Chiral Quark Soliton Model and Nucleon Spin Structure Functions
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Chiral Quark Soliton Model and Nucleon Spin Structure Functions
M. Wakamatsu, Osaka Univ., July 2009, Bled
1. Introduction
2. Basics of Chiral Quark Soliton Model
3. CQSM and Parton Distribution Functions
4. On the role and achievements of CQSM in the DIS physics
5. Chiral-odd twist-3 distribution function
6. Nucleon spin problem revisited : current status
7. Generalized Parton Distributions and Ji’s angular momentum sum rule
8. Semi-empirical analysis of the nucleon spin contents
Plan of talk
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1. Introduction
What is Chiral Quark Soliton Model like ?
What is (or was) Skyrme model ?
Bohr’s collective model in baryon physics !
Bohr’s model of rotational nucleiSkyrme model
CQSM
microscopic basis
Deformed Hartree theory
Cranking quantization
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[1988] D. Diakonov, V. Petrov, and P.Pobylitsa• proposal of the model based on instanton picture of QCD vacuum
(Skyrme model, Hybrid chiral bag model, …… )
• numerical basis for nonperturbative evaluation of nucleon observables including
[1991] M. W. and H. Yoshiki
[1993] M. W. and T. Watabe
vacuum polarization effects ( based on the work by Kahana-Ripka-Soni, 1984 )
- resolution of - problem -
• discovery of novel correction missing in corresponding Skyrme model
[1996 - ] D. Diakonov et al., H. Weigel et al., M. Wakamatsu et al.
• application to Parton Distribution Functions of the nucleon
Compressed history of CQSM
• nucleon spin sum rule : importance of
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2. Basics of Chiral Quark Soliton Model
Basic lagrangian
effective meson action
derivative expansion
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Soliton construction without derivative expansion
M.F. Dirac equation
Hartree condidion
breaks rotational symmetry
Energy of
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Quark hedgehog state
Spin-isospin projection using cranking method (linear response theory)
in the rotating or body-fixed intrinsic frame
2. Evaluate changes of intrinsic quark w.f. and associate changes of observables
by treating this Coriolis coupling as an external perturbation
3. Canonically quantize iso-rotational motion
Underlying dynamical assumption here is the validity of adiabatic treatment
1. Cranked iso-rotation of hedgehog M.F. induces Coriolis coupling
slow collective rotation and fast internal motion !
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Final formula for evaluating nucleon observables
with
and
diagonal sum over occupied states ( = valence + Dirac sea )
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virtual transitions from occupied to nonoccupied states
double sum
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Noteworthy achievements of CQSM for low energy baryon observables
(1) reproduce small quark spin fraction of N consistent with EMC observation !
(2) reproduce large sigma term !
(3) resolve problem of the Skyrme model !
• Still, most low energy baryon observables are insensitive to low energy models !
• We demonstrate that the potential ability of CQSM manifests most clearly
in its predictions for internal partonic structure of the nucleon (or baryons) !
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3. CQSM and Parton Distribution Functions
Field theoretical definition of quark distribution functions
- nucleon matrix element of quark bilinear operator with light-cone separation -
We take account of this nonlocality in space and time in path-integral formalism
Answer in schematic form
where
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Remark on the antiquark distributions (unpolarized distribution)
where
one can prove
for longitudinally polarized distribution
we have
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Factorization theorem
• Standard approach to DIS physics
Soft part is treated as a black box, which should be determined via experiments !
We however believe that, even if this part is completely fixed by experiments, one still wants to know why those PDFs take the form so determined !
• Nonstandard but complementary approach to DIS physics is necessary here to
understand hidden chiral dynamics of soft part, based on models or lattice QCD
reasonable strategy !
4. On the role and achievement of CQSM in Deep-Inelastic-Scattering physics
PDFs
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Merits of CQSM over many other effective models of baryons
parameter-free predictions for PDFs
Default
Lack of explicit gluon degrees of freedom
• only 1 parameter of the model (dynamical quark mass M) was already fixed from low energy phenomenology
• it is a relativistic mean-field theory of quarks, consistent with
• field theoretical nature of the model (nonperturbative inclusion of polarized Dirac-sea quarks) enables reasonable estimation of antiquark distributions.
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• use predictions of CQSM as initial-scale distributions of DGLAP equation• initial energy scale is fixed to be (similarly to the GRV PDF fitting program)
Follow the spirit of empirical PDF fit by Glueck-Reya-Vogt (GRV)
• They start the QCD evolution at the extraordinary low energy scales like
• They found that, even at such low energy scales, one needs nonperturbatively
generated sea-quarks, which may be connected with effects of meson clouds.
How should we use predictions of CQSM ?
Our general strategy
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QCD running coupling constant at next-to-leading order (NLO)
pQCD is barely applicable !
