TMD Evolution

1. TMD Evolution Feng Yuan Lawrence Berkeley National Laboratory

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

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

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

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

6. Sign change of Sivers asymmetry Drell-Yan, π- (190GeV)p COMPASS Q2~16-30GeV2 Q2~3-6GeV2

7. Outlines • General theory background • Applying to single spin asymmetries • Consistent resummation in high enegy • BFKL vs Sudakov

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 (PT,Q)=H(Q) f1(k1T,Q) f2(k2T, Q) S(T) • There is no x- and kt-dependence in the hard factor

9. DGLAP vs CSS • DGLAP for integrated parton distributions • One hard scale (Q)=H(Q/) f1()… • Collins-Soper-Sterman for TMDs • Two scales, large double logs

10. Evolution vs resummation • Any evolution is to resum large logarithms • DGLPA resum single large logarithms • CSS evolution resum double logarithms

11. Sudakov Large Double Logarithms Sudakov, 1956 • Differential cross section depends on Q1, where Q2>>Q12>>2QCD • We have to resum these large logs to make reliable predictions • QT: Dokshitzer, Diakonov, Troian, 78; Parisi Petronzio, 79; Collins, Soper, Sterman, 85 • Threshold: Sterman 87; Catani and Trentadue 89

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

13. Collins-Soper-Sterman Resummation • Introduce a new concept, the Transverse Momentum Dependent PDF • Prove the Factorization in terms of the TMDs (PT,Q)=H(Q) f1(k1T,Q) f2(k2T, Q) S(T) • Large Logs are resummed by solving the energy evolution equation of the TMDs (Collins-Soper 81, Collins-Soper-Sterman 85)

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. Two Large Scales Processes • Very success in applications, • DIS and Drell-Yan at small PT (QT Resum) • DIS and Drell-Yan at large x (Threshold Resum) • Higgs production at small PT 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 Kang, Xiao, Yuan, PRL 11; Rogers et al., PRD, 2012

17. Evolution equations Idilbi-Ji-Ma-Yuan, PRD04 Boer, NPB, 2002

18. Final resum form • Sudakov the same

19. Coefficients at one-loop order

20. Constraints from SIDIS Sun, Yuan, 1308.5003

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

22. Predictions for COMPASS Drell-Yan, π- (190GeV)p COMPASS Q2~16-30GeV2 Q2~3-6GeV2

23. Fermilab Drell-Yan 120GeV proton beam

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

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

26. At very large Q2 (e.g., Z0 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. √S = 500GeV Drell-Yan Q=6GeV Sun, Yuan, 1308.5003

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

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 pAcollisions Complementary to the EIC Missions!!

31. Kt-dependent observables • 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) PJ>>KT KT CSS

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 Mueller, Xiao, Yuan, PRL110,082301 (2013); arXiv:1308.2993

33. Final result • Double logs at one-loop order • Collins-Soper-Sterman resummation

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 Qs is small, back to dilute region • If Qs is large (~Q), we can safely neglect the Sudakov effects

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 Mueller, Xiao, Yuan, PRL110,082301 (2013); arXiv:1308.2993

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

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

38. Two particle correlations in Central dAu collisions • η1~η2~3.2 • Q2sA~0.85A(1/3) Qsp2 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