Adnan Bashir, Michoacán University, Mexico Hadron Form Factors From Schwinger-Dyson Equations Pion Form Factors From Schwinger-Dyson Equations Collaborators: F. Akram, University of Punjab, Pakistan Y.X. Liu, Peking University, China M.R. Pennington, Durham University & JLab, UK J.R. Quintero, Huelva University, Spain A. Raya, Michoacán University, Mexico C.D. Roberts, Argonne National Laboratory, USA P.C. Tandy, Kent State University, USA R. Bermudez, University of Sonora, Mexico L. Chang, University of Adelaide, Australia L.X. Gutiérrez, University of Michoacán, Mexico D. Wilson, Jlab, USA

Contents Conclusions Conclusions Conclusions Conclusions Conclusions Conclusions Conclusions Conclusions Introduction Introduction Introduction Introduction Introduction Introduction Introduction Introduction 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 Quark Propagator: Quark Mass Function Quark Propagator: Quark Mass Function Quark Propagator: Quark Mass Function Quark Propagator: Quark Mass Function Quark-Photon Vertex Quark-Photon Vertex Quark-Photon Vertex Quark-Photon Vertex Pion Electromagnetic Form Factor Pion Electromagnetic Form Factor Pion Electromagnetic Form Factor Pion Electromagnetic Form Factor The Gluon Propagator/Quark Gluon Vertex The Gluon Propagator/Quark Gluon Vertex The Gluon Propagator/Quark Gluon Vertex The Gluon Propagator/Quark Gluon Vertex The Q 2 Evolution of Form Factors: The Q 2 Evolution of Form Factors: The Q 2 Evolution of Form Factors: The Q 2 Evolution of Form Factors: The Q 2 Evolution of Form Factors: The Q 2 Evolution of Form Factors: The Q 2 Evolution of Form Factors: The Q 2 Evolution of Form Factors: Pion to  * Transition Form Factor Pion to  * Transition Form Factor Pion to  * Transition Form Factor Pion to  * Transition Form Factor Mass Function and Form Factors Mass Function and Form Factors Mass Function and Form Factors Mass Function and Form Factors

Introduction Hadronic form factors are intimately related to their Hadronic form factors are intimately related to their internal structure. The challenge of their understanding internal structure. The challenge of their understanding occupies a central place in particle/nuclear physics. occupies a central place in particle/nuclear physics. Hadronic form factors are intimately related to their Hadronic form factors are intimately related to their internal structure. The challenge of their understanding internal structure. The challenge of their understanding occupies a central place in particle/nuclear physics. occupies a central place in particle/nuclear physics. QCD is the established theory of strong interactions QCD is the established theory of strong interactions which is responsible for binding quarks and gluons to which is responsible for binding quarks and gluons to form these hadrons (mesons and baryons). form these hadrons (mesons and baryons). QCD is the established theory of strong interactions QCD is the established theory of strong interactions which is responsible for binding quarks and gluons to which is responsible for binding quarks and gluons to form these hadrons (mesons and baryons). form these hadrons (mesons and baryons). Unraveling hadronic form factors from the basic building Unraveling hadronic form factors from the basic building blocks of QCD is an outstanding problem. blocks of QCD is an outstanding problem. Unraveling hadronic form factors from the basic building Unraveling hadronic form factors from the basic building blocks of QCD is an outstanding problem. blocks of QCD is an outstanding problem. Schwinger-Dyson equations are the fundamental equations Schwinger-Dyson equations are the fundamental equations of QCD and combine its UV and IR behavior. of QCD and combine its UV and IR behavior. Schwinger-Dyson equations are the fundamental equations Schwinger-Dyson equations are the fundamental equations of QCD and combine its UV and IR behavior. of QCD and combine its UV and IR behavior. Thus they provide a platform to study the form factors Thus they provide a platform to study the form factors from small to large photon virtualities, studied at different from small to large photon virtualities, studied at different hadron facilities. hadron facilities. Thus they provide a platform to study the form factors Thus they provide a platform to study the form factors from small to large photon virtualities, studied at different from small to large photon virtualities, studied at different hadron facilities. hadron facilities.

