Contributions to the nucleon spin from lattice QCD calculations

1. Contributions to the nucleon spin from lattice QCD calculations supported by Philipp Hägler Related talks at this conference: Xi.-D. Ji (Monday 12:25h), D. Müller (Today 14:00), M. Diehl (Wednesday 14:00h) and spin physics session on Wednesday afternoon.

2. Overview Nucleon spin sum rule brief outline of computation of nucleon matrix elements in lattice QCD very brief intro to ChPT calculations selected lattice results on • form factors of the energy momentum tensor • spin and OAM contrubutions to the nucleon spin together with chiral extrapolations from QCDSF/UKQCD and LHPC comparison of lattice results with experiment&phenomenology summary/outlook

3. The nucleon spin sumrule and GPDs DVCS Nucleon spin sumrule (Ji PRL 1997) everything is: -gauge-invariant -scale and scheme dependent -measurable moments of GPDs where conservation of momentum rotational symmetry translation invariance conservation of angular momentum momentum sumrule

4. Lattice calculation of nucleon matrix elements = vector-, axialvector-, graviton-, quark spin flip-, „spin-n“ coupling quark propagators compute the path-integral numerically gauge fields/links U concepts methods algorithms machines quarks

5. Calculation of configurations and propagators SGI Altix 4700 at LRZ Garching APEmille at NIC/DESY Zeuthen 7n cluster at JLab Quellen: (http://www.lrz-muenchen.de/wir/einweihungsfeier/bildergalerie/_fotoindex.html) & Jlab webpage

6. Chiral perturbation theory and chiral extrapolations low energy effective field theory of QCD systematic expansion in small masses/momenta/energies baryon chiral perturbation theory (BChPT) covariant baryon ChPT (CBChPT) non-relativistic expansion Dorati, Gail, Hemmert NPA 2008 heavy baryon ChPT (HBChPT) expansion in 1/mN HBChPT with the Δ, small scale expansion (SSE) Bernard, Hemmert, Meißner; e.g. for gA: Hemmert, Procura, Weise PRD 2003 Chen and Ji, PRL 2002 unknown couplings / low energy constants (LECs) have to be determined from experiment and/or lattice HBChPT by Diehl, Manashov, Schäfer EJPA 2006, Ando, Chen, Kao PRD 2006

7. Lattice parameters all results transformed to MS at 4 GeV2 LHPC • domain-wall valence and staggered „Asqtad“ sea quarks („hybrid“ approach) • unquenched, Nf=2+1 • only connected contributions • lattice spacing fixed using the force F1(r) • lattice spacing 0.125 fm • pion masses as low as 350 MeV • NP improved perturbative renormalization arXiv:0705.4295 (to appear in PRD) QCDSF/UKQCD • Wilson fermions with NP clover improvement • unquenched, Nf=2 • only connected contributions • lattice spacing fixed using mN →r0=0.467fm • lattice spacings ~0.1 … 0.07 fm • pion masses as low as 350 MeV • non-perturbative renormalization arXiv:0710.1534 (proceedings)

8. Isovector quark momentum fraction chiral extrapolation based on covariant BChPT by Dorati, Gail, Hemmert NPA 2008 with common parameter global simultaneous chiral 9-parameter fit to ~120 lattice datapoints at finite t HBChPT-fit HB-limit mN → ∞ LHCP PRD 2008  CTEQ6 not yet full O(p3) depends in CBChPT simultaneously on and , in contrast to HBChPT

9. Cross-checks estimate (?) of O(p3)-corrections including comparison of chiral fits to different regions in m chiral fits agree at small and large m simple m3 -term doesn‘t fit lattice data well bending towards chiral limit seems genuine uncertainties dominated by statistical errors

10. Isovector quark momentum fraction comparison QCDSF/UKQCD and LHPC LHPC QCDSF/UKQCD substantial difference in the overall normalization of the lattice data major unresolved issue

11. Gravitomagnetic form factor B20 QCDSF/UKQCD chiral extrapolation based on covariant BChPT by Dorati, Gail, Hemmert NPA 2008 LHPC

12. Isosinglet B20(t) form factor (including quark anomalous gravitomagnetic moment AGM) (p3) small quark AGM based on HBChPT by Diehl, Manashov, Schäfer EJPA 2006, Ando, Chen, Kao PRD 2006 including LHCP PRD 2008 non-linear correlation in t and m

13. Quark angular momentum from extrapolated B20(t→0) based on HBChPT including the Δ by Chen and Ji, PRL 2002 including e.g. quarks carry 40% of total nucleon spin 1/2

