Lessons from $T^{\mu}_{~ \mu}$ on inflation models: two-loop renormalization of $\eta$ in the scalar QED

TL;DR

This paper investigates the quantum behavior of a non-minimal coupling in scalar QED, revealing that its renormalization does not include inhomogeneous terms at two loops, which clarifies its quantum properties in inflation models.

Contribution

It provides the first two-loop renormalization analysis of the non-minimal coupling in scalar QED, showing the absence of inhomogeneous terms at this order.

Findings

No inhomogeneous term in the RGE at two loops

Similar behavior to higher-loop orders in other theories

Clarifies quantum nature of non-minimal coupling in inflation models

Abstract

A non-minimal coupling $\eta$ has been attracting growing interest particularly in the context of inflation models, though its quantum nature is not clear yet. We study the renormalization of a non-minimal coupling in the scalar quantum electrodynamics (QED). We find ${\it no}$ inhomogeneous term of the renormalization group equation (RGE) at the two-loop level. This is similar to other theories, where an inhomogeneous term of the RGE appears only at a higher-loop order: e.g., four-loop order in $\lambda \phi^{4}$ theory.