Manifestly scale-invariant regularization and quantum effective operators

Abstract

Scale invariant theories are often used to address the hierarchy problem, however the regularization of their quantum corrections introduces a dimensionful coupling (dimensional regularization) or scale (Pauli-Villars, etc) which break this symmetry explicitly. We show how to avoid this problem and study the implications of a manifestly scale invariant regularization in (classical) scale invariant theories. We use a dilaton-dependent subtraction function $\mu(\sigma)$ which after spontaneous breaking of scale symmetry generates the usual DR subtraction scale $\mu(\langle\sigma\rangle)$. One consequence is that "evanescent" interactions generated by scale invariance of the action in $d=4-2\epsilon$ (but vanishing in $d=4$), give rise to new, finite quantum corrections. We find a (finite) correction $\Delta U(\phi,\sigma)$ to the one-loop scalar potential for $\phi$ and $\sigma$, beyond the Coleman-Weinberg term. $\Delta U$ is due to an evanescent correction ($\propto\epsilon$) to the field-dependent masses (of the states in the loop) which multiplies the pole ($\propto 1/\epsilon$) of the momentum integral, to give a finite quantum result. $\Delta U$ contains a non-polynomial operator $\sim \phi^6/\sigma^2$ of known coefficient and is independent of the subtraction dimensionless parameter. A more general $\mu(\phi,\sigma)$ is ruled out since, in their classical decoupling limit, the visible sector (of the higgs $\phi$) and hidden sector (dilaton $\sigma$) still interact at the quantum level, thus the subtraction function must depend on the dilaton only. The method is useful in models where preserving scale symmetry at quantum level is important.