Non-Gaussian path integration in self-interacting scalar field theories

In self-interacting scalar field theories kinetic expansion is an alternative way of calculating the generating functional for Green’s functions where the zeroth order non-Gaussian path integral becomes diagonal in $x$-space and reduces to the product of an ordinary integral at each point which can be evaluated exactly. We discuss how to deal with such functional integrals and propose a new perturbative expansion scheme which combines the elements of the kinetic expansion with the usual perturbation theory techniques. It is then shown that, when the cutoff dependences of the bare parameters in the potential are chosen to have a well defined non-Gaussian path integral without the kinetic term, the theory becomes trivial in the continuum limit.

Figure 1

The first order correction to the propagator which arises from $ϵA_2{\phi }^4$ term in the potential (50). The loop integral, which is calculated with the propagator $\Delta$ in (53), diverges like ${\Lambda }^n$. This divergence is compensated by the $ϵ$ factor and the result becomes finite in the $\Lambda \to \infty$ limit.