Abstract
We develop a generally applicable method for constructing functions, $C$, which have properties similar to Zamolodchikov's $C$-function, and are geometrically natural objects related to the theory space explored by non-perturbative functional renormalization group (RG) equations. Employing the Euclidean framework of the Effective Average Action (EAA), we propose a $C$-function which can be defined for arbitrary systems of gravitational, Yang-Mills, ghost, and bosonic matter fields, and in any number of spacetime dimensions. It becomes stationary both at critical points and in classical regimes, and decreases monotonically along RG trajectories provided the breaking of the split-symmetry which relates background and quantum fields is sufficiently weak. Within the Asymptotic Safety approach we test the proposal for Quantum Einstein Gravity in $d>2$ dimensions, performing detailed numerical investigations in $d=4$. We find that the bi-metric Einstein-Hilbert truncation of theory space introduced recently is general enough to yield perfect monotonicity along the RG trajectories, while its more familiar single-metric analog fails to achieve this behavior which we expect on general grounds. Investigating generalized crossover trajectories connecting a fixed point in the ultraviolet to a classical regime with positive cosmological constant in the infrared, the $C$-function is shown to depend on the choice of the gravitational instanton which constitutes the background spacetime. For de Sitter space in 4 dimensions, the Bekenstein-Hawking entropy is found to play a role analogous to the central charge in conformal field theory. We also comment on the idea of a `$\Lambda$-$N$ connection' and the `$N$-bound' discussed earlier.
Highlights
On the similarity of C to the thermodynamical free energy [6], leading to a conjectural ‘F -theorem’ which states that, under certain conditions, the finite part of the free energy of 3-dimensional field theories on S3 decreases along renormalization group (RG) trajectories and is stationary at criticality [7]
We develop a generally applicable method for constructing functions, C, which have properties similar to Zamolodchikov’s C-function, and are geometrically natural objects related to the theory space explored by non-perturbative functional renormalization group (RG) equations
To sum it up we can say that the expected monotonicity of Ck arises under the following conditions: first, the bi-metric version of the Einstein-Hilbert truncation is used, and second, the underlying RG trajectory is split-symmetry restoring
Summary
We develop a generally applicable framework for constructing functions Ck which have properties similar to a C-function, and at the same time are ‘geometrically natural’ objects from the perspective of the theory space explored by the EAA. It would be instructive to know if there exist special backgrounds in which the fluctuations are ‘tame’ such that, for vanishing external source, they amount to only small oscillations about a stable equilibrium, with a vanishing mean: φ ≡ φ = 0 Such distinguished backgrounds Φ ≡ Φsc are referred to as self-consistent (sc) since, if we pick one of those, the expectation value of the field Φ = Φ = Φdoes not get changed by any violent φ-excitations that, generically, can shift the point of equilibrium. Later on in the applications this trivial observation has the important consequence that self-consistent background field configurations Φskc(x) can contain only running coupling constants of level p = 1, that is, the couplings parameterizing the functional Γ1k which is linear in φ.3. The EAA, written as Γk[Φ, Φ ], satisfies the following exact functional equation which governs the ‘extra’ background dependence it has over and above the one which combines with the fluctuations to form the full field Φ:. For a discussion of the modified BRS-Ward identity enjoyed by the EAA we refer to [42] and [17]
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