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Moments of GPDs and the Nucleon Spin Structure from Lattice QCD. Titelpage. Philipp Hägler. affiliations etc. supported by. excellence cluster universe. Overview. GPDs and generalized form factors. local operators and correlators on the lattice. lattice results on lowest moments of GPDs.

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  1. Moments of GPDsand the Nucleon Spin Structurefrom Lattice QCD Titelpage Philipp Hägler affiliations etc supported by excellence cluster universe

  2. Overview GPDs and generalized form factors local operators and correlators on the lattice lattice results on lowest moments of GPDs chiral extrapolations decomposition of the nucleon spin Summary and challenges

  3. GPDs and hadron structure Decomposition of the nucleon spin Ji‘s nucleon spin sum rule (X. Ji, PRL 1997) everything is: -gauge-invariant -scale & scheme dependent -measurable Tomography [Burkardt PRD`00;Ralston, Pire PRD‘02] center of momentum [Diehl,PhH EPJC`05; QCDSF PRL`07`08]

  4. GPDs and non-local operators Higher moments and local operators ... of matrix elements GFFs …of GPDs Ji&Lebed PRD 2000 Ph.H. PLB 2004 first exploratory studies of non-local couplings on the lattice related to TMDs [Musch(Fr 2pm), PhH, Schäfer, Negele] bilocal on light-cone DVCS (– factorization) [Müller at el. `94-, Radyushkin, Ji `97-, Collins `99,`01] for the time being: local couplings

  5. Local quark operators on the lattice discretized operators on a Euclidean space-time lattice LorentzO(4) H(4) n=2 operator renormalization lattice perturbation theory non-perturbatively gluon-ops in singlet sector operator mixing under renormalization non-singlet sector H(4) leading twist-2

  6. Correlation functions on the lattice quark propagators products of quark propagators compute the path-integral using MC methods gauge fields/links U quarks

  7. Lattice simulation details • mixed action approach: DW fermions • on a Asqtad staggered sea for Nf=2+1; • including HYP-smearing • - Ls=16, mres0.1mq • - lattice spacing a ~ 0.124 fm • - volumes of ~2.5 and ~3.5 fm3 • two sink momenta P‘=(0,0,0), (-1,0,0) operator renormalization: # of „measurements“ increased by factor 8 compared to PRD 77 094502 (2008) ongoing efforts within LHPC based on DW fermions (RBC/UKQCD) and improved Wilson fermions (BMW)

  8. n=2 - A, B, C - Form factors of the energy momentum tensor seems to be compatible with large Nc limit – see e.g. Goeke, Polyakov, Vanderhaeghen PiPaNP 2001 disconnected contributions are not included↔ only u-d is „exact“

  9. u+d u-d u-d u+d

  10. O(p3) small quark AGM !? B20 and the anomalous gravitomagnetic moment [Teryaev `99-; Brodsky, Hwang et al. `00-] based on HBChPT by Diehl, Manashov, Schäfer EJPA 2006 Ando, Chen, Kao PRD 2006 LHPC nf=2+1 mixed arXiv:1001.3620 (updating PRD 2008) including non-linear correlation in t and m

  11. O(p3) sizeable and negative !? C20 and the second moment of the D-term [Polyakov&Weiss `99] based on HBChPT by Diehl, Manashov, Schäfer EJPA 2006 Ando, Chen, Kao PRD 2006 LHPC nf=2+1 mixed arXiv:1001.3620 (updating PRD 2008) including non-linear correlation in t and m

  12. chiral extrapolation based on covariant BChPT by Dorati, Gail, Hemmert NPA 2008 with common parameter Global, simultaneous chiral extrapolation of A, B, C LHPC nf=2+1 mixed; arXiv:1001.3620 (updating PRD 2008, 0810.1933) only quark line connected contributions

  13. Stability of the BChPT fit cuts

  14. preliminary LHPC DW (Syritsyn et al.) Quark angular momentum from covariant BChPT extrapolations LHPC arXiv:1001.3620

