TMDs at small x

1. 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

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 probeNLO analysis

8. Gluon initiated Drell-Yan process • M2 >>pT2, TMD factorization, resummed by Collins-Soper equation • S>>M2, Kt factorization, resummed by BFKL equation 1982-1983, 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; 2015 Marzani

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

10. The main strategy 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. 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 functionsinstead of PDFs. Two step evolution: S M2 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. Unpolarized Circular polarization (helicity distribution) Linear polarization Spin dependent gluon distributions

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 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 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. 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 LODumitru & J. Marian 2002 NLOChirilli, 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 parameter free! also verified by the calculation performed in the position space γ*- jet production in pA 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.