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Parameter free predictions ofCQSM for 3 twist-2 PDFs
• unpolarized PDFs
• longitudinally polarized PDFs
• transversities (chiral-odd)
totally different behavior ofthe Dirac-sea contributions
in different PDFs !
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Isoscalar unpolarized PDF
positivity
sea-like soft component
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Isovector unpolarized PDF
- NMC observation -
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parameter free prediction
CQSM
parameter free prediction
ratio in comparison with Fermi-Lab. Drell-Yan data
old fits
new fit
FNAL E866 / NuSea
NA51
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SU(2) : M. W. and T. Kubota, Phys. Rev. D60 (1999) 034022 SU(3) : M. Wakamatsu, Phys. Rev. D67 (2003) 034005
Longitudinally polarized structure functions for p, n, D : (data before 2003)
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New compass data (2005)
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CQSM
New COMPASS and HERMES fits for together with CQSM prediction
Old data
New data
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Isovector longitudinally polarized PDF
This means that antiquarks gives sizable positive contribution to Bjorken S.R.
contradict the HERMES analysis of semi-inclusive DIS data
CQSM predicts
However, HERMES analysis also denies negative strange-quark polarization favored by most global-analysis heavily depending on inclusive DIS data !
• HERMES Collaboration, Phys. Rev. D71 (2005) 012003
A recent new global fit including polarized pp data at RHIC
• D. Florian, R. Sassot, M. Strattmann, W. Vogelsang, hep-ph/0804.0422
A proposal to measure and via polarized Drell-Yan at JPark
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Why is it interesting ?
M.Burkardt and Y.Koike (2002)
What is the physical origin of this delta-function singularity ?
chiral-odd
5. Chiral-odd twist-3 distribution function
pQCD
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with
measures light-cone correlation of scalar type
existence of delta-function singularity in indicates
long-range (infinite-range) correlation of scalar type
disentangling the origin of delta-function singularity in
general definition of
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Within the CQSM, we can analytically confirm this fact
M.W. and Y.Ohnishi, Phys. Rev. D67 (2003) 114011
existence of this infinite-range correlation is inseparably connected with
nontrivial vacuum structure of QCD
spontaneous SB and nonvanishing vacuum quark condensate
Why does vacuum property come into a hadron observable ?
connected with extraordinary nature of scalar quark density in the nucleon
P. Schweitzer, Phys. Rev. D67 (2003) 114010
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CQSM prediction for the scalar quark density of the nucleon
valence
total
Dirac sea
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This in turn dictates that
We thus conclude that
Nonvanishing quark condensate as a signal of the spontaneous SB of
the QCD vacuum is the physical origin of -type singularity in
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Y. Ohnishi and M.W., Phys. Rev D69 (2004) 114002
We find that
where
with
Sophisticated numerical method to treat containing
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numerically
dominant
with this gives
Favors fairly large sigma term
1st moment sum rule for isoscalar
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Dirac seavalence
total
Isovector part of
no singularity at
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Comparison with CLAS semi-inclusive data extracted by Efremov and Schweitzer
Combining isoscalar- and isovector-part of , we can get any of
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To sum up this part
manifestation of nontrivial vacuum structure of QCD in hadron observable
(2) Existence of this singularity will be observed as
(1) delta-function singularity in chiral-odd twist-3 distribution is
violation of sigma-term sum rule of
need more precise experimental information on this quantity in wider range of
especially in small region
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two remarkable recent progresses :
(1) New COMPASS & HERMES analyses
(2) COMPASS, PHENIX, STAR analyses
• Precise measurements of deuteron spin-dependent structure function with 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
fairly precisely determined !
6. Nucleon spin problem revisited : current status
likely to be small, but still with large uncertainties !
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What is our current understanding of the nucleon spin ?
Interesting possibility is to get direct empirical information on through Generalized Parton Distributions (GPDs) appearing in
high-energy DVCS & DVMP processes
The remaining 70 % of nucleon spin should be carried by ,
However, we are in a quite confusing situation concerning the separation of the remaining part.
from Lattice QCD
from direct measurements by RHIC et al.
from Brodsky-Gardner’s argument
What carry the rest of the nucleon spin ?
safe statement !
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7. Generalized Parton Distributions and Ji’s angular momentum sum rule
DVCS and DVMP amplitude dominant in Bjorken limit
Handbag diagram
lower part of Handbag Diagram contains information on nonpertubative
quark-gluon structure of the nucleon, parametrized by 4 GPDs depending
on 3 kinematical variables
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Generalized form factors of the nucleon
energy momentum tensor coupled to graviton
electromagnetic current coupled to photon
Dirac F.F.
Pauli F.F.
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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)
is a measurable quantity, since it is the 2nd moment of GPD
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8. Semi-empirical analysis of nucleon spin contents
We start with Ji’s angular momentum sum rule
where
We also need the isovector combination for flavor decomposition
with the constraint
• M.W. and Y. Nakakoji, Phys. Rev. D77 (2008) 074011/1-15. Phys. Rev. D74 (2006) 054006/1-27.
