Abstract
We derive the full set of beta functions for the marginal essential couplings of projectable Ho\ifmmode \check{r}\else \v{r}\fi{}ava gravity in ($3+1$)-dimensional spacetime. To this end we compute the divergent part of the one-loop effective action in static background with an arbitrary spatial metric. The computation is done in several steps: reduction of the problem to three dimensions, extraction of an operator square root from the spatial part of the fluctuation operator, and evaluation of its trace using the method of universal functional traces. This provides us with the renormalization of couplings in the potential part of the action which we combine with the results for the kinetic part obtained previously. The calculation uses symbolic computer algebra and is performed in four different gauges yielding identical results for the essential beta functions. We additionally check the calculation by evaluating the effective action on a special background with spherical spatial slices using an alternative method of spectral summation. We conclude with a preliminary discussion of the properties of the beta functions and the resulting renormalization group flow, identifying several candidate asymptotically free fixed points.
Highlights
A theory whose bare action does not contain any irrelevant operators under this scaling is power-counting renormalizable, implying that it has fair chances to be perturbatively renormalizable in the strict sense; i.e., all divergences generated within perturbation theory can be absorbed into redefinition of the couplings in the action
In this paper we have obtained the full set of one-loop β-functions for marginal essential coupling constants in projectable Horava gravity (HG)
The results underwent a number of very powerful checks that confirm gauge independence of these beta functions in a wide set of gauge conditions—the cornerstone of the physically invariant content of quantum gauge theories
Summary
The Lagrangian is built out of all local FDiff-invariant operators that can be constructed from these fields and have dimension less than or equal to 2d The latter bound comes from the scaling dimension of the spacetime integration measure 1⁄2dtddx 1⁄4 −2d and ensures that the terms in the action corresponding to relevant (1⁄2O < 2d) and marginal (1⁄2O 1⁄4 2d) operators have nonpositive dimensions. The instability can be eliminated by tuning η 1⁄4 0 or by expanding around a curved vacuum In both cases the theory does not appear to reproduce GR in the low-energy limit, as there is no regime where the dispersion relation of the tt-graviton would have the relativistic form ω2tt ∝ k2. The purpose of this work is to provide β-functions of the remaining couplings in the list (1.12)
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