
Identify 6 leading power gluon TMDs for a transversely polarized target
(8 in total). Among them, 3 gluon TMDs are T-odd distributions.
Mulders, Rodrigues, 2001
Are the T-odd gluon TMDs relevant at small x?

26. Quark target model No 1/x enhancement for T-odd distributions.
Meissner, Metz, Goeke; 2007
No 1/x enhancement for T-odd distributions.

27. Collinear twist-3 contributions
The WW case
Leading 1/x terms cancel out between different contributions.
The Dipole case (in the small x limit)
Boer, Echevarria, Mulders, ZJ; 2016

28. Goes beyond the DGLAP treatment
How to formulate them in the small x formalism?

29. T-odd gluon TMDs & the odderon
Starting point,
Using time reversal invariance and parity symmetry, at small x one obtains,
Schematically,
Nothing but an odderon operator in CGC
Kovchegov, Szymanowski & Wallon 2004
Hatta, Iancu, Itakura & McLerran 2005

30. How to identify the spin correlation in the odderon operator?

31. Valence quark distribution
Odderon in the MV model
Spin independent odderon has been studied in the MV/Dipole model
Kovchegov, Szymanowski & Wallon 2004
Jeon,Venugopalan 2005
Spin dependent odderon in the MV model
Valence quark distribution
ZJ 2013
Impact parameter dependent
valence quark distribution (GPDs)
M. Burkardt 2000, 2003

32. Spin dependent odderon
Caution: The application of the MV to a proton is less well justified.
In the momentum space:
ZJ 2013

33. Two different paramertrizations
Equaling two parmetrizations:
Simple algebra leads to
Boer, Echevarria, Mulders, ZJ; 2016
All of three dipole type T-odd gluon TMDs become identical at small x!

34. SSA in jet production in the backward region of pp collisions
LO Dumitru & J. Marian 2002
NLO Chirilli, Xiao & Yuan 2012
ZJ 2013
Unpolarized gluon distribution
Spin dependent odderon

35. AnDY at RHIC, 2012

36. Elliptic gluon GTMD inside a nucleus

37. GTMD operator definition
Motivations:
tomography picture of nucleon
encode information on parton OAM
See also Bowen's talk and Yoshitaka's talk
In the small x limit:
Hatta, Xiao and Yuan, 2016

38. Dilute limit in the MV model
which is a trivial result,
Taylor expansion leads to

39. Saturation regime less trivial to obtain,
ZJ, 2016
Uniform color source distribution leads to a vanishing eGTMD.
Parametrization of b dependence of Qs

40. Observable Quasi-elastic virtual photon-nucleus scattering,
azimuthal independent cross section
Buchmuller, Gehrmann and Hebecker, 99
Kovchegove and McLerran, 99
cos 2phi modulation
ZJ, 2016

41. Summary At small x: TMD factorization is applicable
Very rich polarization dependent phenomenology
Gluon tomography induced by saturation effect

42. Back up slides

43. Cos2Ф asymmetries in eA and pA collisions
Di-jet in eA
Metz & ZJ 2011
γ*- jet production in pA
parameter free!
also verified by the calculation performed in the position space
Dominguez, Qiu, Xiao & Yuan 2011

44. Courtesy of C. Lorce

45. Physical picture behind the effective TMD factorization
Low jet PT ≤ KT ( only CGC applicable )
four Wilson lines
High jet PT >> KT ( both CGC and TMD applicable )
Gluons can not resolve the internal
structure of the color dipole system.
Collapse to two semi-finite Wilson lines

46. Time reversal invariance & parity symmetry
Starting point,
Due to time reversal invariance and parity symmetry,
Schematically,
Therefore,

47. Summary II Small x physics: Powerfull theoretical tools
Dipole approach ﹠MV model
BFKL and BK equations
CGC and JIMWLK equation …
TMD/Spin physics:
Novel ideas/concepts
T-odd effect
Universality v.s. process dependent TMDs
Color entanglement effect …
The study of hardon/nucleus structure is benefiting
from the joint force of small x physics and spin physics.