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Adnan Bashir Michoacán University, Mexico Michoacán University, Mexico Argonne National Laboratory, USA Kent State University, USA γ * π 0 γ Transition Form Fatctor: QCD & BaBar γ * π 0 γ Transition Form Fatctor: QCD & BaBar March 15, 2012 Jefferson Laboratory, USA

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Contents Pion Electromagnetic Form Factor Pion Electromagnetic Form Factor Pion Electromagnetic Form Factor Pion Electromagnetic Form Factor Pion Electromagnetic Form Factor Pion Electromagnetic Form Factor Pion Electromagnetic Form Factor Pion Electromagnetic Form Factor Conclusions Conclusions Conclusions Conclusions Conclusions Conclusions Conclusions Conclusions Introduction Introduction Introduction Introduction Introduction Introduction Introduction Introduction Schwinger-Dyson Equations Schwinger-Dyson Equations Schwinger-Dyson Equations Schwinger-Dyson Equations Schwinger-Dyson Equations Schwinger-Dyson Equations Schwinger-Dyson Equations Schwinger-Dyson Equations 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 Bethe-Salpeter Amplitude Pion Bethe-Salpeter Amplitude Pion Bethe-Salpeter Amplitude Pion Bethe-Salpeter Amplitude Pion Transition Form Factor Pion Transition Form Factor Pion Transition Form Factor Pion Transition Form Factor Pion Transition Form Factor Pion Transition Form Factor Pion Transition Form Factor Pion Transition Form Factor

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The transition form factor is measured The transition form factor is measured through the process: through the process: The transition form factor is measured The transition form factor is measured through the process: through the process: Introduction

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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). Introduction

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Collinear factorization: Collinear factorization: Collinear factorization: Collinear factorization: Transition form factor is the correlator of two currents : Transition form factor is the correlator of two currents : Transition form factor is the correlator of two currents : Transition form factor is the correlator of two currents : T: hard scattering amplitude with quark gluon sub-processes. T: hard scattering amplitude with quark gluon sub-processes. T: hard scattering amplitude with quark gluon sub-processes. T: hard scattering amplitude with quark gluon sub-processes. is the pion distribution amplitude: is the pion distribution amplitude: is the pion distribution amplitude: is the pion distribution amplitude: In asymptotic QCD: In asymptotic QCD: In asymptotic QCD: In asymptotic QCD:

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Schwinger-Dyson Equations Schwinger-Dyson Equations Schwinger-Dyson Equations Schwinger-Dyson Equations Schwinger-Dyson Equations (SDE) are the fundamental Schwinger-Dyson Equations (SDE) are the fundamental Schwinger-Dyson Equations (SDE) are the fundamental Schwinger-Dyson Equations (SDE) are the fundamental equations of a field theory. equations of a field theory. Schwinger-Dyson Equations (SDE) are the fundamental Schwinger-Dyson Equations (SDE) are the fundamental Schwinger-Dyson Equations (SDE) are the fundamental Schwinger-Dyson Equations (SDE) are the fundamental equations of a field theory. equations of a field theory. 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. SDE have been employed to calculate: SDE have been employed to calculate: SDE have been employed to calculate: SDE have been employed to calculate: SDE have been employed to calculate: SDE have been employed to calculate: SDE have been employed to calculate: SDE have been employed to calculate: P. Maris, C.D. Roberts, Phys. Rev. C56 3369 (1997). P. Maris, P.C. Tandy, Phys. Rev. C62 055204 (2000). D. Jarecke, P. Maris, P.C. Tandy, Phys. Rev. C67 035202 (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. C83062201 (2011). nucleon form factors nucleon form factors nucleon form factors nucleon form factors G. Eichmann, et. al., G. Eichmann, et. al., Phys. Rev. C79 012202 (2009). D. Wilson, L. Chang and C.D. Roberts, Phys. Rev. C85 025205 (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, arXiv:1201.3366[nucl-th]

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Schwinger-Dyson Equations Schwinger-Dyson Equations Schwinger-Dyson Equations Schwinger-Dyson Equations Rainbow-ladder truncation Rainbow-ladder truncation Rainbow-ladder truncation Rainbow-ladder truncation SDE: Full quark propagator: SDE: Full quark propagator: SDE: Full quark propagator: SDE: Full quark propagator:

