Covariance, dynamics and symmetries, and hadron form factors
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1 Covariance, dynamics and symmetries, and hadron form factors Craig D. Roberts Physics Division Argonne National Laboratory Exclusive Reactions at High Momentum Transfer, 21-24May/07, p. 1/30
2 Dichotomy of the Pion Exclusive Reactions at High Momentum Transfer, 21-24May/07, p. 2/30
3 Dichotomy of the Pion How does one make an almost massless particle from two massive constituent-quarks? Exclusive Reactions at High Momentum Transfer, 21-24May/07, p. 2/30
4 Dichotomy of the Pion How does one make an almost massless particle from two massive constituent-quarks? Not Allowed to do it by fine-tuning a potential Current Algebra Must exhibit m 2 π m q Exclusive Reactions at High Momentum Transfer, 21-24May/07, p. 2/30
5 Dichotomy of the Pion How does one make an almost massless particle from two massive constituent-quarks? Not Allowed to do it by fine-tuning a potential Current Algebra Must exhibit m 2 π m q The correct understanding of pion observables; e.g. mass, decay constant and form factors, requires an approach to contain a well-defined and valid chiral limit, and an accurate realisation of dynamical chiral symmetry breaking. Exclusive Reactions at High Momentum Transfer, 21-24May/07, p. 2/30
6 Dichotomy of the Pion How does one make an almost massless particle from two massive constituent-quarks? Not Allowed to do it by fine-tuning a potential Current Algebra Must exhibit m 2 π m q The correct understanding of pion observables; e.g. mass, decay constant and form factors, requires an approach to contain a well-defined and valid chiral limit, and an accurate realisation of dynamical chiral symmetry breaking. Requires detailed understanding of Connection between Current-quark and Constituent-quark masses Exclusive Reactions at High Momentum Transfer, 21-24May/07, p. 2/30
7 Dichotomy of the Pion How does one make an almost massless particle from two massive constituent-quarks? Not Allowed to do it by fine-tuning a potential Current Algebra Must exhibit m 2 π m q The correct understanding of pion observables; e.g. mass, decay constant and form factors, requires an approach to contain a well-defined and valid chiral limit, and an accurate realisation of dynamical chiral symmetry breaking. Requires detailed understanding of Connection between Current-quark and Constituent-quark masses Using DSEs, we ve provided this. Exclusive Reactions at High Momentum Transfer, 21-24May/07, p. 2/30
8 What s the Problem? Exclusive Reactions at High Momentum Transfer, 21-24May/07, p. 3/30
9 Must calculate the hadron s wave function Can t be done using perturbation theory What s the Problem? Exclusive Reactions at High Momentum Transfer, 21-24May/07, p. 3/30
10 What s the Problem? Must calculate the hadron s wave function Can t be done using perturbation theory So what? Same is true of hydrogen atom Exclusive Reactions at High Momentum Transfer, 21-24May/07, p. 3/30
11 What s the Problem? Must calculate the hadron s wave function Can t be done using perturbation theory So what? Same is true of hydrogen atom Differences Exclusive Reactions at High Momentum Transfer, 21-24May/07, p. 3/30
12 What s the Problem? Must calculate the hadron s wave function Can t be done using perturbation theory So what? Same is true of hydrogen atom Differences Here relativistic effects are crucial virtual particles Quintessence of Relativistic Quantum Field Theory Exclusive Reactions at High Momentum Transfer, 21-24May/07, p. 3/30
13 What s the Problem? Must calculate the hadron s wave function Can t be done using perturbation theory So what? Same is true of hydrogen atom Differences Here relativistic effects are crucial virtual particles Quintessence of Relativistic Quantum Field Theory Interaction between quarks the Interquark Potential Unknown throughout > 98% of the pion s/proton s volume Exclusive Reactions at High Momentum Transfer, 21-24May/07, p. 3/30
14 Must calculate the hadron s wave function Can t be done using perturbation theory What s the Problem? So what? Same is true of hydrogen atom Differences Here relativistic effects are crucial virtual particles Quintessence of Relativistic Quantum Field Theory Interaction between quarks the Interquark Potential Unknown throughout > 98% of the pion s/proton s volume Determination of hadrons s wave function requires ab initio nonperturbative solution of fully-fledged relativistic quantum field theory Exclusive Reactions at High Momentum Transfer, 21-24May/07, p. 3/30
15 What s the Problem? Must calculate the hadron s wave function Can t be done using perturbation theory So what? Same is true of hydrogen atom Determination of hadron s wave function requires ab initio nonperturbative solution of fully-fledged relativistic quantum field theory Modern Physics & Mathematics Still quite some way from being able to do that Exclusive Reactions at High Momentum Transfer, 21-24May/07, p. 3/30
16 Dyson-Schwinger Equations Exclusive Reactions at High Momentum Transfer, 21-24May/07, p. 4/30
17 Dyson-Schwinger Equations Well suited to Relativistic Quantum Field Theory Exclusive Reactions at High Momentum Transfer, 21-24May/07, p. 4/30
18 Dyson-Schwinger Equations Well suited to Relativistic Quantum Field Theory Simplest level: Generating Tool for Perturbation Theory Materially Reduces Model Dependence Exclusive Reactions at High Momentum Transfer, 21-24May/07, p. 4/30
19 Dyson-Schwinger Equations Well suited to Relativistic Quantum Field Theory Simplest level: Generating Tool for Perturbation Theory Materially Reduces Model Dependence NonPerturbative, Continuum approach to QCD Exclusive Reactions at High Momentum Transfer, 21-24May/07, p. 4/30
