  We report on the existence and phenomenology of type II critical collapse
within the one-parameter family of SU(2) $\sigma$-models coupled to gravity.
Numerical investigations in spherical symmetry show discretely self-similar
(DSS) behavior at the threshold of black hole formation for values of the
dimensionless coupling constant $\ccbeta$ ranging from 0.2 to 100; at 0.18 we
see small deviations from DSS. While the echoing period $\Delta$ of the
critical solution rises sharply towards the lower limit of this range, the
characteristic mass scaling has a critical exponent $\gamma$ which is almost
independent of $\ccbeta$, asymptoting to $0.1185 \pm 0.0005$ at large
$\ccbeta$. We also find critical scaling of the scalar curvature for
near-critical initial data. Our numerical results are based on an
outgoing-null-cone formulation of the Einstein-matter equations, specialized to
spherical symmetry. Our numerically computed initial-data critical parameters
$p^*$ show 2nd order convergence with the grid resolution, and after
compensating for this variation in $p^*$, our individual evolutions are
uniformly 2nd order convergent even very close to criticality.
