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

2. Contents Conclusions Conclusions Conclusions Conclusions Conclusions Conclusions Conclusions Conclusions Schwinger-Dyson Equations – The Ingredients Schwinger-Dyson Equations – The Ingredients Schwinger-Dyson Equations – The Ingredients Schwinger-Dyson Equations – The Ingredients Schwinger-Dyson Equations – The Ingredients Schwinger-Dyson Equations – The Ingredients Schwinger-Dyson Equations – The Ingredients Schwinger-Dyson Equations – The Ingredients Nucleon Electromagnetic & Transition Form Factors Nucleon Electromagnetic & Transition Form Factors Nucleon Electromagnetic & Transition Form Factors Nucleon Electromagnetic & Transition Form Factors Nucleon Electromagnetic & Transition Form Factors Nucleon Electromagnetic & Transition Form Factors Nucleon Electromagnetic & Transition Form Factors Nucleon Electromagnetic & Transition Form Factors Pion Electromagnetic & Transition Form Factors Pion Electromagnetic & Transition Form Factors Pion Electromagnetic & Transition Form Factors Pion Electromagnetic & Transition Form Factors Pion Electromagnetic & Transition Form Factors Pion Electromagnetic & Transition Form Factors Pion Electromagnetic & Transition Form Factors Pion Electromagnetic & Transition Form Factors Rho and Diquark Form Factors Rho and Diquark Form Factors Rho and Diquark Form Factors Rho and Diquark Form Factors Rho and Diquark Form Factors Rho and Diquark Form Factors Rho and Diquark Form Factors Rho and Diquark Form Factors

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

4. The gluon propagator and the quark-gluon vertex are The gluon propagator and the quark-gluon vertex are The gluon propagator and the quark-gluon vertex are The gluon propagator and the quark-gluon vertex are directly responsible for the quarks to acquire their directly responsible for the quarks to acquire their constituent masses. constituent masses. The gluon propagator and the quark-gluon vertex are The gluon propagator and the quark-gluon vertex are The gluon propagator and the quark-gluon vertex are The gluon propagator and the quark-gluon vertex are directly responsible for the quarks to acquire their directly responsible for the quarks to acquire their constituent masses. constituent masses. Schwinger-Dyson Equation for the Schwinger-Dyson Equation for the The Quark Propagator The Quark Propagator Schwinger-Dyson Equation for the Schwinger-Dyson Equation for the The Quark Propagator The Quark Propagator Schwinger-Dyson Equations – The Ingredients Schwinger-Dyson Equations – The Ingredients Schwinger-Dyson Equations – The Ingredients Schwinger-Dyson Equations – The Ingredients

5. The Gluon Propagator The Gluon Propagator The Gluon Propagator The Gluon Propagator Modern SDE and lattice results support decoupling solution for the gluon propagator. Modern SDE and lattice results support decoupling solution for the gluon propagator. Modern SDE and lattice results support decoupling solution for the gluon propagator. Modern SDE and lattice results support decoupling solution for the gluon propagator. Momentum dependent gluon mass is reminiscent of the momentum dependent quark mass function. Momentum dependent gluon mass is reminiscent of the momentum dependent quark mass function. Momentum dependent gluon mass is reminiscent of the momentum dependent quark mass function. Momentum dependent gluon mass is reminiscent of the momentum dependent quark mass 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. Rodríguez A. Ayala, AB, D. Binosi, M. Cristoforetti, J. Rodríguez hep-ph: arXiv:1208.0795 (2012). AB, C. Lei, I. Cloet, B. El Bennich, Y. Liu, C. Roberts, AB, C. Lei, I. Cloet, B. El Bennich, Y. Liu, C. Roberts, P. Tandy, Comm. Theor. Phys. 58 79-134 (2012) Schwinger-Dyson Equations – The Ingredients Schwinger-Dyson Equations – The Ingredients Schwinger-Dyson Equations – The Ingredients Schwinger-Dyson Equations – The Ingredients

6. 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). Schwinger-Dyson Equations – The Ingredients Schwinger-Dyson Equations – The Ingredients Schwinger-Dyson Equations – The Ingredients Schwinger-Dyson Equations – The Ingredients The Quark-Gluon The Quark-Gluon Vertex One of the 12 form factors One of the 12 form factors The Quark-Gluon The Quark-Gluon Vertex One of the 12 form factors One of the 12 form factors

7. The Quark-Photon Vertex: Schwinger-Dyson Equations – The Ingredients Schwinger-Dyson Equations – The Ingredients Schwinger-Dyson Equations – The Ingredients Schwinger-Dyson Equations – The Ingredients 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. 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.

