Critical phenomena in gravitational collapse with two competing massless matter fields

In the gravitational collapse of matter beyond spherical symmetry, gravitational waves are necessarily present. On the other hand, gravitational waves can collapse to a black hole even without matter. One might therefore wonder whether the interaction and competition between the matter fields and gravitational waves affect critical phenomena at the threshold of black hole formation. For a toy model for this, we study type II critical collapse with two matter fields in spherical symmetry, namely a scalar field and a Yang-Mills field. On their own, both display discrete self-similarity in type II critical collapse, and we can take either one of them as a toy model for gravitational waves. To our surprise, in numerical time evolutions, we find that, for sufficiently good fine-tuning, the scalar field always dominates on sufficiently small scales. We explain our results by the conjectured existence of a ``quasidiscretely self-similar'' (QSS) solution shared by the two fields, equal to the known Yang-Mills critical solution at infinitely large scales and the known scalar field critical solution (the Choptuik solution) at infinitely small scales, with a gradual transition from one field to the other. This QSS solution itself has only one unstable mode and so acts as the critical solution for any mixture of scalar field and Yang-Mills initial data.