Adnan Bashir Michoacán University, Mexico Argonne National Laboratory, USA

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

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

3. 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.

4. 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.

5. 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: arXiv:1208.0795 (2012). It is in accord with the improved GZ-picture.

6. 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).

7. 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.

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

9. Schwinger-Dyson Equations – The Ingredients D.C. Curtis and M.R. Pennington Phys. Rev. D42 4165 (1990) Phenomenology AB, M.R. Pennington Phys. Rev. D50 7679 (1994) A. Kizilersu and M.R. Pennington Phys. Rev. D79 125020 (2009) Gauge Covariance Lattice L. Chang, C.D. Roberts, Phys. Rev. Lett. 103 081601 (2009) 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). Significantly, this last ansatz contains nontrivial factors associated with those tensors whose appearance is solely driven by dynamical chiral symmetry breaking. Quark-photon/ quark-gluon vertex Perturbation Theory Multiplicative Renormalization It yields gauge independent critical coupling in QED. The Quark-Photon Vertex It also reproduces large anomalous magnetic moment for electrons in the infrared.

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

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

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

13. 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).

14. Pion Elastic and Transition Form Factors The transition form factor: H.L.L. Robertes, C.D. Roberts, AB, L.X. Gutiérrez and P.C. Tandy, Phys. Rev. C82, (065202:1-11) 2010. CELLOH.J. Behrend et.al., Z. Phys C49 401 (1991). 0.7 – 2.2 GeV2 CLEOJ. Gronberg et. al., Phys. Rev. D57 33 (1998).1.7 – 8.0 GeV2 The leading twist pQDC calculation was carried out in: BaBarR. 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). BelleS. Uehara et. al., arXiv:1205.3249 [hep-ex] (2012). 4.0 – 40.0 GeV2

15. 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, arXiv:0812---- (2012).

16. 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: 2<Q2<9 GeV2 electromagnetic and transition pion form factors.

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

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

19. 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, (065202:1-11) 2010.

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

21. 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%.

22. 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.

23. 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.

24. Transition 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. In the contact interaction model, the calculation of the transition form factors involves the diagram:

26. Transition First look at: V→ V1V1. Bose symmetry of 2 particles implies:

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