  A new conformally invariant energy for four-dimensional hypersurfaces is
devised. It renders possible the study of a large class of curvature energies,
and we show that their critical points are smooth. As corollaries, we obtain
the regularity of the critical points of the four-dimensional analogues of the
Willmore energy, of the $Q$-curvature energy, but also that Bach-flat
hypersurfaces are smooth, along with relevant estimates.
