1. Xiangdong Ji U. Maryland/ 上海交通大学 Recent progress in understanding the spin structure of the nucleon RIKEN, July 29, 2013 PHENIX Workshop on Physics Prospects with Detector and Accelerator Upgrades

2. Nucleon spin structure is still an unsolved problem ！ The nucleon spin structure continues motivating new theoretical ideas and experiment measurements The nucleon spin structure continues motivating new theoretical ideas and experiment measurements Experimental efforts Experimental efforts …. Polarized PP collision at RHIC (This workshop) COMPASS JLAB and 12 GeV upgrade EIC (A. Deshpande)

3. The canonical (or “local”) approach The nucleon spin structure can be studied through QCD AM operator. The nucleon spin structure can be studied through QCD AM operator. Explicit gauge-invariant and local form, Explicit gauge-invariant and local form, yields the following spin sum rule yields the following spin sum rule where OAM can be measured through GPD sum rules. where OAM can be measured through GPD sum rules.

4. Pros and cons Pros Pros Explicitly gauge invariant Local The sum rule is maximally frame independent Cons Cons Physical interpretation of the terms does not allow free-particle picture Has no explicit connection to partons, in particular, it does not naturally involve the gluon helicity contribution in the infinite momentum frame (IMF).

5. IMF picture of the nucleon The nucleon is mostly probed in high-energy scattering. The nucleon is mostly probed in high-energy scattering. Relative to the probes, the nucleon travels at the speed of light. We learn light-cone wave functions, or parton physics. Relative to the probes, the nucleon travels at the speed of light. We learn light-cone wave functions, or parton physics. According to QCD factorizations, certain observables of the nucleon can be explained in terms of free partons  A fast moving proton is a beam of free quarks and gluons! According to QCD factorizations, certain observables of the nucleon can be explained in terms of free partons  A fast moving proton is a beam of free quarks and gluons! Can one describe the spin structure completely in terms of partons? Can one describe the spin structure completely in terms of partons?

6. Gluon helicity distribution The gluon partons have well-defined helicity ± 1 and densities g ± (x) in wavelength The gluon partons have well-defined helicity ± 1 and densities g ± (x) in wavelength Gluon helicity distribution is Gluon helicity distribution is  g(x) = g + (x) – g - (x) and  G = ʃ dx  g(x) is the fraction of the proton helicity carried in the gluon. 1/2 +1 or -1

7. Implication for the spin structure ΔG is an obvious contribution to the spin of the proton. ΔG is an obvious contribution to the spin of the proton. Can contribute to the quark helicity through axial anomaly (Altarelli & Ross, Carlitz, Collins, & Mueller,…), several units of h-har? Can contribute to the quark helicity through axial anomaly (Altarelli & Ross, Carlitz, Collins, & Mueller,…), several units of h-har? Its contribution to the spin grows like 1/ α S Its contribution to the spin grows like 1/ α S However, there are a number of puzzles associate with this quantity…

8. QCD expression The total gluon helicity ΔG is gauge invariant quantity, and has a complicated expression in QCD factorization (Manohar, 1991) The total gluon helicity ΔG is gauge invariant quantity, and has a complicated expression in QCD factorization (Manohar, 1991) It does not look anything like gluon spin or helicity! Not in any textbook! It does not look anything like gluon spin or helicity! Not in any textbook!

9. Don’t know how to calculate Δ G involves explicit light-cone correlation or real time. No one knows how to calculate this in lattice QCD (Models: RL Jaffe, Chen & Ji) Δ G involves explicit light-cone correlation or real time. No one knows how to calculate this in lattice QCD (Models: RL Jaffe, Chen & Ji) One can consider A+=0 gauge, but no one knows how to fix this gauge in lattice QCD One can consider A+=0 gauge, but no one knows how to fix this gauge in lattice QCD Thus there is no way to confront theory with experiment: Thus there is no way to confront theory with experiment:  G = ʃ dx  g(x) Is there a large contribution from small x? Is there a large contribution from small x?

10. Light-cone gauge In light-cone gauge A + =0, the above expression reduces to a simple form In light-cone gauge A + =0, the above expression reduces to a simple form which is the spin of the photon (gluon) which is the spin of the photon (gluon) (J. D. Jackson, CED), but is not gauge-symmetric: There is no gauge symmetry notion of the gluon spin! (J. D. Jackson, L. Landau & Lifshitz).

