One-loop potential with scale invariance and effective operators

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 $\mu$ of the dimensional regularization is generated after spontaneous scale symmetry breaking, from a subtraction function of the fields, $\mu(\phi,\sigma)$. This function is then uniquely determined from general principles showing that it depends on the dilaton only, with $\mu(\sigma)\sim \sigma$. The result is a scale invariant one-loop potential $U$ for a higgs field $\phi$ and dilaton $\sigma$ that contains an additional {\it finite} quantum correction $\Delta U(\phi,\sigma)$, beyond the Coleman Weinberg term. $\Delta U$ contains new, non-polynomial effective operators like $\phi^6/\sigma^2$ 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 $\phi$ remains small without tuning (of its self-coupling, etc) beyond the initial, classical tuning (of the dilaton coupling) that enforces a hierarchy $\langle\sigma\rangle\gg \langle\phi\rangle$. The approach is useful to models that investigate scale symmetry at the quantum level.