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Titelpage. Lattice calculations of GPDs. Philipp Hägler. affiliations etc. supported by. excellence cluster universe. Overview. Physics. What we can / cannot do on the lattice. x-moments of GPDs: Our A, B, C. Impact on the nucleon spin sum rule. Impact on the transverse size.

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Titelpage

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  1. Titelpage Lattice calculations of GPDs Philipp Hägler affiliations etc supported by excellence cluster universe

  2. Overview Physics What we can / cannot do on the lattice x-moments of GPDs: Our A, B, C Impact on the nucleon spin sum rule Impact on the transverse size Conclusions, open questions, and perspectives

  3. Physics: Concentrate on: Tomography; Transverse size center of momentum Burkardt PRD 2000 Decomposition of the nucleon spin Ji‘s nucleon spin sum rule (X. Ji, PRL 1997) everything is: -gauge-invariant -scale & scheme dependent -measurable

  4. What we can / cannot do on the lattice (presently) + higher-spin couplings spin-2 gravitonal coupling spin-1 helicity-flip coupling spin-1 (axial-)vector coupling ... in principle access to nucleon spin sumrule (for t=0) ! highly non-trivial in practice… flavor decomposition; quark/hadron polarizations; range of momentum transfers straightforward first exploratory studies of non-local couplings on the lattice related to TMDs [Musch, PhH, Schäfer, Negele ] bilocal on light-cone DVCS – factorization [Collins `99,`01] for the time being: local couplings

  5. Generalized form factors and basic sumrules nmax~4 full x()-dependence cannot be reconstructed model-independently quark, anti-quark contributions cannot be separated rotational symmetry translation invariance Noether‘s theorem Ji&Lebed PRD 2000 Ph.H. PLB 2004 conservation of angular momentum conservation of momentum relation to (moments of) GPDs in QCD, consider the energy momentum tensor (and the angular momentum density tensor) n=1  Dirac&Pauli FFs n=2  FFs of the EM-tensor

  6. Lattice QCD calculations of hadron structure disconnected hard, but can be done recent progress by Collins, Bali, Schäfer; Kentucky group = local vector-, axialvector-, quark spin flip-,(spin-2) graviton-, „spin-n“ coupling u-d is safe quark propagators compute the path-integral numerically MC integration gauge fields/links U quarks algorithms concepts methods machines

  7. Lattice QCD calculations of hadron structure systematic „ab initio“-approach, but • statistical errors from MC integration • discretization and finite volume errors/effects • contaminations from excited states • large quark masses • large minimal non-zero momenta approximations can be continuously improved limited by computational and human resources

  8. LHPC nf=2+1 mixed action DW valence+staggered sea A, B, C LHPC nf=2+1 mixed; arXiv:1001.3620 (updating PRD 2008, 0810.1933) u+d u-d B A C A B C u-d u+d disconnected contributions are not included↔ only u-d is „exact“

  9. Chiral extrapolations of A,B,C global simultaneous fits of A, B, C with common parameter <x> + 8 additional free parameters/LECs, to >80 lattice data points in each case (u-d and u+d) LHPC nf=2+1 mixed arXiv:1001.3620 (updating PRD 2008) only quark line connected contributions

  10. Quark angular momentum from covariant BChPT extrapolations LHPC arXiv:1001.3620

  11. Nucleon spin structure and spin sum rule LHPC PRD D 77, 094502 (2008) LHPC PRD 2003 relativistic quark models LHPC nf=2+1 mixed arXiv:1001.3620 (updating PRD 2008)

  12. Nucleon spin structure and spin sum rule LHPC nf=2+1 mixed arXiv:1001.3620 (updating PRD 2008) pioneering lattice calculations by Gadiyak, Ji and Jung in 2001

  13. Contributions to the nucleon spin * compares well with study by Goloskokov&Kroll 2008 [non-singlet, connected only] *

  14. Looks great, but ... momentum fraction of quarks in the nucleon substantial systematic uncertainties J. Zanotti et al. [QCDSF], PoS Lat2009 PhH, Phys.Rep. 2009

  15. Transverse size of the nucleon – basic observations x-dependent; defined at =0  xB=0 cannot be inferred directly (model-independently) from DVCS cross section, i.e. the xB-dependence of slope in t cannot be computed directly in lattice QCD local matrix elements x-moments DVCS (LO) same for distance to spectators „transverse size“ distance to COM Burkardt PRD 2000;... extraction of generalized charge radii requires (dipole) parametrization of t-dependence of GFFs cannot be directly accessed in DVCS

  16. LHPC nf=2+1 mixed; arXiv:1001.3620 (updating PRD 2008, 0810.1933)

  17. Generalized mean square radii of the nucleon correlations in x and b LHPC PRL 93 (2004) LHPC nf=2+1 mixed; arXiv:1001.3620 (updating PRD 2008, 0810.1933) u-d strong correlations in x and b no factorization of GPDs in x and t

  18. Pion mass dependence : Dirac mean square radius SSE ChPT Hemmert, Weise EPJC 02 Göckeler et al PRD 93 1-loop HBChPT Bernard, Kaiser, Meissner NPB 1993 Dirac and Pauli FFs PhH Phys Rept `09

  19. Pion mass dependence of generalized radii for n>1 Diehl, Manashov, Schäfer EPJ A31 (2007) no (logarithmic) divergence in m for n>1 strong effect seen for n=1,...,3 will most likely persist when mmphys in any case

  20. Conclusions, open questions, and perspectives I remarkable progress in lattice studies of lowest moments of GPDs small statistical errors at m~300...350 MeV for many observables disconnected contributions time to address systematic effects and uncertainties on the lattice gluon contributions renormalization (dipole, chiral,...) extrapolations in t; ptbcs most chiral extrapolations still not quantitatively reliable need high statistics lattice results for m250 MeV global, simultaneous chiral fits DVCS measurements at COMPASS, extending the kinematic ranges of HERMES, H1, Zeus, JLab, are crucial for our understanding of nucleon structure

  21. Conclusions, open questions, and perspectives II lattice calculations and phenomenological/experimental studies of GPDs are mostly complementary We observe strong correlations in x and t based on, e.g., generalized radii Transverse size strongly reduced as larger x are approached Precise relevant region in x? What about correlations in ,xB and t ? Interpretation? Lattice results on spin sum rule, quark OAM are exciting and surprising; observe many cancellations strong motivation for further phenomenological and experimental studies Should lattice results be used to constrain GPD-models/parametrizations?

  22. as always, I am indebted to my collaborators M. Altenbuchinger, M. Procura, W. Weise (T3x, TUM) D. Brömmel (Southampton), M. Diehl (DESY), M. Göckeler, Th. Hemmert, A. Schäfer (Regensburg U.) M. Gürtler (TU München) R. Horsley, J. Zanotti (Edinburgh U.) Y. Nakamura (DESY Zeuthen) P. Rakow (Liverpool U.) D. Pleiter, G. Schierholz (DESY Zeuthen) H. Stüben (ZIB) (QCDSF/UKQCD) H. Meyer (CERN), B. Bistrovic, J. Bratt, M. Lin, J.W. Negele, A. Pochinsky, S. Syritsyn (MIT) R.G. Edwards, H.-W. Lin, B. Musch, D.G. Richards (JLab) K. Orginos (W&M) M. Engelhardt (New Mexico) D.B. Renner (DESY Zeuthen), W. Schroers (Berlin) (LHPC) References: LHPC PRD 77, 094502 (2008) LHPC arXiv:1001.3620; QCDSF 0912.0167

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