The soliton of the effective chiral action in the two-point approximation
In this thesis, we study the "two-point approximation" for highly non-local effective actions, in the particular case of the Chiral Soliton Model of the nucleon. The nucleon in this model is regarded as being made of three valence quarks bound together by a meson field in a soliton form. Mesons are treated in mean field theory and the vacuum energy due to one-quark loops is included. The theory is defined with a finite cut-off in momentum space, consistent with an effective theory for the low-energy description of the strong interactions. We use the two-point approximation to calculate the vacuum correction to the chiral soliton energy for a variety of soliton profile functions, investigating the effect of different regularisation schemes. Results are little influenced by the choice of the cut-off, and are within 20% of exact calculations, done with the full inclusion of the Dirac sea. We then perform a dynamical calculation of the chiral soliton by including sea-quark effects self-consistently in the two-point approximation. We find a typical 20% (or less) deviation in the soliton energy from exact calculations. We apply a further "pole" approximation which leads to a significant algebraic simplification in the self-consistent equations. We show, in particular, that a simple numerical fit of the pole form to the two-point cut-off function yields essentially indistinguishable results from the latter. We finally calculate some static nucleon observables in the two-point approximation and find general agreement with exact calculations. In view of the results obtained, we may hope that the pole form of the twopoint approximation may prove to be a generally useful approach to similar problems involving highly non-local actions.