One-loop potential with scale invariance and effective operators

Abstract

We study quantum corrections to the scalar potential in classically scale invariant theories, using a manifestly scale invariant regularization. To this purpose, the subtraction scale $μ$ of the dimensional regularization is generated after spontaneous scale symmetry breaking, from a subtraction function of the fields, $μ(ϕ,σ)$. This function is then uniquely determined from general principles showing that it depends on the dilaton only, with $μ(σ)\sim σ$. The result is a scale invariant one-loop potential $U$ for a higgs field $ϕ$ and dilaton $σ$ that contains an additional {\it finite} quantum correction $ΔU(ϕ,σ)$, beyond the Coleman Weinberg term. $ΔU$ contains new, non-polynomial effective operators like $ϕ^6/σ^2$ whose quantum origin is explained. A flat direction is maintained at the quantum level, the model has vanishing vacuum energy and the one-loop correction to the mass of $ϕ$ remains small without tuning (of its self-coupling, etc) beyond the initial, classical tuning (of the dilaton coupling) that enforces a hierarchy $\langleσ\rangle\gg \langleϕ\rangle$. The approach is useful to models that investigate scale symmetry at the quantum level.