Two-loop scale-invariant scalar potential and quantum effective operators

Abstract

Spontaneous breaking of quantum scale invariance may provide a solution to the hierarchy and cosmological constant problems. In a scale-invariant regularization, we compute the two-loop potential of a Higgs-like scalar ϕ in theories in which scale symmetry is broken only spontaneously by the dilaton (σ). Its VEV ⟨ σ⟩ generates the DR subtraction scale (μ∼ ⟨ σ⟩ ), which avoids the explicit scale symmetry breaking by traditional regularizations (where μ= fixed scale). The two-loop potential contains effective operators of non-polynomial nature as well as new corrections, beyond those obtained with explicit breaking (μ= fixed scale). These operators have the form ϕ6/ σ2, ϕ8/ σ4, etc., which generate an infinite series of higher dimensional polynomial operators upon expansion about ