Spin structure of hadrons from lattice QCD Philipp

Spin structure of hadrons from lattice QCD Philipp Hägler In collaboration with and 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) 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) (QCDSF/UKQCD-collaboration) (LHPC/MILC) supported by Ph. Hägler, COMPASS, 2007 1

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 ↔ tmd. PDFs implications for asymmetries summary Ph. Hägler, COMPASS, 2007 2

Spin structure of the nucleon at a given resolution scale Ph. Hägler, COMPASS, 2007 3

Some fundamental sumrules symmetries Noether‘s theorem conservation laws QCD energy momentum tensor (and angular momentum density tensor) where momentum sumrule vanishing of the anomalous gravitomagnetic moment (Jaffe, Ji-) nucleon spin sumrule Ph. Hägler, COMPASS, 2007 4

Typical processes DIS DVCS Ph. Hägler, COMPASS, 2007 SIDIS exclusive 5

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 Ph. Hägler, COMPASS, 2007 6

Transversity and spin density matrices number of independent quark-nucleon helicity amplitudes→ spin density matrix helicities transversity/tensor/ quark helicity-flip helicities Soffer-bound Ph. Hägler, COMPASS, 2007 7

Introduction to GPDs, continued four transversity / tensor / quark helicity flip GPDs at twist 2 (M. Diehl, EPJ C 19, 2001) Ph. Hägler, COMPASS, 2007 8

Some basic properties of GPDs forward limit form factors of energy momentum tensor Ph. Hägler, COMPASS, 2007 „local“ limit higher Mellin moments 9

Generalized form factors in Lattice QCD Mellin moments nucleon source/sink operator insertion at time t quark-path through lattice Ph. Hägler, COMPASS, 2007 10

Computation of moments of GPDs on the lattice disconnected contributions very expensive, not included so far drop out for u-d Ph. Hägler, COMPASS, 2007 11

Longitudinal spin structure of the nucleon (in collaboration with LHPC/MILC) Ph. Hägler, COMPASS, 2007 12

Quark spin fraction from lattice simulation Gell-Mann-Oakes-Renner chiral extrapolation based on leading order heavy baryon ch. PT Diehl, Manashov&Schäfer, EPJA 2006 LHPC/MILC preliminary Ph. Hägler, COMPASS, 2007 13

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

Isosinglet generalized form factor anomalous gravitomagnetic moment of quarks u-d LHPC/MILC preliminary Ph. Hägler, COMPASS, 2007 15

Anomalous gravitomagnetic form factor of quarks u+d ry ina m LC P LH I /M C chiral extrapolation based on heavy baryon ch. PT O(q²) Ph. Hägler, COMPASS, 2007 li e r p Diehl, Manashov, Schäfer, EPJC 2006 16

Quark spin and OAM contributions to the nucleon spin LHPC/MILC preliminary Ph. Hägler, COMPASS, 2007 17

Quark spin and OAM contributions to the nucleon spin LHPC/MILC preliminary Ph. Hägler, COMPASS, 2007 18

Transverse spin structure of the nucleon (in collaboration with QCDSF/UKQCD) Ph. Hägler, COMPASS, 2007 19

Lowest two moments of the Soffer-bound from lattice QCDSF/UKQCD PLB 627 (2005) Ph. Hägler, COMPASS, 2007 20

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 / Ph. H EPJC 2005 where Ph. Hägler, COMPASS, 2007 21

Transversely polarized quarks transversity basis obtain probability density interpretation for the transversity part of the density matrix multipole-expansion Diehl / Ph. H EPJC 2005 monopole Ph. Hägler, COMPASS, 2007 dipole quadrupole 22

Results for moments of the tensor GPDs QCDSF/UKQCD PLB 2005 QCDSF/UKQCD hep-lat/0612032 the tensor GFFs Ph. Hägler, COMPASS, 2007 QCDSF/UKQCD PLB 2005 QCDSF/UKQCD hep-lat/0612032 are remarkably large 23

Lowest n=1 moments of up- and down-quark densities QCDS/UKQCD hep-lat/0612032 up up up down strong deviations from spherical symmetry Ph. Hägler, COMPASS, 2007 24

