The running coupling constants (in particular, the gravitational one) are studied in asymptotically free GUT’s and in finite GUT’s in curved spacetime, with explicit examples. The running gravitational coupling is used to calculate the leading quantum GUT corrections to the Newtonian potential, which turn out to be of logarithmic form in asymptotically free GUT’s. A comparison with the effective theory for the conformal factor, where leading quantum corrections to the Newtonian potential are again logarithmic, is made. A totally asymptotically free O(N) GUT with quantum higher derivative gravity is then constructed, using the technique of introducing renormalization group (RG) potentials in the space of couplings. RG equations for the cosmological and gravitational couplings in this theory are derived, and solved numerically, showing the influence of higher-derivative quantum gravity on the Newtonian potential. The RG-improved effective gravitational Lagrangian for asymptotically free massive GUT’s is calculated in the strong (almost constant) curvature regime, and the nonsingular de Sitter solution to the quantum-corrected gravitational equations is subsequently discussed. Finally, possible extensions of the results here obtained are briefly outlined.

• Received 20 March 1995

DOI:https://doi.org/10.1103/PhysRevD.52.2202