  The valley method for computing the total high-energy anomalous cross section
$S_{anom}$ is the extension of the optical theorem to the case of
instanton-antiinstanton backgrounds. As a toy model for baryon number violation
in Electroweak theory, we consider a version of the $O(3)$ $\sigma$ model in
which the conformal invariance is broken perturbatively. We show that at a
critical energy the saddle-point values of the instanton size and
instanton-antiinstanton separation bifurcate into complex conjugate pairs. This
nonanalytic behavior signals the breakdown of the valley method at an energy
where $S_{anom}$ is still exponentially suppressed.
  (Figures replaced 5/3/93).