  15. Quark spin and OAM LHPC PRD D 77, 094502 (2008) LHPC PRD 2003 preliminary LHPC DW (Syritsyn et al.) relativistic quark models LHPC nf=2+1 mixed arXiv:1001.3620 (updating PRD 2008)

  16. Nucleon spin structure and spin sum rule LHPC nf=2+1 mixed arXiv:1001.3620 pioneering lattice calculations by Gadiyak, Ji and Jung in 2001

  17. Contributions to the proton spin * [non-singlet, connected only; additional uncertainties due to chiral extrapolations, renormalization] *

  18. Ju, Jd template figure [JLab Hall A PRL`07; HERMES JHEP`08] LHPC arXiv:1001.3620 (this work) LHPC PRD `08 0705.4295 QCDSF (Ohtani et al.) 0710.1534 Goloskokov&Kroll EPJC`09 0809.4126 Wakamatsu 0908.0972 DiFeJaKr EPJC `05 hep-ph/0408173 (Myhrer&)Thomas PRL`08 0803.2775

  19. Conclusions and outlook lattice QCD a great tool to study all sorts of local couplings form factors, magnetic moments,… (axial-) vector coupling „gravitational“ coupling decomposition of the nucleon spin transversity, transverse spin structure of hadrons spin-flip coupling Challenges disconnected diagrams; strange quark contributions; gluon operators; operator mixing;… form factors (radii, magnetic moments, etc.)[H.-W. Lin Wed ~3pm]: systematic uncertainties complementary to experimental efforts at JLab, HERMES, COMPASS; phenomenological and model studies

  20. as always, I am indebted to my collaborators M. Altenbuchinger, B. Musch (→JLab), M. Gürtler (→Regensburg), W. Weise (T39, 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) H. Stüben (ZIB) (QCDSF/UKQCD) B. Bistrovic, J. Bratt, J.W. Negele, A. Pochinsky, S. Syritsyn (MIT) R.G. Edwards, B. Musch, D.G. Richards (JLab) K. Orginos (W&M) M. Engelhardt (New Mexico) G. Fleming, M. Lin (Yale), H.-W. Lin (INT), H. Meyer (Mainz), D.B. Renner (DESY Zeuthen), M. Procura (TUM), W. Schroers (LHPC) References: QCDSF PoS(LAT2006)120, 0710.1534, PRL 98 222001 (2007), PRL 2008 (0708.2249), Brömmel et al EPJC 2007; LHPC PRD 77, 094502 (2008), 0810.1933; 1001.3620; Diehl&Hägler EPJC hep-ph/0504175; Musch et al. 0811.1536; Musch arXiv:0907.2381; PhH, Musch et al. EPL 2009 (arXiv:0908.1283) PhH Phys.Rep. 2010 (0912.5483)

  21. Quark angular momentum emplyoing HBChPT+ results [Chen Ji PRL 2002] LHPC arXiv:1001.3620 BChPT-fit

  22. Lattice QCD vs relativistic quark models – QCD evolution (Wakamatsu 2005; Thomas, PRL 2008)

  23. lattice + evolution Lattice QCD vs relativistic quark models – QCD evolution (Wakamatsu 2005; Thomas, PRL 2008) in coll. with LHPC and M. Altenbuchinger, W. Weise (TUM) relativistic quark model

  24. Lattice in contradiction with well-known (analytical and model) results? requires non-vanishing quark OAM L0 non-zero Sivers effect Brodsky, Hwang et al NPB 2001, PLB 2002 Burkardt, Hwang PRD 2004 relativistic quark models

  25. Comparison with phenomenology, previous lattice studies from Hall A PRL 2007; 0709.0450 outdated old preliminary unpublished Published PRD 2008 arXiv:0705.4295 QCDSF unquenched Lattice 2007 LHPC mixed 2010 disclaimer: JLab band from integral over VGG model, constrained at a single x=-point proceedings arXiv:0710.1534 this work arXiv:1001.3620

