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

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72 6.4. Electromagnetic current operator For a pointlike axial-vector we have µ 1 + = 2; and χ 1 + = 1, which corresponds to an oblate charge distribution. In addition, equations(6.64)–(6.67) with equations(6.72)–(6.74) realise the constraints of reference[BH92], namely, independent of the values of µ 1 + and χ 1 +, the form factors assume the ratios G 1+ E (Q2 ) : G 1+ M (Q2 ) : G 1+ Q (Q2 ) Q2 →∞ = (1 − 2 3 τ 1 +) : 2 : −1 . (6.78) 6.4.3 Coupling to the exchanged quark Diagram 3 depicts a photon coupling to the quark that is exchanged as one diquark breaks up and another is formed. While this is the first two-loop diagram we have described, no new elements appear in its specification: the dressed-quark-photon vertex was discussed in section 6.4.1. The explicit contribution to the vertex is obtained with J ex µ = −1 2 S(k q)∆ i (k d )Γ i (p 1 , k d )S T (q)ˆΓ quT µ (q ′ , q)S T (q ′ )¯Γ jT (p ′ 2 , p d)∆ j (p d )S(p q ) , (6.79) wherein the vertex ˆΓ qu µ appeared in equation(6.54). The full contribution is obtained by summing over the superscripts i, j, which can each take the values 0 + , 1 + . It is noteworthy that the process of quark exchange provides the attraction necessary in the Faddeev equation to bind the nucleon. It also guarantees that the Faddeev amplitude has the correct antisymmetry under the exchange of any two dressed-quarks. This key feature is absent in models with elementary (noncomposite) diquarks. 6.4.4 Scalar ↔ axialvector transition This contribution differs from diagram 2 in expressing the contribution to the nucleons’ form factors owing to an electromagnetically induced transition between scalar and axialvector diquarks: [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) , (6.80) where [ˆΓ dq µ (p d ; k d )] i=j = 0, and [ˆΓ dq µ (p d ; k d )] 1,2 = Γ SA , which is given in equation(6.81), and [ˆΓ dq µ (p d; k d )] 2,1 = Γ AS . Naturally, the diquark propagators match the line to which they are attached. The transition vertex is a rank-2 pseudotensor and can therefore be expressed Γ γα SA (l 1, l 2 ) = −Γ γα AS (l 1, l 2 ) = i M N T (l 1 , l 2 ) ε γαρλ l 1ρ l 2λ , (6.81)
Chapter 6. Nucleon Form Factors in a Covariant Diquark-Quark model 73 where γ, α are, respectively, the vector indices of the photon and axial-vector diquark. For simplicity we proceed under the assumption that T (l 1 , l 2 ) = κ T , (6.82) i.e. a constant, for which a typical value is [OAS00]: κ T ∼ 2 . (6.83) In the nucleons’ rest frame, a outstanding piece of the Faddeev amplitude that describes an axial-vector diquark inside the bound state can be characterised as containing a bystander quark whose spin is antiparallel to that of the nucleon, with the axial-vector diquark’s parallel. The interaction pictured in this diagram does not affect the bystander quark but the transformation of an axial-vector diquark into a scalar effects a flip of the quark spin within the correlation. After this transformation, the spin of the nucleon must be formed by summing the spin of the bystander quark, which is still aligned antiparallel to that of the nucleon, and the orbital angular momentum between that quark and the scalar diquark. 8 This argument, while not sophisticated, does motivate an expectation that diagram 4 will strongly impact on the nucleons’ magnetic form factors. 6.4.5 Seagull contributions The two-loop diagrams 5 and 6 are the so-called “seagull” terms, which appear as partners to diagram 3 and arise because binding in the nucleons’ Faddeev equations is effected by the exchange of nonpointlike diquark correlations [OPS00]. The explicit expression for their contribution to the nucleons’ form factors is given by J sg µ = 1 2 S(k q)∆ i (k d ) ( X i µ (p q, q ′ , k d )S T (q ′ )¯Γ jT (p ′ 2 , p d) − Γ i (p 1 , k d )S T (q) ¯X j µ (−k q, −q, p d ) ) ∆ j (p d )S(p q ) , (6.84) where, again, the superscripts are summed. In equations(6.79) and (6.84) the momenta are q = ˆηP − ηP ′ − p − k , q ′ = ˆηP ′ − ηP − p − k , p 1 = (p q − q)/2 , p ′ 2 = (−k q + q ′ )/2 , p ′ 1 = (p q − q ′ )/2 , p 2 = (−k q + q)/2 . (6.85) 8 A less prominent component of the amplitude has the bystander quark’s spin parallel to that of the nucleon while the axial-vector diquark’s is antiparallel: this q ↑ ⊕ (qq) ↓ 1 + system has one unit of angular momentum. That momentum is absent in the q ↑ ⊕(qq) 0 + system. Other combinations also contribute via diagram 3 but all mediated processes inevitably require a modification of spin and/or angular momentum.
Page 1 and 2:
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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Chapter 3. Remarks on QCD in Coulom
Page 27 and 28: Chapter 3. Remarks on QCD in Coulom
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 77: 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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