Spontaneous violation of dilatation invariance in relativistic field theories

Adopting the point of view that the mass of observed particles is dynamical in origin, the effect of spontaneous breakdown of dilatation invariance is investigated. Neither extra fields nor anomalous dimensions are introduced from the outset so that the Lagrangians in our formalism are strictly invariant against dilatation transformation. The Bethe-Salpeter equation is employed throughout in order to demonstrate that, in the pair approximation, the appearance of massive fields does not contradict the formal invariance of the theory. The result is interpreted in terms of the dimensional transformation previously introduced by the authors. It is shown that the dilatation transformation for the Heisenberg operators turns into the dimensional transformation for the massive asymptotic fields and the inhomogeneous dilatation transformation for the Goldstone boson, and that the very appearance of the inhomogeneous term cancels the extra dimension brought in by the mass. A Goldberger-Treiman-like relation is derived. Finally, it is pointed out that the composite nature of the Goldstone boson in one model (Sec. II) seems to be the distinctive feature leading to a nontrivial scattering matrix.