Adnan Bashir Michoacán University, Mexico Michoacán University, Mexico Argonne National...

Adnan Bashir Adnan Bashir Michoacán Michoacán University, MexicoUniversity, Mexico

Argonne National Laboratory, USAArgonne National Laboratory, USAKent State University, USA Kent State University, USA

From Free Quarks to Nucleon Form From Free Quarks to Nucleon Form FactorsFactors

August 15, 2012August 15, 2012University of South University of South

CarolinaCarolina

ContentsContents

• ConclusionsConclusions• ConclusionsConclusions

• Schwinger-Dyson Equations – The Schwinger-Dyson Equations – The IngredientsIngredients

• Schwinger-Dyson Equations – The Schwinger-Dyson Equations – The IngredientsIngredients

• Nucleon Electromagnetic & Transition Form Nucleon Electromagnetic & Transition Form Factors Factors

• Nucleon Electromagnetic & Transition Form Nucleon Electromagnetic & Transition Form Factors Factors

• Pion Electromagnetic & Transition Form Pion Electromagnetic & Transition Form FactorsFactors

• Pion Electromagnetic & Transition Form Pion Electromagnetic & Transition Form FactorsFactors

• Rho and Diquark Form Factors Rho and Diquark Form Factors • Rho and Diquark Form Factors Rho and Diquark Form Factors

Observing the transition of the hadron from a sea Observing the transition of the hadron from a sea ofofquarks and gluons to the one with valence quarks quarks and gluons to the one with valence quarks alone is an experimental and theoretical alone is an experimental and theoretical challenge. challenge.

Observing the transition of the hadron from a sea Observing the transition of the hadron from a sea ofofquarks and gluons to the one with valence quarks quarks and gluons to the one with valence quarks alone is an experimental and theoretical alone is an experimental and theoretical challenge. challenge.

Schwinger-Dyson equations are the fundamental Schwinger-Dyson equations are the fundamental equationsequationsof QCD and combine its UV and IR behaviour. of QCD and combine its UV and IR behaviour.

Schwinger-Dyson equations are the fundamental Schwinger-Dyson equations are the fundamental equationsequationsof QCD and combine its UV and IR behaviour. of QCD and combine its UV and IR behaviour.

Schwinger-Dyson Equations – The Schwinger-Dyson Equations – The IngredientsIngredients


• The The gluon propagatorgluon propagator and the and the quark-gluon quark-gluon vertexvertex are are directly responsible for the quarks to acquire directly responsible for the quarks to acquire theirtheir constituent massesconstituent masses..

• The The gluon propagatorgluon propagator and the and the quark-gluon quark-gluon vertexvertex are are directly responsible for the quarks to acquire directly responsible for the quarks to acquire theirtheir constituent massesconstituent masses..

Schwinger-Dyson Equation for Schwinger-Dyson Equation for thethe

The Quark PropagatorThe Quark Propagator

Schwinger-Dyson Equation for Schwinger-Dyson Equation for thethe

The Quark PropagatorThe Quark Propagator



The Gluon The Gluon PropagatorPropagator The Gluon The Gluon PropagatorPropagator

Modern SDE and Modern SDE and lattice results support lattice results support decoupling solution for decoupling solution for the gluon propagator.the gluon propagator.

Modern SDE and Modern SDE and lattice results support lattice results support decoupling solution for decoupling solution for the gluon propagator.the gluon propagator.

Momentum dependent gluon mass is reminiscent Momentum dependent gluon mass is reminiscent of the momentum dependent quark mass of the momentum dependent quark mass function.function.

Momentum dependent gluon mass is reminiscent Momentum dependent gluon mass is reminiscent of the momentum dependent quark mass of the momentum dependent quark mass function.function.It is in accord with the improved GZ-picture.It is in accord with the improved GZ-picture.It is in accord with the improved GZ-picture.It is in accord with the improved GZ-picture.

A. Ayala, AB, D. Binosi, M. Cristoforetti, J. A. Ayala, AB, D. Binosi, M. Cristoforetti, J. Rodríguez Rodríguez

hep-ph: arXiv:1208.0795 (2012).hep-ph: arXiv:1208.0795 (2012).

