Dressing the nucleon propagator

The nucleon propagator in the "nested-bubbles" approximation is analyzed. The approximation is built from the minimal set of diagrams which is needed to maintain the unitarity condition under the two-pion production threshold in the two-nucleon Bethe-Salpeter equation. Recursive formulas for subsets of "nested-bubbles" diagrams calculated in the framework of the pseudoscalar interaction are obtained by the use of dispersion relations. We prove that the sum of all the nested bubbles diverges. Moreover, the successive iterations are plagued with ghost poles. We prove that the first approximation-which is the so-called chain approximation-has ghost poles for any nonvanishing coupling constant. In an earlier paper we have shown that ghost poles lead to ghost cuts. These cuts are present in the nested bubbles. Ghost elimination procedures are discussed. Modifications of the nested-bubbles approximation are introduced in order to obtain convergence and in order to eliminate the ghost poles and ghost cuts. In a similar way as in the Lee model, cutoff functions are introduced in order to eliminate the ghost poles. The necessary and sufficient conditions for the absence of ghost poles are formulated and analyzed. The spectral functions of the modified nested bubbles are analyzed and computed. Finally we present a theorem, similar in its form to Levinson's theorem in scattering theory, which enables one to compute in a simple way the number of ghost poles.