Caustics in the Grassmann Integral

It is shown that a simple model of 2N-Grassmann variables with a four-body coupling involves caustics after the integral is converted to a bosonic form with the aid of an auxiliary field. In order to check the validity of the auxiliary field method (AFM), approximations in terms of 1/N near a saddle point and 1/ ³sqrt{N} near a caustic, are carried out. It turns out that even for N=2, the smallest value of N for which a four-body interaction exists, AFM is increasingly accurate as higher order effects are taken into account.