SDE for QCD have been extensively applied to meson SDE for QCD have been extensively applied to meson SDE for QCD have been extensively applied to meson SDE for QCD have been extensively applied to meson spectra and interactions below the masses ~ 1 GeV. spectra and interactions below the masses ~ 1 GeV. SDE for QCD have been extensively applied to meson SDE for QCD have been extensively applied to meson SDE for QCD have been extensively applied to meson SDE for QCD have been extensively applied to meson spectra and interactions below the masses ~ 1 GeV. spectra and interactions below the masses ~ 1 GeV. They have been employed to calculate: They have been employed to calculate: They have been employed to calculate: They have been employed to calculate: They have been employed to calculate: They have been employed to calculate: They have been employed to calculate: They have been employed to calculate: P. Maris, C.D. Roberts, Phys. Rev. C (1997). P. Maris, P.C. Tandy, Phys. Rev. C (2000). D. Jarecke, P. Maris, P.C. Tandy, Phys. Rev. C (2003). the masses, charge radii and decays of mesons the masses, charge radii and decays of mesons the masses, charge radii and decays of mesons the masses, charge radii and decays of mesons elastic pion and kaon form factors elastic pion and kaon form factors elastic pion and kaon form factors elastic pion and kaon form factors pion and kaon valence quark-distribution functions pion and kaon valence quark-distribution functions pion and kaon valence quark-distribution functions pion and kaon valence quark-distribution functions T. Nguyen, AB, C.D. Roberts, P.C. Tandy, T. Nguyen, AB, C.D. Roberts, P.C. Tandy, Phys. Rev. C (2011). nucleon form factors nucleon form factors nucleon form factors nucleon form factors G. Eichmann, et. al., G. Eichmann, et. al., Phys. Rev. C (2009). D. Wilson, L. Chang and C.D. Roberts, Phys. Rev. C (2012). “Collective Perspective on advances in DSE QCD”,AB, L. Chang, I.C. Cloet, B. El Bennich, Y. Liu, C.D. Roberts, P.C. Tandy, Commun. Theor. Phys (2012) Introduction Through SDEs, we can study the structure of hadrons Through SDEs, we can study the structure of hadrons Through SDEs, we can study the structure of hadrons Through SDEs, we can study the structure of hadrons through first principles in the continuum. through first principles in the continuum. Through SDEs, we can study the structure of hadrons Through SDEs, we can study the structure of hadrons Through SDEs, we can study the structure of hadrons Through SDEs, we can study the structure of hadrons through first principles in the continuum. through first principles in the continuum.

Introduction Parity Partners & Parity Partners & Chiral Symmetry Breaking Chiral Symmetry Breaking Parity Partners & Parity Partners & Chiral Symmetry Breaking Chiral Symmetry Breaking

Introduction The QCD Lagrangian: The QCD Lagrangian: The QCD Lagrangian: The QCD Lagrangian:

0.2 fm0.02 fm0.002 fm 1000 MeV 5 MeV QCD We can trace the origin of 98% of the luminous matter to QCD interactions. We can trace the origin of 98% of the luminous matter to QCD interactions.Introduction Asymptotic Freedom Asymptotic Freedom Asymptotic Freedom Asymptotic Freedom Infrared Slavery Infrared Slavery

Introduction

Simplest SDE - Simplest SDE - quark propagator: quark propagator: Simplest SDE - Simplest SDE - quark propagator: quark propagator: The Quark Propagator The Quark Propagator The Quark Propagator The Quark Propagator

The quark The quark propagator: The quark The quark propagator: Quark mass is a Quark mass is a function of momentum, function of momentum, dropping as 1/p 2 in the dropping as 1/p 2 in the ultraviolet. Quark mass is a Quark mass is a function of momentum, function of momentum, dropping as 1/p 2 in the dropping as 1/p 2 in the ultraviolet. Higgs mechanism is Higgs mechanism is almost irrelevant to the almost irrelevant to the infrared enhancement of infrared enhancement of quark mass. quark mass. Higgs mechanism is Higgs mechanism is almost irrelevant to the almost irrelevant to the infrared enhancement of infrared enhancement of quark mass. quark mass. The Quark Propagator The Quark Propagator The Quark Propagator The Quark 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 et. al. Phys. Rev. D (2012). A. Ayala et. al. Phys. Rev. D (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 (2012) Gluon Propagator: Gluon Propagator: Gluon Propagator: Gluon Propagator: A. Bashir, A. Raya, J. Rodrigues-Quintero, A. Bashir, A. Raya, J. Rodrigues-Quintero, Phys. Rev. D (2013). I.L. Bogolubsky, et. al. Phys. Lett. B (2009). I.L. Bogolubsky, et. al. Phys. Lett. B (2009).

