  It has been suggested that electroweak symmetry breaking in the Standard
Model may be natural if the Standard Model merges into a conformal field theory
(CFT) at short distances. In such a scenario the Higgs mass would be protected
from quantum corrections by the scale invariance of the CFT. In order for the
Standard Model to merge into a CFT at least one new ultraviolet (UV) scale is
required at which the couplings turn over from their usual Standard Model
running to the fixed point behavior. We argue that the Higgs mass is sensitive
to such a turn-over scale even if there are no associated massive particles and
the scale arises purely from dimensional transmutation. We demonstrate this
sensitivity to the turnover scale explicitly in toy models. Thus if scale
invariance is responsible for Higgs mass naturalness, then the transition to
CFT dynamics must occur near the TeV scale with observable consequences at
colliders. In addition, the UV fixed point theory in such a scenario must be
interacting because logarithmic running near a free fixed point constitutes
hard breaking of scale invariance and spoils the Higgs mass protection.
