Ultraviolet behavior of four-dimensional σ models coupled to gravity

In a nonlinear σ model coupled to gravity we compute all one-loop divergences including contributions from both virtual gravitons and scalars ${\mathrm{\phi }}^{\mathit{i}}$ which are valued on an internal manifold M. The calculation is performed in arbitrary curved backgrounds and it is useful not only for studying the renormalization properties of this particular gravity theory but also for constructing the effective σ-model Lagrangian incorporating the dominant one-loop effects near the Planck scale. The latter may be of relevance to low-energy effective theories inspired by some string models. We find that the geometry of the manifold M plays no important role in improving the ultraviolet behavior of the model. As regards the σ-model effective-Lagrangian calculation we have found that a large portion of the dominant radiative effects can be absorbed by a redefinition of the four-dimensional metric ${\mathit{g}}_{\mathrm{\mu }\nu }$, the rest inducing renormalizations of the manifold metric ${\mathit{scrG}}_{\mathit{i}\mathit{j}}$ and the potential V not depending on the four-dimensional geometry. Up to terms logarithmic in the momentum cutoff scale Λ having the structure (${\mathrm{\partial }}_{\mathrm{\Gamma }}$φ${)}^4$, (${\mathit{scrD}}_{\mathrm{\mu }}$ ${\mathit{scrD}}_{\nu }$φ${)}^2$ the resulting one-loop effective theory is of the same form as the tree-level Lagrangian with the renormalized metric ${\mathit{scrG}}_{\mathit{i}\mathit{j}}$ and potential V.