Spin structure of hadrons from lattice QCD

1. Spin structure of hadrons from lattice QCD Philipp Hägler and B. Bistrovic, J. Bratt, J.W. Negele, A. Pochinsky (MIT) R.G. Edwards, D.G. Richards, K. Orginos (Jlab) M. Engelhardt (New Mexico) G. Fleming (Yale) B. Musch (TUM) D.B. Renner (Arizona) W. Schroers (DESY Zeuthen) (LHPC/MILC) In collaboration with D. Brömmel, M. Diehl (DESY) M. Göckeler, A. Schäfer (Regensburg U.) R. Horsley, J. Zanotti (Edinburgh U.) P. Rakow (Liverpool U.) D. Pleiter, G. Schierholz (DESY Zeuthen) (QCDSF/UKQCD-collaboration) supported by

2. Overview introduction to generalized parton distributions (GPDs) computation of moments of (G)PDs in lattice QCD selected lattice results on • quark spin fraction • form factors of the energy momentum tensor • transverse spin densities relation GPDs ↔ tmdPDFs implications for asymmetries summary

3. Spin structure of the nucleon at a given resolution scale

4. Some fundamental sumrules rotational symmetry translation invariance Noether‘s theorem symmetries conservation laws Noether‘s theorem conservation of angular momentum conservation of momentum QCD energy momentum tensor (and angular momentum density tensor) in QCD, consider the energy momentum tensor (and the angular momentum density tensor) where momentum sumrule vanishing of the anomalous gravitomagnetic moment (Jaffe,Ji-) nucleon spin sumrule

5. Typical processes DVCS DIS SIDIS find a process like deep ineleastic scattering (DIS) or deeply virtual compton scattering which (DVCS) experiment is sensitive to the nucleon matrix elements we are interested in factorizes contributions from higher twist exclusive complicated convolutions of numerous variables and non-perturbative observables

6. Brief introduction to GPDs Müller,Robaschik,Geyer, Dittes,Horejsi,1994 Ji, 1997, Radyushkin, 1997 four unpolarized and polarized generalized distributions at twist 2 vector axial-vector

7. Transversity and spin density matrices number of independent quark-nucleon helicity amplitudes→ spin density matrix transversity/tensor/ quark helicity-flip helicities Soffer-bound helicities

8. Introduction to GPDs, continued four transversity / tensor / quark helicity flip GPDs at twist 2 (M. Diehl, EPJ C19, 2001)

9. Some basic properties of GPDs „local“ limit forward limit form factors of energy momentum tensor higher Mellin moments transverse spin distributions in transverse coordinate space M. Diehl, Ph.H.

10. Generalized form factors in Lattice QCD Mellin moments nucleon source/sink requires inversion of huges matrices operator insertion at time t quark-path through lattice

11. Computation of moments of GPDs on the lattice disconnected contributions very expensive, not included so far drop out for u-d

12. Longitudinal spin structure of the nucleon (in collaboration with LHPC/MILC)

13. Quark spin fraction from lattice simulation Gell-Mann-Oakes-Renner chiral extrapolation based on leading order heavy baryon chPT Diehl, Manashov&Schäfer, EPJA 2006 LHPC/MILC preliminary

14. Isosinglet momentum fraction of quarks in the nucleon u+d LHPC/MILC preliminary chiral extrapolation based on covariant chPT O(p²) Dorati, Gail, Hemmert 2007

15. Isosinglet generalized form factor anomalous gravitomagnetic moment of quarks u-d LHPC/MILC preliminary

16. Anomalous gravitomagnetic form factor of quarks u+d LHPC/MILC preliminary chiral extrapolation based on heavy baryon chPT O(q²) Diehl, Manashov, Schäfer, EPJC 2006

17. Quark spin and OAM contributions to the nucleon spin LHPC/MILC preliminary

18. Quark spin and OAM contributions to the nucleon spin LHPC/MILC preliminary

19. Transverse spin structure of the nucleon (in collaboration with QCDSF/UKQCD)

20. Lowest two moments of the Soffer-bound from lattice QCD QCDSF/UKQCD PLB 627 (2005)

21. Transverse spin structure of the nucleon let‘s add an additional degree of freedom distance of the quark to the center of momentum of the nucleon in the transverse plane where Diehl / PhH EPJC 2005 where

22. Transversely polarized quarks transversity basis obtain probability density interpretation for the transversity part of the density matrix multipole-expansion Diehl / PhH EPJC 2005 quadrupole monopole dipole

23. Results for moments of the tensor GPDs QCDSF/UKQCD PLB 2005 QCDSF/UKQCD PLB 2005 QCDSF/UKQCD hep-lat/0612032 QCDSF/UKQCD hep-lat/0612032 the tensor GFFs are remarkably large

24. Lowest n=1 moments of up- and down-quark densities QCDS/UKQCD hep-lat/0612032 up up up down down down strong deviations from spherical symmetry