14. Quark spin, OAM and total angular momentum QCDSF/UKQCD overall agreement within errors LHPC see also pioneering lattice calculations by Gadiyak, Ji and Jung 2001

15. Towards the decomposition of the nucleon spin QCDSF unquenched Lattice 2007 gluon spin contribution (e.g. COMPASS/CERN) Ji spin sumrule lattice „graviton-like coupling“ in lattice QCD exp/ pheno be aware of systematic uncertainties of lattice simulations Jlab Hall A PRL 2007

16. Towards the decomposition of the nucleon spin substantial uncertainties due to model dependence uncertainties will be reduced in future → more HERMES data, JLab 12 GeV upgrade be aware of systematic uncertainties of lattice simulations HERMES 0802.2499

17. Summary&Outlook lattice simulations provide new insights into structure of hadrons direct access to quark angular momentum and nucleon spin sumrule ChPT essential for extrapolation to the physical point (near) future: studies of GPDs of the rho (M. Gürtler in collaboration with QCDSF/UKQCD) spin structure of a spin-1 state complementary to enormous experimental efforts at e.g. HERMES, COMPASS, JLab

18. Many thanks to my collaborators • M. Göckeler, M. Ohtani • Schäfer (Regensburg U.) • M. Gürtler (TUM) • R. Horsley, J. Zanotti (Edinburgh U.) • P. Rakow (Liverpool U.) • D. Pleiter, Y. Nakamura, • G. Schierholz (DESY Zeuthen) • W. Schroers (National Taiwan U.) • (QCDSF/UKQCD) B. Bistrovic, J. Bratt, J.W. Negele, A. Pochinsky (MIT) R.G. Edwards, D.G. Richards (JLab) K. Orginos (W&M) M. Engelhardt (New Mexico) G. Fleming (Yale) B. Musch (TUM) D.B. Renner (Arizona) W. Schroers (DESY Zeuthen) (LHPC) arXiv:0710.1534 (proceedings) arXiv:0705.4295 (to appear in PRD)

20. Covariant baryon chiral perturbation theory for nucleon EM-FFs and FFs of the energy momentum tensor fully compatible with MS-HBChPT nucleon form factors FFs of the energy momentum tensor Nucl. Phys. A 798 (2008) 96 covariant BChPT HBChPT includes Choice of renormalization scheme Becher, Leutwyler 1999 Dorati, Gail, Hemmert NPA 2008 Gail&Hemmert (tpb) nucleon form factors FFs of the energy momentum tensor

21. Lattice parameters – LHPC/MILC operator renormalization:

22. Lattice parameters – QCDSF/UKQCD

23. Generalized form factors and moments of GPDs for n=2 # of GFFs moments of GPDs FT

24. Computation of moments of GPDs on the lattice disconnected contributions very expensive, not included so far drop out for u-d systematic uncertainty in isosinglet quantities large quark masses finite lattice spacing finite volume

25. Overview 1: The GFFs A,B and C disconnected contributions are not included↔only u-d is „exact“ in summary compatible with large Nc limit – see e.g. Goeke, Polyakov and Vanderhaeghen PiPaNP 2001

26. Quark spin and OAM contributions to the nucleon spin • HERMES 2007 results for Δ

27. Quark spin and OAM contributions to the nucleon spin • HERMES 2007 results for Δ

28. At finite momentum transfer

29. Example 2: Isosinglet quark momentum fraction chiral extrapolation based on CBChPT by Dorati, Gail, Hemmert 2007 with common parameter global simultaneous chiral 9-parameter fit to ~120 lattice datapoints at finite t HBChPT-fit HB-limit mN → ∞  CTEQ6 warning: only a part of CBChPT (p3)-corrections are known only depends on , in contrast to isovector channel

30. At finite momentum transfer

31. Example 2: Isosinglet C20(t) form factor (p3) (p3) chiral extrapolation based on CBChPT by Dorati, Gail, Hemmert 2007 including non-linear dependence on (t,m) warning: only a part of CBChPT (p3)-corrections are known

32. The GFFs A,B and C LHPC QCDSF/UKQCD

33. Quark spin, OAM and total angular momentum QCDSF/UKQCD LHPC overall agreement within errors in particular Lu+d0

34. Conclusions two different dynamical lattice simulations one message?! (CTEQ6) (beta-decay & HERMES PRL 2007) • disclaimer: additional uncertainties arising from • missing disconnected contributions, • - discretization effects, • - finite size effects, • - chiral extrapolation (in particular for B20)