Renormalization of interacting scalar field theory in three-dimensional curved spacetime

The renormalizability of a real scalar field with a polynomial self-interaction of degree six on a general curved three-dimensional background spacetime is considered. The method used is the background-field expansion of the effective action. We show that the theory is finite (after regularization) at one- and three-loop order. Complete counterterms are found at two-loop order. A partial analysis at four-loop order is presented to analyze the gravitationally dependent counterterms. A previous paper dealing with a special case of the theory considered here is shown to be incorrect