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The renormalization group for non-renormalizable theories: Einstein gravity with a scalar field

We develop a renormalization-group formalism for non-renormalizable theories and apply it to Einstein gravity theory coupled to a scalar field with the Lagrangian $L=\sqrt{g} [R U(ϕ)-{1/2} G(ϕ) g^{μν} \partial_μϕ\partial_νϕ- V(ϕ)]$, where $U(ϕ), G(ϕ)$ and $V(ϕ)$ are arbitrary functions of the scalar field. We calculate the one-loop counterterms of this theory and obtain a system of renormalization-group equations in partial derivatives for the functions $U, G$ and $V$ playing the role of generalized charges which substitute for the usual charges in multicharge theories. In the limit of a large but slowly varying scalar field and small spacetime curvature this system gives the asymptotic behaviour of the generalized charges compatible with the conventional choice of these functions in quantum cosmological applications. It also demonstrates in the over-Planckian domain the existence of the Weyl-invariant phase of gravity theory asymptotically free in gravitational and cosmological constants.