Semiclassical and inverse scattering methods, previously restricted to two-dimensional models, are extended to four-dimensional scalar field theories. The necessary trace identities are derived, mode sums are carried out, and renormalization is explicitly demonstrated. The four-dimensional $\mathrm{O}\mathrm{}\left(N\right)\mathrm{}$ model is used to illustrate these techniques. A renormalized expression for bound-state energies of this model, parametrized by the scattering data of a three-dimensional Schrödinger equation, is obtained. This expression contains nontrivial quantum fluctuation effects, and corresponds to the leading-order term of a $\frac{1}{N}$ expansion. In this model, stable $n$-body bound states do not seem to appear, even though a two-body bound state has been found in a $\frac{1}{N}$ expansion of the Green's functions. Possible reasons for this failure to bind are discussed.

• Received 10 May 1976

DOI:https://doi.org/10.1103/PhysRevD.14.1977