Quantum implications of a scale invariant regularization

Abstract

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 φ) by the associated Goldstone mode (dilaton σ), which enables a scale-invariant regularization and whose vacuum expectation value σcopyright generates the subtraction scale (μ). While the hidden (σ) and visible sector (φ) are classically decoupled in d=4 due to an enhanced Poincaré symmetry, they interact through (a series of) evanescent couplings ϵ, dictated by the scale invariance of the action in d=4-2ϵ. At the quantum level, these couplings generate new corrections to the potential, as scale-invariant nonpolynomial effective operators φ2n+4/σ2n. These are comparable in size to "standard" loop corrections and are important for values of φ close to σcopyright. 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 σcopyright, and the effective potential becomes that of a renormalizable theory with explicit scale symmetry breaking by the DR scheme (of μ=constant).