Ultraviolet completion without symmetry restoration
We show that it is not possible to UV complete certain low-energy effective theories with spontaneously broken spacetime symmetries by embedding them into linear sigma models, that is, by adding "radial" modes and restoring the broken symmetries. When such a UV completion is not possible, one can still raise the cutoff up to arbitrarily higher energies by adding fields that transform nonlinearly under the broken symmetries, that is, new Goldstone bosons. However, this (partial) UV completion does not necessarily restore any of the broken symmetries. We illustrate this point by considering a concrete example in which a combination of spacetime and internal symmetries is broken down to a diagonal subgroup. Along the way, we clarify a recently proposed interpretation of inverse Higgs constraints as gauge-fixing conditions.