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The role of orbital angular momentum
in the internal spin structure of the nucleon
M. Wakamatsu (Osaka University) : PACSPIN-07
based on collaboration with Y. Nakakoji, H. Tsujimoto
Plan of Talk
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
1. Current status of nucleon spin problem
Early stage proposals to explain very small quark spin fraction
(I) Gluon spin hypothesis
(A) naive claim
The question remains
why
！
OK if
(B) axial-anomaly of QCD
but need very large
!
(II) Quark orbital angular momentum hypothesis
unfavored ?
is small, because
is large !
(III) Gluon orbital angular momentum hypothsis
no serious consideration until recently！
Two remarkable recent progresses :
(1) New COMPASS & HERMES analyses
• Precise measurements of deuteron spin-dependent structure function
high statistics, especially at lower x region
(2) COMPASS, PHENIX, STAR analyses
• COMPASS : quasi-real photoproduction of high-
hadron pairs
• 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
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 ?
Importance of quark orbital angular momentum
- Large
chiral soliton picture of the nucleon -

Skyrme model
(Ellis-Karliner-Brodsky, 1988)

Chiral Quark Soliton Model (Wakamatsu-Yoshiki, 1991)
In particular, since the latter is an effective quark theory
Spin S.R
Collective quark motion generating
rotating M.F. of hedgehog shape
We need more direct empirical information on
new recent development
possibility of direct measurement of
through Generalized Parton Distributions (GPDs)
appearing in high-energy DVCS & DVMP processes
Ji’s angular momentum sum rule
2. Role of CQSM in nucleon structure function (DIS) physics
Hard part :
Factorization
Perturbative QCD
Black Box
Soft part :
Lattice QCD
• most promising in the long run
- still at incomplete stage • continuum limit & chiral limit ?
• only lower moments of PDF
• physical interpretation ?
Nonpurturbative QCD
Effective models of QCD
So many !
Necessary condition of good model,
which has predictive power ?
• able to explain many observables
with less parameters !
Advantages of Chiral Quark Soliton Model
• only 1 parameter of the model (dynamical quark mass ) was
already fixed from low energy phenomenology
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
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
On the Applicability of pQCD ?
NLO
pQCD is barely applicable ?
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
sea-like soft component
positivity
Isovector unpolarized PDF
Isoscalar longitudinally polarized PDF
New COMPASS data
New COMPASS and HERMES fits for
in comparison with CQSM prediction
CQSM
[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
• HERMES Collabotation, Phys. Rev. D71 (2005) 012003
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
Transversities vs. longitudinally polarized PDF : CQSM predictions
• M. Wakamatsu, arXiv:0705.2917 [hep/ph]
not so small
global fit
global fit
[global fit] M. Anselmino et. al., Phys. Rev. D75 (2007) 054032.
3. CQSM analyses of unpolarized GPDs
natural spin decomposition in Breit frame
corresponds to Sachs decomposition of electromagnetic F.F.
1st and 2nd moment sum rules
magnetic moment desity
in Feynman x-space
canonical part
anomalous part
quark number dist.
angular momentum density
in Feynman x-space
canonical part
quark momentum dist.
anomalous part
CQSM predictions for GPDs
So far, only the forward limit
was calculated.
(A) Isovector channel
forward limit of GPDs
• M.W. and H. Tsujimoto, Phys. Rev. D71 (2005) 074001
• M.W. and Y. Nakakoji, Phys. Rev. D74 (2006) 054006
(B) Isoscalar channel
forward limit of GPDs
• J. Ossmann et. al., Phys. Rev D70 (2005) 034011
• See also M.W. and Y. Nakakoji above
(A) Isovector magnetic moment distribution :
magnetic moment dist.
in Feynman x-space
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
significant contribution to the 1st moment
but no contribution to the 2nd moment
, it gives a
Interpretation of sharp peak of

Since partons with
around
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
in the transverse direction
validity of claimed picture may be confirmed by investigating
dependence of
(B) Isoscalar magnetic moment distribution :
positivity of antiquark dist.
anomalous part
no net Dirac sea contribution
Dirac sea contribution
to
small valence contribution
valence contribution
cancel !
4. Model independent prediction for nucleon spin contents
important observation (it is a sound fact)
total nucleon anomalous gravitomagnetic moment (AGM) vanishes !
It follows from
two possibilities
anomalous gravitomagnetic form factor
within the CQSM
Lattice QCD
LHPC2005
CQSM
LHPC2005
QCDSF2004
1st important observation

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.
We are then led to surprisingly simple relations :
• 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
,
(II) 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 operators !
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)
Key quantity is quark AGM :
sensitive to models !
known
known
Ji’s angular momentum sum rule
unknown
known
Lattice and CQSM predictions for Isovector AGM :
LHPC2005
ChPT extrapolation
LHPC2007 (new)
QCDSF2004
CQSM2006
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
but
, there is a crossover where
• 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
in Feynman momentum x-space

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
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
Sivers function
and its lowest
-moment
within the diquark model in the light-front formalism
Main assumptions
anomalous magnetic moment distribution
• scalar diquark
• final state interation necessary for Sivers mechanism is one-gluon-exchange
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
Basic model laglangian
with
and
After obtaining self-consistent soliton solutions for several values of
, we calculate nucleon observables in question.
New COMPASS QCD fits at NLO
New HERMES QCD fits at NLO
s-quark polarization
Reasoning to show smallness of
small !
probably smaller !
Remember that