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 of down-quarks is anti-aligned with nucleon spin assume : down-quarks in an „unpolarized“ nucleon for up- and down-quarks =1/2( + high density in the upper half plane d-quarks „unpolarized“ down-quarks in a polarized nucleon d-quarks Ph. Hägler, COMPASS, 2007 ) 1/2 = + high density in the lower half plane 25

Relation between GPDs and tmd. PDFs GPDs tmd. PDFs conjecture / hypothesis Burkardt PRD 72 (2005) Ph. Hägler, COMPASS, 2007 asymmetries 26

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) Ph. Hägler, COMPASS, 2007 27

Lattice QCD prediction for azimuthal asymmetries strongly distorted densities of transversely polarized quarks in an unpolarized nucleon lead to prediction of a large Boer-Mulders effect with the same sign for up and down quarks cos(2 Ф) asymmetries in SIDIS (JLab, COMPASS) could be observed in unpolarized Drell-Yan (GSI-FAIR/PANDA) Ph. Hägler, COMPASS, 2007 approved Jlab proposal PR 12 -06 -112 28

First results on the spin structure of pions from lattice QCD longitudinal spin structure is trivial pion spin structure? what about the transverse spin structure of the pion? but is non-zero? YES n=1 n=2 the pion has a very suprising non-trivial transverse spin structure! Ph. Hägler, COMPASS, 2007 29

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 Ph. Hägler, COMPASS, 2007 30

Ph. Hägler, COMPASS, 2007 31

n=2 moments of up- and down-quark densities up up up down same pattern as before for n=1 densities are strongly deformed strong correlations of coordinate and spin degrees of freedom Ph. Hägler, COMPASS, 2007 32

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 assume : up-quarks d-quarks Ph. Hägler, COMPASS, 2007 spin of down-quarks is anti-aligned with nucleon spin for up- and down-quarks high density in the upper half plane high density in the lower half plane 33

=1/2( and since for down-quarks in an „unpolarized“ nucleon d-quarks Ph. Hägler, COMPASS, 2007 ) 1/2 high density in the upper half plane however for „unpolarized“ down-quarks in a polarized nucleon d-quarks + = + high density in the lower half plane 34

Sivers asymmetry measured by HERMES pion production from positrons on a transversely polarized hydrogen target Sivers asymmetry HERMES collaboration PRL 2005 Ph. Hägler, COMPASS, 2007 35

chiral extrapolation based on ch. PT including the -resonance in a finite volume Gell-Mann-Oakes-Renner Hemmert/Procura/Weise PRD 2003 Wollenweber, diploma thesis, TUM 2005 QCDSF/UKQCD, PRD 2006 Ph. Hägler, COMPASS, 2007 36

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

Computation of moments of GPDs on the lattice, continued finally, equate the lattice ratio 3 pt/2 pts and the continuum parametrization for LHPC/SESAM PRD 2003 different momenta, indices and projectors simultaneously (for a given momentum transfer 2 t) Ph. Hägler, COMPASS, 2007 extract the GFFs from overdetermined set of equations 38

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Overview – computation of GPDs on the Lattice choose nucleon source/sink insert complete set of states+ use translation operators parametrize matrix element of local ops in terms of generalized form factors finally, we equate the lattice ratio 3 pt/2 pts and the continuum parametrization for different momenta and indices simultaneously at a given momentum transfer 2 t Ph. Hägler, COMPASS, 2007 this gives an (overdetermined) set of linear equations which is solved to obtain the GFFs 40

Computation of moments of GPDs on the lattice (1) define 3 pt-function and choose operator (2) choose sink/source operators (3) get rid of all-to-all propagators Ph. Hägler, COMPASS, 2007 41

(4) (5) (6) renormalize transfer matrix formalism parametrize matrix element of local ops in terms of generalized form factors (7) (8) finally, equate the lattice ratio 3 pt/2 pts and the continuum parametrization for different momenta, indices and projectors simultaneously (for a given momentum transfer 2 t) Ph. Hägler, COMPASS, 2007 extract the GFFs from overdetermined set of equations enjoy 42

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