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, LAMBDA(chi), which must be less than of order 4pif/square-root 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 4pif/square-root 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 LAMBDA(chi) falls as N increases, the masses of resonances will decrease relative to f(pi), at least as fast as 1/square-root N and argue that the estimates of ''oblique'' corrections from technicolor obtained by scaling from QCD are untrustworthy.