20 Dyson-Schwinger Equations Well suited to Relativistic Quantum Field Theory Simplest level: Generating Tool for Perturbation Theory Materially Reduces Model Dependence NonPerturbative, Continuum approach to QCD Hadrons as Composites of Quarks and Gluons Exclusive Reactions at High Momentum Transfer, 21-24May/07, p. 4/30
21 Dyson-Schwinger Equations Well suited to Relativistic Quantum Field Theory Simplest level: Generating Tool for Perturbation Theory Materially Reduces Model Dependence NonPerturbative, Continuum approach to QCD Hadrons as Composites of Quarks and Gluons Qualitative and Quantitative Importance of: Dynamical Chiral Symmetry Breaking Generation of fermion mass from nothing Quark & Gluon Confinement Coloured objects not detected, not detectable? Exclusive Reactions at High Momentum Transfer, 21-24May/07, p. 4/30
22 Dyson-Schwinger Equations Well suited to Relativistic Quantum Field Theory Simplest level: Generating Tool for Perturbation Theory Materially Reduces Model Dependence NonPerturbative, Continuum approach to QCD Hadrons as Composites of Quarks and Gluons Qualitative and Quantitative Importance of: Dynamical Chiral Symmetry Breaking Generation of fermion mass from nothing Quark & Gluon Confinement Coloured objects not detected, not detectable? Understanding InfraRed (long-range) behaviour of α s (Q 2 ) Exclusive Reactions at High Momentum Transfer, 21-24May/07, p. 4/30
23 Dyson-Schwinger Equations Well suited to Relativistic Quantum Field Theory Simplest level: Generating Tool for Perturbation Theory Materially Reduces Model Dependence NonPerturbative, Continuum approach to QCD Hadrons as Composites of Quarks and Gluons Qualitative and Quantitative Importance of: Dynamical Chiral Symmetry Breaking Generation of fermion mass from nothing Quark & Gluon Confinement Coloured objects not detected, not detectable? Method yields Schwinger Functions Propagators Exclusive Reactions at High Momentum Transfer, 21-24May/07, p. 4/30
24 Dyson-Schwinger Equations Well suited to Relativistic Quantum Field Theory Simplest level: Generating Tool for Perturbation Theory Materially Reduces Model Dependence NonPerturbative, Continuum approach to QCD Hadrons as Composites of Quarks and Gluons Qualitative and Quantitative Importance of: Dynamical Chiral Symmetry Breaking Generation of fermion mass from nothing Quark & Gluon Confinement Coloured objects not detected, not detectable? Cross-Sections built from Schwinger Functions Exclusive Reactions at High Momentum Transfer, 21-24May/07, p. 4/30
25 Persistent Challenge Exclusive Reactions at High Momentum Transfer, 21-24May/07, p. 5/30
26 Persistent Challenge Infinitely Many Coupled Equations Σ = γ D S Γ Exclusive Reactions at High Momentum Transfer, 21-24May/07, p. 5/30
27 Persistent Challenge Infinitely Many Coupled Equations Solutions are Schwinger Functions (Euclidean Green Functions) Exclusive Reactions at High Momentum Transfer, 21-24May/07, p. 5/30
28 Persistent Challenge Infinitely Many Coupled Equations Solutions are Schwinger Functions (Euclidean Green Functions) Not all are Schwinger functions are experimentally observable but all are same VEVs measured in Lattice-QCD simulations... opportunity for comparisons at pre-experimental level... cross-fertilisation Exclusive Reactions at High Momentum Transfer, 21-24May/07, p. 5/30
29 Persistent Challenge Infinitely Many Coupled Equations Solutions are Schwinger Functions (Euclidean Green Functions) Coupling between equations necessitates truncation Weak coupling expansion Perturbation Theory Exclusive Reactions at High Momentum Transfer, 21-24May/07, p. 5/30
30 Persistent Challenge Infinitely Many Coupled Equations Solutions are Schwinger Functions (Euclidean Green Functions) Coupling between equations necessitates truncation Weak coupling expansion Perturbation Theory Not useful for the nonperturbative problems in which we re interested Exclusive Reactions at High Momentum Transfer, 21-24May/07, p. 5/30
31 Persistent Challenge Infinitely Many Coupled Equations Solutions are Schwinger Functions (Euclidean Green Functions) There is at least one systematic nonperturbative, symmetry-preserving truncation scheme H.J. Munczek Phys. Rev. D 52 (1995) 4736 Dynamical chiral symmetry breaking, Goldstone s theorem and the consistency of the Schwinger-Dyson and Bethe-Salpeter Equations A. Bender, C. D. Roberts and L. von Smekal, Phys. Lett. B 380 (1996) 7 Goldstone Theorem and Diquark Confinement Beyond Rainbow Ladder Approximation Exclusive Reactions at High Momentum Transfer, 21-24May/07, p. 6/30
32 Persistent Challenge Infinitely Many Coupled Equations Solutions are Schwinger Functions (Euclidean Green Functions) There is at least one systematic nonperturbative, symmetry-preserving truncation scheme Has Enabled Proof of EXACT Results in QCD Exclusive Reactions at High Momentum Transfer, 21-24May/07, p. 6/30
33 Persistent Challenge Infinitely Many Coupled Equations Solutions are Schwinger Functions (Euclidean Green Functions) There is at least one systematic nonperturbative, symmetry-preserving truncation scheme Has Enabled Proof of EXACT Results in QCD And Formulation of Practical Phenomenological Tool to Illustrate Exact Results Exclusive Reactions at High Momentum Transfer, 21-24May/07, p. 6/30
34 Persistent Challenge Infinitely Many Coupled Equations Solutions are Schwinger Functions (Euclidean Green Functions) There is at least one systematic nonperturbative, symmetry-preserving truncation scheme Has Enabled Proof of EXACT Results in QCD And Formulation of Practical Phenomenological Tool to Make Predictions with Readily Quantifiable Errors Exclusive Reactions at High Momentum Transfer, 21-24May/07, p. 6/30
