1. SSAs at small x and the the anomalous magnetic moment Jian Zhou Regensburg University Based on: Phys.Rev. D89 (2014) 074050. ZJ arXiV:1308.4961 A. Schafer and ZJ ECT, Trento, 2014

2. Outline:  Tri-gluon correlations and their asymptotic behavior  Polarized odderon & the anomalous magnetic moment  Summary

3. Transverse single spin asymmetris pp π L R

4.  TMD factorization approach : quark Siverse distribution gluon Siverse distribution  Collinear twist-3 approach : quark-gluon correlation (ETQS function) tri-gluon correlations We focuse on tri-gluon correlations, in particular, its asymptotic behavior at small x TMD factorization V.S. Collinear twist-3 approach

5. Operator defination for tri-gluon correlations Ji 1992 Kang & Qiu 2008 Beppu, Koike, Tanaka & Yoshida 2010 Receive contributions from:

6. Differ from the existing results Kang & Qiu 2008 Use the different paramertization for tri-gluon correlations Braun, Manashov & Pirnay 2009  Diagram b  Diagrams c, d, e More work will be needed to settle this issue !  Diagram a

7. Within the leading logarithm lnx accuracy, we made approximations Symmetry relations Beppu, Koike, Tanaka & Yoshida 2010 Schafer, ZJ 2013 The asymptotic behavior at small x

8.  Phenomenological implications I. Gluon Sivers function could change sign at small x II. Gluon Sivers function dies out quickly with decreasing x III. C-odd tri-gluon correlation dominates SSA at small x A UT → Constant, when x → 0 ?

9. The odderon origin of SSAs at small x Ahmedov, Akushevich, Kuraev & Ratcliffe 1999 Ahmedov, Antonov, Bartos, Kuraev & Zemlyanaya 2003 kovchegov & Sievert 2012 Heavy fermion model CGC calculation Spin independent odderon !

10. Elastic hadron-hadron scattering  In Regge theory, an object exchanged called pomeron, C=+1  In QCD, pomeron modeled as two gluon exchange(Low-Nussinov Pomeron)  BFKL pomeron, resuming leading higher order corrections log(s) Leading to the growth of cross section with energy

11. Odderon Regge exchange with vacuum quantum numbers except for C=-1 Lukaszuk & Niolescu,1974 Responsible for the difference between proton-proton and proton-antiproton scatterings. In PQCD, exchange of three gluon in a colorless C=-1 state BKP equation Bartels 1980, Kwiecinski & Praszalowicz 1980 BLV Solution: high energy intercept is exactly 1 Bartels, Lipatov & Vacca 2000

12.  Formulation in the CGC Hatta, Iancu, Itakura & McLerran 2005  Formulation in Mueller’s dipole model Kovchegov, Szymanowski & Wallon 2004 Wilson line

13. The odderon origin of SSAs kovchegov & Sievert 2012 CGC calculation The interaction between Spin depenent LC wave function & the odderon from unpolarized nucleon/nucleus

14.  Spin dependent odderon (transversely polarized proton)

15. Odderon in the MV model Mclerran, Venugopalan 1994 Jeon, Venugopalan 2004 Valence quark distribution The expectation value of the odderon operator Jeon,Venugopalan 2005

16. Odderon in the MV model M. Burkardt 2000, 2003 M. Burkardt 2003, and also see Alessandro’s talk ZJ 2013  Impact parameter dependent valence quark distribution  An earlier attempt to connect SSA phenomena to GPD E has been made.  Remark: The application of the MV model to proton may not be well justified.

17. SSA in jet production in the backward region of pp collisions Unpolarized gluon distribution Polarized odderon LO Dumitru & J. Marian 2002 NLO Chirilli, Xiao & Yuan 2012

18.  Remark 1  Remark 2  Remark 3 BLV solution

19. AnDY at RHIC, 2012

20. SSA for open charm production in SIDIS  Collinear twist-3 approach Kang, Qiu 2008 Beppu, Koike, Tanaka & Yoshida 2010 Hard part ( transversely polarized photon) LO Mueller 1999; Mclerran & Venugopalan 1999 NLO Balitsky & Chirilli 2011; Beuf 2012

21.  SSA in open charm production and DY/direct photon K T moment of the odderon & C-odd tri-gluon correlation Kang, Qiu, Vogelsang & Yuan 2008 Koike & Yoshida 2011, 2012 Twist-3 approach

22. Summary  The C-odd tri-gluon correlation dominates SSA at small x The gluon Sivers distribution is irrelevant to SSA phenomenology at small x  Identify a spin dependent odderon, propotional to the anomalous magnetic moment the moment of polarized odderon related to the C-odd tri-gluon correlation according to the BLV solution, Outlook The CGC formalism the twist-3 apporach in the dilute limit ?