Transcript 一般化されたパートン分布関数

Chiral Quark Soliton Model and Nucleon Spin Structure Functions
M. Wakamatsu, Osaka Univ., July 2009, Bled
Plan of talk
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
1. Introduction
What is Chiral Quark Soliton Model like ?
What is (or was) Skyrme model ?
Bohr’s collective model in baryon physics !
Skyrme model
Bohr’s model of rotational nuclei
microscopic basis
Deformed Hartree theory
CQSM
Cranking quantization
Compressed history of CQSM
[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, …… )
[1991] M. W. and H. Yoshiki
• numerical basis for nonperturbative evaluation of nucleon observables including
vacuum polarization effects ( based on the work by Kahana-Ripka-Soni, 1984 )
• nucleon spin sum rule : importance of
[1993] M. W. and T. Watabe
• discovery of novel
correction missing in corresponding Skyrme model
- resolution of
- problem -
[1996 - ] D. Diakonov et al., H. Weigel et al., M. Wakamatsu et al.
• application to Parton Distribution Functions of the nucleon
2. Basics of Chiral Quark Soliton Model
Basic lagrangian
effective meson action
derivative expansion
Soliton construction without derivative expansion
M.F. Dirac equation
breaks rotational symmetry
Energy of
Hartree condidion
Quark hedgehog state
Spin-isospin projection using cranking method (linear response theory)
1. Cranked iso-rotation of hedgehog M.F. induces Coriolis coupling
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
slow collective rotation
and fast internal motion !
Final formula for evaluating nucleon observables
with
and
diagonal sum over occupied states ( = valence + Dirac sea )
double sum
virtual transitions from occupied to nonoccupied states
Noteworthy achievements of CQSM for low energy baryon observables
(1) reproduce small quark spin fraction of N consistent with EMC observation !
(2) reproduce large
(3) resolve
sigma term !
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) !
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
Remark on the antiquark distributions (unpolarized distribution)
where
one can prove
for longitudinally polarized distribution
we have
4. On the role and achievement of CQSM in Deep-Inelastic-Scattering physics
• Standard approach to DIS physics
Factorization theorem
PDFs
Soft part is treated as a black box, which should be determined via experiments !
reasonable strategy !
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
Merits of CQSM over many other effective models of baryons
• 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.
• only 1 parameter of the model (dynamical quark mass M) was already fixed
from low energy phenomenology
parameter-free predictions for PDFs
Default
Lack of explicit gluon degrees of freedom
How should we use predictions of CQSM ?
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.
Our general strategy
• 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)
QCD running coupling constant at next-to-leading order (NLO)
pQCD is barely applicable !
Parameter free predictions of
CQSM for 3 twist-2 PDFs
• unpolarized PDFs
• longitudinally polarized PDFs
• transversities (chiral-odd)
totally different behavior of
the Dirac-sea contributions
in different PDFs !
Isoscalar unpolarized PDF
sea-like soft component
positivity
Isovector unpolarized PDF
- NMC observation -
ratio in comparison with Fermi-Lab. Drell-Yan data
NA51
FNAL E866 / NuSea
old fits
CQSM
parameter free prediction
new fit
parameter free prediction
Longitudinally polarized structure functions for p, n, D : (data before 2003)
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)
New COMPASS and HERMES fits for
together with CQSM prediction
CQSM
New data
Old data
Isovector longitudinally polarized PDF
CQSM predicts
This means that antiquarks gives sizable positive contribution to Bjorken S.R.
contradict the HERMES analysis of semi-inclusive DIS data
• HERMES Collaboration, Phys. Rev. D71 (2005) 012003
However, HERMES analysis also denies negative strange-quark polarization
favored by most global-analysis heavily depending on inclusive DIS data !
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
5. Chiral-odd twist-3 distribution function
chiral-odd
Why is it interesting ?
pQCD
M.Burkardt and Y.Koike (2002)
What is the physical origin of this delta-function singularity ?
disentangling the origin of delta-function singularity in
general definition of
with
measures light-cone correlation of scalar type
existence of delta-function singularity in
indicates
long-range (infinite-range) correlation of scalar type
Within the CQSM, we can analytically confirm this fact
M.W. and Y.Ohnishi, Phys. Rev. D67 (2003) 114011
P. Schweitzer, Phys. Rev. D67 (2003) 114010
existence of this infinite-range correlation is inseparably connected with
nontrivial vacuum structure of QCD
spontaneous cSB 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
CQSM prediction for the scalar quark density of the nucleon
total
valence
Dirac sea
This in turn dictates that
We thus conclude that
Nonvanishing quark condensate as a signal of the spontaneous cSB of
the QCD vacuum is the physical origin of
-type singularity in
Sophisticated numerical method to treat
containing
Y. Ohnishi and M.W., Phys. Rev D69 (2004) 114002
We find that
where
with
1st moment sum rule for isoscalar
numerically
dominant
with
this gives
Favors fairly large
sigma term
Isovector part of
total
valence
no singularity at
Dirac sea
Combining isoscalar- and isovector-part of
, we can get any of
Comparison with CLAS semi-inclusive data extracted by Efremov and Schweitzer
To sum up this part
(1) delta-function singularity in chiral-odd twist-3 distribution
is
manifestation of nontrivial vacuum structure of QCD in hadron observable
(2) Existence of this singularity will be observed as
violation of
sigma-term sum rule of
need more precise experimental information on this quantity in wider range of
especially in small
region
6. Nucleon spin problem revisited : current status
two remarkable recent progresses :
(1) New COMPASS & HERMES analyses
• Precise measurements of deuteron spin-dependent structure function
with high statistics, especially at lower x region
fairly precisely determined !
(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
likely to be small, but still with large uncertainties !
What is our current understanding of the nucleon spin ?
The remaining 70 % of nucleon spin should be carried by
,
safe statement !
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 ?
Interesting possibility is to get direct empirical information on
through Generalized Parton Distributions (GPDs) appearing in
high-energy DVCS & DVMP processes
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
Generalized form factors of the nucleon
Dirac F.F.
Pauli F.F.
electromagnetic current coupled to photon
energy momentum tensor coupled to graviton
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
8. Semi-empirical analysis of nucleon spin contents
• M.W. and Y. Nakakoji, Phys. Rev. D77 (2008) 074011/1-15.
Phys. Rev. D74 (2006) 054006/1-27.
We start with Ji’s angular momentum sum rule
where
with the constraint
We also need the isovector combination for flavor decomposition
Since the momentum fractions
are already well determined
phenomenologically, we are left with two empirically unknowns
theoretical information on isovector
satisfactory agreement between the predictions of CQSM and lattice QCD
Old Lattice
New Lattice
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 !
1st important observation
( of our semi-empirical analysis )
The quark- and gluon- momentum fractions,
and
, are
scale-dependent quantities, but they are empirically fairly precisely known.
In fact, MRST2004 & CTEQ5 QCD fits give almost the same numbers
for those between
[Ex.]
well-known solution of LO evolution equation
asymptotic limit
with
Scale dependencies of quark and gluon momentum fraction at NLO
evolve down to
low-energy scale
using
NLO evolution eq.
2nd important observation
and
( due to Xiangdong. Ji )
obey exactly the same evolution equation !
[Reason] forming spatial moments of
and
does not change the
short-distance singularity of the operators !
The evolution equations at NLO may be used to estimate
at any energy scale !
as well as
Scale dependencies of quark and gluon total angular momentum
proportionality !
total angular momentum fraction at the nonperturbative scale

