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We continue the study of the nonrelativistic short-distance completions of anaturally light Higgs, focusing on the interplay between the gauge symmetriesand the polynomial shift symmetries. We investigate the naturalness ofnonrelativistic scalar quantum electrodynamics with a dynamical criticalexponent $z=3$ by computing leading power law divergences to the scalarpropagator in this theory. We find that power law divergences exhibit a morerefined structure in theories that lack boost symmetries. Finally, in this toymodel, we show that it is possible to preserve a fairly large hierarchy betweenthe scalar mass and the high energy naturalness scale across 7 orders ofmagnitude, while accommodating a gauge coupling of order 0.1.

Nonrelativistic Yang-Mills Theory for a Naturally Light Higgs Boson