35 Dressed-Quark Propagator Exclusive Reactions at High Momentum Transfer, 21-24May/07, p. 7/30
36 Dressed-Quark Propagator S(p) = Z(p 2 ) iγ p + M(p 2 ) Σ = γ D S Γ Gap Equation Exclusive Reactions at High Momentum Transfer, 21-24May/07, p. 7/30
37 Dressed-Quark Propagator S(p) = Z(p 2 ) iγ p + M(p 2 ) Gap Equation Gap Equation s Kernel Enhanced on IR domain IR Enhancement of M(p 2 ) 10 1 Σ = γ D S Γ 10 0 M(p 2 ) (GeV) b quark c quark s quark u,d quark chiral limit M 2 (p 2 ) = p p 2 (GeV 2 ) Exclusive Reactions at High Momentum Transfer, 21-24May/07, p. 7/30
38 Dressed-Quark Propagator S(p) = Z(p 2 ) iγ p + M(p 2 ) Gap Equation Gap Equation s Kernel Enhanced on IR domain IR Enhancement of M(p 2 ) 10 1 Σ = γ D S Γ Euclidean Constituent Quark Mass: M E f : p2 = M(p 2 ) 2 flavour u/d s c b M E m ζ M(p 2 ) (GeV) b quark c quark s quark u,d quark chiral limit M 2 (p 2 ) = p p 2 (GeV 2 ) Exclusive Reactions at High Momentum Transfer, 21-24May/07, p. 7/30
39 Dressed-Quark Propagator Longstanding Prediction of Dyson-Schwinger Equation Studies Exclusive Reactions at High Momentum Transfer, 21-24May/07, p. 8/30
40 Dressed-Quark Propagator Longstanding Prediction of Dyson-Schwinger Equation Studies E.g., Dyson-Schwinger equations and their application to hadronic physics, C. D. Roberts and A. G. Williams, Prog. Part. Nucl. Phys. 33 (1994) 477 Exclusive Reactions at High Momentum Transfer, 21-24May/07, p. 8/30
41 Dressed-Quark Propagator Longstanding Prediction of Dyson-Schwinger Equation Studies Long used as basis for efficacious hadron physics phenomenology E.g., Dyson-Schwinger equations and their application to hadronic physics, C. D. Roberts and A. G. Williams, Prog. Part. Nucl. Phys. 33 (1994) 477 Exclusive Reactions at High Momentum Transfer, 21-24May/07, p. 8/30
42 Dressed-Quark Propagator Longstanding Prediction of Dyson-Schwinger Equation Studies Long used as basis for efficacious hadron physics phenomenology Electromagnetic pion form-factor and neutral pion decay width, C. D. Roberts, Nucl. Phys. A 605 (1996) 475 E.g., Dyson-Schwinger equations and their application to hadronic physics, C. D. Roberts and A. G. Williams, Prog. Part. Nucl. Phys. 33 (1994) 477 Exclusive Reactions at High Momentum Transfer, 21-24May/07, p. 8/30
43 Quenched-QCD Dressed-Quark Propagator M(p) 0.5 Z(p) p (GeV) M(p) p (GeV) Exclusive Reactions at High Momentum Transfer, 21-24May/07, p. 9/30
44 2002 M(p) 0.5 Quenched-QCD Dressed-Quark Propagator Z(p) p (GeV) M(p) p (GeV) data: Quenched Lattice Meas. Bowman, Heller, Leinweber, Williams: he-lat/ Exclusive Reactions at High Momentum Transfer, 21-24May/07, p. 9/30
45 2002 M(p) 0.5 Quenched-QCD Dressed-Quark Propagator Z(p) p (GeV) M(p) p (GeV) data: Quenched Lattice Meas. Bowman, Heller, Leinweber, Williams: he-lat/ current-quark masses: 30 MeV, 50 MeV, 100 MeV Exclusive Reactions at High Momentum Transfer, 21-24May/07, p. 9/30
46 2002 M(p) 0.5 Quenched-QCD Dressed-Quark Propagator Z(p) p (GeV) M(p) p (GeV) data: Quenched Lattice Meas. Bowman, Heller, Leinweber, Williams: he-lat/ current-quark masses: 30 MeV, 50 MeV, 100 MeV Curves: Quenched DSE Cal. Bhagwat, Pichowsky, Roberts, Tandy nu-th/ Exclusive Reactions at High Momentum Transfer, 21-24May/07, p. 9/30
47 2002 M(p) 0.5 Quenched-QCD Dressed-Quark Propagator Z(p) p (GeV) M(p) p (GeV) data: Quenched Lattice Meas. Bowman, Heller, Leinweber, Williams: he-lat/ current-quark masses: 30 MeV, 50 MeV, 100 MeV Curves: Quenched DSE Cal. Bhagwat, Pichowsky, Roberts, Tandy nu-th/ Linear extrapolation of lattice data to chiral limit is inaccurate Exclusive Reactions at High Momentum Transfer, 21-24May/07, p. 9/30
48 Hadrons Established understanding of two- and three-point functions Exclusive Reactions at High Momentum Transfer, 21-24May/07, p. 10/30
49 Hadrons Established understanding of two- and three-point functions What about bound states? Exclusive Reactions at High Momentum Transfer, 21-24May/07, p. 10/30
50 Hadrons Without bound states, Comparison with experiment is impossible Exclusive Reactions at High Momentum Transfer, 21-24May/07, p. 10/30
51 Hadrons Without bound states, Comparison with experiment is impossible They appear as pole contributions to n 3-point colour-singlet Schwinger functions Exclusive Reactions at High Momentum Transfer, 21-24May/07, p. 10/30
52 Hadrons Without bound states, Comparison with experiment is impossible Bethe-Salpeter Equation QFT Generalisation of Lippmann-Schwinger Equation. Exclusive Reactions at High Momentum Transfer, 21-24May/07, p. 10/30
53 Hadrons Without bound states, Comparison with experiment is impossible Bethe-Salpeter Equation QFT Generalisation of Lippmann-Schwinger Equation. What is the kernel, K? Exclusive Reactions at High Momentum Transfer, 21-24May/07, p. 10/30
54 Hadrons Without bound states, Comparison with experiment is impossible Bethe-Salpeter Equation QFT Generalisation of Lippmann-Schwinger Equation. What is the kernel, K? or Exclusive Reactions at High Momentum Transfer, 21-24May/07, p. 10/30
55 What is the Long-Range Potential? Exclusive Reactions at High Momentum Transfer, 21-24May/07, p. 11/30
56 What is the Long-Range Potential? Exclusive Reactions at High Momentum Transfer, 21-24May/07, p. 11/30
57 Bethe-Salpeter Kernel Exclusive Reactions at High Momentum Transfer, 21-24May/07, p. 12/30
58 Bethe-Salpeter Kernel Axial-vector Ward-Takahashi identity P µ Γ l 5µ(k;P) = S 1 (k + ) 1 2 λl fiγ λl fiγ 5 S 1 (k ) M ζ iγ l 5(k;P) iγ l 5(k;P)M ζ QFT Statement of Chiral Symmetry Exclusive Reactions at High Momentum Transfer, 21-24May/07, p. 12/30
59 Bethe-Salpeter Kernel Axial-vector Ward-Takahashi identity P µ Γ l 5µ(k;P) = S 1 (k + ) 1 2 λl fiγ λl fiγ 5 S 1 (k ) M ζ iγ l 5(k;P) iγ l 5(k;P)M ζ Satisfies BSE Satisfies DSE Exclusive Reactions at High Momentum Transfer, 21-24May/07, p. 12/30
