Equivalence Between Four-Fermion and Yukawa Coupling, and theZ3=0Condition for Composite Bosons

The principal objective of the present work is the derivation of conditions for equivalence (A) between a four-Fermi theory with pseudoscalar coupling and a Yukawa theory with pseudoscalar bosons, and (B) in the corresponding vector case. In general, two conditions are found to be both necessary and sufficient. The first is a condition for the existence of an appropriate boson bound state in the four-Fermi theory. The second condition is that the boson wave function renormalization constant $Z_3$ in the Yukawa theory be equal to zero. We first derive our results by consideration of fermion-fermion scattering in the chain approximation, and proceed afterwards to prove them valid to all orders in perturbation theory. We also discuss the degenerate vacuum theories of Nambu and Jona-Lasinio and of Bjorken in the chain approximation. In each of these four-Fermi theories, the existence of boson bound states (massless pseudoscalar and massive scalar bosons in the former case, massless vector bosons in the latter) follows automatically from self-consistency conditions. Hence to have equivalence to Yukawa theories in which the bosons are described by elementary fields, we need only impose the $Z_3=0\mathrm{}$ conditions on the Yukawa theories. Finally, we comment on Birula's theory of quantum electrodynamics without electromagnetic field. It is to be stressed that we deal throughout with full-scale relativistic field theories of physical consequence.