Presentaci

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From Free Quarks to Nucleon Form Factors
Adnan Bashir Michoacán University, Mexico
Argonne National Laboratory, USA Kent State
University, USA
August 15, 2012 University of South Carolina
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Contents

• Schwinger-Dyson Equations The Ingredients

• Pion Electromagnetic Transition Form Factors

• Rho and Diquark Form Factors

• Nucleon Electromagnetic Transition Form
  Factors

• Conclusions

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Schwinger-Dyson Equations The Ingredients
Schwinger-Dyson equations are the fundamental
equations of QCD and combine its UV and IR
behaviour.
Observing the transition of the hadron from a sea
of quarks and gluons to the one with valence
quarks alone is an experimental and theoretical
challenge.
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Schwinger-Dyson Equations The Ingredients
Schwinger-Dyson Equation for the The Quark
Propagator

• The gluon propagator and the quark-gluon vertex
  are
• directly responsible for the quarks to acquire
  their
• constituent masses.

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Schwinger-Dyson Equations The Ingredients
The Gluon Propagator
Modern SDE and lattice results support decoupling
solution for the gluon propagator.
AB, C. Lei, I. Cloet, B. El Bennich, Y. Liu, C.
Roberts, P. Tandy, Comm. Theor. Phys. 58 79-134
(2012)
Momentum dependent gluon mass is reminiscent of
the momentum dependent quark mass function.
A. Ayala, AB, D. Binosi, M. Cristoforetti, J.
Rodríguez hep-ph arXiv1208.0795 (2012).
It is in accord with the improved GZ-picture.
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Schwinger-Dyson Equations The Ingredients
The Quark-Gluon Vertex One of the 12 form
factors
J. Skullerud, P. Bowman, A. Kizilersu, D.
Leinweber, A. Williams, J. High Energy Phys. 04
047 (2003)
M. Bhagwat, M. Pichowsky, C. Roberts, P. Tandy,
Phys. Rev. C68 015203 (2003).
AB, L. Gutiérrez, M. Tejeda, AIP Conf. Proc. 1026
262 (2008).
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Schwinger-Dyson Equations The Ingredients
The Quark-Photon Vertex
In studying the elastic or transition form
factors of hadrons, it is the photon which probes
its constituents, highlighting the importance of
the quark-photon vertex.
Fortunately, both the quark-photon the
quark-gluon vertices require the same number of
basis tensors (12) for their description. So a
unified approach is possible.
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Schwinger-Dyson Equations The Ingredients
Quark-Photon Vertex (Ward-Takahashi identity)
The Ward identity is then invoked
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Schwinger-Dyson Equations The Ingredients
Significantly, this last ansatz contains
nontrivial factors associated with those tensors
whose appearance is solely driven by dynamical
chiral symmetry breaking.
It yields gauge independent critical coupling in
QED.
The Quark-Photon Vertex
It also reproduces large anomalous magnetic
moment for electrons in the infrared.
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Schwinger-Dyson Equations The Ingredients
Bethe Salpeter Amplitude
Goldberger-Triemann relations
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Schwinger-Dyson Equations The Ingredients
The quark propagator, electron-photon vertex and
the Bethe Salpeter Amplitude provide the
ingredients for the pion form factor
calculations.
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Schwinger-Dyson Equations The Ingredients
Contact interaction
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Pion Elastic and Transition Form Factors
Transition region for the electromagnetic pion
form factor may be accessible with the high
energy electron beam proposed for the 12 GeV
upgrade at JLab.
G.P. Lepage, and S.J. Brodsky, Phys. Rev. D22,
2157 (1980).
L. Gutiérrez, AB, I.C. Cloet, C.D. Roberts,
Phys. Rev. C81 065202 (2010).
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Pion Elastic and Transition Form Factors
H.L.L. Robertes, C.D. Roberts, AB, L.X. Gutiérrez
and P.C. Tandy, Phys. Rev. C82, (0652021-11)
2010.
CELLO H.J. Behrend et.al., Z. Phys C49 401
(1991). 0.7 2.2 GeV2
CLEO J. Gronberg et. al., Phys. Rev. D57 33
(1998). 1.7 8.0 GeV2
The leading twist pQDC calculation was carried
out in
BaBar R. Aubert et. al., Phys. Rev. D80 052002
(2009). 4.0 40.0 GeV2
G.P. Lepage, and S.J. Brodsky, Phys. Rev. D22,
2157 (1980).
Belle S. Uehara et. al., arXiv1205.3249 hep-ex
(2012). 4.0 40.0 GeV2
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Pion Elastic and Transition Form Factors
The pattern of chiral symmetry breaking dictates
the momentum dependence of physical observables.
F. Akram, AB, L. Gutiérrez, B. Masud, J.
Quintero, C. Calcaneo, M. Tejeda, arXiv0812----
(2012).
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Pion Elastic and Transition Form Factors
When do we expect perturbation theory to set in?

