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

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

[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

2. Basics of Chiral Quark Soliton Model

Basic lagrangian

effective meson action

derivative expansion

Soliton construction without derivative expansion

M.F. Dirac equation

Hartree condidion

breaks rotational symmetry

Energy of

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 !

Final formula for evaluating nucleon observables

with

and

diagonal sum over occupied states ( = valence + Dirac sea )

virtual transitions from occupied to nonoccupied states

double sum

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) !

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

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

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.

• 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

QCD running coupling constant at next-to-leading order (NLO)

pQCD is barely applicable !

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 !

Isoscalar unpolarized PDF

positivity

sea-like soft component

Isovector unpolarized PDF

- NMC observation -

parameter free prediction

CQSM

parameter free prediction

ratio in comparison with Fermi-Lab. Drell-Yan data

old fits

new fit

FNAL E866 / NuSea

NA51

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)

New compass data (2005)

CQSM

New COMPASS and HERMES fits for together with CQSM prediction

Old data

New data

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

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

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

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

CQSM prediction for the scalar quark density of the nucleon

valence

total

Dirac sea

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

Y. Ohnishi and M.W., Phys. Rev D69 (2004) 114002

We find that

where

with

Sophisticated numerical method to treat containing 　

numerically

dominant

with this gives

Favors fairly large sigma term

1st moment sum rule for isoscalar

Dirac seavalence

total

Isovector part of

no singularity at

Comparison with CLAS semi-inclusive data extracted by Efremov and Schweitzer

Combining isoscalar- and isovector-part of 　 　　　　　　　 , we can get any of

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

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 !

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 !

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

energy momentum tensor coupled to graviton

electromagnetic current coupled to photon

Dirac F.F.

Pauli F.F.

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

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

satisfactory agreement between the predictions of CQSM and lattice QCD

theoretical information on isovector

New Lattice

Old 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

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

Scale dependencies of quark and gluon momentum fraction at NLO

evolve down tolow-energy scale

using

NLO evolution eq.

[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 )

Scale dependencies of quark and gluon total angular momentum

proportionality !

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 !

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

flavor decomposition of quark total angular momentum

small isconsistent withlattice QCDprediction !

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

scale dependence of quark OAM

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.

• 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

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 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.

• 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.

[Backup Slides]

Isoscalar longitudinally polarized PDF

New COMPASS data

deuteron

sign change inlow region !

nonlocalitycorrection !

• 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

auxiliary field method

with

nonlinear constraint (by hand)

reparametrization

relation with NJL model