  Previous results on fermion chirality-flipping four-point functions are
extended to $SU(N)$ gauge theories. The problem is purely non-perturbative, and
it is approached by truncating the Schwinger-Dyson hierarchy. The large-$N$
limit also simplifies the problem substantially. The resulting equation is
solved numerically by relaxation techniques and an estimate of the critical
coupling and momentum behavior is obtained. We also comment on the behavior of
chirality-flipping $2n$-point functions for general $n$.
