Renormalization group flows and fixed points for a scalar field in curved space with nonminimal F (φ)R coupling

Using covariant methods, we construct and explore the Wetterich equation for a nonminimal coupling F(φ)R of a quantized scalar field to the Ricci scalar of a prescribed curved space. This includes the often considered nonminimal coupling ξφ2R as a special case. We consider the truncations without and with scale- and field-dependent wave-function renormalization in dimensions between four and two. Thereby the main emphasis is on analytic and numerical solutions of the fixed point equations and the behavior in the vicinity of the corresponding fixed points. We determine the nonminimal coupling in the symmetric and spontaneously broken phases with vanishing and nonvanishing average fields, respectively. Using functional perturbative renormalization group methods, we discuss the leading universal contributions to the RG flow below the upper critical dimension d=4.