Abstract
Unimodular Gravity is normally assumed to be equivalent to General Relativity for all matters but the character of the Cosmological Constant. Here we discuss this equivalence in the presence of a non-minimally coupled scalar field. We show that when we consider gravitation to be dynamical in a QFT sense, quantum corrections can distinguish both theories if the non-minimal coupling is non-vanishing. In order to show this, we construct a path integral formulation of Unimodular Gravity, fixing the complicated gauge invariance of the theory and computing all one-loop divergences. We find a combination of the couplings in the Lagrangian to which we can assign a physical meaning. It tells whether quantum gravitational phenomena can be ignored or not at a given energy scale. Its renormalization group flow differs depending on if it is computed in General Relativity or Unimodular Gravity.
Highlights
One of the most everlasting problems in theoretical physics is the Cosmological Constant problem [1, 2] — the question of why our Universe is currently evolving according to the presence of a very small cosmological constant, corresponding to MP2Λ ∼ 10−46 GeV4, where MP ∼ 1019 GeV is the Planck mass
If we think of an Effective Field Theory (EFT) setting, the cosmological constant receives contributions proportional to the cut-off of the theory, which encodes the ignorance about the UV degrees of freedom [5]
This hierarchy problem can be solved by the inclusion of a very fine-tuned counter-term, it raises a question about the sensitivity of low energy observables to high energy degrees of freedom and poses a problem for the viability of the EFT, where separation of scales is critical
Summary
One of the most everlasting problems in theoretical physics is the Cosmological Constant problem [1, 2] — the question of why our Universe is currently evolving according to the presence of a very small cosmological constant, corresponding to MP2Λ ∼ 10−46 GeV4, where MP ∼ 1019 GeV is the Planck mass. If we think of an Effective Field Theory (EFT) setting, the cosmological constant receives contributions proportional to the cut-off of the theory, which encodes the ignorance about the UV degrees of freedom [5] This means that in a gravitational theory described at low energies by General Relativity (GR), we expect corrections of the form δ(MP2Λ) ∼ MP4, which are clearly much larger than the observed value of the cosmological constant. The numerical results of both works differ, something which may be a gauge artefact, their physical conclusion is the same — the cosmological constant does not renormalize and UG is one-loop finite Since in both works the theory is taken in vacuum, it is not possible to have access to any physical observable in order to compare the dynamics of UG with that of GR. We add two small appendices describing the computation in GR — appendix A — and the discussion of divergences in UG in vacuum, in appendix B
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