Abstract
We discuss the nature of quantum field theories involving gravity that are classically scale-invariant. We show that gravitational radiative corrections are crucial in the determination of the nature of the vacuum state in such theories, which are renormalisable, technically natural, and can be asymptotically free in all dimensionless couplings. In the pure gravity case, we discuss the role of the Gauss-Bonnet term, and we find that Dimensional Transmutation (DT) \`a la Coleman-Weinberg leads to extrema of the effective action corresponding to nonzero values of the curvature, but such that these extrema are local maxima. In even the simplest extension of the theory to include scalar fields, we show that the same phenomenon can lead to extrema that are local minima of the effective action, with both non-zero curvature and non-zero scalar vacuum expectation values, leading to spontaneous generation of the Planck mass. Although we find an asymptotically free (AF) fixed point exists, unfortunately, no running of the couplings connect the region of DT to the basin of attraction of the AF fixed point. We also find there remains a flat direction for one of the conformal modes. We suggest that in more realistic models AF and DT could be compatible, and that the same scalar vacuum expectation values could be responsible both for DT and for spontaneous breaking of a Grand Unified gauge group.
Highlights
In principle, such models may provide an ultraviolet (UV) completion of Einstein gravity
In the pure gravity case, we discuss the role of the Gauss-Bonnet term, and we find that Dimensional Transmutation (DT) `a la Coleman-Weinberg leads to extrema of the effective action corresponding to nonzero values of the curvature, but such that these extrema are local maxima
We have presented a number of new formal results for classically scaleinvariant, renormalizable gravity models
Summary
Because we are interested in classically scale invariant theories in four dimensions, the action for pure gravity will contain the quadratic invariants given in eq (1.1) this is not the action that has been the starting point for analyses of this theory [1,2,3,4]. Any definition for continuous n that reduces to this linear combination as n → 4 should suffice This enables the definition of renormalized operators and couplings in four dimensions, at which point, one may rewrite G = R∗R∗ = ∇μBμ locally, using the special properties of the curvature tensor in four dimensions, such as the Bianchi identities. It represents a generalization to the quantized R2-gravity case of the candidate a-function proposed by Cardy [25] as manifesting a 4dimensional c-theorem Results for this anomaly coefficient (without quantizing gravity) include a non-zero 5-loop contribution involving four quartic scalar couplings [26] and nonzero three loop contributions involving gauge and Yukawa couplings [27].
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