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Schwinger-Dyson Equations Schwinger-Dyson Equations Schwinger-Dyson Equations Schwinger-Dyson Equations Contact interaction: Contact interaction: Contact interaction: Contact interaction:

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Bethe-Salpeter amplitude for the pion: Bethe-Salpeter amplitude for the pion: Bethe-Salpeter amplitude for the pion: Bethe-Salpeter amplitude for the pion: Goldberger-Triemann relations: Goldberger-Triemann relations: Schwinger-Dyson Equations Schwinger-Dyson Equations Schwinger-Dyson Equations Schwinger-Dyson Equations

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P. Maris and C.D. Roberts, Phys. Rev. C58 3659-3665 (1998). Thus the pseudo-vector Thus the pseudo-vector component of the BS- component of the BS- amplitude dictates the amplitude dictates the transition of the pion transition of the pion form factor to the form factor to the perturbative limit. perturbative limit. Thus the pseudo-vector Thus the pseudo-vector component of the BS- component of the BS- amplitude dictates the amplitude dictates the transition of the pion transition of the pion form factor to the form factor to the perturbative limit. perturbative limit. Pion Form Factor: Pion Form Factor: Pion Form Factor: Pion Form Factor: Schwinger-Dyson Equations Schwinger-Dyson Equations Schwinger-Dyson Equations Schwinger-Dyson Equations

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For the contact interaction: For the contact interaction: For the contact interaction: For the contact interaction: Schwinger-Dyson Equations Schwinger-Dyson Equations Schwinger-Dyson Equations Schwinger-Dyson Equations Employing a proper time regularization scheme, one can Employing a proper time regularization scheme, one can ensure (i) confinement, (ii) axial vector Ward ensure (i) confinement, (ii) axial vector Ward Takahashi identity is satisfied and (iii) the corresponding Takahashi identity is satisfied and (iii) the corresponding Goldberger-Triemann relations are obeyed: Goldberger-Triemann relations are obeyed: Employing a proper time regularization scheme, one can Employing a proper time regularization scheme, one can ensure (i) confinement, (ii) axial vector Ward ensure (i) confinement, (ii) axial vector Ward Takahashi identity is satisfied and (iii) the corresponding Takahashi identity is satisfied and (iii) the corresponding Goldberger-Triemann relations are obeyed: Goldberger-Triemann relations are obeyed:

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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,. AB, R. Bermudez, L. Chang, C.D. Roberts, arXiv:1112.4847 [nucl-th]. Phenomenology Gauge Covariance Lattice Multiplicative Renormalization Perturbation Theory Quark-photon/ quark-gluon vertex Schwinger-Dyson Equations Schwinger-Dyson Equations Schwinger-Dyson Equations Schwinger-Dyson Equations 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. Quark-photon Vertex Quark-photon Vertex Quark-photon Vertex Quark-photon Vertex

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

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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
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We can dress quark-photon vertex: We can dress quark-photon vertex: We can dress quark-photon vertex: We can dress quark-photon vertex: We can dress quark-photon vertex: We can dress quark-photon vertex: We can dress quark-photon vertex: We can dress quark-photon vertex: H.L.L. Robertes, C.D. Roberts, A. Bashir, L.X. Gutiérrez and P.C. Tandy, H.L.L. Robertes, C.D. Roberts, A. Bashir, 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: Pion Electromagnetic Form Factor

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Dressed quark-photon vertex with ρ-pole on the time- Dressed quark-photon vertex with ρ-pole on the time- Dressed quark-photon vertex with ρ-pole on the time- Dressed quark-photon vertex with ρ-pole on the time- like axis does not alter the asymptotic behaviour of the like axis does not alter the asymptotic behaviour of the form factor for large space-like momenta. form factor for large space-like momenta. Dressed quark-photon vertex with ρ-pole on the time- Dressed quark-photon vertex with ρ-pole on the time- Dressed quark-photon vertex with ρ-pole on the time- Dressed quark-photon vertex with ρ-pole on the time- like axis does not alter the asymptotic behaviour of the like axis does not alter the asymptotic behaviour of the form factor for large space-like momenta. form factor for large space-like momenta. H.L.L. Robertes, C.D. Roberts, A. Bashir, L.X. Gutiérrez and P.C. Tandy, Phys. Rev. C82, (065202:1-11) 2010. Pion Electromagnetic Form factor

