Graduiertenkolleg Freiburg 24-02-2007 Graduiertenkolleg Freiburg 24-02-2007

The nucleon as non-The nucleon as non-topological chiral topological chiral

solitonsoliton

Klaus Goeke Klaus Goeke Ruhr-Universität Bochum, Theoretische Physik

II

Hadronenphysik

Applications of the Chiral Quark Soliton Modelto current topical experiments and lattice data

Verbundforschung BMBF Transregio/SFB Bonn-Bochum-Giessen

ContentsContents Chiral Quark Soliton ModelChiral Quark Soliton Model

Quantum ChromodynamicsQuantum Chromodynamics Relativistic Mean Field DescriptionRelativistic Mean Field Description

Parton distributions, transversity, magnetic Parton distributions, transversity, magnetic moments (HERMES, COMPASS)moments (HERMES, COMPASS)

Strange magnetic form factorsStrange magnetic form factors Experiments A4 G0 SAMPLE HAPPEXExperiments A4 G0 SAMPLE HAPPEX

Lattice QCD and extrapolation to small mLattice QCD and extrapolation to small m Form factors of energy momentum tensorForm factors of energy momentum tensor

Distributions of (angular) momentum in nucleonDistributions of (angular) momentum in nucleon Distribution of pressure and shear in the nucleonDistribution of pressure and shear in the nucleon

Summary and conclusions Summary and conclusions

AuthorsAuthors

Anatoli Efremov (Dubna)Anatoli Efremov (Dubna) Hyun-Chul Kim (Busan)Hyun-Chul Kim (Busan) Andreas Metz (Bochum)Andreas Metz (Bochum) Jens Ossmann (Bochum)Jens Ossmann (Bochum) Maxim Polyakov (Bochum)Maxim Polyakov (Bochum) Peter Schweitzer (Bochum)Peter Schweitzer (Bochum) Antonio Silva (Coimbra)Antonio Silva (Coimbra) Diana Urbano (Coimbra/Porto)Diana Urbano (Coimbra/Porto) Gil-Seok Yang (Bochum/Busan)Gil-Seok Yang (Bochum/Busan)

Quantum Quantum Chromo Chromo

dynamicsdynamics

Has problems with the chiral limit

Constructed to work in the chiral limit

Chiral Quark Soliton ModelNucleon

Baryon –Octet –

Decuplet -Antidecuplet

SU(3)

QCDQCD Lattice TechniquesLattice Techniques Aim: exactAim: exact T T infinite infinite V V infinite infinite a a zero zero Pion mass > 500 GeV Pion mass > 500 GeV Wilson Clover Wilson Clover

StaggeredStaggered (Un)quenched(Un)quenched Extraction of Extraction of

dimensional quantitiesdimensional quantities ExpensiveExpensive

Effective ModelsEffective Models ApproximateApproximate Certain physical Certain physical

regionregion Pion mass = 140 MeVPion mass = 140 MeV Identification of Identification of

relevant degrees of relevant degrees of freedomfreedom

InexpensiveInexpensive

Chiral Symmetry of QCDChiral Symmetry of QCD

(2) : ' expu uA AV

d d

SU i

Light Systems: QCD in chiral Limit, QCD-Quarkmasses zero ~ 0

QCD 2

1( )

4a aL F F i A

g

5(2) : ' expu uA AA

d d

SU i

Global QCD-Symmetries Lagrangean invariant under:

Multiplets: 8, 10, 10

No multipletts Symmetry

spontaneousl broken

Dynamic mass generation Pions as massless Goldstone bosons

Simplest effective LagrangeanSimplest effective Lagrangean

( )effL i MU

( )effL i M

QSM 5( ) ( ) exp( ( ) )A AiL i MU U x x

f

Chiral Quark Soliton Model (ChQSM):Pseudo-scalar pion-

Kaon-Goldstone field

Invariant: flavour vector transformation

Not invariant: flavour axial transformation

Invariant: flavour vector transformation and axial transformation U(x) must transform properly U(x) exists

† 4

Partition function :

exp ( )QSMZ DU D D d xL x

Scattering of light quarks at randomly distributed Instantons (fluctuations of the gluon field with topological properties)

