Distinction between Composite and Elementary Particles in a Model Field Theory

The Zachariasen model field theory is examined with a view to understanding the behavior of field theory with a nontrivial vertex function in the limit of vanishing renormalization constants $Z_1$ and $Z_3$. Two formulations are given for the model, which deals with two scalar relativistic fields $A$ and $B$ with allowed interactions $A\to B+\overline{B}$ and $B+\overline{B}\to B+\overline{B}$. The model is defined: (1) by specialization of spectral representations for the $A$-particle propagator and $\overline{B}\mathrm{BA}$ vertex functions to allow only elastic unitarity and no crossing in $\overline{B}B$ scattering amplitude; (2) in terms of selected ($\overline{B}B$ chain) diagrams generated by the Fermi interaction ${\lambda }_0\overline{B}B\overline{B}B$ and the Yukawa interaction $g_0\overline{B}\mathrm{BA}$. In both formulations, expressions are derived for the renormalization constants $Z_1$, $Z_3$, and $\delta {\mu }^2$ which generalize the work of earlier authors, by allowing poles in the vertex function and inverse propagator, and also allow a complete treatment of the combined (Fermi and Yukawa) interactions. The Zachariasen model is found to be such that when $Z_3\to 0$: (1) $Z_1\to 0$ and $\frac{Z_1^{}{}_{}{}^2}{{\mathrm{Z}}_3}\to 0$, $g_0^{}{}_{}{}^2\to 0$, and $\delta {\mu }^2$ is finite; (2) the poles of the scattering amplitude are no longer propagator poles (elementary particles) but are vertex poles (bound states); (3) the theory is found to develop a redundant zero. The behavior is strikingly different from the earlier models considered, and the significance of this is pointed out. Bounds are obtained on the renormalized coupling constant despite poles of the vertex function, and the distinction between a $Z_3\ne 0$ elementary particle theory and a $Z_3=0\mathrm{}$ elementary particle or a bound state is examined with reference to Levinson's theorem on the high-energy behavior of phase shifts.