Longitudinal diffeomorphisms obstruct the protection of vacuum energy

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.