60 Bethe-Salpeter Kernel Axial-vector Ward-Takahashi identity P µ Γ l 5µ(k;P) = S 1 (k + ) 1 2 λl fiγ λl fiγ 5 S 1 (k ) M ζ iγ l 5(k;P) iγ l 5(k;P)M ζ Satisfies BSE Satisfies DSE Kernels must be intimately related Exclusive Reactions at High Momentum Transfer, 21-24May/07, p. 12/30
61 Bethe-Salpeter Kernel Axial-vector Ward-Takahashi identity P µ Γ l 5µ(k;P) = S 1 (k + ) 1 2 λl fiγ λl fiγ 5 S 1 (k ) M ζ iγ l 5(k;P) iγ l 5(k;P)M ζ Satisfies BSE Satisfies DSE Kernels must be intimately related Relation must be preserved by truncation Exclusive Reactions at High Momentum Transfer, 21-24May/07, p. 12/30
62 Bethe-Salpeter Kernel Axial-vector Ward-Takahashi identity P µ Γ l 5µ(k;P) = S 1 (k + ) 1 2 λl fiγ λl fiγ 5 S 1 (k ) M ζ iγ l 5(k;P) iγ l 5(k;P)M ζ Satisfies BSE Satisfies DSE Kernels must be intimately related Relation must be preserved by truncation Nontrivial constraint Exclusive Reactions at High Momentum Transfer, 21-24May/07, p. 12/30
63 Bethe-Salpeter Kernel Axial-vector Ward-Takahashi identity P µ Γ l 5µ(k;P) = S 1 (k + ) 1 2 λl fiγ λl fiγ 5 S 1 (k ) M ζ iγ l 5(k;P) iγ l 5(k;P)M ζ Satisfies BSE Satisfies DSE Kernels must be intimately related Relation must be preserved by truncation Failure Explicit Violation of QCD s Chiral Symmetry Exclusive Reactions at High Momentum Transfer, 21-24May/07, p. 12/30
64 Radial Excitations & Chiral Symmetry Exclusive Reactions at High Momentum Transfer, 21-24May/07, p. 13/30
65 (Maris, Roberts, Tandy nu-th/ ) Radial Excitations & Chiral Symmetry f H m 2 H = ρ H ζ M H Exclusive Reactions at High Momentum Transfer, 21-24May/07, p. 13/30
66 (Maris, Roberts, Tandy nu-th/ ) Radial Excitations & Chiral Symmetry f H m 2 H = ρh ζ M H Mass 2 of pseudoscalar hadron Exclusive Reactions at High Momentum Transfer, 21-24May/07, p. 13/30
67 (Maris, Roberts, Tandy nu-th/ ) Radial Excitations & Chiral Symmetry f H m 2 H = ρ H ζ M H M H := tr flavour [ M (µ) {T H, ( T H) t }] = m q1 +m q2 Sum of constituents current-quark masses e.g., T K+ = 1 ( 2 λ 4 + iλ 5) Exclusive Reactions at High Momentum Transfer, 21-24May/07, p. 13/30
68 (Maris, Roberts, Tandy nu-th/ ) Radial Excitations & Chiral Symmetry f H m 2 H = ρ H ζ M H f H p µ = Z 2 Λ q 1 2 tr { (T H ) t γ5 γ µ S(q + )Γ H (q;p)s(q ) Pseudovector projection of BS wave function at x = 0 Pseudoscalar meson s leptonic decay constant i S } π f k µ π A µ 5 = i Γ 5 i(τ/2)γ µ γ 5 k i S Exclusive Reactions at High Momentum Transfer, 21-24May/07, p. 13/30
69 (Maris, Roberts, Tandy nu-th/ ) Radial Excitations & Chiral Symmetry iρ H ζ = Z 4 Λ q f H m 2 H = ρ H ζ M H 1 2 tr { (T H ) t γ5 S(q + )Γ H (q;p)s(q ) Pseudoscalar projection of BS wave function at x = 0 i S } π ρ π P 5 = i Γ 5 i(τ/2) γ 5 k i S Exclusive Reactions at High Momentum Transfer, 21-24May/07, p. 13/30
70 (Maris, Roberts, Tandy nu-th/ ) Radial Excitations & Chiral Symmetry f H m 2 H = ρ H ζ M H Light-quarks; i.e., m q 0 f H f 0 H & ρh ζ qq 0 ζ f 0 H, Independent of m q Hence m 2 H = qq 0 ζ (f 0 H )2 m q... GMOR relation, a corollary Exclusive Reactions at High Momentum Transfer, 21-24May/07, p. 13/30
71 Höll, Krassnigg, Roberts nu-th/ Radial Excitations & Chiral Symmetry f H m 2 H = ρ H ζ M H Valid for ALL Pseudoscalar mesons Exclusive Reactions at High Momentum Transfer, 21-24May/07, p. 14/30
72 Höll, Krassnigg, Roberts nu-th/ Radial Excitations & Chiral Symmetry f H m 2 H = ρ H ζ M H Valid for ALL Pseudoscalar mesons ρ H finite, nonzero value in chiral limit, M H 0 Exclusive Reactions at High Momentum Transfer, 21-24May/07, p. 14/30
73 Höll, Krassnigg, Roberts nu-th/ Radial Excitations & Chiral Symmetry f H m 2 H = ρ H ζ M H Valid for ALL Pseudoscalar mesons ρ H finite, nonzero value in chiral limit, M H 0 radial excitation of π-meson, m 2 π n 0 > m 2 π n=0 = 0, in chiral limit Exclusive Reactions at High Momentum Transfer, 21-24May/07, p. 14/30
74 Höll, Krassnigg, Roberts nu-th/ Radial Excitations & Chiral Symmetry f H m 2 H = ρ H ζ M H Valid for ALL Pseudoscalar mesons ρ H finite, nonzero value in chiral limit, M H 0 radial excitation of π-meson, m 2 π n 0 > m 2 π n=0 = 0, in chiral limit f H = 0 ALL pseudoscalar mesons except π(140) in chiral limit Exclusive Reactions at High Momentum Transfer, 21-24May/07, p. 14/30
75 Höll, Krassnigg, Roberts nu-th/ Radial Excitations & Chiral Symmetry f H m 2 H = ρ H ζ M H Valid for ALL Pseudoscalar mesons ρ H finite, nonzero value in chiral limit, M H 0 radial excitation of π-meson, m 2 π n 0 > m 2 π n=0 = 0, in chiral limit f H = 0 ALL pseudoscalar mesons except π(140) in chiral limit Dynamical Chiral Symmetry Breaking Goldstone s Theorem impacts upon every pseudoscalar meson Exclusive Reactions at High Momentum Transfer, 21-24May/07, p. 14/30
76 McNeile and Michael he-la/ Radial Excitations & Lattice-QCD Exclusive Reactions at High Momentum Transfer, 21-24May/07, p. 15/30
77 McNeile and Michael he-la/ Radial Excitations & Lattice-QCD When we first heard about [this result] our first reaction was a combination of that is remarkable and unbelievable. Exclusive Reactions at High Momentum Transfer, 21-24May/07, p. 15/30
78 McNeile and Michael he-la/ Radial Excitations & Lattice-QCD When we first heard about [this result] our first reaction was a combination of that is remarkable and unbelievable. CLEO: τ π(1300) + ν τ f π1 < 8.4 MeV Diehl & Hiller he-ph/ Exclusive Reactions at High Momentum Transfer, 21-24May/07, p. 15/30
79 McNeile and Michael he-la/ Radial Excitations & Lattice-QCD When we first heard about [this result] our first reaction was a combination of that is remarkable and unbelievable. CLEO: τ π(1300) + ν τ f π1 < 8.4 MeV Diehl & Hiller he-ph/ f π /f π Lattice-QCD check: , a 0.1 fm, two-flavour, unquenched f π 1 f π = (93) not improved NP improved Expt. bound ( r 0 m π ) 2 Exclusive Reactions at High Momentum Transfer, 21-24May/07, p. 15/30