8. Quark-Photon Vertex: (Ward-Takahashi identity) The Ward identity is then invoked: Schwinger-Dyson Equations – The Ingredients Schwinger-Dyson Equations – The Ingredients Schwinger-Dyson Equations – The Ingredients Schwinger-Dyson Equations – The Ingredients

9. 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 (1990) D.C. Curtis and M.R. Pennington Phys. Rev. D42 4165 (1990) A. Kizilersu and M.R. Pennington Phys. Rev. D79 125020 (2009) A. Kizilersu and M.R. Pennington Phys. Rev. D79 125020 (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 Gauge Covariance Lattice Multiplicative Renormalization Perturbation Theory Quark-photon/ quark-gluon vertex 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. It also reproduces large anomalous magnetic moment for electrons in the infrared. The Quark-Photon Vertex The Quark-Photon Vertex The Quark-Photon Vertex The Quark-Photon Vertex Schwinger-Dyson Equations – The Ingredients Schwinger-Dyson Equations – The Ingredients Schwinger-Dyson Equations – The Ingredients Schwinger-Dyson Equations – The Ingredients

10. Goldberger-Triemann relations: Goldberger-Triemann relations: Schwinger-Dyson Equations – The Ingredients Schwinger-Dyson Equations – The Ingredients Schwinger-Dyson Equations – The Ingredients Schwinger-Dyson Equations – The Ingredients Bethe Salpeter Amplitude:

11. The quark propagator, electron-photon vertex and the Bethe Salpeter Amplitude provide the ingredients for the pion form factor calculations. The quark propagator, electron-photon vertex and the Bethe Salpeter Amplitude provide the ingredients for the pion form factor calculations. The quark propagator, electron-photon vertex and the Bethe Salpeter Amplitude provide the ingredients for the pion form factor calculations. The quark propagator, electron-photon vertex and the Bethe Salpeter Amplitude provide the ingredients for the pion form factor calculations. Schwinger-Dyson Equations – The Ingredients Schwinger-Dyson Equations – The Ingredients Schwinger-Dyson Equations – The Ingredients Schwinger-Dyson Equations – The Ingredients

12. Contact interaction: Contact interaction: Contact interaction: Contact interaction: Schwinger-Dyson Equations – The Ingredients Schwinger-Dyson Equations – The Ingredients Schwinger-Dyson Equations – The Ingredients Schwinger-Dyson Equations – The Ingredients

13. Transition region for the electromagnetic pion form factor Transition region for the electromagnetic pion form factor may be accessible with the high energy electron beam may be accessible with the high energy electron beam proposed for the 12 GeV upgrade at JLab. proposed for the 12 GeV upgrade at JLab. Transition region for the electromagnetic pion form factor Transition region for the electromagnetic pion form factor may be accessible with the high energy electron beam may be accessible with the high energy electron beam proposed for the 12 GeV upgrade at JLab. 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). L. Gutiérrez, AB, I.C. Cloet, C.D. Roberts, Phys. Rev. C81 065202 (2010). Pion Elastic and Transition Form Factors Pion Elastic and Transition Form Factors Pion Elastic and Transition Form Factors Pion Elastic and Transition Form Factors

14. The transition form factor: The transition form factor: The transition form factor: The transition form factor: CELLO CELLO H.J. Behrend et.al., Z. Phys C49 401 (1991). 0.7 – 2.2 GeV 2 CLEO CLEO J. Gronberg et. al., Phys. Rev. D57 33 (1998). 1.7 – 8.0 GeV 2 BaBar BaBar R. Aubert et. al., Phys. Rev. D80 052002 (2009). 4.0 – 40.0 GeV 2 The leading twist pQDC calculation was carried out in: The leading twist pQDC calculation was carried out in: The leading twist pQDC calculation was carried out in: The leading twist pQDC calculation was carried out in: G.P. Lepage, and S.J. Brodsky, Phys. Rev. D22, 2157 (1980). Belle Belle S. Uehara et. al., arXiv:1205.3249 [hep-ex] (2012). 4.0 – 40.0 GeV 2 H.L.L. Robertes, C.D. Roberts, AB, L.X. Gutiérrez and P.C. Tandy, Phys. Rev. C82, (065202:1-11) 2010. Pion Elastic and Transition Form Factors Pion Elastic and Transition Form Factors Pion Elastic and Transition Form Factors Pion Elastic and Transition Form Factors

