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 $\frac{4\pi f}{\sqrt{N}}$ where $f$ is the Goldstone-boson decay constant and $N$ is the number of flavors. The chiral expansion therefore breaks down at or below $\frac{4\pi f}{\sqrt{N}}$. 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 $\frac{1}{\sqrt{N}}$ and argue that the estimates of "oblique" corrections from technicolor obtained by scaling from QCD are untrustworthy.