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 $langlesigmarangle$ generates the subtraction scale ($mu$). While the hidden ($sigma$) and visible sector ($phi$) are classically decoupled in $d=4$ due to an enhanced Poincare symmetry, they interact through (a series of) evanescent couplings $proptoepsilon^k$, ($kgeq 1$), dictated by the scale invariance of the action in $d=4-2epsilon$. 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 $langlesigmarangle$. 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 $langlesigmarangle$ and the effective potential becomes that of a renormalizable theory with explicit scale symmetry breaking by the usual DR scheme (of $mu=$constant).