Analyticity, crossing symmetry, and the limits of chiral perturbation theory

The chiral Lagrangian for Goldstone-boson scattering is a power-series expansion in numbers of derivatives. Each successive term is suppressed by powers of a scale, Λ_χ, which must be less than of order 4πf√N where f is the Goldstone-boson decay constant and N is the number of flavors. The chiral expansion therefore breaks down at or below 4πf√N. Because of crossing symmetry, some "isospin" channels will deviate from their low-energy behavior well before they approach the scale at which their low-energy amplitudes would violate unitarity. The breakdown of the chiral expansion is associated with the appearance of physical states other than Goldstone bosons. We speculate that, since the bound on Λ_χ falls as N increases, the masses of resonances will decrease relative to f_π at least as fast as 1√N and argue that the estimates of "oblique" corrections from technicolor obtained by scaling from QCD are untrustworthy.