Quantum implications of a scale invariant regularisation

Understand

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 $d=4-2\epsilon$.