Approximate solutions in the model $\mathscr L_{\mathrm{int}}=h^2\psi^2\varphi^2$ and equations for Green's functions on paths

The introduction of auxiliary fields $A_i(x)$ ($i=1,2$) reduces the solution of the mode with $\mathscr L_{\mathrm{int}}=h^2\psi^2\varphi^2$ to the finding of solutions in the theory with the interaction $\mathscr L_{\mathrm{int}}=-h\psi^2(x)A_1(x)-h\varphi^2(x)A_2(x)$ and subsequent functional averaging over the fields $A_i(x)$. In the framework of the approximation that enables one to allow partly for the contributions from the vacuum polarization in the model $-h\varphi^2(x)A_2(x)$, the corresponding solutions in the theory $h^2\psi^2\varphi^2$ are investigated for the Green's functions and scattering amplitudes.