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We develop a renormalization-group formalism for non-renormalizable theoriesand apply it to Einstein gravity theory coupled to a scalar field with theLagrangian $L=\sqrt{g} [R U(\phi)-{1/2} G(\phi) g^{\mu\nu} \partial_{\mu}\phi\partial_{\nu}\phi- V(\phi)]$, where $U(\phi), G(\phi)$ and $V(\phi)$ arearbitrary functions of the scalar field. We calculate the one-loop countertermsof this theory and obtain a system of renormalization-group equations inpartial derivatives for the functions $U, G$ and $V$ playing the role ofgeneralized charges which substitute for the usual charges in multichargetheories. In the limit of a large but slowly varying scalar field and smallspacetime curvature this system gives the asymptotic behaviour of thegeneralized charges compatible with the conventional choice of these functionsin quantum cosmological applications. It also demonstrates in theover-Planckian domain the existence of the Weyl-invariant phase of gravitytheory asymptotically free in gravitational and cosmological constants.Comment: 54 pages, preprint Alberta Thy-6-93, late

Renormalization group for nonrenormalizable theories: Einstein gravity with a scalar field