80 McNeile and Michael he-la/ Radial Excitations & Lattice-QCD When we first heard about [this result] our first reaction was a combination of that is remarkable and unbelievable. CLEO: τ π(1300) + ν τ f π1 < 8.4 MeV Diehl & Hiller he-ph/ f π /f π Lattice-QCD check: , a 0.1 fm, two-flavour, unquenched f π 1 f π = (93) not improved NP improved Expt. bound ( r 0 m π ) 2 Full ALPHA formulation is required to see suppression, because PCAC relation is at the heart of the conditions imposed for improvement (determining coefficients of irrelevant operators) Exclusive Reactions at High Momentum Transfer, 21-24May/07, p. 15/30
81 McNeile and Michael he-la/ Radial Excitations & Lattice-QCD When we first heard about [this result] our first reaction was a combination of that is remarkable and unbelievable. CLEO: τ π(1300) + ν τ f π1 < 8.4 MeV Diehl & Hiller he-ph/ f π /f π Lattice-QCD check: , a 0.1 fm, two-flavour, unquenched f π 1 f π = (93) not improved NP improved Expt. bound ( r 0 m π ) 2 The suppression of f π1 is a useful benchmark that can be used to tune and validate lattice QCD techniques that try to determine the properties of excited states mesons. Exclusive Reactions at High Momentum Transfer, 21-24May/07, p. 15/30
82 Pion... J = 0 but... Orbital angular momentum is not a Poincaré invariant. However, if absent in a particular frame, it will appear in another frame related via a Poincaré transformation. Exclusive Reactions at High Momentum Transfer, 21-24May/07, p. 16/30
83 Pion... J = 0 but... Nonzero quark orbital angular momentum is thus a necessary outcome of a Poincaré covariant description. Exclusive Reactions at High Momentum Transfer, 21-24May/07, p. 16/30
84 Pion... J = 0 but... Pseudoscalar meson Bethe-Salpeter amplitude χ π (k;p) = γ 5 [ie πn (k;p) + γ P F πn (k;p) γ k k P G πn (k;p) + σ µν k µ P ν H πn (k;p)] Exclusive Reactions at High Momentum Transfer, 21-24May/07, p. 16/30
85 Pion... J = 0 but... Pseudoscalar meson Bethe-Salpeter amplitude χ π (k;p) = γ 5 [ie πn (k;p) + γ P F πn (k;p) γ k k P G πn (k;p) + σ µν k µ P ν H πn (k;p)] J = 0... but while E and F are purely L = 0 in the rest frame, the G and H terms are associated with L = 1. Thus a pseudoscalar meson Bethe-Salpeter wave function always contains both S- and P -wave components. Exclusive Reactions at High Momentum Transfer, 21-24May/07, p. 16/30
86 Pion... J = 0 J = 0... but while E and F are purely L = 0 in the rest but... frame, the G and H terms are associated with L = 1. Thus a pseudoscalar meson Bethe-Salpeter wave function always contains both S- and P -wave components. Introduce mixing angle θ π such that χ π cos θ π L = 0 + sin θ π L = 1 Exclusive Reactions at High Momentum Transfer, 21-24May/07, p. 16/30
87 Pion... J = 0 J = 0... but while E and F are purely L = 0 in the rest but... frame, the G and H terms are associated with L = 1. Thus a pseudoscalar meson Bethe-Salpeter wave function always contains both S- and P -wave components. Introduce mixing angle θ π such that χ π cos θ π L = 0 + sin θ π L = 1 θ πn=0 (degrees) n meson mass (GeV) θ πn (degrees) meson mass (GeV) n=0 Exclusive Reactions at High Momentum Transfer, 21-24May/07, p. 16/30
88 Pion... J = 0 J = 0... but while E and F are purely L = 0 in the rest but... frame, the G and H terms are associated with L = 1. Thus a pseudoscalar meson Bethe-Salpeter wave function always contains both S- and P -wave components. Introduce mixing angle θ π such that χ π cos θ π L = 0 + sin θ π L = 1 L is significant in the neighbourhood of the chiral limit, and decreases with increasing current-quark mass. θ πn=0 (degrees) meson mass (GeV) n= n meson mass (GeV) θ πn (degrees) Exclusive Reactions at High Momentum Transfer, 21-24May/07, p. 16/30
89 New Challenges Exclusive Reactions at High Momentum Transfer, 21-24May/07, p. 17/30
90 New Challenges Next Steps... Applications to excited states and axial-vector mesons, e.g., will improve understanding of confinement interaction between light-quarks. Exclusive Reactions at High Momentum Transfer, 21-24May/07, p. 17/30
91 New Challenges Next Steps... Applications to excited states and axial-vector mesons, e.g., will improve understanding of confinement interaction between light-quarks. Move on to the problem of a symmetry preserving treatment of hybrids and exotics. Exclusive Reactions at High Momentum Transfer, 21-24May/07, p. 17/30
92 New Challenges Another Direction... Also want/need information about three-quark systems Exclusive Reactions at High Momentum Transfer, 21-24May/07, p. 17/30
93 New Challenges Another Direction... Also want/need information about three-quark systems With this problem... current expertise at approximately same point as studies of mesons in Exclusive Reactions at High Momentum Transfer, 21-24May/07, p. 17/30
94 New Challenges Another Direction... Also want/need information about three-quark systems With this problem... current expertise at approximately same point as studies of mesons in Namely... Model-building and Phenomenology, constrained by the DSE results outlined already. Exclusive Reactions at High Momentum Transfer, 21-24May/07, p. 17/30
95 Nucleon EM Form Factors: A Précis Höll, Kloker, et al.: nu-th/ & nu-th/ Exclusive Reactions at High Momentum Transfer, 21-24May/07, p. 18/30
96 Nucleon EM Form Factors: A Précis Höll, Kloker, et al.: nu-th/ & nu-th/ Interpreting expts. with GeV electromagnetic probes requires Poincaré covariant treatment of baryons Exclusive Reactions at High Momentum Transfer, 21-24May/07, p. 18/30
