Gentles, Andrew James (1996) Nonperturbative propagators in axial gauge QCD. University of Southampton, Doctoral Thesis.
Abstract
This thesis describes a study of the nonperturbative quark propagator in axial gauge QCD within the context of a truncated set of Dyson-Schwinger equations. It is shown how the Ward-Takahashi identity determines the longitudinal component of the quark-gluon vertex and how the introduction of a transverse contribution is strongly constrained by forcing it to respect the requirements of multiplicative renormalisability. A condition is derived which defines a critical value of the strong coupling, αc, marking the breakdown of chiral symmetry.
A specific form of the gluon propagator, which is softened in the infrared limit, is employed in a numerical study of the chiral symmetric phase of this model of QCD for a variety of choices for the arbitrary component of the transverse quark-gluon vertex when the angular integrations inherent in the DSE are performed by an approximate analytic method or a direct numerical one. Critical couplings determined in these schemes show surprisingly little variation. The dependence of αc on the infrared behaviour of the gluon propagator is also investigated and it is found that the critical coupling decreases as the propagator becomes more singular.
The renormalisation of the massive phase is considered in some detail, revealing that in a momentum cutoff with finite regulator Λ2, it is inconsistent to simultaneously set both the bare and renormalised quark masses to zero. An explicit set of renormalisation conditions is derived which are subsequently applied to the cases of explicit and dynamical chiral symmetry breaking. Solutions are recovered for super-critical couplings and their stability is examined through the Cornwall-Jackiw-Tomboulis effective action.
Coupled Dyson-Schwinger-Bethe-Salpeter equation phenomenology is then employed to find an expression for the weak pion decay constant fπ in terms of the functions appearing in the dressed quark propagator. This is used to predict the value of the strong coupling at the renormalisation point μ = 0.8 GeV, which is below the energy scale currently amenable to lattice gauge theory simulations. It is indicated into what areas the technique can be extended and a brief summary of covariant gauge studies is presented.
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