1. EIC, Nucleon Spin Structure, Lattice QCD Xiangdong Ji University of Maryland

2. Outline  EIC and partons  Spin structure of the nucleon  lattice QCD and parton physics  Nothing to say about TMDs, Wigner distributions, GTMDs here.

3. Nature of QCD bound states  The nucleon is a strongly interacting, relativistic bound states.  Understanding such a system theoretically is presently beyond the capability of the best minds in the world.  It posts a great intellectual challenge to the QCD community (theorists and experimentalists).

4. What an EIC can do?  Studying partonic content of a QCD bound state (nucleon and nucleus)  The probe is hard and relativistic, the nucleon is measured in the infinite momentum frame (light-cone correlations)  Note that the wave function is frame-dependent, the DIS naturally selects the IMF.  Structure and probe physics can be separated nicely (factorization)!

5. Partons: good and bad  Good: constructing the bound state properties in terms of those of the individual particles, such as the momentum, part of the spin.  Bad: not all quantities can be done this way Mass, contains twist-2, 3 and 4 components No Lorentz symmetric physical picture, transverse polarization, longitudinal polarization, different components of a four vector, etc.

6. Why Parton pictures for the nucleon spin? The spin structure of the nucleon?  Theorists: any frame, any gauge..  Experimenters: Infinite momentum frame (IFM): how the nucleon spin is made of parton constituents, this is the frame where the proton structure are probed!

7. Summary  The spin parton picture only exists for transverse polarization, in which the GPD’s provide the transverse polarization density (JI)  For the nucleon helicity, the sum rule involves twist-three effects, need more theoretical development.(Jaffe & Manohar)  The partonic AM sum rule is not related to TMDs. Ji, Xiong, Yuan, Phys.Rev.Lett. 109 (2012) 152005

8. Matrix element of AM density 

9. Transverse polarization vs spin  Transverse angular momentum (spin) operator J ┴ does not commute with the longitudinal momentum operator P+.  Thus, transverse polarized proton is NOT in the transverse spin eigenstate!  Transverse polarized nucleon is in the eigenstate of the transverse Pauli-Lubanski vector W ┴  W ┴ contains two parts P - J + ┴ and P + J - ┴

10. Two parts of the transverse pol  While the first part is leading in the parton interpretation for the AM, the second part is subleading.  However, the good news is that the second part is related to the the first part through Lorentz symmetry.  Even the component of is not all leading in light-cone. Only the second term has leading light- cone Fock component. However, the first term again can be related to the second by Lorentz symmetry.

11. Light-cone picture for S_perp  Burkardt (2005), Important point:  Sum rule works for transverse pol. if a parton picture works, one has to show that a parton of momentum x will carry angular momentum x(q(x)+E(x)). We can shown this in PRL paper

12. comments 

13. 

14. EIC will do a great job for helicities  \Delta q (x), better understanding about the sea and strange quarks  \Delta g(x), better measurement through two jet and evolution.  However, OAM is harder but not impossible.

15. References  Ji, Xiong, Yuan, Phys.Rev.Lett. 109 (2012) 152005; Phys.Rev. D88 (2013) 1, 014041, Phys.Lett. B717 (2012) 214-218  Hatta, Phys. Lett. B708, 186 (2012); Hatta, Tanaka and Yoshda, JHEP, 1302, 003 (2013).

16. Helicity sum rule involving twist-three 

17. Parton transverse momentum and transverse coordinates  Why twist-three? Because the AM operator involves parton transverse momentum!  To develop a parton picture, we need partons with transverse d.o.f.

18. A simpler parton picture?  Presence of A ┴ in the covariant derivative spoils simple parton picture! Let’s get rid of it.  One can work in the fixed gauge A + =0 （ jaffe & manohar) All operator are bilinear in fields, which leads to a spin decomposition in light-cone gauge

19. OAM density and relation to twist-three GPD  OAM density Bashingsky, Jaffe, Hagler, Schaefer…  Partonic OAM can be related to the matrix elements of twist-three GPDs (Hatta, Ji, Yuan)  Thus the orbital AMO density in LC is measurable in experiment!  Possibly program for EIC with twist-three GPDs, need more work.

20. Lattice QCD and partons  Lattice simulation is the only systematic approach to calculate non-perturbaive QCD physics at present.  Lattice cannot solve all non-perturbative QCD problems. Particularly it has not been very ineffective in computing parton physics. This has been a serious problem.  However, this situation might have been changed recently.

21. Light-front correlations  Parton physics is related to light-front correlations, which involve real time.  In the past the only approach has been to compute moments, which quickly become intractable for higher ones.  Light-front quantization, although a natural theoretical tools, has not yield a systematic approximation.

22. Recent development  X. Ji, Phys. Rev. Lett. 110, 262002 (2013).  X. Ji, J. Zhang, and Y. Zhao, Phys. Rev. Lett. 111, 112002 (2013).  Y. Hatta, X. Ji, and Y. Zhao, Phys. Rev. D89 (2014) 085030  X. Ji, J. Zhang, X. Xiong, Phys. Rev. D90 (2014) 014051  H. W. Lin, Chen, Cohen and Ji, arXiv:1402.1462  J. W. Qiu and Y. Q. Ma, arXiv:1404.6860  X. Ji, P. Sun, F. Yuan, arXiv:1406.0320

23. Solution of the problem  Step 1: Light-cone correlations can be approximated by “off but near” light-cone “space-like” correlations  Step 2: Any space-like separation can be made simultaneous by suitably choosing the Lorentz frame.  +  - t x Step1 Step2

24. Justification for step 1  Light-cone correlations arise from when hadrons travel at the speed-of-light or infinite momentum  A hadron travels at large but finite momentum should not be too different from a hadron traveling at infinite momentum. The difference is related to the size of the momentum P But physics related to the large momentum P can be calculated in perturbation theory according to asymptotic freedom.

25. Consequences for step 2  Equal time correlations can be calculated using Wilson’s lattice QCD method.  Thus all light-cone correlations can in the end be treated with Monte Carlo simulations.

26. Effective field theory language 

27. Infrared factorization  Infrared physics of O(P,a) is entirely captured by the parton physics o(µ). In particular, it contains all the collinear divergence when P gets large.  Z contains all the lattice artifact (scheme dependence), but only depends on the UV physics, can be calculated in perturbation theory  Factorization can be proved to all orders in perturbation theory (Qiu et al.)

28. A Euclidean quasi- distribution  Consider space correlation in a large momentum P in the z-direction. Quark fields separated along the z-direction The gauge-link along the z-direction The matrix element depends on the momentum P.  3  0 Z 0

29. Taking the limit P-> ∞ first  After renormalizing all the UV divergences, one has the standard quark distribution! One can prove this using the standard OPE One can also see this by writing |P˃ = U(Λ(p)) |p=0> and applying the boost operator on the gauge link.  3  0  +  -

30. Finite but large P 

31. Practical considerations  For a fixed x, large P z means large k z, thus, as P z gets larger, the valence quark distribution in the z-direction get Lorentz contracted, z ~1/k z.  Thus one needs increasing resolution in the z- direction for a large-momentum nucleon. Roughly speaking: a L /a T ~ γ

32. z x,y γ=4 Large P

33. Power of the approach  Gluon helicity distribution and total gluon spin  Generalized parton distributions  Transverse-momentum dependent parton distributions  Light-cone wave functions  Higher twist observables

34. Opportunities for both EIC and lattice  Lattice allows calculating TMDs and GPDs at experimental kinematic points, x, k_perp, etc. This entails powerful comparison between exp and theory.  Lattice calculations are hard, but not impossible…