Since the momentum fractions are already well determinedphenomenologically, we are left with two empirically unknowns
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satisfactory agreement between the predictions of CQSM and lattice QCD
theoretical information on isovector
New Lattice
Old Lattice
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theoretical information on
Lattice QCD
• QCDSF-UKQCD (2007)
• LHPC (2007)
: covariant BchPT
: HBChPT
very sensitive to the chiral extrapolation method !
CQSM
only a reasonable bound can be given (due to lack of gluon field)
In the following, we treat as an unknown quantity within this range !
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1st important observation
The quark- and gluon- momentum fractions, and , are
In fact, MRST2004 & CTEQ5 QCD fits give almost the same numbers
[Ex.]
well-known solution of LO evolution equation
asymptotic limit with
( of our semi-empirical analysis )
scale-dependent quantities, but they are empirically fairly precisely known.
for those between
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Scale dependencies of quark and gluon momentum fraction at NLO
evolve down tolow-energy scale
using
NLO evolution eq.
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[Reason] forming spatial moments of and does not change the
short-distance singularity of the operators !
and obey exactly the same evolution equation !
The evolution equations at NLO may be used to estimate as well as
at any energy scale !
2nd important observation
( due to Xiangdong. Ji )
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Scale dependencies of quark and gluon total angular momentum
proportionality !
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quarks and gluons respectively carry about 80% and 20% of
total angular (and linear) momentum of the nucleon
total angular momentum fraction at the nonperturbative scale
quarks and gluons respectively carry about 65% and 35% of
total angular momentum of the nucleon
The truth would lie between these two limiting cases !
we conjecture that here comes from gluon OAM not from !
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Once is known, we can determine the quark OAM through
Since is approximatelyscale independent, we use herecentral fit of HERMES analysis :
is a rapidly decreasingfunctions of !
One observes that
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flavor decomposition of quark total angular momentum
small isconsistent withlattice QCDprediction !
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prominent features
isovector dominance of quark OAM !
scale indep.
Information on quark OAM, can be obtained by subtractingthe known information on intrinsic quark polarizations
Neglecting error bars, for simplicity, we have at
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scale dependence of quark OAM
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Note that is a decreasing (increasing) function of , since
decreasing func. scale indep.
Since from evolution equation, we then find that
This is really an surprising conclusion, since it means that the isovectorcombination of quark OAM in the asymptotic limit is solely determinedby the neutron beta-decay coupling constant ! Why ?
This mysterious conclusion is an inevitable consequence of the following twotheoretical postulates :
• The definition of via Ji’s angular momentum sum rule :
• The observation that and obey the same evolution equation.
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• F. Ellinghaus et. al., Eur. Phys. J. C46 (2006) 729.
• F. Ellinghaus, arXiv:0710.5768.
HERMES Collaboration
hard exclusive production on the transversely polarized hydrogen target
JLab Hall A Collaboration
• M. Mazous et. al., Phys. Rev. Lett. 99 (2007) 242501.
analysis of DVCS and Bethe-Heitler processes on the deuteron
• Z. Ye, hep-ex/0606061.
Comparison with obtained from GPD analyses
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GPD extraction of
our semi-empirical estimate
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Summary on nucleon spin problem
• Accepting the observation that the intrinsic quark spin carries only about 1/3 of the total nucleon spin, what carry the rest of it ?
message from our analysis
The answer drastically depends on the energy scale of observation !
• At the relatively high-energy scale around
• The decomposition of into and is gauge-dependent, and has large uncertainties.
• Still, our phenomenological analysis indicates that relatively large at high energy is a consequence of partial cancellation of large and positive and negative with moderate magnitudes.
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• At the low energy scale of nonperturbative QCD around , we get a very different picture :
Importance of quark OAM is consistent with the nucleon picture of CQSM !
• Unexpected finding is concerned with flavor decomposition of the quark OAM in the asymptotic limit : we have found that
• A precise determination of and , especially their scale dependencies is of vital importance to check our scenario on the nucleon spin contents, since and are basically scale-independent and already known !
beta-decay coupling const.
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[Backup Slides]
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Isoscalar longitudinally polarized PDF
New COMPASS data
deuteron
sign change inlow region !
nonlocalitycorrection !
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• Recently, Anselmino et al., succeeded to get a first empirical information on the transversities from the combined global analysis of the azimuthal asymmetries in semi-inclusive DIS scatterings measured by HERMES and COMPASS groups, and those in processes by the Belle Collaboration.
Brief comment on transversities
• Their results, although with large uncertainties, already indicates a remarkable qualitative difference between transversities and longitudinally polarized PDFs such that
No (or less) spin crisis in the tensor channel !
• Our theoretical analysis indicates that the cause of this feature can be traced
back to the relation
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auxiliary field method
with
nonlinear constraint (by hand)
reparametrization
relation with NJL model