Quantum implications of a scale invariant regularisation

We study scale invariance at the quantum level (three loops) in a perturbative approach. For a scale-invariant classical theory the scalar potential is computed at three-loop level while keeping manifest this symmetry. Spontaneous scale symmetry breaking is transmitted at quantum level to the visible sector (of $\phi$) by the associated Goldstone mode (dilaton $\sigma$) which enables a scale-invariant regularisation and whose vev $\langle\sigma\rangle$ generates the subtraction scale ($\mu$). While the hidden ($\sigma$) and visible sector ($\phi$) are classically decoupled in $d=4$ due to an enhanced Poincar\'e symmetry, they interact through (a series of) evanescent couplings $\propto\epsilon^k$, ($k\geq 1$), dictated by the scale invariance of the action in $d=4-2\epsilon$. At the quantum level these couplings generate new corrections to the potential, such as scale-invariant non-polynomial effective operators $\phi^{2n+4}/\sigma^{2n}$ and also log-like terms ($\propto \ln^k \sigma$) restoring the scale-invariance of known quantum corrections. The former are comparable in size to "standard" loop corrections and important for values of $\phi$ close to $\langle\sigma\rangle$. For $n=1,2$ the beta functions of their coefficient are computed at three-loops. In the infrared (IR) limit the dilaton fluctuations decouple, the effective operators are suppressed by large $\langle\sigma\rangle$ and the effective potential becomes that of a renormalizable theory with explicit scale symmetry breaking by the "usual" DR scheme (of $\mu=$constant).