1. Xiangdong Ji University of Maryland/SJTU Physics of gluon polarization Jlab, May 9, 2013

2. Gluon polarization Ever since the EMC “spin crisis”, the gluon polarization has been one of the most important pursuits of hadron physics community Ever since the EMC “spin crisis”, the gluon polarization has been one of the most important pursuits of hadron physics community HERMES COMPASS RHIC SPIN “HERA-N” “EIC…”

3. Physics arguments 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,…) Can contribute to the quark helicity through axial anomaly (Altarelli & Ross, Carlitz, Collins, & Mueller,…) 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 theoretical puzzles about it! However, there are a number of theoretical puzzles about it!

4. Axial Anomaly It was argued that Δq probed by DIS is not entirely due to the quark contribution. There is a gluon anomaly contribution. This contribution is proportional to ( α S /2π)ΔG It was argued that Δq probed by DIS is not entirely due to the quark contribution. There is a gluon anomaly contribution. This contribution is proportional to ( α S /2π)ΔG For the anomaly to have a large contribution For the anomaly to have a large contribution ΔG must be on the order few unit of hbar. ΔG must be on the order few unit of hbar. Thus, ΔG could be large even at non- perturbative scale. Thus, ΔG could be large even at non- perturbative scale.

5. Physics: Feynman parton picture A fast moving proton is a beam of free quarks and gluons. A fast moving proton is a beam of free quarks and gluons. 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

6. 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!

7. 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).

8. 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?

9. A LL from RHIC 2009 9

10. Electric field of a charge

11. A moving charge

12. Gauge potential

13. Observations Although the transverse part of the vector potential is gauge invariant, the separately E ┴ does not transform properly, under Loretez transformation, and is not a physical observable (X. Chen et al, x. Ji, PRL) Although the transverse part of the vector potential is gauge invariant, the separately E ┴ does not transform properly, under Loretez transformation, and is not a physical observable (X. Chen et al, x. Ji, PRL) E ┴ generated from E ║ from Lorentz boost. A lorentz-transformed E has different decomposition E = E ┴ + E ║ in different frames. There is no charge that separately responds to E ┴ and E ║

14. 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) Thus gauge-invariant A ┴ appears to be now physical which generates the E ┴ & B. Thus gauge-invariant A ┴ appears to be now physical which generates the E ┴ & B.

15. Gauge invariant photon helicity X. Chen et al (PRL, 09’) proposed that a gauge invariant photon angular momentum can be defined as X. Chen et al (PRL, 09’) proposed that a gauge invariant photon angular momentum can be defined as ExA ┴ ExA ┴ This is not an observable when the system move at finite momentum because this is only a part of the contribution which cannot be measured separately. However, it becomes an observable in the IMF when Weizsacker-William’s picture is true! However, it becomes an observable in the IMF when Weizsacker-William’s picture is true!

16. 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 ┴

17. 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

18. 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.

19. 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 anomalous dimension. This is exactly photon (gluon) helicity calculated in QCD factorization! Has the correct anomalous dimension.

20. Subtlety of the limiting procedures There are two possible limits, There are two possible limits, Taking p-> ∞ before UV regularization (physical case, light-cone) Taking UV regularization before p-> ∞ (practical calculation, time-independent) Two limits get the same IR physics Two limits get the same IR physics One can get one limit from the other by a perturbative matching condition, Z. One can get one limit from the other by a perturbative matching condition, Z.

21. 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

22. 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.

23. x-dependence? X-dependence of a parton distribution has been very difficult to calculate in the past. The only approach is through the local moments. X-dependence of a parton distribution has been very difficult to calculate in the past. The only approach is through the local moments. However, it is very difficult to calculate higher moments numerically. However, it is very difficult to calculate higher moments numerically. It will be nice to find a way to directly calculate the x-dependence on lattice It will be nice to find a way to directly calculate the x-dependence on lattice

24. arXiv1305.1539

25. observation Usual parton distribution Usual parton distribution Consider instead Consider instead

26. Relationship The matching condition is perturbative The matching condition is perturbative The correction is power-suppressed. For practical calculation, a momentum of 5 GeV might be good enough. The correction is power-suppressed. For practical calculation, a momentum of 5 GeV might be good enough.

28. The extension of the approach GPDs GPDs TMDs TMDs

29. The extension of the approach Wigner distribution Wigner distribution Light-cone amplitudes Light-cone amplitudes Light-cone wave functions Light-cone wave functions Higher-twists…. Higher-twists….

30. Conclusions We find the gluon helicity measured in high- energy scattering is just EXA ┴ in the large momentum limit, We find the gluon helicity measured in high- energy scattering is just EXA ┴ in the large momentum limit, This gives the gauge invariant and physically manifest notion of the total gluon helicity This gives practical way to calculate ΔG We find a practical way how to calculate light-cone distributions: We find a practical way how to calculate light-cone distributions: PDFs, TMDs, GPDs, HTs, LCWFs, LCDAs, etc… PDFs, TMDs, GPDs, HTs, LCWFs, LCDAs, etc… ten years from now there will be a lot of lattice result. ten years from now there will be a lot of lattice result.