Quantum implications of a scale invariant regularization

We study scale invariance at the quantum level in a perturbative approach. For a scale-invariant classical theory, the scalar potential is computed at a three-loop level while keeping manifest this symmetry. Spontaneous scale symmetry breaking is transmitted at a quantum level to the visible sector (of $\varphi$) by the associated Goldstone mode (dilaton $\sigma$), which enables a scale-invariant regularization and whose vacuum expectation value $⟨\sigma ⟩$ generates the subtraction scale ($\mu$). While the hidden ($\sigma$) and visible sector ($\varphi$) are classically decoupled in $d=4$ due to an enhanced Poincaré symmetry, they interact through (a series of) evanescent couplings $\propto \epsilon$, 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, as scale-invariant nonpolynomial effective operators ${\varphi }^{2n+4}/{\sigma }^{2n}$. These are comparable in size to “standard” loop corrections and are important for values of $\varphi$ close to $⟨\sigma ⟩$. For $n=1$, 2, the beta functions of their coefficient are computed at three loops. In the IR limit, dilaton fluctuations decouple, the effective operators are suppressed by large $⟨\sigma ⟩$, and the effective potential becomes that of a renormalizable theory with explicit scale symmetry breaking by the DR scheme (of $\mu =\text{constant}$).