The phase of scalar field wormholes at one loop in the path integral formulation for Euclidean quantum gravity

We here calculate the one-loop approximation to the Euclidean Quantum Gravity coupled to a scalar field around the classical Carlini and Miji\'c wormhole solutions. The main result is that the Euclidean partition functional $Z_{EQG}$ in the ``little wormhole'' limit is real. Extension of the CM solutions with the inclusion of a bare cosmological constant to the case of a sphere $S^4$ can lead to the elimination of the destabilizing effects of the scalar modes of gravity against those of the matter. In particular, in the asymptotic region of a large 4-sphere, we recover the Coleman's $\exp \left (\exp \left ({1\over \lambda_{eff}}\right )\right )$ peak at the effective cosmological constant $\lambda_{eff}=0$, with no phase ambiguities in $Z_{EQG}$.