Instanton model of vacuum Random matrix theory Effective momentum dependent quark mass

ChQSM (Diakonov,Petrov)

Similar to scattering of electrons at impurities in a solid state

ChQSM - parametersChQSM - parameters

QSM 0 5( ) ( ) exp( ( ) )A AiL i m MU U x x

f

0

2

Regularization: Proper Time, Pauli-Villars regularization

SU(2): Lagrangean: , ,

SU(2): Physics: 93 , 139 ,

(3) : In addition 180 and Witten's embedding (2) (3)

a

cutoff

c proton

s

m M

f MeV m MeV r

SU m MeV SU SU

nd perturbative treatment in collective quantization

0

Numbers:

420 , 15 , 600cutoffM MeV m MeV MeV

3(250 )MeV

Chiral Quark Soliton Chiral Quark Soliton Practice Practice

Partition Function:

exp totaleffZ D S

† 4

Partition function :

exp ( )QSMZ DU D D d xL x

QSM 0 5( ) ( ) exp( ( ) )A AiL i m MU U x x

f

0 - Stationary phase approx ( )

Selfconsistent mean field ( ) - Iterative procedure

totaleff

c c

c

SN

x

Bound valence quarks

Polarized Dirac Sea

Relativistic selfconsistent mean Relativistic selfconsistent mean fieldfield

5( ) ~

i i i

A Ai i

i occ

i MU

x

5( ) exp( ( ) )A AiU x x

f

Selfconsistent Soliton:

( )x

x

0

Selfconsistent mean field ( )

Iterative procedure

totaleff

c

c

S

x

ChQSM: Parton distributiosChQSM: Parton distributios

Fitted to data Fitted to data

Selfconsistently fulfilled: QCD-sum rules, positivity, Soffer-bounds, forward limits of GPDs, etc.

Azimuthal Azimuthal asymmetries asymmetries

transversal targettransversal target

Quark

unpol

quark

Meson unpol

Distr.

Fragm.

ChQSM: Transversity distribution

ChQSM: Transversity Parton ChQSM: Transversity Parton Distribution FunctionDistribution Function

Positive, close to Soffer bound

HERMES SIDIS-data for HERMES SIDIS-data for proton proton Favoure

d: positiv

0/ / /1 1 1 1

/ /1 1 1

... 2

...

fav u d u

unf d u

H H H H

H H H

COMPASS SIDIS-data for COMPASS SIDIS-data for deuteron deuteron

BELLEBELLE

Transversity distribution: Transversity distribution: Facts Facts

Chiral Quark Soliton Model

Parity violating electron Parity violating electron scatteringscattering

0( )Z q

Magnetic moments of octet baryons Magnetic moments of octet baryons SU(3) SU(3)

pp 2.4002.400 2.7932.793

nn -1.651-1.651 -1.913-1.913

LambdLambdaa

-0.652-0.652 -0.613-0.613

Sigma-Sigma- -0.958-0.958 -1.16-1.16

Sigma-Sigma-00

0.6750.675 --

SigmaSigma++

2.3092.309 2.4582.458

Xi-Xi- -0.606-0.606 -0.651-0.651

Xi-0Xi-0 -1.450-1.450 -1.250-1.250

particle ChQSM experiment

Strange Form Factors FStrange Form Factors F11 and F and F22

0 5( ) ( ) exp( ( ) )A Aeff

iL i m MU U x x

f

Hedgehog: ( ) (| |)| |

AA xx P x

x

1( ) exp( )

K

P r rr

m m

Strange weak, Strange weak, electric, electric,

magnetic form magnetic form factorsfactors

Axial and strange axial form factors Axial and strange axial form factors

2sAG Q

3 2AG Q

Experiment: 1.26

Parity violating Parity violating asymmetriesasymmetries

Polarized eP-scattering, circularly polarized electrons, positive and negative helicities

PVA

1 (3) 0

1 0

1(1 )

2

0.41 0.24 0.06 0.14 (Zhu et al.)

e p NC sA A A A A A A

A A

G G G R G R G

R R

Parity violating Parity violating asymmetries of asymmetries of

proton proton

SAMPLE

HAPPEX

A4

Parity violating Parity violating asymmetries: G0 forward asymmetries: G0 forward

anglesangles

Prediction (backward

angles)

A4, G0: Parity violating e-A4, G0: Parity violating e-scatt.scatt.