quarks and gluons respectively carry about 80% and 20% of
total angular (and linear) momentum of the nucleon

we conjecture that
here comes from gluon OAM not from

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 !
!
Once
is known, we can determine the quark OAM through
Since
is approximately
scale independent, we use here
central fit of HERMES analysis :
One observes that
is a rapidly decreasing
functions of
!
flavor decomposition of quark total angular momentum
small
is
consistent with
lattice QCD
prediction !
Information on quark OAM,
can be obtained by subtracting
the known information on intrinsic quark polarizations
scale indep.
Neglecting error bars, for simplicity, we have at
prominent features
isovector dominance of quark OAM !
scale dependence of quark OAM
Note that
is a decreasing (increasing) function of
decreasing func.
Since
, since
scale indep.
from evolution equation, we then find that
This is really an surprising conclusion, since it means that the isovector
combination of quark OAM in the asymptotic limit is solely determined
by the neutron beta-decay coupling constant ! Why ?
This mysterious conclusion is an inevitable consequence of the following two
theoretical postulates :
• The definition of
• The observation that
via Ji’s angular momentum sum rule :
and
obey the same evolution equation.
Comparison with
obtained from GPD analyses
HERMES Collaboration
• F. Ellinghaus et. al., Eur. Phys. J. C46 (2006) 729.
• F. Ellinghaus, arXiv:0710.5768.
• Z. Ye, hep-ex/0606061.
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
GPD extraction of
our semi-empirical estimate
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
has large uncertainties.
into
and
is gauge-dependent, and
• 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.
• 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
beta-decay coupling const.
• 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 !
[Backup Slides]
Isoscalar longitudinally polarized PDF
deuteron
New COMPASS data
sign change in
low region !
nonlocality
correction !
Brief comment on transversities
• 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.
• Their results, although with large uncertainties, already indicates a
remarkable qualitative difference between transversities and
longitudinally polarized PDFs such that
• Our theoretical analysis indicates that the cause of this feature can be traced
back to the relation
No (or less) spin crisis in the tensor channel !
relation with NJL model
auxiliary field method
with
nonlinear constraint (by hand)
reparametrization