15. The pattern of chiral symmetry breaking dictates the The pattern of chiral symmetry breaking dictates the momentum dependence of physical observables. momentum dependence of physical observables. The pattern of chiral symmetry breaking dictates the The pattern of chiral symmetry breaking dictates the momentum dependence of physical observables. momentum dependence of physical observables. F. Akram, AB, L. Gutiérrez, B. Masud, J. Quintero, C. Calcaneo, M. Tejeda, arXiv:0812---- (2012). Pion Elastic and Transition Form Factors Pion Elastic and Transition Form Factors Pion Elastic and Transition Form Factors Pion Elastic and Transition Form Factors

16. 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 (Q/2) among quarks as BSA is peaked at zero relative momentum. Momentum transfer Q is primarily shared equally (Q/2) among quarks as BSA is peaked at zero relative momentum. Momentum transfer Q is primarily shared equally (Q/2) among quarks as BSA is peaked at zero relative momentum. Momentum transfer Q is primarily shared equally (Q/2) among quarks as BSA is peaked at zero relative momentum. Jlab 12GeV: 2<Q 2 <9 GeV 2 electromagnetic and transition pion form factors. Pion Elastic and Transition Form Factors Pion Elastic and Transition Form Factors Pion Elastic and Transition Form Factors Pion Elastic and Transition Form Factors

17. ργρ Elastic Form Factors: ργρ Elastic Form Factors: ργρ Elastic Form Factors: ργρ Elastic Form Factors: Rho Form Factors Electromagnetic current of a vector meson is: Electromagnetic current of a vector meson is: Electromagnetic current of a vector meson is: Electromagnetic current of a vector meson is: Bose symmetry and charge conjugation yields: Bose symmetry and charge conjugation yields: Bose symmetry and charge conjugation yields: Bose symmetry and charge conjugation yields:

18. Rho Form Factors ργρ Elastic Form Factors: ργρ Elastic Form Factors: ργρ Elastic Form Factors: ργρ Elastic Form Factors: Within the impulse approximation & the contact interaction Within the impulse approximation & the contact interaction model: Within the impulse approximation & the contact interaction Within the impulse approximation & the contact interaction model:

19. The quark-photon vertex can be dressed as: The quark-photon vertex can be dressed as: The quark-photon vertex can be dressed as: The quark-photon vertex can be dressed as: The quark-photon vertex can be dressed as: The quark-photon vertex can be dressed as: The quark-photon vertex can be dressed as: The quark-photon vertex 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: The corresponding IBS-equation thus yields: The corresponding IBS-equation thus yields: The corresponding IBS-equation thus yields: The corresponding IBS-equation thus yields: Rho Form Factors

20. Electric, magnetic Electric, magnetic & quadrupole form & quadrupole form factors Electric, magnetic Electric, magnetic & quadrupole form & quadrupole form factors ργρ Elastic ργρ Elastic Form Factors: Form Factors: ργρ Elastic ργρ Elastic Form Factors: Form Factors: ργπ transition form factor is very similar to γ * πγ ργπ transition form factor is very similar to γ * πγ ργπ transition form factor is very similar to γ * πγ ργπ transition form factor is very similar to γ * πγ

21. Faddeev equation for a baryon. Faddeev equation for a baryon. Faddeev equation for a baryon. Faddeev equation for a baryon. G. Eichmann, Phys. Rev. D84, 014014 (2011). Faddeev equation in the quark diquark picture reproduces Faddeev equation in the quark diquark picture reproduces nucleon masses to within 5%. nucleon masses to within 5%. Faddeev equation in the quark diquark picture reproduces Faddeev equation in the quark diquark picture reproduces nucleon masses to within 5%. nucleon masses to within 5%. Nucleon – The Diquark Picture

