Perturbative evaluation of interacting scalar fields on a curved manifold with boundary

The effects of quantum corrections to a conformally invariant scalar field theory on a curved manifold of positive constant curvature with boundary are considered in the context of a renormalization procedure. The renormalization of the theory to second order in the scalar self-coupling pursued herein involves explicit calculations of up to third loop-order and reveals that, in addition to the renormalization of the scalar self-coupling and scalar field, the removal of all divergences necessitates the introduction of conformally non-invariant counterterms proportional to R(PHI)2 and K(PHI)2 in the bare scalar action as well as counterterms proportional to RK2, R2 and RK in the gravitational action. The substantial backreaction effects and their relevance to the renormalization procedure are analysed.