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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