22. In the diquark picture of the nucleon, the calculation of its In the diquark picture of the nucleon, the calculation of its electromagnetic and transition form factors requires the electromagnetic and transition form factors requires the knowledge of the diquarks & their interaction with photons. knowledge of the diquarks & their interaction with photons. In the diquark picture of the nucleon, the calculation of its In the diquark picture of the nucleon, the calculation of its electromagnetic and transition form factors requires the electromagnetic and transition form factors requires the knowledge of the diquarks & their interaction with photons. knowledge of the diquarks & their interaction with photons. In a color singlet baryon, any 2 quarks are necessarily In a color singlet baryon, any 2 quarks are necessarily in a 3(bar) color state. in a 3(bar) color state. In a color singlet baryon, any 2 quarks are necessarily In a color singlet baryon, any 2 quarks are necessarily in a 3(bar) color state. in a 3(bar) color state. Color algebra of the BS equation reveals the gluon exchange Color algebra of the BS equation reveals the gluon exchange is attractive in this channel, forming confined diquarks. is attractive in this channel, forming confined diquarks. Color algebra of the BS equation reveals the gluon exchange Color algebra of the BS equation reveals the gluon exchange is attractive in this channel, forming confined diquarks. is attractive in this channel, forming confined 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. Nucleon – The Diquark Picture

23. A nucleon primarily consists of scalar and axial vector A nucleon primarily consists of scalar and axial vector diquarks because they have the same parity as the nucleon. diquarks because they have the same parity as the nucleon. A nucleon primarily consists of scalar and axial vector A nucleon primarily consists of scalar and axial vector diquarks because they have the same parity as the nucleon. diquarks because they have the same parity as the 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 & transition form To calculate the nucleon electromagnetic & transition form factors, one needs to evaluate the diquark elastic and factors, one needs to evaluate the diquark elastic and transition form factors. transition form factors. To calculate the nucleon electromagnetic & transition form To calculate the nucleon electromagnetic & transition form factors, one needs to evaluate the diquark elastic and factors, one needs to evaluate the diquark elastic and transition form factors. transition form factors. Moreover, they have parity opposite to that of the nucleon. Moreover, they have parity opposite to that of the nucleon. To get the parity correct, non-zero quark angular To get the parity correct, non-zero quark angular momentum of the quark has to be invoked. So they can be momentum of the quark has to be invoked. So they can be ignored in the description of the nucleon (ground state). ignored in the description of the nucleon (ground state). Moreover, they have parity opposite to that of the nucleon. Moreover, they have parity opposite to that of the nucleon. To get the parity correct, non-zero quark angular To get the parity correct, non-zero quark angular momentum of the quark has to be invoked. So they can be momentum of the quark has to be invoked. So they can be ignored in the description of the nucleon (ground state). ignored in the description of the nucleon (ground state).

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

25. Transition The nucleon primarily consists of scalar and axial vector diquarks and N(1535) of its parity partners. The nucleon primarily consists of scalar and axial vector diquarks and N(1535) of its parity partners. The nucleon primarily consists of scalar and axial vector diquarks and N(1535) of its parity partners. 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 In the contact interaction model, the calculation of the transition form factors involves the diagram: transition form factors involves the diagram: In the contact interaction model, the calculation of the In the contact interaction model, the calculation of the transition form factors involves the diagram: transition form factors involves the diagram:

26. Transition First look at: V → V 1 V 1. First look at: V → V 1 V 1. First look at: V → V 1 V 1. First look at: V → V 1 V 1. Bose symmetry of Bose symmetry of 2 particles implies: 2 particles implies: Bose symmetry of Bose symmetry of 2 particles implies: 2 particles implies:

27. Transition Moreover, the vector current conservation implies: Moreover, the vector current conservation implies: Moreover, the vector current conservation implies: Moreover, the vector current conservation implies: It reduces the independent form factors to two. For the It reduces the independent form factors to two. For the on shell vector bosons: on shell vector bosons: It reduces the independent form factors to two. For the It reduces the independent form factors to two. For the on shell vector bosons: on shell vector bosons: Ongoing...

28. 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. 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. 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. 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 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. useful results which can be compared and contrasted with full QCD calculation and experiment. A fully consistent treatment of the contact interaction model is simple to implement and can help us provide 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. 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 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. momentum dependent interaction will then be implemented. 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 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. momentum dependent interaction will then be implemented.