Xiangdong Ji University of Maryland/SJTU

1. XiangdongJi 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 • HERMES • COMPASS • RHIC SPIN • “HERA-N” • “EIC…”

3. Physics arguments • 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,…) • Its contribution to the spin grows like 1/αS • 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 (αS /2π)ΔG • For the anomaly to have a large contribution ΔG must be on the order few unit of hbar. • 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. • The gluon partons have well-defined helicity ± 1 and densities g±(x) in wavelength • Gluon helicity distribution is g(x) = g+(x) – g-(x) and • G = ʃdx 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) • 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 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 • ΔG 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 • Thus there is no way to confront theory with experiment: G = ʃdx g(x) Is there a large contribution from small x?

9. ALL from RHIC 2009

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) • 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 • E┴ become physically meaningul • The E┴ & B fields appear to be that of the free radiation • Weizsacker-William equivalent photon approximation (J. D. Jackson) • 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 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!

16. Theorem • 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) is just the IMF limit of the matrix element of ExA┴

17. QCD case • A gauge potential can be decomposed into longitudinal and transverse parts (R.P. Treat,1972), • The transverse part is gauge covariant, • 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 • Anomalous dimension coincides with X. Chen et al.

19. Taking large P limit • 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.

20. Subtlety of the limiting procedures • 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 • 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! • To get large momentum nucleon, one has to have a fine lattice in the z-direction: P ~ 1/a • To separate excited states of the moving nucleon, one also needs fine lattice spacing in the time direction. • 322X642

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 • Since the experimental data says, how much ΔG sits at very small x? • 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. • 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

24. arXiv1305.1539

25. observation • Usual parton distribution • Consider instead

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

28. The extension of the approach • GPDs • TMDs

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

30. Conclusions • 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: PDFs, TMDs, GPDs, HTs, LCWFs, LCDAs, etc… ten years from now there will be a lot of lattice result.