Quantum implications of a scale invariant regularisation

TL;DR

This paper investigates quantum scale invariance at three loops, revealing how a dilaton field enables a scale-invariant regularisation and generates non-polynomial operators, impacting the effective potential and symmetry breaking.

Contribution

It introduces a three-loop computation of the scalar potential maintaining manifest scale invariance and explores how evanescent couplings induce new quantum corrections.

Findings

Quantum scale invariance is preserved at three loops.

Evanescent couplings generate non-polynomial operators.

In the IR, the theory reduces to a renormalizable form with explicit scale breaking.

Abstract

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).