Collective phenomena inλφ4field theory treated in the random-phase approximation

We investigate $\lambda {\phi }^4$ theory in the random-phase approximation as a prototypic model for understanding the formation of bound states (here 2-meson bound states) and the nature of the effective interaction between such states. We give the random-phase approximation a functional-derivative interpretation which allows us to determine the effective expansion parameter of the random-phase approximation to be $\left(\frac{{{\lambda }_R}^2}{N}\right)D_R\left(q^2\right)$, where $D_R\left(q^2\right)$ is the propagator of the collective mode. The field theory in this approximation has both mass and coupling-constant renormalization, and we derive expressions for unrenormalized and renormalized 2-, 4-, and 6-point functions for the original scalar field, and determine the 4-point function for bound-state—bound-state scattering as well as the effective coupling between bound states. We show the relation between the randomphase approximation and the $\mathrm{O}\mathrm{}\left(N\right)\mathrm{}$ model for large $N$ and prove that without single-field symmetry breaking there is a range of renormalized coupling constant where there is no ghost.