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

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6 2.1. Chiral symmetry of the QCD Lagrangian generators of SU(3), which are the well-known Gell-Mann matrices: A µ αβ = λa αβ 2 (Aa ) µ , α, β = 1, 2, ..., N c , a = 1, ..., (N 2 c − 1) , where Einstein’s summation convention is understood and N c is the number of colours. In order to discuss the symmetries of the quark fields we introduce left- and righthanded fields Ψ f R = 1 2 (1 + γ 5)Ψ f , Ψ f L = 1 2 (1 − γ 5)Ψ f . Since γ 5 anticommutes with all gamma matrices this results in the decomposition Ψ f iγ µ D µ Ψ f = Ψ f Riγ µ D µ Ψ f R + Ψf Liγ µ D µ Ψ f L . (2.4) Regarding the Lagrangian density this means that in the limit of zero current quark masses left- and right-handed quarks propagate independently from each other. The Lagrangian is invariant under unitary transformations of the left- and right-handed quark field operators, which do not depend on each other. These transformations form the group U L (N f ) × U R (N f ). For the purpose of discussing the conserved charges we decompose this group in a first step into semisimple Lie sub-groups: U L (N f ) × U R (N f ) ∼ = U L+R (1) × U L−R (1) × SU L (N f ) × SU R (N f ) . (2.5) The symmetry transformation of the subgroup U L+R (1) yields the conservation of baryon number whereas the divergence of the U L−R (1)-current does not vanish even in the massless because of the Adler-Bell-Jackiw anomaly [Ynd, Pok]. Chiral symmetry is the invariance under the transformations SU L (N f ) × SU R (N f ). It implies the conservation of 2(N 2 f − 1) charges. The corresponding currents are JLµ r (x) = Ψf L (x)γ t r fg µ 2 Ψg L , (2.6) JRµ r (x) = Ψf R (x)γ t r fg µ 2 Ψg R , (2.7) f, g = 1, ..., N f , r = 1, ..., Nf 2 − 1 . (2.8)
Chapter 2. Chiral Symmetry Breaking in QCD 7 The matrices t r are a basis of the fundamental representation of SU(N f ). The resulting charges ∫ Q r L = ∫ Q r R = d 3 xJ r L0(x, t) , (2.9) d 3 xJR0 r (x, t) , (2.10) fulfil the same commutation relations as the generators of the Lie algebra of SU L (N f ) × SU R (N f ). Under parity transformations they transform into each other, PQ L P −1 = Q R , which explains the name chiral transformations. The parity eigenstates are obtained by examining the diagonal subgroups SU L+R (N f ) and SU L−R (N f ) and the corresponding currents: V r µ (x) = Jr Rµ (x) + Jr Lµ (x) = Ψ f (x)γ µ t r fg 2 Ψg , (2.11) A r µ (x) = Jr Rµ (x) − Jr Lµ (x) = Ψ f (x)γ µ γ 5 t r fg 2 Ψg . (2.12) They transform as vector and an axial-vector respectively under parity transformations. Their time components fulfil the equal time commutation relations [V0 r s (x, t), V0 (y, t)] = ifrst V0 t (x, t)δ(3) (x − y) , (2.13) [V r 0 (x, t), A s 0(y, t)] = if rst A t 0(x, t)δ (3) (x − y) , (2.14) [A r 0 (x, t), As 0 (y, t)] = ifrst V t 0 (x, t)δ(3) (x − y) . (2.15) In these equations f rst are the antisymmetric structure constants of the Lie algebra of SU(N f ). The set of commutation relations is known as the current algebras. Since the left hand sides of the relations are quadratic in the currents and the right hand sides are linear, non-linear constraints for the currents follow. In particular their normalisation is determined by these equations. They also give rise (via the current algebra sum rules [IZ, CL, TW]) to relations between different cross sections of lepton-nucleon scattering, which are in good agreement with experiment. Explicit breaking of chiral symmetry is caused by the fermionic mass terms in the Lagrangian (2.1). Nevertheless chiral symmetry is a good approximation concerning the light quarks due to the fact that their current quark masses are small compared to the typical energy scale of strong interaction. For the strange quark the concept of an approximate
Page 1 and 2: The QCD Quark Propagator in Coulomb
Page 3 and 4: 3 Zusammenfassung In der vorliegend
Page 5 and 6: Contents 1 Prologue 1 2 Chiral Symm
Page 7 and 8: Chapter 1 Prologue Quantum Chromo D
Page 9 and 10: Chapter 1. Prologue 3 an ultraviole
Page 11: Chapter 2 Chiral Symmetry Breaking
Page 15 and 16: Chapter 2. Chiral Symmetry Breaking
Page 21 and 22: Chapter 3 Remarks on QCD in Coulomb
Page 23 and 24: 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
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Chapter 6. Nucleon Form Factors in
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Chapter 7 Epilogue In this thesis w
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Appendix A Units and Conventions In
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Appendix B Gauge potential, field s
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Appendix C. Numerical method employ
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Bibliography 99 [Alk88] Alkofer, Re
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Bibliography 101 [CMZ02] Cucchieri,
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Bibliography 103 [GO03] [GOZ05] [Gr
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Bibliography 105 [Mar04] Maris, Pie
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Bibliography 107 [PS] Peskin, Micha
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The QCD Quark Propagator in Coulomb Gauge and - Institut fÃ¼r Physik

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