AB, C. Lei, I. Cloet, B. El Bennich, Y. Liu, C. AB, C. Lei, I. Cloet, B. El Bennich, Y. Liu, C. Roberts, Roberts,

P. Tandy, Comm. Theor. Phys. 58 79-134 (2012) P. Tandy, Comm. Theor. Phys. 58 79-134 (2012)



J. Skullerud, P. Bowman, A. Kizilersu, D. Leinweber, A. Williams, J. High J. Skullerud, P. Bowman, A. Kizilersu, D. Leinweber, A. Williams, J. High Energy Phys. 04 047 (2003) Energy Phys. 04 047 (2003)

M. Bhagwat, M. Pichowsky, C. Roberts, P. Tandy, Phys. Rev. C68 015203 M. Bhagwat, M. Pichowsky, C. Roberts, P. Tandy, Phys. Rev. C68 015203 (2003).(2003).AB, L. Gutiérrez, M. Tejeda, AIP Conf. Proc. 1026 262 (2008).AB, L. Gutiérrez, M. Tejeda, AIP Conf. Proc. 1026 262 (2008).



The Quark-The Quark-GluonGluonVertexVertex

One of the 12 One of the 12 form factorsform factors

The Quark-The Quark-GluonGluonVertexVertex

One of the 12 One of the 12 form factorsform factors

The Quark-Photon Vertex:The Quark-Photon Vertex:



Fortunately, both the quark-photon & the quark-Fortunately, both the quark-photon & the quark-gluongluonvertices require the same number of basis tensors vertices require the same number of basis tensors (12)(12)for their description. So a unified approach is for their description. So a unified approach is possible.possible.

In studying the elastic or transition form factors In studying the elastic or transition form factors ofofhadrons, it is the photon which probes its hadrons, it is the photon which probes its constituents,constituents,highlighting the importance of the quark-photon highlighting the importance of the quark-photon vertex. vertex.

Quark-Photon VertexQuark-Photon Vertex: (Ward-Takahashi : (Ward-Takahashi identity)identity)

The Ward identity is then invoked:The Ward identity is then invoked:



AB, M.R. Pennington Phys. Rev. D50 7679 (1994) AB, M.R. Pennington Phys. Rev. D50 7679 (1994)

D.C. Curtis and M.R. Pennington Phys. Rev. D42 4165 D.C. Curtis and M.R. Pennington Phys. Rev. D42 4165 (1990)(1990)

A. Kizilersu and M.R. Pennington Phys. Rev. D79 125020 A. Kizilersu and M.R. Pennington Phys. Rev. D79 125020 (2009)(2009) L. Chang, C.D. Roberts, Phys. Rev. Lett. 103 081601 (2009) L. Chang, C.D. Roberts, Phys. Rev. Lett. 103 081601 (2009)

AB, C. Calcaneo, L. Gutiérrez, M. Tejeda, Phys. Rev. D83 033003 (2011)AB, C. Calcaneo, L. Gutiérrez, M. Tejeda, Phys. Rev. D83 033003 (2011)

AB, R. Bermudez, L. Chang, C.D. Roberts Phys. Rev. C85 045205 (2012).AB, R. Bermudez, L. Chang, C.D. Roberts Phys. Rev. C85 045205 (2012).

Phenomenology

GaugeCovariance

Lattice

MultiplicativeRenormalization

PerturbationTheory

Quark-photon/ quark-gluon

vertex

Significantly, this last ansatz contains nontrivial Significantly, this last ansatz contains nontrivial factors associated with those tensors whose factors associated with those tensors whose appearance is solely driven by dynamical chiral appearance is solely driven by dynamical chiral symmetry breaking.symmetry breaking.It yields gauge independent critical coupling in It yields gauge independent critical coupling in QED.QED.It also reproduces large anomalous magnetic It also reproduces large anomalous magnetic moment for electrons in the infrared.moment for electrons in the infrared.

The Quark-Photon The Quark-Photon VertexVertexThe Quark-Photon The Quark-Photon VertexVertex



Goldberger-Goldberger-Triemann Triemann

relations:relations:

Goldberger-Goldberger-Triemann Triemann

relations:relations:



Bethe Salpeter Amplitude:Bethe Salpeter Amplitude:

The The quark propagator,quark propagator, electron-photonelectron-photon vertex vertex and and the Bethe Salpeter Amplitude the Bethe Salpeter Amplitude provide the provide the ingredients for the pion form factor calculations. ingredients for the pion form factor calculations.

The The quark propagator,quark propagator, electron-photonelectron-photon vertex vertex and and the Bethe Salpeter Amplitude the Bethe Salpeter Amplitude provide the provide the ingredients for the pion form factor calculations. ingredients for the pion form factor calculations.



