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 / 0511028

What about ?

“Evidence for the Absence of Gluon Orbital Angular Momentum

in the Nucleon”, S.J. Brodsky and S. Gardner, hep-ph / 0608219.

• 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) 034022

• SU(3) : M. Wakamatsu, Phys. Rev. D67 (2003) 034005

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 part anomalous part

familiar unpolarized PDF

and its canonical part in CQSM

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 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

increasing

Scale dependencies of