An illustrative model for non-perturbative scattering in theories with weak coupling

A simple two-dimensional model with one scalar field and a slightly metastable vacuum is used as a test ground to analyze non-perturbative multiparticle scattering at high energy induced by a non-trivial saddle point configuration in the euclidean path integral analogous to the instanton-induced multiparticle processes in the electroweak theory. Existence of an additional small parameter in the theory makes it possible to find the behavior of the scattering amplitudes up to the analog of the sphaleron energy. In particular it is illustrated that large quantum corrections arising in the existing instanton calculations of such amplitudes develop because the quantization is performed near the stationary configuration which is not distorted by the sources of external particles. It is explicitly shown that once the proper saddle point configuration in the presence of external sources is found the quantization around it yields only small corrections. Also the explicit solution shows what happens to some known first quantum effects when they are incorporated in the non-perturbative result.