  To guarantee the stability of the cosmological constant sector against
radiative corrections coming from quantum matter fields, one of the most
natural ingredients to invoke is the symmetry under scale transformations of
the gravitational field. Previous attempts to follow this path have
nevertheless failed in providing a consistent picture. Here we point out that
this failure is intimately tied up to an assumption which is typically embedded
in modern studies of the gravitational interaction: invariance under the full
group of diffeomorphisms. We base the discussion on the gravitational theory
known as Weyl transverse gravity. While leading to the same classical solutions
as general relativity, and so to the same classical phenomenology, we show that
in the presence of quantum matter (i) the degeneracy between these theories is
broken: general relativity exhibits the well-known cosmological constant
problem while in Weyl transverse gravity the cosmological constant sector is
protected due to gravitational scale invariance, and (ii) this is possible as
the result of abandoning the assumption of full diffeomorphism invariance,
which permits to circumvent classic results on scale-invariance anomalies and
guarantees that gravitational scale invariance survives quantum corrections.
Both results signal new directions in the quest of finding an ultraviolet
completion of gravity.