Contact Contact interaction:interaction:

Contact Contact interaction:interaction:



Transition region for the Transition region for the electromagnetic pion electromagnetic pion form factorform factormay be accessible with the high energy electron may be accessible with the high energy electron beambeamproposed for the 12 GeV upgrade at JLab. proposed for the 12 GeV upgrade at JLab.

Transition region for the Transition region for the electromagnetic pion electromagnetic pion form factorform factormay be accessible with the high energy electron may be accessible with the high energy electron beambeamproposed for the 12 GeV upgrade at JLab. proposed for the 12 GeV upgrade at JLab.

G.P. Lepage, and S.J. Brodsky, G.P. Lepage, and S.J. Brodsky, Phys. Rev. D22, 2157 Phys. Rev. D22, 2157 (1980).(1980).

L. Gutiérrez, AB, I.C. Cloet, C.D. Roberts, Phys. Rev. C81 065202 (2010).L. Gutiérrez, AB, I.C. Cloet, C.D. Roberts, Phys. Rev. C81 065202 (2010).

Pion Elastic and Transition Form Pion Elastic and Transition Form FactorsFactors


The The transition form factortransition form factor: : The The transition form factortransition form factor: :

CELLOCELLO H.J. Behrend et.al., Z. Phys C49 401 (1991). 0.7 – 2.2 GeV2 CLEOCLEO J. Gronberg et. al., Phys. Rev. D57 33 (1998). 1.7 – 8.0 GeV2 BaBarBaBar R. Aubert et. al., Phys. Rev. D80 052002 (2009). 4.0 – 40.0 GeV2

The leading twist pQDC calculation was carried The leading twist pQDC calculation was carried out in: out in: The leading twist pQDC calculation was carried The leading twist pQDC calculation was carried out in: out in:

G.P. Lepage, and S.J. Brodsky, G.P. Lepage, and S.J. Brodsky, Phys. Rev. D22, 2157 Phys. Rev. D22, 2157 (1980).(1980).BelleBelle S. Uehara et. al., arXiv:1205.3249 [hep-ex] (2012). 4.0 – 40.0

GeV2

H.L.L. Robertes, C.D. Roberts, AB, L.X. H.L.L. Robertes, C.D. Roberts, AB, L.X. Gutiérrez and P.C. Tandy, Gutiérrez and P.C. Tandy, Phys. Rev. Phys. Rev. C82, (065202:1-11) 2010.C82, (065202:1-11) 2010.



The pattern of chiral symmetry breaking The pattern of chiral symmetry breaking dictates dictates the the momentum dependence of physical observables. momentum dependence of physical observables.

The pattern of chiral symmetry breaking The pattern of chiral symmetry breaking dictates dictates the the momentum dependence of physical observables. momentum dependence of physical observables.

F. Akram, AB, L. Gutiérrez, B. Masud, J. Quintero, C. Calcaneo, M. Tejeda, F. Akram, AB, L. Gutiérrez, B. Masud, J. Quintero, C. Calcaneo, M. Tejeda, arXiv:0812---- (2012).arXiv:0812---- (2012).



When do we expect perturbation theory to set in? When do we expect perturbation theory to set in? When do we expect perturbation theory to set in? When do we expect perturbation theory to set in?

Perturbative

Momentum transfer Q is primarily shared equally Momentum transfer Q is primarily shared equally (Q/2) among quarks as BSA is peaked at zero (Q/2) among quarks as BSA is peaked at zero relative momentum. relative momentum.

Momentum transfer Q is primarily shared equally Momentum transfer Q is primarily shared equally (Q/2) among quarks as BSA is peaked at zero (Q/2) among quarks as BSA is peaked at zero relative momentum. relative momentum.

Jlab 12GeV: 2<Q2<9 GeV2 electromagnetic and transition pion form factors.



