Inverse scattering transform and instantons of four-dimensional Yukawa and PHI4 theories

The inverse scattering method is applied here to four-dimensional effective actions containing the logarithm of a functional Dirac determinant obtained by integrating over the fermions in a Yukawa model. Our method consists in replacing the bosonic fields, assumed spherically symmetric, by the scattering data of the radial Dirac equation. This gives a closed form for the determinant in terms of scattering data (SD). The full effective action takes a closed form in terms of these SD in two cases. Analytic solutions of the saddle-point equation are then derived. These solutions give information about the large-order behaviour of perturbative expansions. In the first case, the two coupling constants g(PHI4) and lambda(anti psipsiphi) are related through g=lambda2. Infinitely many instantons then exist, and are computed explicitly. They are shown to spoil the Borel summability of the perturbative expansion, for this bosonic effective action, in the unique coupling constant gsub(R). The second case corresponds to the pure bosonic coupling gsub(R)PHI4. In this case we reconstruct by IST the previously known instanton. (orig.)