Perturbative
Momentum transfer Q is primarily shared equally
(Q/2) among quarks as BSA is peaked at zero
relative momentum.
Jlab 12GeV 2ltQ2lt9 GeV2 electromagnetic and
transition pion form factors.
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Rho Form Factors
??? Elastic Form Factors
Electromagnetic current of a vector meson is
Bose symmetry and charge conjugation yields
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Rho Form Factors
??? Elastic Form Factors
Within the impulse approximation the contact
interaction model
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Rho Form Factors

• The quark-photon vertex can be dressed as

• The corresponding IBS-equation thus yields

H.L.L. Robertes, C.D. Roberts, AB, L.X. Gutiérrez
and P.C. Tandy, Phys. Rev. C82, (0652021-11)
2010.
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Rho Form Factors
??? Elastic Form Factors
Electric, magnetic quadrupole form factors
??p transition form factor is very similar to
?p?
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Nucleon The Diquark Picture
Faddeev equation for a baryon.
G. Eichmann, Phys. Rev. D84, 014014 (2011).
Faddeev equation in the quark diquark picture
reproduces nucleon masses to within 5.
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Nucleon The Diquark Picture
In a color singlet baryon, any 2 quarks are
necessarily in a 3(bar) color state.
Color algebra of the BS equation reveals the
gluon exchange is attractive in this channel,
forming confined diquarks.
Each meson has a diquark partner which is
non-point like with finite radial extent
comparable to mesons.
In the diquark picture of the nucleon, the
calculation of its electromagnetic and transition
form factors requires the knowledge of the
diquarks their interaction with photons.
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Nucleon The Diquark Picture
A nucleon primarily consists of scalar and
axial vector diquarks because they have the
same parity as the nucleon.
Pseudo-scalar and vector diquarks are heavy.
Moreover, they have parity opposite to that of
the nucleon. To get the parity correct,
non-zero quark angular momentum of the quark
has to be invoked. So they can be ignored in
the description of the nucleon (ground state).
To calculate the nucleon electromagnetic
transition form factors, one needs to evaluate
the diquark elastic and transition form
factors.
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Transition current quark-diquark picture of the
nucleon
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The nucleon primarily consists of scalar and
axial vector diquarks and N(1535) of its parity
partners.
In the contact interaction model, the calculation
of the transition form factors involves the
diagram
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Transition
First look at V? V1V1.
Bose symmetry of 2 particles implies
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Transition
Moreover, the vector current conservation implies
It reduces the independent form factors to two.
For the on shell vector bosons
Ongoing...
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Conclusions
Dynamical chiral symmetry breaking and the
momentum dependence of the quark mass function in
QCD have experimental signals which enable us to
differentiate its predictions from others.
A fully consistent treatment of the contact
interaction model is simple to implement and can
help us provide useful results which can be
compared and contrasted with full QCD calculation
and experiment.
A program to provide electromagnetic as well
transition form factors for mesons, diquarks and
nucleons is in progress within the simple contact
interaction model. The momentum dependent
interaction will then be implemented.