The World data for GsM and GsE from The World data for GsM and GsE from A4, HAPPEX and SAMPLE A4, HAPPEX and SAMPLE

19) ChQSM 21) Lewis et al. 20) Lyubovitskij et al.16) Park + Weigel 17) Hammer et al. 22) Leinweber et al. 18) Hammer + Musolf

The World data for GsM and GsE from The World data for GsM and GsE from A4, HAPPEX and SAMPLE + A4, HAPPEX and SAMPLE +

HAPPEX(2005)HAPPEX(2005)19) ChQSM 21) Lewis et al. 20) Lyubovitskij et al.16) Park + Weigel 17) Hammer et al. 22) Leinweber et al. 18) Hammer + Musolf

preliminary

Data combined from parity-violating Data combined from parity-violating electron-scattering and neutrino- and electron-scattering and neutrino- and anti-neutrino scattering (Pate et al.) anti-neutrino scattering (Pate et al.)

2( )sMG Q

Data combined from parity-violating Data combined from parity-violating electron-scattering and neutrino- and electron-scattering and neutrino- and anti-neutrino scattering (Pate et al.) anti-neutrino scattering (Pate et al.)

2( )sEG Q

Data combined from parity-violating Data combined from parity-violating electron-scattering and neutrino- and electron-scattering and neutrino- and anti-neutrino scattering (Pate et al.) anti-neutrino scattering (Pate et al.)

2( )sAG Q

Strange Form factorsStrange Form factors

Experiments: SAMPLE HAPPEX A4 G0Experiments: SAMPLE HAPPEX A4 G0 Parity violating e-scattParity violating e-scatt -scattering-scattering ChQSM works well for all form factorsChQSM works well for all form factors Only approach with Only approach with ss>0>0 Experiments with large error barsExperiments with large error bars Clear predictions for A4, G0Clear predictions for A4, G0 Theory with large error barsTheory with large error bars

Experiment - TheoryExperiment - Theory

Experiment

QCDLattice Gauge

QCD Chiral

Perturb. Th.

Chiral Quarksoliton model

500m MeV 300m MeV

100 1000MeV m MeV

Nucleon mass: mNucleon mass: m--dependencedependence

One fit parameter

Quenched vs. Unquenched Quenched vs. Unquenched

MILC LQCD-data MILC LQCD-data

Extrapolation to small mExtrapolation to small mby by ChPT and ChQSMChPT and ChQSM

Extrapolation to small mExtrapolation to small mby by ChPT and ChQSMChPT and ChQSM

Energy Momentum Tensor of QCD: Energy Momentum Tensor of QCD: New form factors New form factors

2}{

1ˆ2 4

Q G

a a

T T

iT G G g G

NO

1

2

2

( )ˆ' ' {2

+ ( ) }( ) )5

)

(

( ()Q

N N

Q

N

QQ p p i p pp T p N p

M M

gd t C t g N

J t

p

M t

m

Lorentz decomposition: 1 ' '

2p p p p p

12Formfactors equally fundamental ( ) ( ) : ( )QQ QJ t dt tM

d

-term: Polyakov and Weissd

DVCS and Form factors of energy-DVCS and Form factors of energy-momentum tensor of QCDmomentum tensor of QCD

1

1

12

2 1

1

12

2 1

1

, , , , ( ) ( )

4, , ( ) ( )

5

4, , 2 ( ) ( ) ( )

5

q q q Q

q q

q Q Q

q

q Q Q Q

q

dxx H x t E x t J t J t

dxx H x t M t d t

dxx E x t J t M t d t

Sum rule of Ji

1

2

0

1

2

0

Forward limit:

( 0) ( ) ( )

( 0) ( )

Q

q

G

M t dxx q x q x

M t dxxG x

Forward limit:

( 0) contribution of total angular momentum

of quarks to total angular momentom of nucleon.

