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We report on the existence and phenomenology of type II critical collapsewithin 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 thedimensionless coupling constant $\ccbeta$ ranging from 0.2 to 100; at 0.18 wesee small deviations from DSS. While the echoing period $\Delta$ of thecritical solution rises sharply towards the lower limit of this range, thecharacteristic mass scaling has a critical exponent $\gamma$ which is almostindependent of $\ccbeta$, asymptoting to $0.1185 \pm 0.0005$ at large$\ccbeta$. We also find critical scaling of the scalar curvature fornear-critical initial data. Our numerical results are based on anoutgoing-null-cone formulation of the Einstein-matter equations, specialized tospherical symmetry. Our numerically computed initial-data critical parameters$p^*$ show 2nd order convergence with the grid resolution, and aftercompensating for this variation in $p^*$, our individual evolutions areuniformly 2nd order convergent even very close to criticality.Comment: 23 pages, includes 10 postscript figure files, uses REVTeX, epsf,  psfrag, and AMS math fonts (amstex + amssymb); to appear in PRD15. Summary of  revisions from v2: fix wrong formula in figure 6 caption and y-axis label,  also minor wording changes and update publication status of refs 5-

physical review. d. particles, fields, gravitation, and cosmology/physical review. d. particles and fields

Type II critical collapse of a self-gravitating nonlinear<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>σ</mml:mi></mml:math>model

Type II critical collapse of a self-gravitating nonlinearσmodel