1. 11/5/20151 Energy Evolution of Sivers asymmetry in Hard Processes Feng Yuan Lawrence Berkeley National Laboratory

2. Outlines General theory background Implement the TMD evolution from low Q SIDIS to Drell-Yan Match to high Q Drell-Yan/W/Z Collins asymmetries 11/5/20152

3. Hard processes In the context of this talk, the hard processes means low transverse momentum hard processes  Semi-inclusive DIS at low pt  Drell-Yan/W/Z production  Higgs production  … 11/5/20153

4. 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 11/5/20154

5. DGLAP vs CSS DGLAP for integrated parton distributions  One hard scale  (Q)=H(Q/  ) f 1 (  )… CSS for TMDs  Two scales, large double logs 11/5/20155

6. Evolution vs resummation Any evolution is to resum large logarithms DGLPA resum single large logarithms CSS evolution resum double logarithms 11/5/20156

7. Energy Evolution CS evolution for TMD distribution/fragmentation functions, scheme-dependent  Collins-Soper 81, axial gauge  Ji-Ma-Yuan 04, Feynman gauge, off-light  Collins 11, cut-off  SCET, quite a few CSS evolution on the cross sections  TMD factorization implicit 11/5/20157

8. Energy dependence Collins-Soper Evolution, 1981 Collins-Soper-Sterman, 1985 Boer, 2001 Idilbi-Ji-Ma-Yuan, 2004 Kang-Xiao-Yuan, 2011 Collins 2010 Aybat-Collins-Rogers-Qiu, 2011 Aybat-Prokudin-Rogers,2012 Idilbi, et al., 2012 11/5/20158

9. Semi-inclusive DIS Fourier transform Evolution 11/5/20159

10. Calculate at small-b Sudakov 11/5/201510

11. b * -prescription and non- perturbative form factor b * always in perturbative region This will introduce a non-perturbative form factors Generic behavior 11/5/201511

12. Rogers et al. Calculate the structure at two Q, Relate high Q to low Q Low Q parameterized as Gaussian 11/5/201512

13. BLNY form factors Fit to Drell-Yan and W/Z boson production 11/5/201513 b max =0.5GeV -1

14. BLNY form can’t describe SIDIS Log Q dependence is so strong, leading to a≈0.08 at HERMES energy Hermes data require a≈0.2 11/5/201514 BLNY will be even Worse Any modification will Introduce new problem

15. It could be that the functional form is not adequate to describe large-b physics  In particular, for \ln Q term (see follows) Or evolution has to be reconsidered in the relative (still perturbative) low Q range around HERMES/COMPASS  Q>~Q 0 ~1/b * ~2GeV (for b max =0.5GeV -1 ) 11/5/201515

16. One solution: back to old way Parameterize at scale Q 0 11/5/201516

17. Limitations It’s an approximation: both Q 0 and Q are restricted to a limited range, definitely not for W/Z boson  Log(Q 0 b) in the evolution kernel Do not have correct behavior at small-b (could be improved), will have uncertainties at large pt x-dependence is not integrated into the formalism 11/5/201517

18. Advantages There is no Landau pole singularity in the integral Almost parameter-free  No Q-dependent non-perturbative form factor  Gaussian assumption at lower scale Q 0 11/5/201518

19. Almost parameter-free prediction SIDIS Drell-Yan in similar x-range 11/5/201519

20. Fit to Sivers asymmetries With the evolution effects taken into account. Not so large Q difference 11/5/201520

21. Systematics of the SIDIS experiments are well understood Q range is large to apply perturbative QCD Sivers functions are only contributions to the observed asymmetries 11/5/201521

22. Predictions at RHIC About a factor of 2 reduction, as compared to previous order of magnitude difference 11/5/201522

23. Cross checks Re-fit Rogers et al’s parameterization to the pt-distributions, and calculate the SSA, in similar range Assume a simple Gaussian for both SIDIS and Drell-Yan (Schweitzer et al.), and again obtain similar size SSA for Drell-Yan 11/5/201523

24. Match to higher Q Extract the transverse momentum-moment of the Sivers function, and use the b * prescription and resummation, and again obtain similar size of SSA for Drell-Yan This can be used to calculate the asymmetries up to W/Z boson production 11/5/201524

25. High energies 11/5/201525 w/o evolution b * -prescription with evolution Z boson Q=5.5GeV P T (GeV)

26. Collins asymmetries E c.m. ≈10GeV, di-hadron azimuthal asymmetric correlation in e+e- annihilation 11/5/201526

27. Collins asymmetries in SIDIS asd 11/5/201527

28. Test the evolution at BEPC E c.m. =4.6GeV, di-hadron in e + e - annihilation BEPC-(Beijing electron-positron collider) 11/5/201528

29. It is extremely important to test this evolution effect EIC will be perfect, because Q coverage Anselm Vossen also suggests to do it at BELLE with ISR with various Q possible 11/5/201529

30. Conclusion We evaluate the energy dependence for Sivers asymmetries in hard processes, from HERMES/COMPASS to typical Drell- Yan process The same evolution procedure consistently describes the Collins asymmetries from HERMES/COMPASS and BELLE Further tests are needed to nail down this issue 11/5/201530