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Pattern of DCSB & Experimental Signatures

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QCD based prediction QCD based prediction through SDE through SDE QCD based prediction QCD based prediction through SDE through SDE H.L.L. Robertes, C.D. Roberts, AB, L.X. Gutiérrez and P.C. Tandy, Phys. Rev. C82, (065202:1-11) 2010. Pattern of DCSB & Experimental Signatures Quark propagator: quark mass function Quark propagator: quark mass function Quark propagator: quark mass function Quark propagator: quark mass function Dressed quark-photon vertex Dressed quark-photon vertex Dressed quark-photon vertex Dressed quark-photon vertex Pion Bethe-Salpeter amplitude Pion Bethe-Salpeter amplitude Pion Bethe-Salpeter amplitude Pion Bethe-Salpeter amplitude PQCD prediction PQCD prediction PQCD prediction PQCD prediction Contact interaction Contact interaction Contact interaction Contact interaction

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Pattern of DCSB & Experimental Signatures

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Other transition form factors are in accordance with Other transition form factors are in accordance with Other transition form factors are in accordance with Other transition form factors are in accordance with asymptotic QCD: asymptotic QCD: Other transition form factors are in accordance with Other transition form factors are in accordance with Other transition form factors are in accordance with Other transition form factors are in accordance with asymptotic QCD: asymptotic QCD:

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Could anything conceivably go astray in the experiment? Could anything conceivably go astray in the experiment? Could anything conceivably go astray in the experiment? Could anything conceivably go astray in the experiment? A possible erroneous way to extract the pion transition A possible erroneous way to extract the pion transition form factor from the data could be the problem of form factor from the data could be the problem of π 0 -π 0 subtraction. π 0 -π 0 subtraction. A possible erroneous way to extract the pion transition A possible erroneous way to extract the pion transition form factor from the data could be the problem of form factor from the data could be the problem of π 0 -π 0 subtraction. π 0 -π 0 subtraction. There is another channel which is the crossed channel of virtual Compton scattering on a pion. Diehl et. al. Phys. Rev. D62 073014 (2000). The misinterpretation of some events (where the second π 0 is not seen) may be different at different Q 2. Pattern of DCSB & Experimental Signatures

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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. 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. In a fully consistent treatment of pion in SDE QCD (static properties, elastic and transition form factors), the asymptotic limit of the product Q 2 G(Q 2 ), is not exceeded at any finite value of spacelike momentum transfer and is in disagreement with BaBar data for large momenta. In a fully consistent treatment of pion in SDE QCD (static properties, elastic and transition form factors), the asymptotic limit of the product Q 2 G(Q 2 ), is not exceeded at any finite value of spacelike momentum transfer and is in disagreement with BaBar data for large momenta. In a fully consistent treatment of pion in SDE QCD (static properties, elastic and transition form factors), the asymptotic limit of the product Q 2 G(Q 2 ), is not exceeded at any finite value of spacelike momentum transfer and is in disagreement with BaBar data for large momenta. In a fully consistent treatment of pion in SDE QCD (static properties, elastic and transition form factors), the asymptotic limit of the product Q 2 G(Q 2 ), is not exceeded at any finite value of spacelike momentum transfer and is in disagreement with BaBar data for large momenta. A fully consistent treatment of the contact interaction model produces results for pion elastic and transition form factors that are in striking disagreement with experiment. A fully consistent treatment of the contact interaction model produces results for pion elastic and transition form factors that are in striking disagreement with experiment. A fully consistent treatment of the contact interaction model produces results for pion elastic and transition form factors that are in striking disagreement with experiment. A fully consistent treatment of the contact interaction model produces results for pion elastic and transition form factors that are in striking disagreement with experiment.

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