1. 1 In collaboration with L. Gamberg, Z. Kang, H. Xing Based on Phys.Lett. B743 (2015) 112-120, arXiv:1412.3401 Quasi-parton distribution functions: a study in the di-quark spectator model

2. Outline of the talk 2 Advances in computationally enabled understanding of the nucleon structure. Applicability of quasi-PDFs Basics and phenomenological success Unpolarized distribution, helicity and transversity. Analytic limits Guidance on the nucleon boost for good approximation Soffer inequality. Analytic and numerical results

3. 3 We will focus on the unpolarized distribution f 1, the helicity distribution g 1 and the transversity distribution h 1 Motivation Quark distribution functions P. Mulders et al. 1995

4. Global analysis Note: spin is along y direction M. Echevarria et al, 2014 C. Aidala et al, 2014, P. Sun et al, 2014, U. D’Alesio et al, 2014 M. Anselmino et al, 2008

5. Lattice QCD for TMDs T. Bhattacharya et al. in preparation B. Musch et al., 2011 X. Ji. 2013, Ma et al. 2014  New formulation of quasi-PDFs for lattice evaluation  Equal time correlators, suitable for lattice evaluation C. Alexandrou et al. 2014

6. Spectator di-quark model A. Bacchetta et al. 2008 R. Jakob et al. 1997, L. Gamberg et al. 2003  Truncates the sum over all possible final states to a single di-quark final state. Initially contained only scalar di-quarks  Axial-vector di-quarks different possibilities for formfactors For PDFs For Quasi-PDFs

7. Phenomenological success of the spectator di-quark model A. Bacchetta et al. 2008  Example ZEUS2002 GRSV2000 Qualitative and sometimes quantitative agreement with global analysis

8. Calculation of PDFs and quasi-PDFs  Standard PDFs  Quasi-PDFs  Standard PDFs  Quasi PDFs

9. Analytic computations of f 1  The spectator di-quark model, tree level calculation an the dipole form factors make the calculation straightforward Invariant mass Result While in slightly different notation agree with A. Bacchetta et al. 2008

10. Analytic computations of the quasi-f 1  Notation  Invariant mass  To find

11. Summary of quasi-PDF results (unintegrated)

12. The collinear unpolarized, helicity and transversity distributions

13. Numerical studies of PDFs and quasi-PDFs The unpolarized distribution L. Gamberg et al. 2014 P z ~1 GeV never works (not surprisingly). Many non-perturbative parameters are of O(1 GeV), Λ X, M X … P z > 2 GeV the apprximations (quasi-PDFs) work considerably better

14. Helicity and Transversity Similar results

15. Quantifying deviations: small to moderate x  For x <0.4 at P z ~ 2 GeV quasi-PDFs approximate PDFs to 30%, For P z ~ 4 GeV the deviation is only 10%

16. Deviations at large x For x ~0.7 at P z ~ 2 GeV quasi-PDFs deviate for PDFs a lot – factor of 2-3 For P z ~ 4 GeV the deviation is ~50%. Clearly much larger boost is needed for large values of Bjorken x See the approximation / limit

17. Sum rules and bounds  Arise form symmetries, kinematics, etc. With (quasi-)unpolarized, helicity and transversity it is easy to check the Soffer inequaity X. Artru al. 2008 A. Bacchetta et al. 2000 Soffer bound  Generally bounds do not hold for quasi-PDFs. The approach to the sum rule is similar

18. Conclusions 18 Global fits to TMDs with evolution. Collinear PDFs constrained for a long time. One can obtain analytic results for the dipole form factor (both scalar and axial diquarks). Show that in the limit of Pz-> infinity they reduce to the PDFs “Traditional” lattice approach. New proposition to use quasi-PDFs. Lattice can benefit from guidance at what boost the quasi-PDFs approach PDFs For small to moderate x, P z ~ 2 GeV approximated to ~30%. For large x the approximation is worse, 50% for P z ~ 4 GeV Generally does not hold for quasi-PDFs