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Many years ago Weinberg formulated a definition of ``naturalness'' foreffective theories: if an effective theory is to make sense, coefficients mustnot change too much when the cutoff scale is changed by a factor of order 1. Asan example, we consider simple field theories in which an $O(N)$ symmetryspontaneously breaks to $O(N-1)$. We show that in these theories Weinberg'scriterion for a natural effective theory may be applied directly to the$S$-matrix; it implies that the scale of new physics, beyond the Goldstonebosons, may not be too large: there is always a particle or a cut of mass belowor about $4 \pi f / \sqrt{N}$. We discuss the range of convergence of theexpansion of the chiral Lagrangian. It appears to be impossible to construct anunderlying theory of the type considered here that fails to satisfy Weinberg'scriterion.Comment: 25 pages including figures in pictex. text uses harvmac. BUHEP-93-14,  HUTP-93/A01

Implications of naturalness in effective field theory on the masses of resonances