  In the present paper, we revisit gravitational theories which are invariant
under TDiffs -- transverse (volume preserving) diffeomorphisms and global scale
transformations. It is known that these theories can be rewritten in an
equivalent diffeomorphism-invariant form with an action including an
integration constant (cosmological constant for the particular case of
non-scale-invariant unimodular gravity). The presence of this integration
constant, in general, breaks explicitly scale invariance and induces a runaway
potential for the (otherwise massless) dilaton, associated with the determinant
of the metric tensor. We show, however, that if the metric carries mass
dimension $\left[\text{GeV}\right]^{-2}$, the scale invariance of the system is
preserved, unlike the situation in theories in which the metric has mass
dimension different from $-2$. The dilaton remains massless and couples to
other fields only through derivatives, without any conflict with observations.
We observe that one can define a specific limit for fields and their
derivatives (in particular, the dilaton goes to zero, potentially related to
the small distance domain of the theory) in which the only singular terms in
the action correspond to the Higgs mass and the cosmological constant. We
speculate that the self-consistency of the theory may require the regularity of
the action, leading to the absence of the bare Higgs mass and cosmological
constant, whereas their small finite values may be generated by nonperturbative
effects.
