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EIC, Nucleon Spin Structure, Lattice QCD

EIC, Nucleon Spin Structure, Lattice QCD. Xiangdong Ji University of Maryland. Outline. EIC and partons Spin structure of the nucleon lattice QCD and parton physics Nothing to say about TMDs, Wigner distributions, GTMDs here. Nature of QCD bound states.

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EIC, Nucleon Spin Structure, Lattice QCD

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  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. 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.

  14. 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).

  15. Helicity sum rule involving twist-three

  16. 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.

  17. 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

  18. 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.

  19. 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.

  20. 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.

  21. 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

  22. 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. 𝜉- 𝜉+ t Step1 Step2 x

  23. 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.

  24. 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.

  25. Effective field theory language

  26. 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.)

  27. 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. 𝜉0 0 Z 𝜉3

  28. 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. 𝜉- 𝜉0 𝜉+ 𝜉3

  29. Finite but large P

  30. Practical considerations • For a fixed x, large Pz means large kz, thus, as Pz gets larger, the valence quark distribution in the z-direction get Lorentz contracted, z ~1/kz. • Thus one needs increasing resolution in the z-direction for a large-momentum nucleon. Roughly speaking: aL/aT ~ γ

  31. γ=4 x,y Large P z

  32. 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

  33. 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…

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