97 Nucleon EM Form Factors: A Précis Höll, Kloker, et al.: nu-th/ & nu-th/ Interpreting expts. with GeV electromagnetic probes requires Poincaré covariant treatment of baryons Covariant dressed-quark Faddeev Equation Exclusive Reactions at High Momentum Transfer, 21-24May/07, p. 18/30
98 Nucleon EM Form Factors: A Précis Höll, Kloker, et al.: nu-th/ & nu-th/ Interpreting expts. with GeV electromagnetic probes requires Poincaré covariant treatment of baryons Covariant dressed-quark Faddeev Equation Excellent mass spectrum (octet and decuplet) Easily obtained: ( 1 N H H [M exp H Mcalc H ]2 [M exp H ]2 ) 1/2 = 2% Exclusive Reactions at High Momentum Transfer, 21-24May/07, p. 18/30
99 Nucleon EM Form Factors: A Précis Höll, Kloker, et al.: nu-th/ & nu-th/ Interpreting expts. with GeV electromagnetic probes requires Poincaré covariant treatment of baryons Covariant dressed-quark Faddeev Equation Excellent mass spectrum (octet and decuplet) Easily obtained: ( 1 N H [M exp H Mcalc H ]2 [M exp H H ]2 (Oettel, Hellstern, Alkofer, Reinhardt: ) 1/2 = 2% nucl-th/ ) Exclusive Reactions at High Momentum Transfer, 21-24May/07, p. 18/30
100 Nucleon EM Form Factors: A Précis Höll, Kloker, et al.: nu-th/ & nu-th/ Interpreting expts. with GeV electromagnetic probes requires Poincaré covariant treatment of baryons Covariant dressed-quark Faddeev Equation Excellent mass spectrum (octet and decuplet) Easily obtained: ( 1 N H H But is that good? [M exp H Mcalc H ]2 [M exp H ]2 ) 1/2 = 2% Exclusive Reactions at High Momentum Transfer, 21-24May/07, p. 18/30
101 Nucleon EM Form Factors: A Précis Höll, Kloker, et al.: nu-th/ & nu-th/ Interpreting expts. with GeV electromagnetic probes requires Poincaré covariant treatment of baryons Covariant dressed-quark Faddeev Equation Excellent mass spectrum (octet and decuplet) Easily obtained: ( 1 N H H [M exp H Mcalc H ]2 [M exp H ]2 ) 1/2 = 2% But is that good? Cloudy Bag: δm π loop + = 300 to 400 MeV! Exclusive Reactions at High Momentum Transfer, 21-24May/07, p. 18/30
102 Nucleon EM Form Factors: A Précis Höll, Kloker, et al.: nu-th/ & nu-th/ Interpreting expts. with GeV electromagnetic probes requires Poincaré covariant treatment of baryons Covariant dressed-quark Faddeev Equation Excellent mass spectrum (octet and decuplet) Easily obtained: ( 1 N H H [M exp H Mcalc H ]2 [M exp H ]2 ) 1/2 = 2% But is that good? Cloudy Bag: δm π loop + = 300 to 400 MeV! Critical to anticipate pion cloud effects Roberts, Tandy, Thomas, et al., nu-th/ Exclusive Reactions at High Momentum Transfer, 21-24May/07, p. 18/30
103 Faddeev equation Exclusive Reactions at High Momentum Transfer, 21-24May/07, p. 19/30
104 Faddeev equation p q p d a Ψ P = p q p d Γ a q Γ b b Ψ P Exclusive Reactions at High Momentum Transfer, 21-24May/07, p. 19/30
105 Faddeev equation p q p d a Ψ P = p q p d Γ a q Γ b b Ψ P Linear, Homogeneous Matrix equation Yields wave function (Poincaré Covariant Faddeev Amplitude) that describes quark-diquark relative motion within the nucleon Scalar and Axial-Vector Diquarks... In Nucleon s Rest Frame Amplitude has... s, p & d wave correlations Exclusive Reactions at High Momentum Transfer, 21-24May/07, p. 19/30
106 Diquark correlations QUARK-QUARK Exclusive Reactions at High Momentum Transfer, 21-24May/07, p. 20/30
107 Same interaction that describes mesons also generates three coloured quark-quark correlations: blue red, blue green, green red Diquark correlations Confined... Does not escape from within baryon. Scalar is isosinglet, Axial-vector is isotriplet DSE and lattice-qcd m [ud]0 + = m (uu)1 + = m (ud) 1 + = m (dd) 1 + = QUARK-QUARK Exclusive Reactions at High Momentum Transfer, 21-24May/07, p. 20/30
108 Harry Lee Pions and Form Factors Exclusive Reactions at High Momentum Transfer, 21-24May/07, p. 21/30
109 Harry Lee Pions and Form Factors Dynamical coupled-channels model... Analyzed extensive JLab data... Completed a study of the (1236) Meson Exchange Model for πn Scattering and γn πn Reaction, T. Sato and T.-S. H. Lee, Phys. Rev. C 54, 2660 (1996) Dynamical Study of the Excitation in N(e, e π) Reactions, T. Sato and T.-S. H. Lee, Phys. Rev. C 63, /1-13 (2001) Exclusive Reactions at High Momentum Transfer, 21-24May/07, p. 21/30
110 Harry Lee Pions and Form Factors Dynamical coupled-channels model... Analyzed extensive JLab data... Completed a study of the (1236) Meson Exchange Model for πn Scattering and γn πn Reaction, T. Sato and T.-S. H. Lee, Phys. Rev. C 54, 2660 (1996) Dynamical Study of the Excitation in N(e, e π) Reactions, T. Sato and T.-S. H. Lee, Phys. Rev. C 63, /1-13 (2001) Pion cloud effects are large in the low Q 2 region. 3 Ratio of the M1 form factor in γn transition and proton dipole form factor G D. Solid curve is G M (Q2 )/G D (Q 2 ) including pions; Dotted curve is G M (Q 2 )/G D (Q 2 ) without pions Dressed Bare Q 2 (GeV/c) 2 Exclusive Reactions at High Momentum Transfer, 21-24May/07, p. 21/30
111 Harry Lee Pions and Form Factors Dynamical coupled-channels model... Analyzed extensive JLab data... Completed a study of the (1236) Meson Exchange Model for πn Scattering and γn πn Reaction, T. Sato and T.-S. H. Lee, Phys. Rev. C 54, 2660 (1996) Dynamical Study of the Excitation in N(e, e π) Reactions, T. Sato and T.-S. H. Lee, Phys. Rev. C 63, /1-13 (2001) Pion cloud effects are large in the low Q 2 region. 3 Ratio of the M1 form factor in γn transition and proton dipole form factor G D. 2 Solid curve is G M (Q2 )/G D (Q 2 ) including pions; Dotted curve is G M (Q 2 )/G D (Q 2 ) without pions. 1 Dressed Quark Core Bare Responsible for only 2/3 of result at small Q 2 Dominant for Q 2 >2 3 GeV Q 2 (GeV/c) 2 Exclusive Reactions at High Momentum Transfer, 21-24May/07, p. 21/30
