A four-dimensional scalar field theory with quartic and of higher-power interactions suffers the triviality issue at the quantum level. This is due to coupling constants that, contrary to the physical expectations, seem to grow without a bound with energy. Since this problem concerns the high- energy domain, interaction with a quantum gravitational field may provide a natural solution to it. In this paper we address this problem considering a scalar field theory with a general analytic potential having ${\mathbb{Z}}_2$ symmetry and interacting with a quantum gravitational field. The dynamics of the latter is governed by the cosmological constant and the Einstein-Hilbert term, both being the lowest and next-to-lowest terms of the effective theory of quantum gravity. Using the Vilkovisky-DeWitt method we calculate the one-loop correction to the scalar field effective action. We also derive the gauge-independent one-loop beta functions for all the scalar field couplings in the minimal subtraction scheme. We find that the leading gravitational corrections act in the direction of asymptotic freedom. Moreover, assuming both the Newton and cosmological constants have nonzero fixed point values, we find asymptotically free Halpern-Huang potentials.

• Received 1 October 2012

DOI:https://doi.org/10.1103/PhysRevD.87.024026