The QCD Quark Propagator in Coulomb Gauge and - Institut fÃ¼r Physik

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70 6.4. Electromagnetic current operator 6.4.2 Coupling to the diquark In this section we treat diagram 2. It depicts the photon coupling directly to a diquark correlation. The explicit expression is [ˆΓdq ij Jµ dq = ∆ i (p d ) µ (p d; k d )] ∆ j (k d )S(k q )(2π) 4 δ 4 (p − k + ηQ) , (6.59) where [ˆΓ dq µ (p d ; k d )] ij = diag[Q 0 +Γ 0+ µ , Q 1 +Γ 1+ µ ], with Q 0 + = 1/3 and Γ 0+ µ is given in equation(6.62), and Q 1 + = diag[4/3, 1/3, −2/3] where Γ 1+ µ is given in equation(6.64). Naturally, the diquark propagators match the line to which they are attached. In the case of a scalar correlation, the general form of the diquark-photon vertex is Γ 0+ µ (l 1 , l 2 ) = 2 k µ f + (k 2 , k · Q, Q 2 ) + Q µ f − (k 2 , k · Q, Q 2 ) , (6.60) and it must satisfy the Ward-Takahashi identity: Q µ Γ 0+ µ (l 1 , l 2 ) = Π 0+ (l 2 1) − Π 0+ (l 2 2) , Π JP (l 2 ) = {∆ JP (l 2 )} −1 . (6.61) The evaluation of scalar diquark elastic electromagnetic form factors in reference[Mar04] is a first step toward calculating this vertex. However, in providing only an on-shell component, it is insufficient for our requirements. We therefore adapt equation(6.57) to this case and write Γ 0+ µ (l 1, l 2 ) = k µ ∆ Π 0 +(l2 1 , l2 2 ) , (6.62) which is the minimal ansatz that satisfies equation(6.61), is completely determined by quantities introduced already and is free of kinematic singularities. It implements f − ≡ 0, which is a requirement for elastic form factors, and guarantees a valid normalisation of electric charge, i.e. lim l ′ →l Γ0+ µ (l ′ d , l) = 2 l µ (l 2 ) l2 ∼0 = 2 l dl 2 Π0+ µ , (6.63) owing to equation(6.47). We have to keep in mind that we factored the fractional diquark charge, which therefore appears subsequently in our calculations as a simple multiplicative factor. For the case in which the struck diquark correlation is axial-vector and the scattering is elastic, the vertex assumes the form [HP99]: 7 Γ 1+ µαβ (l 1, l 2 ) = − 3∑ i=1 Γ [i] µαβ (l 1, l 2 ) , (6.64) 7 If the scattering is inelastic the general form of the vertex involves eight scalar functions [SD64]. Absent further constraints and input, we ignore the additional structure in this ansatz.
Chapter 6. Nucleon Form Factors in a Covariant Diquark-Quark model 71 with (T αβ (l) = δ αβ − l α l β /l 2 ) This vertex satisfies: Γ [1] µαβ (l 1, l 2 ) = (l 1 + l 2 ) µ T αλ (l 1 ) T λβ (l 2 ) F 1 (l 2 1, l 2 2) , (6.65) Γ [2] µαβ (l 1, l 2 ) = [T µα (l 1 ) T βρ (l 2 ) l 1ρ + T µβ (l 2 ) T αρ (l 1 ) l 2ρ ] F 2 (l 2 1 , l2 2 ) , (6.66) Γ [3] µαβ (l 1, l 2 ) = − 1 (l 2m 2 1 + l 2 ) µ T αρ (l 1 ) l 2ρ T βλ (l 2 ) l 1λ F 3 (l 2 1, l 2 2) . 1 + (6.67) l 1α Γ 1+ µαβ(l 1 , l 2 ) = 0 = Γ 1+ µαβ(l 1 , l 2 ) l 2β , (6.68) which is a general requirement of the elastic electromagnetic vertex of axial-vector bound states and guarantees that the interaction does not induce a pseudoscalar component in the axial-vector correlation. We note that the electric, magnetic and quadrupole form factors of an axial-vector bound state are expressed [HP99] G 1+ E (Q2 ) = F 1 + 2 3 τ 1 + G1+ Q (Q2 ) , τ 1 + = Q2 4 m 2 1 + (6.69) G 1+ M(Q 2 ) = −F 2 (Q 2 ) , (6.70) G 1+ Q (Q2 ) = F 1 (Q 2 ) + F 2 (Q 2 ) + (1 + τ 1 +) F 3 (Q 2 ) . (6.71) Extant knowledge of the form factors in equations(6.64)–(6.67) is limited and thus one has little information about even this rudimentary vertex model. Hence, we employ the following ansätze: F 1 (l 2 1 , l2 2 ) = ∆ Π 1+(l2 1 , l2 2 ) , (6.72) F 2 (l 2 1, l 2 2) = −F 1 + (1 − τ 1 +) (τ 1 +F 1 + 1 − µ 1 +) d(τ 1 +) (6.73) F 3 (l 2 1 , l2 2 ) = − (χ 1 + (1 − τ 1 +) d(τ 1 +) + F 1 + F 2 ) d(τ 1 +) , (6.74) with d(x) = 1/(1+x) 2 . This construction ensures a valid electric charge normalisation for the axial-vector correlation, lim l ′ →l Γ1+ µαβ (l′ , l) = T αβ (l) d (l 2 ) l2 ∼0 = T dl 2 Π1+ αβ (l) 2 l µ , (6.75) owing to equation(6.47), and current conservation The diquark’s static electromagnetic properties follow: lim l 2 →l 1 Q µ Γ 1+ µαβ (l 1, l 2 ) = 0 . (6.76) G 1+ E (0) = 1 , G1+ M (0) = µ 1 + , G1+ Q (0) = −χ 1 + . (6.77)
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The QCD Quark Propagator in Coulomb
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3 Zusammenfassung In der vorliegend
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Contents 1 Prologue 1 2 Chiral Symm
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Chapter 1 Prologue Quantum Chromo D
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Chapter 1. Prologue 3 an ultraviole
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Chapter 2 Chiral Symmetry Breaking
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Chapter 2. Chiral Symmetry Breaking
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Chapter 3 Remarks on QCD in Coulomb
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Chapter 3. Remarks on QCD in Coulom
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Page 41 and 42: Chapter 4 The Quark Dyson-Schwinger
Page 43 and 44: Chapter 4. The Quark Dyson-Schwinge
Page 55 and 56: Chapter 5. Meson Observables, Diqua
Page 61 and 62: Chapter 6. Nucleon Form Factors in
Page 75: Chapter 6. Nucleon Form Factors in
Page 97 and 98: Chapter 7 Epilogue In this thesis w
Page 99 and 100: Appendix A Units and Conventions In
Page 101 and 102: Appendix B Gauge potential, field s
Page 103 and 104: Appendix C. Numerical method employ
Page 105 and 106: Bibliography 99 [Alk88] Alkofer, Re
Page 107 and 108: Bibliography 101 [CMZ02] Cucchieri,
Page 109 and 110: Bibliography 103 [GO03] [GOZ05] [Gr
Page 111 and 112: Bibliography 105 [Mar04] Maris, Pie
Page 113 and 114: Bibliography 107 [PS] Peskin, Micha
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The QCD Quark Propagator in Coulomb Gauge and - Institut fÃ¼r Physik

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