J. Skullerud, P. Bowman, A. Kizilersu, D. Leinweber, A. Williams, J. High Energy Phys (2003) M. Bhagwat, M. Pichowsky, C. Roberts, P. Tandy, Phys. Rev. C (2003). AB, L. Gutiérrez, M. Tejeda, AIP Conf. Proc (2008). 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 The Quark-Gluon Vertex The Quark-Gluon Vertex The Quark-Gluon Vertex The Quark-Gluon Vertex

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

AB, M.R. Pennington Phys. Rev. D (1994) AB, M.R. Pennington Phys. Rev. D (1994) D.C. Curtis and M.R. Pennington Phys. Rev. D (1990) D.C. Curtis and M.R. Pennington Phys. Rev. D (1990) A. Kizilersu and M.R. Pennington Phys. Rev. D (2009) A. Kizilersu and M.R. Pennington Phys. Rev. D (2009) L. Chang, C.D. Roberts, Phys. Rev. Lett (2009) L. Chang, C.D. Roberts, Phys. Rev. Lett (2009) AB, C. Calcaneo, L. Gutiérrez, M. Tejeda, Phys. Rev. D (2011) AB, C. Calcaneo, L. Gutiérrez, M. Tejeda, Phys. Rev. D (2011) AB, R. Bermudez, L. Chang, C.D. Roberts, Phys. Rev. C85, (2012). AB, R. Bermudez, L. Chang, C.D. Roberts, Phys. Rev. C85, (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. Quark-photon Vertex Quark-photon Vertex Quark-photon Vertex Quark-photon Vertex The Quark-Photon Vertex The Quark-Photon Vertex The Quark-Photon Vertex The Quark-Photon Vertex It also reproduces large anomalous magnetic moment for quarks in infrared. Rocío Bermúdez, Luis Albino : Quark-Gluon Vertex.

The Q 2 Evolution of Form Factors The Q 2 Evolution of Form Factors The Q 2 Evolution of Form Factors The Q 2 Evolution of Form Factors 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.

The Q 2 Evolution of Form Factors Nobel Prize 2008: Nobel Prize 2008: “for the discovery of the mechanism of spontaneous “for the discovery of the mechanism of spontaneous broken symmetry in subatomic physics” broken symmetry in subatomic physics” Nobel Prize 2008: Nobel Prize 2008: “for the discovery of the mechanism of spontaneous “for the discovery of the mechanism of spontaneous broken symmetry in subatomic physics” broken symmetry in subatomic physics” quark-anti-quark

Contact interaction: Contact interaction: Contact interaction: Contact interaction: The Q 2 Evolution of Form Factors The Q 2 Evolution of Form Factors The Q 2 Evolution of Form Factors The Q 2 Evolution of Form Factors

Within the rainbow ladder truncation, the elastic Within the rainbow ladder truncation, the elastic electromagnetic pion form factor: electromagnetic pion form factor: Within the rainbow ladder truncation, the elastic Within the rainbow ladder truncation, the elastic electromagnetic pion form factor: electromagnetic pion form factor: The pattern of chiral symmetry breaking dictates the The pattern of chiral symmetry breaking dictates the momentum dependence of the elastic pion form factor. momentum dependence of the elastic pion form factor. The pattern of chiral symmetry breaking dictates the The pattern of chiral symmetry breaking dictates the momentum dependence of the elastic pion form factor. momentum dependence of the elastic pion form factor. L. Gutiérrez, AB, I.C. Cloet, C.D. Roberts, Phys. Rev. C (2010). L. Gutiérrez, AB, I.C. Cloet, C.D. Roberts, Phys. Rev. C (2010). F. Akram, AB, L. Gutiérrez, B. Masud, J. Quintero, C. Calcaneo, M. Tejeda, Phys Rev. D (2013). [QED] Pion Electromagnetic Form Factor

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 to  * Transition Form Factor Pion to  * Transition Form Factor Pion to  * Transition Form Factor Pion to  * Transition Form Factor 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 C (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. D (2009). 4.0 – 40.0 GeV 2 Leading twist asymptotic QDC calculation was carried out in: Leading twist asymptotic QDC calculation was carried out in: Leading twist asymptotic QDC calculation was carried out in: Leading twist asymptotic QDC calculation was carried out in: G.P. Lepage, and S.J. Brodsky, Phys. Rev. D22, 2157 (1980).

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 C (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. D (2009). 4.0 – 40.0 GeV 2 The leading twist asymptotic QCD calculation: The leading twist asymptotic QCD calculation: The leading twist asymptotic QCD calculation: The leading twist asymptotic QCD calculation: G.P. Lepage, and S.J. Brodsky, Phys. Rev. D22, 2157 (1980). Belle Belle S. Uehara et. al., arXiv: [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) Pion to  * Transition Form Factor Pion to  * Transition Form Factor Pion to  * Transition Form Factor Pion to  * Transition Form Factor