  26. Isovector axial vector coupling constant (required for L=J-Σ/2) emplyoing SSE (HBChPT+) results [Procura, Hemmert, Musch, Weise PRD 2007, QCDSF PRD 2006] LHPC arXiv:1001.3620 compare to LHPC PRL 96 502001 (2006)

  27. Isosinglet quark spin fraction (required for L=J-Σ/2) employing HBChPT by Diehl, Manashov, Schäfer EJPA 2006; Ando, Chen, Kao PRD 2006 LHPC arXiv:1001.3620 HERMES PRD 2007

  28. Covariant chiral perturbation theory for A, B, C Covariant BChPT calculation by Dorati, Gail, Hemmert NPA 2008 similar results for B, C Including the dependence on the squared momentum transfer t with common parameter

  29. Polarized momentum fraction employing HBChPT results [Chen Ji PLB 2001] LHPC arXiv:1001.3620 DSSV PRL 2008

  30. Form factors of the energy momentum tensor isovector quark momentum fraction compared to LHPC PRD 77 094502 (2008) LHPC arXiv:1001.3620 HBChPT-fit

  31. Momentum fraction of quarks in the nucleon substantial systematic uncertainties

  32. Correlations between momenta, positions, spins Correlations in

  33. Reminder: Generalized mean square radii of the nucleon correlations in x and b LHPC nf=2+1 mixed preliminary (updating PRD 2008) strong correlations in x and b no factorization of GPDs in x and t

  34. Transverse momentum dependent PDFs correlations in x and k ≈ factorization of tmdPDFs in x and k Musch et al. nf=2+1 mixed tbp and PoS LC2008 no visible correlations in l P and l2

  35. Spin structure of the pion but is non-zero? quark transverse spin in x-direction up-quarks in a + lattice calculations of quark spin-flip couplings QCDSF nf=2 Clover, PRL 2008 QCDSF nf=2 Clover PRL 2008 Is the pion spin structure trivial? pion spin sumrule the pion has a non-trivial transverse spin structure!

  36. Lattice QCD propaganda BMW (Dürr et al.) Nature 2009

  37. Lattice QCD propaganda Davies, Lepage et al. PRL 2004 Davies, Lepage et al. PRL 2008

  38. Invariant amplitudes related to quadrupole deformations („pretzelosity“) up down

  39. Intrinsic transverse momentum densities of the nucleon down up down up LC quark model: Pasquini et al PRD 2008 Musch et al. tbp

  40. Form factors of the energy momentum tensor and fundamental sumrules translation invariance conservation of momentum rotational symmetry conservation of angular momentum momentum sumrule vanishing of the anomalous gravitomagnetic moment Ji‘s nucleon spin sum rule everything is: -gauge-invariant -scale and scheme dependent -measurable where graviton-coupling spin-2 coupling

  41. multipole-expansion Diehl / PhH EPJC 2005 quadrupole monopole dipole Transversely polarized quarks in transversely polarized nucleons probability density for transversely polarized quarks in a transversely polarized proton result: „multipole-expansion“ Diehl / PhH EPJC 2005 quadrupole monopole dipole

  42. Transverse spin densities in the proton charge distribution 1.2 fm up lattice calculations of quark spin-flip couplings down QCDSF nf=2 Clover PRL 2007 quark transverse spin in x-direction up „Femto-photography“ of quarks in the proton (Pire&Ralston PRD 2002) down strongly deformed transverse spin densities

  43. Charge density in the +-baryon (spin-3/2) Alexandrou et al. nf=0(2) Wilson, PRD 2009

  44. 1-loop ChPT r-pole+polyn. monopole fit Overview of results for the pion charge radius

  45. Nucleon anomalous magnetic moment (anomalous) magnetic moment

  46. Local quark operators on the lattice leading twist-2 discretized operators on a Euclidean space-time lattice LorentzO(4) H(4) n=2 operator renormalization lattice perturbation theory non-perturbatively gluon-ops in singlet sector operator mixing under renormalization non-singlet sector H(4)

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