QJ t

Energy momentum tensor: PropertiesEnergy momentum tensor: Properties

0

3

03

00

0

' Breit-Frame: 0

, ', 0 ,2 2

, energy density

, momentum density

, stress tensor: pressure, shear

Q ir Q

i

ij

p p

dT r s e p s T p s

p

T r s

T r s

T r s

OOOOOOOOOOOOO O

23

2

1(0) ( )

( ))3

(

2 3

1( )

Q Q i j ijNij

i jQ ij ijij

md d rT r r r r

s rr pr

Tr

rr

( ) = pressur

( ) = s e r

e

h as r

p r

Energy density Energy density EE(r) in (r) in ChQSM ChQSM

At the physical point (m=140 MeV) is the energy-density in the centre of the nucleon

13x the energy density of nuclear matter30.13 /NM GeV fm

Angular momentum density Angular momentum density JJ(r) of quarks (spin + (r) of quarks (spin +

orbital)orbital)

2

1Nucleon: 2 2 2 1

2

Chiral Quark Soliton Model

at m 140 and 2

2 0.75 and 2 0.25

QCD-Lattice Calculations:

2 0.60 0.70

Q G

Q G

Q

J J

MeV Q GeV

J J

J

Pressure and Shear Pressure and Shear Distribution inside the Distribution inside the

nucleon nucleon Pressure at r=0 is 10-100 times

higher than in a neutron star

Integral =0

Shear distribution (surface Shear distribution (surface tension) of the nucleon tension) of the nucleon

Nucleon Liquid drop (softened) surface tension

Form factors of the Form factors of the

energy momentum energy momentum

tensortensor2

2

(0)( )

1dip

FF t

tM

ChQSM vs. Lattice-ChQSM vs. Lattice-QCD for MQCD for M22

QQ(t) and (t) and extrapolation to small extrapolation to small

mm2

2

(0)( )

1dip

FF t

tM

ChQSM vs. Lattice-ChQSM vs. Lattice-QCD for MQCD for M22

QQ(t) and (t) and extrapolation to small extrapolation to small

mm

Message of ChQSM:Linear extrapolation of

Lattice-QCD datafor MM22

QQ(t) and J(t) and JQQ(t)(t) works well

ChQSM vs. Lattice-QCD for dChQSM vs. Lattice-QCD for d11QQ(t) and (t) and

extrapolation to small mextrapolation to small m

1 ( 0)Qd t

ChQSM vs. Lattice-QCD for dChQSM vs. Lattice-QCD for d11QQ(t) and (t) and

extrapolation to small mextrapolation to small m

1 ( 0)Qd t Message of ChQSM:Linear extrapolation of

Lattice-QCD datafor dd11

QQ(t) does (t) does NOTNOT work well

2

2

(0)( )

1dip

FF t

tM

Summary and ConclusionsSummary and Conclusions Chiral Quark Soliton ModelChiral Quark Soliton Model

Simplest quark model with proper global symmetriesSimplest quark model with proper global symmetries Relativistic mean field approachRelativistic mean field approach Spontaneous chiral symmetry breakingSpontaneous chiral symmetry breaking Valence quarks and polarized Dirac seaValence quarks and polarized Dirac sea

Parton distributions: Transversity (HERMES, COMPASS, Parton distributions: Transversity (HERMES, COMPASS, BELLE), Sivers-Function, Collins-Fragmentation-FunctionBELLE), Sivers-Function, Collins-Fragmentation-Function

Strange form factorsStrange form factors Magnetic, electric, axial, etc., asymmetriesMagnetic, electric, axial, etc., asymmetries Experiments A4 G0 SAMPLE HAPPEXExperiments A4 G0 SAMPLE HAPPEX

Lattice QCD and extrapolation to small mLattice QCD and extrapolation to small m

Agreement with LQCD at large mAgreement with LQCD at large m

Useful guideline for extrapolation to physicalUseful guideline for extrapolation to physical m m

Form factors of energy momentum tensorForm factors of energy momentum tensor Distributions of (angular) momentum in nucleonDistributions of (angular) momentum in nucleon Distribution of pressure and shear in the nucleonDistribution of pressure and shear in the nucleon