25. Interpretation of the observed deformation patterns in parts based on M. Burkardts interpretation of distortions we know from longitudinal polarized quarks/nucleons: spin of up-quarks is aligned with nucleon spin spin of down-quarks is anti-aligned with nucleon spin assume : for up- and down-quarks down-quarks in an „unpolarized“ nucleon =1/2( + )1/2 high density in the upper half plane d-quarks „unpolarized“ down-quarks in a polarized nucleon = +  high density in the lower half plane d-quarks

26. Relation between GPDs and tmdPDFs GPDs tmdPDFs conjecture / hypothesis Burkardt PRD 72 (2005) asymmetries

27. Implications for asymmetries up-quark FSI M. Burkardt 2003/2005 expect sizeable effect with opposite signs for up- and down-quarks (Sivers-effect) sizeable single spin asymmetries in semi-inclusive deep ineleastic scattering, seen by HERMES (PRL, 2005)

28. Lattice QCD prediction for azimuthal asymmetries large Boer-Mulders effect with the same sign for up and down quarks strongly distorted densities of transversely polarized quarks in an unpolarized nucleon lead to prediction of a cos(2Ф) asymmetries in SIDIS (JLab, COMPASS) could be observed in unpolarized Drell-Yan (GSI-FAIR/PANDA) approved Jlab proposal PR12-06-112

29. what about the transverse spin structure of the pion? but is non-zero? preliminary preliminary YES n=2 n=1 the pion has a very suprising non-trivial transverse spin structure! First results on the spin structure of pions from lattice QCD longitudinal spin structure is trivial pion spin structure?

30. Summary small total orbital angular momentum of quarks small for separate up- and down-quark OAM contributions sizeable ~20% up- and down-quark densities strongly distorted for transversely polarized quarks in transversely polarized nucleons important consequences for asymmetries in SDIS and Drell-Yan (following arguments by M. Burkardt) unpolarized quarks in a transversely polarized nucleon: significant SSAs (seen by e.g. HERMES, PRL 2005) transversely polarized quarks in an unpolarized nucleon: significant azimuthal asymmetries („lattice QCD prediction“, to be confirmed by e.g. Jlab, COMPASS, GSI-FAIR/PANDA measurements) the pion has a non-trivial (transverse) spin structure

32. n=2 moments of up- and down-quark densities up up up down down down same pattern as before for n=1 densities are strongly deformed strong correlations of coordinate and spin degrees of freedom

33. Interpretation of the observed deformation patterns in parts based on M. Burkardts interpretation of distortions we know from longitudinal polarized quarks/nucleons: spin of up-quarks is aligned with nucleon spin spin of down-quarks is anti-aligned with nucleon spin assume : for up- and down-quarks high density in the upper half plane up-quarks high density in the lower half plane d-quarks

34. =1/2( + )  1/2 and since for down-quarks in an „unpolarized“ nucleon high density in the upper half plane d-quarks = +  however for „unpolarized“ down-quarks in a polarized nucleon high density in the lower half plane d-quarks

35. Sivers asymmetry measured by HERMES pion production from positrons on a transversely polarized hydrogen target Sivers asymmetry HERMES collaboration PRL 2005 cos(2Ф) in SIDIS Boer-Mulders effect unpol. DY lattice-prediction expect large negative

36. chiral extrapolation based on chPT including the -resonance in a finite volume Hemmert/Procura/Weise PRD 2003 Wollenweber, diploma thesis, TUM 2005 Gell-Mann-Oakes-Renner QCDSF/UKQCD, PRD 2006

37. Isovector momentum fraction of quarks in the nucleon u-d LHPC/MILC preliminary chiral extrapolation based on covariant chPT O(p²) Dorati, Gail, Hemmert 2007

38. Computation of moments of GPDs on the lattice, continued finally, equate the lattice ratio 3pt/2pts and the continuum parametrization for different momenta, indices and projectors simultaneously (for a given momentum transfer2 t) extract the GFFs from overdetermined set of equations LHPC/SESAM PRD 2003

40. Overview – computation of GPDs on the Lattice insert complete set of states+ use translation operators choose nucleon source/sink parametrize matrix element of local ops in terms of generalized form factors finally, we equate the lattice ratio 3pt/2pts and the continuum parametrization for different momenta and indices simultaneously at a given momentum transfer2 t this gives an (overdetermined) set of linear equations which is solved to obtain the GFFs

41. Computation of moments of GPDs on the lattice (1) define 3pt-function and choose operator (2) choose sink/source operators (3) get rid of all-to-all propagators

42. (4) (5) renormalize parametrize matrix element of local ops in terms of generalized form factors transfer matrix formalism (6) (7) finally, equate the lattice ratio 3pt/2pts and the continuum parametrization for different momenta, indices and projectors simultaneously (for a given momentum transfer2 t) extract the GFFs from overdetermined set of equations (8) enjoy