Abstract
In a planar approximation to a Yukawa-type g psi * psi phi field theory with scalar fields psi and phi the author studies the Bethe-Salpeter (BS) equation for the scattering amplitude of the psi field in the case of vanishing psi wavefunction renormalisation constant Z2=O. Due to the asymptotic behaviour of the non-canonical Psi propagator, given by the corresponding Dyson-Schwinger equation for Z2=O, the Neumann series of the BS equation diverges for Euclidean values of the invariants and all masses m2, mu 2>O. Being responsible for this divergence, only the asymptotic part of the propagator is subsequently retained in the BS equation. Using in the Euclidean metric an exactly soluble high-energy version of the BS equation and treating the difference as a perturbation, he derives a new but equivalent integral equation for the scattering amplitude. By contraction-mapping arguments he obtains existence and multiplicity results for solutions of this transformed equation. The asymptotic behaviour of these solutions is rigorously established and found to be oscillating.
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