112 Results: Nucleon and Masses Exclusive Reactions at High Momentum Transfer, 21-24May/07, p. 22/30
113 Results: Nucleon and Masses Mass-scale parameters (in GeV) for the scalar and axial-vector diquark correlations, fixed by fitting nucleon and masses Set A fit to the actual masses was required; whereas for Set B fitted mass was offset to allow for π-cloud contributions set M N M m 0 + m 1 + ω 0 + ω 1 + A =1/(0.45 fm) 0.59=1/(0.33 fm) B =1/(0.35 fm) 0.63=1/(0.31 fm) m 1 + : M A N = 1.15 GeV; MB N = 1.46 GeV Exclusive Reactions at High Momentum Transfer, 21-24May/07, p. 22/30
114 Results: Nucleon and Masses Mass-scale parameters (in GeV) for the scalar and axial-vector diquark correlations, fixed by fitting nucleon and masses Set A fit to the actual masses was required; whereas for Set B fitted mass was offset to allow for π-cloud contributions set M N M m 0 + m 1 + ω 0 + ω 1 + A =1/(0.45 fm) 0.59=1/(0.33 fm) B =1/(0.35 fm) 0.63=1/(0.31 fm) m 1 + : M A N = 1.15 GeV; MB N = 1.46 GeV Axial-vector diquark provides significant attraction Exclusive Reactions at High Momentum Transfer, 21-24May/07, p. 22/30
115 Results: Nucleon and Masses Mass-scale parameters (in GeV) for the scalar and axial-vector diquark correlations, fixed by fitting nucleon and masses Set A fit to the actual masses was required; whereas for Set B fitted mass was offset to allow for π-cloud contributions set M N M m 0 + m 1 + ω 0 + ω 1 + A =1/(0.45 fm) 0.59=1/(0.33 fm) B =1/(0.35 fm) 0.63=1/(0.31 fm) m 1 + : M A N = 1.15 GeV; MB N = 1.46 GeV Constructive Interference: diquark + µ π Exclusive Reactions at High Momentum Transfer, 21-24May/07, p. 22/30
116 Angular Momentum Rest Frame M. Oettel, et al. nucl-th/ Crude estimate based on magnitudes probability for a u-quark to carry the proton s spin is P u 80 %, with P u 5 %, P d 5 %, P d 10 %. Hence, by this reckoning 30% of proton s rest-frame spin is located in dressed-quark angular momentum. Exclusive Reactions at High Momentum Transfer, 21-24May/07, p. 23/30
117 Nucleon-Photon Vertex Exclusive Reactions at High Momentum Transfer, 21-24May/07, p. 24/30
118 M. Oettel, M. Pichowsky and L. von Smekal, nu-th/ terms... constructed systematically... current conserved automatically Nucleon-Photon Vertex for on-shell nucleons described by Faddeev Amplitude Exclusive Reactions at High Momentum Transfer, 21-24May/07, p. 24/30
119 M. Oettel, M. Pichowsky and L. von Smekal, nu-th/ terms... constructed systematically... current conserved automatically Nucleon-Photon Vertex for on-shell nucleons described by Faddeev Amplitude Q P f i Ψ Ψ P f i P f i Ψ Ψ P f i Q P f Γ i Ψ Ψ P f i Γ Q P f i Ψ Ψ P f i axial vector scalar Q P f i Ψ Ψ P f i Γ X µ Q P f X Q µ i Ψ Ψ P f i Exclusive Reactions at High Momentum Transfer, 21-24May/07, p. 24/30 Γ
120 Form Factor Ratio: GE/GM / G M p µ p G E p Rosenbluth precision Rosenbluth polarization transfer polarization transfer Q 2 [GeV 2 ] Exclusive Reactions at High Momentum Transfer, 21-24May/07, p. 25/30
121 Combine these elements... Form Factor Ratio: GE/GM / G M p µ p G E p Rosenbluth precision Rosenbluth polarization transfer polarization transfer Q 2 [GeV 2 ] Exclusive Reactions at High Momentum Transfer, 21-24May/07, p. 25/30
122 Combine these elements... Dressed-Quark Core 2 Form Factor Ratio: GE/GM / G M p µ p G E p Rosenbluth precision Rosenbluth polarization transfer polarization transfer Q 2 [GeV 2 ] Exclusive Reactions at High Momentum Transfer, 21-24May/07, p. 25/30
123 Combine these elements... Dressed-Quark Core Ward-Takahashi Identity preserving current p p µ p G E / G M Form Factor Ratio: Rosenbluth precision Rosenbluth polarization transfer polarization transfer GE/GM Q 2 [GeV 2 ] Exclusive Reactions at High Momentum Transfer, 21-24May/07, p. 25/30
124 Combine these elements... Dressed-Quark Core Ward-Takahashi Identity preserving current Anticipate and Estimate Pion p p µ p G E / G M Cloud s Contribution Form Factor Ratio: Rosenbluth precision Rosenbluth polarization transfer polarization transfer GE/GM Q 2 [GeV 2 ] Exclusive Reactions at High Momentum Transfer, 21-24May/07, p. 25/30
125 Combine these elements... Dressed-Quark Core Ward-Takahashi Identity preserving current Anticipate and Estimate Pion / G M p µ p G E p Cloud s Contribution Form Factor Ratio: covariant Fadeev result Rosenbluth precision Rosenbluth polarization transfer polarization transfer GE/GM Q 2 [GeV 2 ] Exclusive Reactions at High Momentum Transfer, 21-24May/07, p. 25/30
126 Combine these elements... Dressed-Quark Core Ward-Takahashi Identity preserving current Anticipate and Estimate Pion / G M p µ p G E p Cloud s Contribution All parameters fixed in other applications... Not varied. 2 Form Factor Ratio: covariant Fadeev result Rosenbluth precision Rosenbluth polarization transfer polarization transfer GE/GM Q 2 [GeV 2 ] Exclusive Reactions at High Momentum Transfer, 21-24May/07, p. 25/30
127 Combine these elements... Dressed-Quark Core Ward-Takahashi Identity preserving current Anticipate and Estimate Pion / G M p µ p G E p Cloud s Contribution All parameters fixed in other applications... Not varied. 2 Form Factor Ratio: covariant Fadeev result Rosenbluth precision Rosenbluth polarization transfer polarization transfer GE/GM Q 2 [GeV 2 ] Agreement with Pol. Trans. data at Q 2 > 2 GeV 2 Exclusive Reactions at High Momentum Transfer, 21-24May/07, p. 25/30
128 Combine these elements... Dressed-Quark Core Ward-Takahashi Identity preserving current Anticipate and Estimate Pion / G M p µ p G E p Cloud s Contribution All parameters fixed in other applications... Not varied. 2 Form Factor Ratio: covariant Fadeev result Rosenbluth precision Rosenbluth polarization transfer polarization transfer GE/GM Q 2 [GeV 2 ] Agreement with Pol. Trans. data at Q 2 > 2 GeV 2 Correlations in Faddeev amplitude quark orbital angular momentum essential to that agreement Exclusive Reactions at High Momentum Transfer, 21-24May/07, p. 25/30
