Longitudinal diffeomorphisms obstruct the protection of vacuum energy

Abstract

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.

Weyl relativity: a novel approach to Weyl's ideas

“…Therefore, while using a background volume element is certainly reminiscent of Weyl-transverse (or unimodular) gravity [5], the scope of the present construction is much larger from both conceptual and practical perspectives. A clear formal difference lies in the internal structure of the Ricci scalarR used to define the action (17), which displays a genuine gauge invariance due to the introduction of the covariant derivative with respect to A a .…”

“…(39). For fermion fields a similar definition exists for the vierbein field [5]. For instance, the action for a free massive scalar field would be given by…”

“…Moreover, we wonder whether these two theories could be made formally more similar to one another, for instance by sharing the same symmetry: conformal invariance (recall that the Einstein-Maxwell theory, though somewhat surprisingly due to the massless character of the excitations described, is not conformally invariant). This paper revises Weyl's ideas and their criticisms at the light of recent new developments by the authors [4,5] that are ultimately related to the theory known as unimodular gravity, or its extension Weyl-transverse gravity 2 [6][7][8][9][10][11][12][13][14][15][16][17]; see for recent developments related either with this theory or the core physical ideas behind it.…”

BeyondΛCDM: Problems, solutions, and the road ahead

“…2, so we will simply mention that some attempts to solve (or at least alleviate) it have resorted to modifying GR, e.g. IR modifications exhibiting degravitating solutions [169,86,170,171], or the presence of a Weyl symmetry [172,173,174]. The existence of black hole or big bang singularities indicate a failure of GR in the high-curvature regime.…”

Some solutions for one of the cosmological constant problems

“…The contributions to the vacuum energy from the quantum corrections from the matters are absorbed into a redefinition of a scalar field and the quantum corrections become irrelevant to the dynamics. We have also considered some extensions of this model, and we may construct models which may describe the realistic evolution of the universe (33). We also investigate the properties of the topological models which appear as simple models.…”