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 $4\pi f/\sqrt{N}$ where $f$ is the Goldstone boson decay constant and $N$ is the number of flavors. The chiral expansion therefore breaks down at or below $4 \pi f/\sqrt{N}$. We argue that the breakdown of the chiral expansion is associated with the appearance of physical states other than Goldstone bosons. 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. We argue that the estimates of ``oblique'' corrections from technicolor obtained by scaling from QCD are untrustworthy.