129 Combine these elements... Dressed-Quark Core Ward-Takahashi Identity preserving current Anticipate and Estimate Pion / G M p µ p G E p Cloud s Contribution All parameters fixed in other applications... Not varied. Form Factor Ratio: covariant Fadeev result Rosenbluth precision Rosenbluth polarization transfer polarization transfer GE/GM Q 2 [GeV 2 ] Agreement with Pol. Trans. data at Q 2 > 2 GeV 2 Correlations in Faddeev amplitude quark orbital angular momentum essential to that agreement Predict Zero at Q 2 6.5GeV 2 Exclusive Reactions at High Momentum Transfer, 21-24May/07, p. 25/30
130 Neutron Form Factors / G M n µ n G E n nucl-ex Q 2 [GeV 2 ] Exclusive Reactions at High Momentum Transfer, 21-24May/07, p. 26/30
131 Neutron Form Factors Expt. Madey, et al. nu-ex/ n n µ n G E / G M nucl-ex Q 2 [GeV 2 ] Exclusive Reactions at High Momentum Transfer, 21-24May/07, p. 26/30
132 Neutron Form Factors Expt. Madey, et al. nu-ex/ Calc. Bhagwat, et / G M n al. nu-th/ µ p G n E (Q2 ) G n M (Q2 ) µ n G E n nucl-ex = r2 n 6 Q2 Valid for r 2 n Q2 < Q 2 [GeV 2 ] Exclusive Reactions at High Momentum Transfer, 21-24May/07, p. 26/30
133 Neutron Form Factors Expt. Madey, et al. nu-ex/ Calc. Bhagwat, et / G M n al. nu-th/ µ p G n E (Q2 ) G n M (Q2 ) = r2 n 6 Q2 Valid for r 2 n Q2 < 1 µ n G E n nucl-ex Q 2 [GeV 2 ] No sign yet of a zero in G n E (Q2 ), even though calculation predicts G p E (Q2 6.5 GeV 2 ) = 0 Data to Q 2 = 3.4 GeV 2 is being analysed (JLab E02-013) Exclusive Reactions at High Momentum Transfer, 21-24May/07, p. 26/30
134 Epilogue Exclusive Reactions at High Momentum Transfer, 21-24May/07, p. 27/30
135 Epilogue Exclusive Reactions at High Momentum Transfer, 21-24May/07, p. 27/30
136 Epilogue DCSB exists in QCD. Exclusive Reactions at High Momentum Transfer, 21-24May/07, p. 27/30
137 Epilogue DCSB exists in QCD. It is manifest in the dressed light-quark propagator. It impacts dramatically upon observables. Exclusive Reactions at High Momentum Transfer, 21-24May/07, p. 27/30
138 Epilogue DCSB exists in QCD. It is manifest in the dressed light-quark propagator. It impacts dramatically upon observables. Confinement Exclusive Reactions at High Momentum Transfer, 21-24May/07, p. 27/30
139 Epilogue DCSB exists in QCD. It is manifest in the dressed light-quark propagator. It impacts dramatically upon observables. Confinement Can be realised in dressed propagators of elementary excitations Observables can be used to explore model realisations Exclusive Reactions at High Momentum Transfer, 21-24May/07, p. 27/30
140 Epilogue DCSB exists in QCD. It is manifest in the dressed light-quark propagator. It impacts dramatically upon observables. Confinement Can be realised in dressed propagators of elementary excitations Observables can be used to explore model realisations An excellent way to test conjectures and constrain the possibilities Exclusive Reactions at High Momentum Transfer, 21-24May/07, p. 27/30
141 Contents 1. Dichotomy of the Pion 2. Dyson-Schwinger Equations 3. Persistent Challenge 4. Dressed-Quark Propagator 5. Quenched-QCD Dressed-Quark 6. Hadrons 7. Bethe-Salpeter Kernel 8. Excitations& Chiral Symmetry 9. Radial Excitations 10. Radial Excitations& Lattice-QCD 11. Pion J = Nucleon EM Form Factors 13. Faddeev equation 14. Diquark correlations 15. Pions and Form Factors 16. Results: Nucleon & Masses 17. Angular Momentum 18. Nucleon-Photon Vertex 19. Form Factor Ratio: GE/GM 20. Neutron Form Factors 21. Parametrising diquark properties 22. Contemporary Reviews Exclusive Reactions at High Momentum Transfer, 21-24May/07, p. 28/30
142 Parametrising diquark properties Exclusive Reactions at High Momentum Transfer, 21-24May/07, p. 29/30
143 Parametrising diquark properties Dressed-quark... fixed by DSE and Meson Studies... Burden, Roberts, Thomson, Phys. Lett. B 371, 163 (1996) Exclusive Reactions at High Momentum Transfer, 21-24May/07, p. 29/30
144 Parametrising diquark properties Dressed-quark... fixed by DSE and Meson Studies... Burden, Roberts, Thomson, Phys. Lett. B 371, 163 (1996) Non-pointlike scalar and pseudovector colour-antitriplet diquark correlations described by Bethe-Salpeter amplitudes... width for each ω J P Confining propagators... mass for each m J P Exclusive Reactions at High Momentum Transfer, 21-24May/07, p. 29/30
145 Parametrising diquark properties Dressed-quark... fixed by DSE and Meson Studies... Burden, Roberts, Thomson, Phys. Lett. B 371, 163 (1996) Non-pointlike scalar and pseudovector colour-antitriplet diquark correlations described by Bethe-Salpeter amplitudes... width for each ω J P Confining propagators... mass for each m J P Widths fixed by asymptotic freedom condition d dk 2 ( 1 m 2 J P F(K 2 /ω 2 J P ) ) 1 = 1 ω 2 J = 1 P 2 m2 J, P K 2 =0 Only two parameters; viz., diquark masses : m J P Exclusive Reactions at High Momentum Transfer, 21-24May/07, p. 29/30
146 Contemporary Reviews Dyson-Schwinger Equations: Density, Temperature and Continuum Strong QCD C.D. Roberts and S.M. Schmidt, nu-th/ , Prog. Part. Nucl. Phys. 45 (2000) S1 The IR behavior of QCD Green s functions: Confinement, DCSB, and hadrons... R. Alkofer and L. von Smekal, he-ph/ , Phys. Rept. 353 (2001) 281 Dyson-Schwinger equations: A Tool for Hadron Physics P. Maris and C.D. Roberts, nu-th/ , Int. J. Mod. Phys. E 12 (2003) pp Infrared properties of QCD from Dyson-Schwinger equations. C. S. Fischer, he-ph/ , J. Phys. G 32 (2006) pp. R253-R291 Exclusive Reactions at High Momentum Transfer, 21-24May/07, p. 30/30
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