Dimensional renormalization of scalar field theory in curved space-time

The regularization and renormalization of an interacting scalar field φ in a curved spacetime background is performed by the method of continuation to n dimensions. In addition to the familiar counter terms of the flat-space theory, c-number, “vacuum” counter terms must also be introduced. These involve zero, first, and second powers of the Reimann curvature tensor R_αβψδ. Moreover, the renormalizability of the theory requires that the Lagrange function couple φ² to the curvature scalar R with a coupling constant η. The coupling η must obey an inhomogeneous renormalization group equation, but otherwise it is an arbitrary, free parameter. All the counter terms obey renormalization group equations which determine the complete structure of these quantities in terms of the residues of their simple poles in n − 4. The coefficient functions of the counter terms determine the construction of φ² and φ⁴ in terms of renormalized composite operators 1, [φ²], and [φ⁴]. Two of the counter terms vanish in conformally flat space-time. The others may be computed from the theory in purely flat space-time. They are determined, in a rather intricate fashion, by the additive renormalizations for two-point functions of [φ²] and [φ⁴] in Minkowski space-time. In particular, using this method, we compute the leading divergence of the R² interaction which is of fifth order in the coupling constant λ.