the role of orbital angular momentum in the internal spin structure of the nucleon

The role of orbital angular momentum

in the internal spin structure of the nucleon

based on collaboration with Y. Nakakoji, H. Tsujimoto

1. Current status of nucleon spin problem

2. Role of CQSM in nucleon structure function physics

3. CQSM analysis of unpolarized GPD

4. Model independent analysis of nucleon spin contents

5. Flavor decomposition of nucleon spin (model dependent)

6. Summary

Plan of Talk

M. Wakamatsu (Osaka University) : PACSPIN-07

Early stage proposals to explain very small quark spin fraction

(I) Gluon spin hypothesis

(A) naive claim

(B) axial-anomaly of QCD

(II) Quark orbital angular momentum hypothesis

(III) Gluon orbital angular momentum hypothsis

no serious consideration until recently ！

OK if The question remains why ！

1. Current status of nucleon spin problem

unfavored ?　　　 is small, because 　　　 is large !

but need very large !

Two remarkable recent progresses :

(1) New COMPASS & HERMES analyses

(2) COMPASS, PHENIX, STAR analyses

• Precise measurements of deuteron spin-dependent structure function high statistics, especially at lower x region

• PHENIX : neutral pion double longitudinal spin asymmetry in the p-p collisions• STAR : double longitudinal spin asymmetry in inclusive jet production

in polarized p-p collision

• COMPASS : quasi-real photoproduction of high- hadron pairs

Still totally unknown are and !

Recent interesting observation concerning

“Evidence for the Absence of Gluon Orbital Angular Momentum

in the Nucleon”, S.J. Brodsky and S. Gardner, Phys. Let. B643

The Sivers mechanism for the single-spin asymmetry in the

unpolarized lepton scattering from a transversely polarized nucleon

is driven by the orbital angular momentum of quarks and gluons.

They argued that small single-spin asymmetry on the deuteron

target measured by the COMPASS collaboration is an

indication of small gluon OAM !.

If true, what remains is alone ?

Skyrme model (Ellis-Karliner-Brodsky, 1988)

Chiral Quark Soliton Model (Wakamatsu-Yoshiki, 1991)

Importance of quark orbital angular momentum

Collective quark motion generating rotating M.F. of hedgehog shape

In particular, since the latter is an effective quark theory

- Large chiral soliton picture of the nucleon -

Spin S.R

appearing in high-energy DVCS & DVMP processes

Ji’s angular momentum sum rule

new recent development

through Generalized Parton Distributions (GPDs)

We need more direct empirical information on

possibility of direct measurement of

Factorization

Hard part :

Soft part :

Perturbative QCD

Nonpurturbative QCD

Lattice QCD Effective models of QCD

• most promising in the long run

　 - still at incomplete stage -

• continuum limit & chiral limit ?

• only lower moments of PDF

• physical interpretation ?

So many !

Necessary condition of good model,

which has predictive power ?

• able to explain many observables

with less parameters !

2. Role of CQSM in nucleon structure function (DIS) physics

Black Box

Advantages of Chiral Quark Soliton Model

parameter-free predictions for PDFs

• a nucleon is a composite of valence quarks and infinitely many

Dirac sea quarks moving in a slowly rotating M.F. of hedgehog shape

• field theoretical nature of the model (proper inclusiuon of polarized

Dirac-sea quarks) enables reasonable estimation of antiquark dist.

Default

Lack of explicit gluon degrees of freedom

• only 1 parameter of the model (dynamical quark mass ) was

already fixed from low energy phenomenology

How to use predictions of this low energy model for parton distributions ?

We follow the spirit of

* M. Glueck, E. Reya, and A. Vogt, Z. Phys. C67 (1995) 433

They start the QCD evolution at the extraordinary low energy scales like

Even at such low energy scales, their PDF fit turns out to need

nonperturbatively generated sea-quarks (and some gluons)

which may be connected with the effects of

meson clouds

Our general strategy

• use predictions of CQSM as initial-scale distributions of DGLAP eq.

for flavor SU(2) CQSM

for flavor SU(3) CQSM

• initial energy scale is fixed to be

pQCD is barely applicable ?

On the Applicability of pQCD ?

NLO

Parameter free predictions of the CQSM : 3 twist-2 PDFs

Transversities [3rd twist-2 PDF]

• Totally different behavior of Dirac-sea contributions in different PDFs !

Isoscalar unpolarized PDF

positivity

sea-like soft component

Isovector unpolarized PDF

Isoscalar longitudinally polarized PDF

New COMPASS data

CQSM

New COMPASS and HERMES fits for in comparison with CQSM prediction

[old]

Isovector longitudinally polarized PDF

CQSM predicts

This means that antiquarks gives sizable positive contribution to Bjorken S.R.

denied by the HERMES analysis of semi-inclusive DIS data

However, HERMES analysis also denies negative strange-quark polarization

favored by the global-analysis heavily depending on inclusive DIS data !

We need more complete understanding of

spin-dependent fragmentation mechanism

• HERMES Collabotation, Phys. Rev. D71 (2005) 012003

Transversities vs. longitudinally polarized PDF : CQSM predictions

• M. Wakamatsu, arXiv:0705.2917 [hep/ph]

not so small

[global fit] M. Anselmino et. al., Phys. Rev. D75 (2007) 054032.

global fit

global fit

natural spin decomposition in Breit frame

corresponds to Sachs decomposition of electromagnetic F.F.

3. CQSM analyses of unpolarized GPDs

magnetic moment desity

in Feynman x-space

angular momentum density

in Feynman x-space

canonical part anomalous part

canonical part anomalous part

quark number dist.

quark momentum dist.

