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

More documents

Recommendations

Info

34 3.6. The confinement scenario of Gribov and Zwanziger The confinement scenario is nowadays stated in the following way [AG06]. The Coulomb potential V C ∝ 〈M −1 [A](−∆)M −1 [A]〉 (3.94) is long range and thereby enhanced for small momenta because M −1 [A] is long range. This follows from entropy considerations: Since the dimension of configuration space is quite large, it is reasonable that most of the configurations are located close to the horizon (just as the volume measure r d−1 dr of a sphere in d dimensions is enhanced near the radius). This suggests that the near-zero eigenvalues of the Faddeev-Popov operator are dominating at large quark separations. As already mentioned these questions have been examined via lattice Monte Carlo simulations in Coulomb gauge, which found that V C (R) does indeed rise linearly [NS06]. Furthermore the infrared divergent Coulomb energy of an isolated charge comes about by the mechanism suggested by Gribov and Zwanziger, namely a large density of eigenvalues of the Faddeev-Popov operator near the zero eigenvalue [GOZ05]. There is also a connection to a different confinement picture. If centre vortices are removed from lattice configurations, the Coulomb energy is not confining any more and the Faddeev-Popov eigenvalue distribution resembles that of the abelian theory. Together with the fact that centre vortices are field configurations lying on the boundary ∂Λ of the fundamental modular region if they are gauge transformed to minimal Coulomb gauge, this suggests a close relationship between these two confinement scenarios. On the contrary it also turns out that the Coulomb string tension is roughly three times larger than the string tension of the Wilson potential, so the behaviour of the Coulomb interaction energy cannot be the whole story describing confinement. An interesting investigation (e.g. in [SK06]) is to try to construct physical states in Coulomb gauge, whose energy is below that of a quark-antiquark pair plus their Coulomb field by adding constituent gluons. As a quark and an antiquark separate, they might pull out between them a “chain” of constituent gluons, where each gluon in this chain is supposed to be bound to its nearest neighbours by Coulomb interaction. This picture has the name “gluon chain model” [GT02], and it encourages an imagination of the QCD flux tube as a kind of discretised string. Alternatively such a picture might arise also from a recent worldsheet formulation of gauge theory quantised in light-cone gauge [Tho02].
Chapter 4 The Quark Dyson-Schwinger Equation In this chapter we can use the insight we gained in the last one. The profit concerns mainly the parametrization of the gluon propagator. Its all-dominant part is the Coulomb potential, for which we employ an analytic ansatz. We start by deriving the quark Dyson- Schwinger equation with two different methods. We solve it numerically in two different truncations and compare the results. 4.1 From the QCD action to the gap equation In this section we derive the quark Dyson-Schwinger equation from the QCD action. Because this is not rigorously possible in Coulomb gauge, we choose for this demonstration a linear covariant gauge and Euclidean space-time 1 . The starting point is the renormalised Lagrangian [AS01], which reads ( −∂ 2 δ µν − L QCD = Z 3 1 2 Aa µ ( 1 Z 3 ξ − 1 ) ∂ µ ∂ ν ) A a ν + ˜Z 3 ¯c a ∂ 2 c a + ˜Z 1 gf abc ¯c a ∂ µ (A c µ cb ) − Z 1 gf abc (∂ µ A a ν ) Ab µ Ac ν (4.1) + Z 4 1 4 g2 f abe f cde A a µA b νA c µA d ν + Z 2 ¯q ( − ∂/ + Z m m ) q − Z 1F ig ¯qγ µ t a q A a µ . Employing the usual notation for the quark, gluon and ghost fields and using the notation ξ for the gauge parameter, this equation defines all multiplicative renormalisation constants. In the partiton function of course the source terms A a µ ja µ 1 Appropriately we use the conventions in A.2 . + ηq + qη + σc + cσ (4.2) 35
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 and 12: Chapter 2 Chiral Symmetry Breaking
Page 13 and 14: 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 39: Chapter 3. Remarks on QCD in Coulom
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 91 and 92:
Chapter 6. Nucleon Form Factors in
Page 93 and 94:
Page 95 and 96:
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
show all

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

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?