ργρργρ Elastic Form Elastic Form Factors:Factors: ργρργρ Elastic Form Elastic Form Factors:Factors:

Rho Form FactorsRho Form Factors

Electromagnetic current of a vector Electromagnetic current of a vector meson is:meson is:

Electromagnetic current of a vector Electromagnetic current of a vector meson is:meson is:

Bose symmetry and charge conjugation Bose symmetry and charge conjugation yields:yields:

Bose symmetry and charge conjugation Bose symmetry and charge conjugation yields:yields:


ργρργρ Elastic Form Elastic Form Factors:Factors: ργρργρ Elastic Form Elastic Form Factors:Factors: Within the impulse approximation & the contact Within the impulse approximation & the contact

interactioninteractionmodel:model:

Within the impulse approximation & the contact Within the impulse approximation & the contact interactioninteraction

model:model:

• The The quark-photon vertexquark-photon vertex can be dressed as: can be dressed as: • The The quark-photon vertexquark-photon vertex can be dressed as: can be dressed as:

H.L.L. Robertes, C.D. Roberts, AB, L.X. Gutiérrez and P.C. Tandy, H.L.L. Robertes, C.D. Roberts, AB, L.X. Gutiérrez and P.C. Tandy, Phys. Rev. C82, (065202:1-11) 2010.

• The corresponding IBS-equation thus yields: The corresponding IBS-equation thus yields: • The corresponding IBS-equation thus yields: The corresponding IBS-equation thus yields:



Electric, Electric, magnetic magnetic & quadrupole & quadrupole formformfactorsfactors

Electric, Electric, magnetic magnetic & quadrupole & quadrupole formformfactorsfactors

ργρργρ Elastic Elastic Form Form Factors:Factors:

ργρργρ Elastic Elastic Form Form Factors:Factors:

ργπ transition form factorργπ transition form factor is very similar to is very similar to γγ**πγ πγ ργπ transition form factorργπ transition form factor is very similar to is very similar to γγ**πγ πγ

Faddeev equationFaddeev equation for a baryon. for a baryon. Faddeev equationFaddeev equation for a baryon. for a baryon. G. Eichmann, G. Eichmann, Phys. Rev. D84, 014014 (2011).Phys. Rev. D84, 014014 (2011).

Faddeev equation in the quark diquark picture Faddeev equation in the quark diquark picture reproducesreproducesnucleon masses to within 5%.nucleon masses to within 5%.

Faddeev equation in the quark diquark picture Faddeev equation in the quark diquark picture reproducesreproducesnucleon masses to within 5%.nucleon masses to within 5%.

Nucleon – The Diquark PictureNucleon – The Diquark Picture

In the diquark picture of the nucleon, the In the diquark picture of the nucleon, the calculation of itscalculation of itselectromagnetic and transition form factors electromagnetic and transition form factors requires therequires theknowledge of the diquarks & their interaction with knowledge of the diquarks & their interaction with photons. photons.

In the diquark picture of the nucleon, the In the diquark picture of the nucleon, the calculation of itscalculation of itselectromagnetic and transition form factors electromagnetic and transition form factors requires therequires theknowledge of the diquarks & their interaction with knowledge of the diquarks & their interaction with photons. photons.

In a color singlet baryon, any 2 quarks are In a color singlet baryon, any 2 quarks are necessarilynecessarily in a 3(bar) color state. in a 3(bar) color state.

In a color singlet baryon, any 2 quarks are In a color singlet baryon, any 2 quarks are necessarilynecessarily in a 3(bar) color state. in a 3(bar) color state.

Color algebra of the BS equation reveals the gluon Color algebra of the BS equation reveals the gluon exchangeexchange is attractive in this channel, forming confined is attractive in this channel, forming confined diquarks.diquarks.

Color algebra of the BS equation reveals the gluon Color algebra of the BS equation reveals the gluon exchangeexchange is attractive in this channel, forming confined is attractive in this channel, forming confined diquarks.diquarks. Each meson has a diquark partner Each meson has a diquark partner which is non-point like with finite which is non-point like with finite radial extent comparable to mesons.radial extent comparable to mesons.

Each meson has a diquark partner Each meson has a diquark partner which is non-point like with finite which is non-point like with finite radial extent comparable to mesons.radial extent comparable to mesons.



A nucleon primarily consists of scalar and axial A nucleon primarily consists of scalar and axial vectorvector diquarks because they have the same parity as the diquarks because they have the same parity as the nucleon. nucleon.

A nucleon primarily consists of scalar and axial A nucleon primarily consists of scalar and axial vectorvector diquarks because they have the same parity as the diquarks because they have the same parity as the nucleon. nucleon. Pseudo-scalar and vector diquarks are heavy. Pseudo-scalar and vector diquarks are heavy. Pseudo-scalar and vector diquarks are heavy. Pseudo-scalar and vector diquarks are heavy.

To calculate the nucleon electromagnetic & To calculate the nucleon electromagnetic & transition formtransition form factors, one needs to evaluate the diquark elastic factors, one needs to evaluate the diquark elastic and and transition form factors.transition form factors.

