Effective Lagrangian and the back-reaction problem in a self-interacting $O(N)$ scalar theory in curved spacetime

A derivation of the one-loop effective Lagrangian in the self-interacting $O(N)$ scalar theory, in slowly varying gravitational fields, is presented (using $\zeta$-regularization and heat-kernel techniques). The result is given in terms of the expansion in powers of the curvature tensors (up to quadratic terms) and their derivatives, as well as in derivatives of the background scalar field (up to second derivatives). The renormalization group improved effective Lagrangian is studied, what gives the leading-log approach of the whole perturbation theory. An analysis of the effective equations (back-reaction problem) on the static hyperbolic spacetime $\reals^2 \times H^2/\Gamma$ is carried out for the simplest version of the theory: $m^2=0$ and $N=1$. The existence of the solution $\reals^2 \times H^2/\Gamma$, induced by purely quantum effects, is shown.