1. 6/1/20161 TMD Evolution Feng Yuan Lawrence Berkeley National Laboratory

2. TMDs: center piece of nucleon structure 2 Long. Momentum distributions Nucleon Spin 3D imaging Transverse-momentum-dependent and Generalized PDFs QCD: Factorization, Universality, Evolution, Lattice, …

3. kt-dependence crucial to the saturation 3 TMDs at small-x

4. TMDs in valence region Quark Sivers function leads to an azimuthal asymmetric distribution of quark in the transverse plane 6/1/20164 Alex Prokudin @EIC-Whitepaper

5. Evolution is crucial to strength the TMD probes 5 Two particle correlations from pp to dAu Evolution? Saturation?

6. Sign change of Sivers asymmetry 6/1/20166 Drell-Yan, π - (190GeV)p Q 2 ~3-6GeV 2 Q 2 ~16-30GeV 2 COMPASS

7. Outlines General theory background Applying to single spin asymmetries Consistent resummation in high enegy  BFKL vs Sudakov 6/1/20167

8. Collinear vs TMD factorization TMD factorization is an extension and simplification to the collinear factorization Extends to the region where collinear fails Simplifies the kinematics  Power counting, correction 1/Q neglected  (P T,Q)=H(Q) f 1 (k 1T,Q) f 2 (k 2T, Q) S( T )  There is no x- and kt-dependence in the hard factor 6/1/20168

9. DGLAP vs CSS DGLAP for integrated parton distributions  One hard scale  (Q)=H(Q/  ) f 1 (  )… Collins-Soper-Sterman for TMDs  Two scales, large double logs 6/1/20169

10. Evolution vs resummation Any evolution is to resum large logarithms DGLPA resum single large logarithms CSS evolution resum double logarithms 6/1/201610

11. 11 Sudakov Large Double Logarithms Differential cross section depends on Q 1, where Q 2 >>Q 1 2 >>  2 QCD We have to resum these large logs to make reliable predictions  Q T : Dokshitzer, Diakonov, Troian, 78; Parisi Petronzio, 79; Collins, Soper, Sterman, 85  Threshold: Sterman 87; Catani and Trentadue 89 Sudakov, 1956

12. 12 How Large of the Resummation effectsResum NLO Kulesza, Sterman, Vogelsang, 02

13. 13 Collins-Soper-Sterman Resummation Introduce a new concept, the Transverse Momentum Dependent PDF Prove the Factorization in terms of the TMDs  (P T,Q)=H(Q) f 1 (k 1T,Q) f 2 (k 2T, Q) S( T ) Large Logs are resummed by solving the energy evolution equation of the TMDs ( Collins-Soper 81, Collins-Soper-Sterman 85 )

14. 14 CSS Formalism (II) K and G obey the renormalization group eq. The large logs will be resummed into the exponential form factor  A,B,C functions are perturbative calculable. (Collins-Soper-Sterman 85)

15. 15 Two Large Scales Processes Very success in applications,  DIS and Drell-Yan at small P T (Q T Resum)  DIS and Drell-Yan at large x (Threshold Resum)  Higgs production at small P T or large x  Thrust distribution  Jet shape function  … ResBos: Nadolsky, et al., PRD 2003 CSS resummation built in

16. Single Transverse Spin Asymmetry Separate the singular and regular parts TMD factorization in b-space 6/1/201616 Kang, Xiao, Yuan, PRL 11; Rogers et al., PRD, 2012

17. Evolution equations 6/1/201617 Idilbi-Ji-Ma-Yuan, PRD04 Boer, NPB, 2002

18. Final resum form Sudakov the same 6/1/201618

19. Coefficients at one-loop order 6/1/201619

20. Constraints from SIDIS 6/1/201620 Sun, Yuan, 1308.5003

21. 21 DIS and Drell-Yan Initial state vs. final state interactions “Universality”: QCD prediction HERMES/C OMPASS ** ** Drell-Yan DIS

22. Predictions for COMPASS 6/1/201622 Drell-Yan, π - (190GeV)p Q 2 ~3-6GeV 2 Q 2 ~16-30GeV 2 COMPASS

23. Fermilab Drell-Yan 120GeV proton beam 6/1/201623

24. Few words on Drell-Yan at RHIC Never been measured before at a collider  Fixed target  W/Z at Tevatron/LHC Understand the x-evolution of the TMDs, saturation?  Compared to that from HERA 6/1/201624

25. 25 Drell-Yan at Fixed Target Q T spectrum from E288, PRD23,604(81) Valence region

26. 26 At very large Q 2 (e.g., Z 0 and W boson), No longer a Gaussian

27. Predictions at RHIC Additional theory uncertainties: x-dependence of the TMDs comes from a fit to fixed target drell-yan and w/z production at Tevatron ---Nadolsky et al. 6/1/201627 √S = 500GeV Drell-Yan Q=6GeV Sun, Yuan, 1308.5003

28. 6/1/201628 √S = 510GeV -0.06 Rapidity of W Pt(GeV) y=0

29. 29 QCD evolution reduces the asymmetries about a factor of 3 for W/Z as compared to Drell-Yan

30. Uniqueness of forward RHIC physics Investigate the sign change of Sivers asymmetries and the associated QCD evolution effects in Drell-Yan and W SSAs Mapping out the saturation physics in di- hadron and single-hadron production in forward pA collisions Complementary to the EIC Missions!! 6/1/201630

31. Kt-dependent observables 6/1/201631 CSS P J >>K T KTKT Hard processes probe the kt-dependent gluon distributions directly Saturation phenomena manifest in the observables Xiao,Yuan, et al, PRL106, 022301 (2011) PRL105, 062001 (2010)

32. Resummation: Sudakov vs BFKL Sudakov double logs can be re-summed in the small-x saturation formalism Radiated gluon momentum Soft gluon, α~β<<1 Collinear gluon, α~1, β<<1 Small-x collinear gluon, 1- β<<1, α  0  Rapidity divergence 6/1/201632 Mueller, Xiao, Yuan, PRL110,082301 (2013); arXiv:1308.2993

33. Final result Double logs at one-loop order Collins-Soper-Sterman resummation 6/1/201633

34. Comments Sudakov double logs can be re-summed consistently in the small-x formalism Kinematics of double logs and small-x evolution are well separated  Soft vs collinear gluons If Q s is small, back to dilute region If Q s is large (~Q), we can safely neglect the Sudakov effects 6/1/201634

35. Sudakov leading double logs: general hard processes Each incoming parton contributes to a half of the associated color factor  Initial gluon radiation, aka, TMDs Soft gluon radiation in collinear calculation also demonstrates this rule  Sterman, et al  Sub-leading logs will be much complicated, usually a matrix form 6/1/201635 Mueller, Xiao, Yuan, PRL110,082301 (2013); arXiv:1308.2993

36. all order factorization 6/1/201636 Similar calculations for pp collisions: Zhu HX, et al., PRL110 (2013) 082001

37. Dijet azimuthal correlation at colliders 6/1/201637 PRL 94, 221801 (2005) preliminary Peng Sun, et al. LO NLL- resummation will be extended to di-hadrons,

38. Two particle correlations in Central dAu collisions η 1 ~ η 2 ~3.2 Q 2 sA ~0.85A (1/3) Q sp 2 38 Stasto,Xiao,Yuan,PLB716,430(2012)

39. Conclusions TMDs are important tool to investigate the partonic structure of nucleon/nucleus, and the associated QCD dynamics Although complicated, the evolution effects have been well understood  Provide solid ground for phen. Applications  Unique place to study QCD 6/1/201639