Lessons from $T^μ_{~ μ}$ on inflation models: two-loop renormalization of $η$ in the scalar QED

A non-minimal coupling $η$ 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 $λφ^{4}$ theory.