To calculate the nucleon electromagnetic & To calculate the nucleon electromagnetic & transition formtransition form factors, one needs to evaluate the diquark elastic factors, one needs to evaluate the diquark elastic and and transition form factors.transition form factors.

Moreover, they have parity opposite to that of the Moreover, they have parity opposite to that of the nucleon.nucleon. To get the parity correct, non-zero quark angularTo get the parity correct, non-zero quark angular momentum of the quark has to be invoked. So they momentum of the quark has to be invoked. So they can becan be ignored in the description of the nucleon (ground ignored in the description of the nucleon (ground state).state).

Moreover, they have parity opposite to that of the Moreover, they have parity opposite to that of the nucleon.nucleon. To get the parity correct, non-zero quark angularTo get the parity correct, non-zero quark angular momentum of the quark has to be invoked. So they momentum of the quark has to be invoked. So they can becan be ignored in the description of the nucleon (ground ignored in the description of the nucleon (ground state).state).

TransitionTransition

Transition current: quark-diquark picture of the Transition current: quark-diquark picture of the nucleon:nucleon:Transition current: quark-diquark picture of the Transition current: quark-diquark picture of the nucleon:nucleon:


The nucleon primarily consists of scalar and The nucleon primarily consists of scalar and axial vector diquarks and N(1535) of its axial vector diquarks and N(1535) of its parity partners.parity partners.

The nucleon primarily consists of scalar and The nucleon primarily consists of scalar and axial vector diquarks and N(1535) of its axial vector diquarks and N(1535) of its parity partners.parity partners.

In the contact interaction model, the calculation In the contact interaction model, the calculation of theof thetransition form factors involves the diagram:transition form factors involves the diagram:

In the contact interaction model, the calculation In the contact interaction model, the calculation of theof thetransition form factors involves the diagram:transition form factors involves the diagram:


First look at: V→ VFirst look at: V→ V11VV11..First look at: V→ VFirst look at: V→ V11VV11..

Bose symmetry of Bose symmetry of 2 particles implies:2 particles implies:Bose symmetry of Bose symmetry of 2 particles implies:2 particles implies:


Moreover, the vector current conservation Moreover, the vector current conservation implies:implies:Moreover, the vector current conservation Moreover, the vector current conservation implies:implies:

It reduces the independent form factors to two. It reduces the independent form factors to two. For theFor theon shell vector bosons:on shell vector bosons:

It reduces the independent form factors to two. It reduces the independent form factors to two. For theFor theon shell vector bosons:on shell vector bosons:

Ongoing...

ConclusionsConclusions

Dynamical chiral symmetry breaking and the Dynamical chiral symmetry breaking and the momentum dependence of the quark mass momentum dependence of the quark mass function in QCD have experimental signals which function in QCD have experimental signals which enable us to differentiate its predictions from enable us to differentiate its predictions from others.others.

Dynamical chiral symmetry breaking and the Dynamical chiral symmetry breaking and the momentum dependence of the quark mass momentum dependence of the quark mass function in QCD have experimental signals which function in QCD have experimental signals which enable us to differentiate its predictions from enable us to differentiate its predictions from others.others.A fully consistent treatment of the contact A fully consistent treatment of the contact interaction model is simple to implement and can interaction model is simple to implement and can help us providehelp us provideuseful results which can be compared and useful results which can be compared and contrasted with full QCD calculation and contrasted with full QCD calculation and experiment.experiment.

A fully consistent treatment of the contact A fully consistent treatment of the contact interaction model is simple to implement and can interaction model is simple to implement and can help us providehelp us provideuseful results which can be compared and useful results which can be compared and contrasted with full QCD calculation and contrasted with full QCD calculation and experiment.experiment.A program to provide electromagnetic as well A program to provide electromagnetic as well transition form factors for mesons, diquarks and transition form factors for mesons, diquarks and nucleons is in progress within the simple contact nucleons is in progress within the simple contact interaction model. Theinteraction model. Themomentum dependent interaction will then be momentum dependent interaction will then be implemented.implemented.

A program to provide electromagnetic as well A program to provide electromagnetic as well transition form factors for mesons, diquarks and transition form factors for mesons, diquarks and nucleons is in progress within the simple contact nucleons is in progress within the simple contact interaction model. Theinteraction model. Themomentum dependent interaction will then be momentum dependent interaction will then be implemented.implemented.

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