My understanding of quantum field theory is that an interacting two-point function of spin-0 bosons will have the form: $ \frac{i}{p^2-m_0^2-\Sigma(p)}$, where $\Sigma(p)$ is the self-energy, the sum of 1-particle irreducible diagrams. In general, the self-energy can be complex, with the real part providing a renormalization of the physical mass of the particle, and the imaginary part giving a finite lifetime or decay rate to the particle.
My question is if it is possible in general to tell if a given action will give rise to an imaginary component of the self-energy without actually computing loop diagrams. For example, would it be possible to tell that $\phi^4$ theory does or does not have an imaginary self-energy component from a priori reasoning? Moreover, could you tell if a given diagram would have an imaginary part without computing it explicitly?
Another answer on stack exchange suggested that complex self-energies arise from effective field theories, but I was unsure why that would be the case.
