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.
Highlights
There has recently been a renewed interest in the scale invariance symmetry to address the hierarchy or the cosmological constant problems
Scale symmetry is not a symmetry of the real world since it requires that no dimensionful parameters be present in the Lagrangian
The anomalous breaking of scale symmetry is, in general, expected. This is because regularization of the loop corrections breaks this symmetry explicitly, either by introducing a dimensionful coupling as in dimensional regularization (DR) or a mass scale (Pauli-Villars, cutoff regularizations, etc)
Summary
The quantum correction to the mass of φ that is due to ΔU remains small without additional tuning of the couplings Using this symmetry-preserving regularization and spontaneous breaking of scale symmetry, one can address the hierarchy problem at higher loops. After introducing the model (Sec. II) we present the scale-invariant result of the one-loop potential for a general subtraction function (Sec. III); this function is shown to depend on the dilaton only μ 1⁄4 zσ (Sec. IV).
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