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

TL;DR

This paper investigates the renormalization group flows and fixed points of a scalar field with nonminimal coupling to curvature in curved space, using covariant functional methods to analyze fixed points and phase behavior.

Contribution

It develops covariant Wetterich equations for nonminimal scalar-curvature couplings and explores their fixed points in various dimensions with analytic and numerical methods.

Findings

Identifies fixed points for nonminimal coupling in different phases.

Analyzes the behavior of the RG flow near fixed points.

Provides insights into universal contributions below four dimensions.

Abstract

Using covariant methods, we construct and explore the Wetterich equation for a non-minimal coupling $F(\phi)R$ of a quantized scalar field to the Ricci scalar of a prescribed curved space. This includes the often considered non-minimal coupling $\xi \phi^2 R$ 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 non-minimal coupling in the symmetric and spontaneously broken phases with vanishing and non-vanishing average fields, respectively. Using functional perturbative renormalization group methods, we discuss the leading universal contributions to the RG flow below the upper critical dimension $d=4$.