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Schwinger-Dyson Approach in Hadron Physics and QCD Confinement

This presentation explores the utilization of Schwinger-Dyson Equations in hadron physics and QCD to understand concepts like QCD confinement, strong coupling, asymptotic freedom, and pQCD. The discussion extends to QED Bound State Equations and P.V. Axial Ward Identity in studying effective interaction strength, quark mass functions, chiral symmetry breaking, and more. Examining the SDE/BSE approach at ANL/KSU, the focus is on pion/vector mesons, electromagnetic form factors, and the behavior of quarks in different energy ranges. Gauge invariance, multiplicative renormalizability, and the bridging of SDE/BSE from theory to experiment are also highlighted.

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Schwinger-Dyson Approach in Hadron Physics and QCD Confinement

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  1. Schwinger-Dyson approach to hadron physics Goecke, Fischer & Williams

  2. QCD confinement strong coupling 1 strong QCD 0 -15 0 10 r (m) asymptotic freedom strong QCD pQCD

  3. Schwinger-Dyson Equations . . . . QED

  4. Bound State Equations Schwinger-Dyson Equations P V Axial Ward Identity QCD

  5. effective interaction strength fp , mp Maris & Tandy p2 GeV2 10-3 103

  6. quark mass function  SB < qq >0 ~ - (240 MeV)3 chiral m 0 = s > 1 Williams, Fischer, P Maris & Roberts

  7. SDE/BSE – ANL/KSU v MV (GeV) pion/vector mesons CP-PACS q 2 MP (GeV2) q P V

  8. electromagnetic formfactors constant mass quark propagator running mass 103 10-3 p2 (GeV)2 q q q q q qq N*

  9. Pion electromagnetic formfactor Maris, Tandy

  10. Pion electromagnetic formfactor g* p+ n p Further extend Fp(Q2) w/ 12 GeV • Measure F up to 6 (GeV/c)2 to probe onset of pQCD • +/- measurements to test t-channel dominance of L • Q2 = 0.30 (GeV/c)2 close to pion pole to compare to +e elastic

  11. Pion electromagnetic formfactor g* p+ n p Further extend Fp(Q2) w/ 12 GeV • Measure F up to 6 (GeV/c)2 to probe onset of pQCD • +/- measurements to test t-channel dominance of L • Q2 = 0.30 (GeV/c)2 close to pion pole to compare to +e elastic

  12. effective interaction strength Qin, Chang, Liu, Roberts, Wilson p2 GeV2 10-3 103

  13. effective interaction strength constant mass running mass p2 GeV2 10-3 103

  14. Schwinger-Dyson Equations • remove divergences (eg. quadratic div.) • (ii) ensure correct gauge dependence (eg. transversality of boson) . . . . Consistent truncation Gauge Invariance & Multiplicative Renormalizibility QED

  15. Gauge Invariance Fermion propagator m -1 -1 = k p k p q m - -1 -1 wavefunction renormalisation mass function

  16. Gauge Invariance m m q -1 -1 = k p k p k p q m qm 0 1,2,..,8 Ward-Green-Takahashi

  17. Gauge Invariance m m q -1 -1 = k p k p k p q m Ball & Chiu Ward-Green-Takahashi Ball & Chiu

  18. Gauge Invariance m m q -1 -1 = k p k p k p q m Ward-Green-Takahashi Goecke, Fischer & Williams step I

  19. Testing for gauge invariance invariant not invariant little invariant

  20. Fermion propagator gm gm F(p) a = 0.5 F(p) a = 1 - -1 -1 p2/L2 wavefunction renormalisation CP BC CP mass function BC

  21. Fermion propagator gm CP R BC p2/L2 - -1 -1 wavefunction renormalisation mass function

  22. Schwinger-Dyson Equations QED . . . . Gauge Invariance & Multiplicative Renormalizibility Kizilersu & P

  23. Transverse components at O (a) Kizilersu, Reenders & P Davydychev

  24. Transverse vertex Kizilersu & P R. Williams, K, Sizer, P & A.G. Williams

  25. Unquenched Massless renormalised at m2 < L2: a=0.2, z : varying Kizilersu et al

  26. Unquenched Massless renormalised at m2 < L2: a=0.2, z : varying Kizilersu et al

  27. Light by Light gauge invariance is NOT optional

  28. Light by Light gauge invariance is NOT optional

  29. Fermion propagator mass gauge dependent  = cutoff

  30. Gauge Invariance and Multiplicative Renormalizibility -1 -1 - CP vertex mass almostgauge independent

  31. Slavnov-Taylor Identity axial gauges BBZ Schwinger-Dyson Equations QCD

  32. Schwinger-Dyson Equations Slavnov-Taylor Identity QCD covariant gauges

  33. Building bridges SDE/BSE in the continuum

  34. Building bridges SDE/BSE in the continuum connects the lattice to the continuum

  35. Building bridges SDE/BSE in the continuum connects theory to experiment

  36. Extra Slides

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