1st and 2nd moment sum rules

CQSM predictions for GPDs

(A) Isovector channel

(B) Isoscalar channel

• M.W. and H. Tsujimoto, Phys. Rev. D71 (2005) 074001

• J. Ossmann et. al., Phys. Rev D70 (2005) 034011

forward limit of GPDs

forward limit of GPDs

So far, only the forward limit was calculated.

• M.W. and Y. Nakakoji, Phys. Rev. D74 (2006) 054006

• See also M.W. and Y. Nakakoji above

magnetic moment dist.

in Feynman x-space

(A) Isovector magnetic moment distribution :

a prominent feature of CQSM prediction for

The contribution of deformed Dirac sea quarks has a large

and sharp peak around Since this large Dirac-sea contribution to

is nearly symmetric with respect to , it gives a

significant contribution to the 1st moment

but no contribution to the 2nd moment

Since partons with are at rest in the longitudinal direction

its large contribution to the first moment must come

from transverse motion of quarks and antiquarks

If one remembers the important role of pion clouds in the

isovector magnetic moment of the nucleon, the above

transverse motion can be interpreted as simulating

pionic quark-antiquark excitation with long-range tail

Interpretation of sharp peak of around

in the transverse direction

validity of claimed picture may be confirmed by investigating

dependence of

positivity of antiquark dist.

(B) Isoscalar magnetic moment distribution :

anomalous part

no net Dirac sea contribution

small valence contribution to

Dirac sea contribution valence contribution

cancel !

two possibilities

total nucleon anomalous gravitomagnetic moment (AGM) vanishes !

important observation (it is a sound fact)

It follows from

4. Model independent prediction for nucleon spin contents

anomalous gravitomagnetic form factor

CQSMLHPC2005

Lattice QCD

within the CQSM

LHPC2005

QCDSF2004

1st important observation

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.

• O.V. Teryaev, hep-ph/0004376 ; hep-ph/0612205

2nd important observation

(I) The quark- and gluon- momentum fractions, and ,

are empirically fairly precisely determined.

In fact, MRST2004 & CTEQ5 QCD fits give almost the same

numbers for these quantities below

[Reason] forming spatial moments of and does not

change the short-distance singularity of the operators !

(II) The above proportionality relations holds scale-independently,

since the evolution equations for and

are exactly the same !

The above evolution equations at NLO may be used to estimate

and at lower energy scales !

Evolve down

MRST2004 evolved down to

Gluons carry about 20% of linear and total angular momentum

fraction even at this low energy scale of nonperturbative QCD ! We conjecture that this comes from gluon OAM not from !

This statement is not inconsistent with the recent observation by

Brodsky and Gardener, since what would be related to Sivers

mechanism is the anomalous part of gluon OAM.

On the other hand, our postulated identity, implies

that gluon OAM comes totally from its canonical orbital motion,

not from the anomalous contribution related to GPD

Nucleon spin contents extracted from

cross over around

CQSM

5. Flavor decomposition of nucleon spin (model dependent)

Ji’s angular momentum sum rule

known known

known

Key quantity is quark AGM :

unknown

sensitive to models !

LHPC2005

QCDSF2004

CQSM2006

LHPC2007 (new)

ChPT extrapolation

Lattice and CQSM predictions for Isovector AGM :

Most recent LHPC results on

• Ph. Hagler et. al., arXiv : 0705.429 [hep-lat]

Main conclusion at

[Cf.] MRST & New COMPASS, HERMES with

Assuming

Transverse target-spin asymmetry of exclusive production on proton

• Hermes Collaboration, arXiv:0707.2486 [hep-ex]

far from conclusive yet !

6. Summary and Conclusion

: long-lasting dispute over this issue.

Based only upon

smallness of flavor singlet quark AGM :

model independent estimate for nucleon spin contents !

Around , there is a crossover where

but

• Flavor decomposition cannot be performed quantitatively yet, since it

depend on highly model-dependent quantity

Still we can conclude that

As increases, decreases rapidly, but the magnitude of

remains large such that even around the

a scale of few GeV.

• This peculiar property of the quark OAMs comes from the way of

their defintion through Ji’s angular momentum as well as the relation

are interesting themselves,

since they give distributions of anomalous magnetic moments

anomalous magnetic moment distribtion may also be related to

Sivers function measured by SSA of semi-inclusive reactions ?

origin of AMM & AGM & OAM of composite particle

in Feynman momentum x-space

We hope rapid progress of experimental GPD studies !

[Appendix]

Intimate relation between Sivers function and anomalous magnetic distribution ?

• Z. Lu and I. Schmidt, hep-ph / 0611158

within the diquark model in the light-front formalism

anomalous magnetic moment distribution

•　 scalar diquark

• final state interation necessary for Sivers mechanism is one-gluon-exchange

Sivers function and its lowest -moment

Main assumptions

LO evolution equation

Request for future Lattice QCD studies

More refined check of the relation

A) Larger lattice space, higher statistics, etc.

B) Stability againt the variation of pion mass, ……

More reliable evaluation of

A) Simulation with smaller pion mass

or

B) Reliable chiral extrapolation ( ex., by using chiral PT )

On the pion mass dependence of observables

with

and

After obtaining self-consistent soliton solutions for several values of

, we calculate nucleon observables in question.

Basic model laglangian

New COMPASS QCD fits at NLO

New HERMES QCD fits at NLO

s-quark polarization

Remember that

small !

probably smaller !

Reasoning to show smallness of