Generalized Form Factors, Generalized Patron Distributions, and Spin Contents of the Nucleon with Y. Nakakoji (Osaka Univ.) 1. Introduction - Still unsolved fundamental puzzle in hadron physics - If intrinsic quark spin carries little of total nucleon spin What carries the rest of nucleon spin ? Nucleon Spin Puzzle : ( EMC measurement, 1988 )

It is meaningless to talk about the spin contents of the nucleon without reference to the energy scale of observation When we talk about nucleon spin contents naively, we should regard that we are thinking of it at low energy scale of nonperturbative QCD grows rapidly as increases, even though it is small at low energy scale decreases rapidly to compensate the increase of (I) Scale dependence of polarized PDF two cautions when discussing nucleon spin contents

(II) Factorization scheme dependence of polarized PDF 2 popular factorization schemes for longitudinally polarized PDF Nonperturbative aspect is totally left unknown ! - arising from axial-anomaly of QCD - Perturbative aspect of axial-anomaly is fully understood no predictive power for the magnitude of

Compatible with Naïve Quark Model ? See, however, the recent COMPASS measurement ! Empirical PDF fits based on the two factorization schemes Relation between the scheme and the AB scheme MSbar scheme AB scheme

In fact, COMPASS group pointed out that, based on around, a value of of about 3 would be required to obtain the expected of the order of 0.6. incompatible with the recent COMPASS data ! “Gluon polarization in the nucleon from quasi-real photoproduction of high- hadron pairs”, hep-ex /

What about ? “Evidence for the Absence of Gluon Orbital Angular Momentum in the Nucleon”, S.J. Brodsky and S. Gardner, hep-ph / The Sivers mechanism for the single-spin asymmetry in the unpolarized lepton scattering from a transversely polarized nucleon is driven by the OAM of quarks and gluons. They claim that small single-spin asymmetry measured by the COMPASS collaboration on the deuteron target is an indication of small gluon OAM !. What remains is alone ?

Skyrme model (Ellis-Karliner-Brodsky, 1988) Chiral Quark Soliton Model (Wakamatsu-Yoshiki, 1991) importance of quark orbital angular momentum collective motion of quarks in rotating hegdehog M.F. dominance of quark OAM - chiral soliton picture of the nucleon -

CQSM well reproduces and at at the leading order QCD assuming CQSM predicts at model energy scale around

SU(2) : M. W. and T. Kubota, Phys. Rev. D60 (1999) SU(3) : M. Wakamatsu, Phys. Rev. D67 (2003)

New compass data (2005)

We repeat the question again : only experiments can settle the despute ! appearing in high-energy DVCS & DVMP processes Ji’s angular momentum sum rules Recent noteworthy development is possibility of direct measurement of through Generalized Parton Distributions (GPDs)

2. Generalized form factors and Ji’s angular momentum sum rule Ji’s angular momentum sum rule where - momentum fraction carried by quarks and gluons - quark and gluon contribution to the nucleon anomalous gravitomagnetic moment (AGM)

and are well determined from empirical PDF fits (Example) MRST004 or CTEQ5 since fundamentally important to know ! : total nucleon AGM identically vanishes ! important observation

two possibilities The 1st possibility, i.e. the absence of the net quark contribution to the anomalous magnetic moment of the nucleon : seems to be favored by LHPC & QCDSF lattice simulations ( as well as CQSM )

CQSM LHPC AGM form factor AGM

natural spin decomposition in Breit frame corresponds to Sachs decomposition of electromagnetic F.F. 3. Relation of GFF with unpolarized GPDs

Mellin moment sum rules relevant to our discussion (1) 1st moment (magnetic moment) sum rule (2) 2nd moment (angular momentum) sum rule unintegrated Ji’s sum rule canonical partanomalous part familiar unpolarized PDF

and its canonical part in CQSM

anomalous part no Dirac sea contribution small valence contribution to Dirac sea contribution valence contribution cancel !

model independet prediction for nucleon spin contents We are then led to surprisingly simple relations : Not only CQSM but also LHPC & QCDSF lattice simulations indicate smallness of quark AGM In the following, we assume smallness of and set them 0, for simplicity.

The quark- and gluon-momentum fractions, and, are empirically well determined. (ex) MRST2004, CTEQ5 PDF fits The above proportionality relations holds scale-independently, since the evolution equations for and are exactly the same ! [Reason] forming spatial moments of and does not change the short-distance singularity of the oparotors ! Important facts

There also exist phenomenological fits for the longitudinally polarized PDF, which contains information on and, although with larger uncertainties, compared with unpolarized case. These are enough to determine quark and gluon OAM as function of energy scale, through the relations : (ex) LSS2005, DNS2005, GRSV2000, AAC PDF fits

The nucleon spin contents at

If evolved down to low energy model scale increasing slowly increasing Nearly half of nucleon spin comes from quark OAM ! around

4. Summary and Conclusion : long-lasting dispute over this issue. Based only upon Ji’s sum rule : Empirical PDF information evolved down to LE scale around 600MeV - model independent conclusion - absence of flavor singlet quark AGM :

For obtaining more definite conclusion, experimental extraction of unpolarized spin-flip GPD (forward limit) is indispensable are interesting themselves, since they give distributions of anomalous magnetic moments More detailed information would be obtained from impact-parameter dependent parton distributions origin of AMM & AGM & OAM of composite particle in Feynman momentum x -space

[ Addendum ] A conjecture on the net quark contribution to nucleon AGM Note that, in CQSM small ! probably small !

Dirac sea valence CQSM prediction for isosinglet

CQSM prediction for isovector GPD

Pionic interpretation of central peak due to polarized Dirac sea may be confirmed by investigating dependence of

Further support may be obtained by investigating the canonical part of NMC observation

Scale dependencies of decreasing increasing

Scale dependencies of