Self-Consistent, Nondegenerate Multiplets

The restrictions of self-consistency are investigated for two sets of interacting particles—vector and scalar (or pseudoscalar)—with unequal masses. Self-consistency is studied within a field-theoretic framework and within a bootstrap framework. It is assumed that solutions exist when the particles of a given set have equal masses and the coupling constants are proportional to the structure constants of $S{U}_3$. The unequal-mass case is studied by perturbing the equal-mass solutions and retaining only first-order terms in the mass and coupling-constant shifts. It is found that fully self-consistent solutions do not exist in either case, but it it is seen how such solutions can come about in the field-theoretic case. In the bootstrap analysis it is very difficult to understand how self-consistent solutions develop unless hidden identities are satisfied.