11. A LL from RHIC 2009 11

12. Jaffe-Manohar Spin Sum Rule Consider the free-field form of the QCD AM operator Consider the free-field form of the QCD AM operator Every term has a simple interpretation, but Except the first, others are not gauge invariant Take it to the IMF and fix the light-cone gauge, one gets the partonic AM sum rule, Take it to the IMF and fix the light-cone gauge, one gets the partonic AM sum rule,

13. Two important questions Although the gauge and IMF make the parton picture clear for the spin, but why these choices are physically relevant? Although the gauge and IMF make the parton picture clear for the spin, but why these choices are physically relevant? Recall that there is a naturally gauge-invariant formulation for parton physics in QCD. This naturalness is the key result of OPE and factorization theorems. How to measure and calculate the relevant quantities in the J-M sum rule? How to measure and calculate the relevant quantities in the J-M sum rule?

14. A new development

15. Gauge-symmetric but non-physical

16. A new observation (X.Ji, Y. Zhao,J.Zhang, 2013)

17. Electric field of a charge

18. A moving charge

19. Gauge potential

20. Large momentum limit As the charge velocity approaches the speed of light, E ┴ >>E ║, B ~ E ┴, thus As the charge velocity approaches the speed of light, E ┴ >>E ║, B ~ E ┴, thus E ┴ become physically meaningul The E ┴ & B fields appear to be that of the free radiation Weizsacker-William equivalent photon approximation (J. D. Jackson) Weizsacker-William equivalent photon approximation (J. D. Jackson) Gauge-invariant A ┴ appears to be now physical, which generates the E ┴ & B. Gauge-invariant A ┴ appears to be now physical, which generates the E ┴ & B.

21. Theorem Thus, one would expect that the total gluon helicity ΔG must be the matrix element of ExA ┴ in a large momentum nucleon. Thus, one would expect that the total gluon helicity ΔG must be the matrix element of ExA ┴ in a large momentum nucleon. X. Ji, J. Zhang, and Y. Zhao (arXiv:1304.6708) X. Ji, J. Zhang, and Y. Zhao (arXiv:1304.6708) is just the IMF limit of the matrix element is just the IMF limit of the matrix element of ExA ┴ of ExA ┴

22. QCD case QCD case A gauge potential can be decomposed into longitudinal and transverse parts (R.P. Treat,1972), A gauge potential can be decomposed into longitudinal and transverse parts (R.P. Treat,1972), The transverse part is gauge covariant, The transverse part is gauge covariant, In the IMF, the gauge-invariant gluon spin becomes In the IMF, the gauge-invariant gluon spin becomes

23. One-loop example The result is frame-dependent, with log dependences on the external momentum The result is frame-dependent, with log dependences on the external momentum Anomalous dimension coincides with X. Chen et al. Anomalous dimension coincides with X. Chen et al.

24. Taking large P limit If one takes P-> ∞ first before the loop integral, one finds If one takes P-> ∞ first before the loop integral, one finds This is exactly photon (gluon) helicity calculated in QCD factorization! Has the correct scale evolution. This is exactly photon (gluon) helicity calculated in QCD factorization! Has the correct scale evolution.

25. Subtlety of limiting procedure

26. Lattice QCD ExA ┴ is perfectly fit for lattice QCD calculation of ΔG! ExA ┴ is perfectly fit for lattice QCD calculation of ΔG! To get large momentum nucleon, one has to have a fine lattice in the z-direction: To get large momentum nucleon, one has to have a fine lattice in the z-direction: P ~ 1/a P ~ 1/a To separate excited states of the moving nucleon, one also needs fine lattice spacing in the time direction. To separate excited states of the moving nucleon, one also needs fine lattice spacing in the time direction. 32 2 X64 2 32 2 X64 2

27. z x,y γ=4

28. What a lattice calculation of ΔG implies? Settles if axial anomaly plays an important role in the quark helicity measurement, by determining how large is ΔG Settles if axial anomaly plays an important role in the quark helicity measurement, by determining how large is ΔG Since the experimental data says, Since the experimental data says, how much ΔG sits at very small x? how much ΔG sits at very small x? How much the gluon helicity contributes to the proton helicity at small scale. How much the gluon helicity contributes to the proton helicity at small scale.

29. Orbital angular momentum

30. Matching condition for OAM Calculate OAM at finite momentum Calculate OAM at finite momentum Extract MS-bar matrix elements by solving Extract MS-bar matrix elements by solving the matching conditions the matching conditions

31. Measuring orbital contribution Parton OAM contribution can be related twist-three GPD’s Parton OAM contribution can be related twist-three GPD’s Y. Hatta, Phys. Lett. B708 (2012) 186-190; Y. Hatta and S. Yoshida, JHEP 1210 (2012) 080 Ji, Xiong, Yuan, Phys. Rev. Lett. 109, 152005 (2012); to appear in PRD ， 2013 Need some DVCS like process (Xiong et al, to be published) Need some DVCS like process (Xiong et al, to be published)

32. X-dependence

34. Ken Wilson (1936-2013)

35. Conclusions