The transition form factor: The transition form factor: The transition form factor: The transition form factor: Belle II will have 40 times more luminosity. Belle II will have 40 times more luminosity. Belle II will have 40 times more luminosity. Belle II will have 40 times more luminosity. Belle II will have 40 times more luminosity. Belle II will have 40 times more luminosity. Belle II will have 40 times more luminosity. Belle II will have 40 times more luminosity. Precise measurements at large Q 2 will provide a stringent Precise measurements at large Q 2 will provide a stringent constraint on the pattern of chiral symmetry breaking. constraint on the pattern of chiral symmetry breaking. Precise measurements at large Q 2 will provide a stringent Precise measurements at large Q 2 will provide a stringent constraint on the pattern of chiral symmetry breaking. constraint on the pattern of chiral symmetry breaking. Vladimir Savinov: Vladimir Savinov: 5 th Workshop of the APS 5 th Workshop of the APS Topical Group on Hadronic Topical Group on Hadronic Physics, April Physics, April Vladimir Savinov: Vladimir Savinov: 5 th Workshop of the APS 5 th Workshop of the APS Topical Group on Hadronic Topical Group on Hadronic Physics, April Physics, April Pion to  * Transition Form FactorC Pion to  * Transition Form FactorC Pion to  * Transition Form FactorC Pion to  * Transition Form FactorC

Pion to  * Transition Form FactorC Pion to  * Transition Form FactorC Pion to  * Transition Form FactorC Pion to  * Transition Form FactorC Precise calculations with different interactions (p 2 ) -α at Precise calculations with different interactions (p 2 ) -α at increasing Q 2 will provide a stringent constraint on the increasing Q 2 will provide a stringent constraint on the pattern of chiral symmetry breaking. pattern of chiral symmetry breaking. Precise calculations with different interactions (p 2 ) -α at Precise calculations with different interactions (p 2 ) -α at increasing Q 2 will provide a stringent constraint on the increasing Q 2 will provide a stringent constraint on the pattern of chiral symmetry breaking. pattern of chiral symmetry breaking.

Double tagging? Double tagging? Double tagging? Double tagging? Double tagging? Double tagging? Double tagging? Double tagging? Probing the (p 2 ) -α dependence can be neater. Probing the (p 2 ) -α dependence can be neater. Probing the (p 2 ) -α dependence can be neater. Probing the (p 2 ) -α dependence can be neater. Probing the (p 2 ) -α dependence can be neater. Probing the (p 2 ) -α dependence can be neater. Probing the (p 2 ) -α dependence can be neater. Probing the (p 2 ) -α dependence can be neater. Vladimir Savinov Vladimir Savinov Vladimir Savinov Vladimir Savinov Pion to  * Transition Form FactorC Pion to  * Transition Form FactorC Pion to  * Transition Form FactorC Pion to  * Transition Form FactorC

Conclusions A systematic framework based upon the QCD equations A systematic framework based upon the QCD equations of motion (SDE) and its symmetries is required to chart of motion (SDE) and its symmetries is required to chart out and comprehend the Q 2 evolution of these form out and comprehend the Q 2 evolution of these form factors and make predictions. factors and make predictions. A systematic framework based upon the QCD equations A systematic framework based upon the QCD equations of motion (SDE) and its symmetries is required to chart of motion (SDE) and its symmetries is required to chart out and comprehend the Q 2 evolution of these form out and comprehend the Q 2 evolution of these form factors and make predictions. factors and make predictions. The large Q 2 evolution of the pion form factors, their experimental evaluation and theoretical predictions are likely to provide us with deep understanding of the The large Q 2 evolution of the pion form factors, their experimental evaluation and theoretical predictions are likely to provide us with deep understanding of the pattern of DCSB and confinement of the fundamental pattern of DCSB and confinement of the fundamental degrees of freedom, namely quarks and gluons. degrees of freedom, namely quarks and gluons. The large Q 2 evolution of the pion form factors, their experimental evaluation and theoretical predictions are likely to provide us with deep understanding of the The large Q 2 evolution of the pion form factors, their experimental evaluation and theoretical predictions are likely to provide us with deep understanding of the pattern of DCSB and confinement of the fundamental pattern of DCSB and confinement of the fundamental degrees of freedom, namely quarks and gluons. degrees of freedom, namely quarks and gluons. Predictions based upon the contact interaction, QCD SDE Predictions based upon the contact interaction, QCD SDE as well as the intermediate power laws (p 2 ) -  can provide experimentalist with a platform to compare and contrast future experimental results. Mesons, diquarks, baryons!!! as well as the intermediate power laws (p 2 ) -  can provide experimentalist with a platform to compare and contrast future experimental results. Mesons, diquarks, baryons!!! Predictions based upon the contact interaction, QCD SDE Predictions based upon the contact interaction, QCD SDE as well as the intermediate power laws (p 2 ) -  can provide experimentalist with a platform to compare and contrast future experimental results. Mesons, diquarks, baryons!!! as well as the intermediate power laws (p 2 ) -  can provide experimentalist with a platform to compare and contrast future experimental results. Mesons, diquarks, baryons!!!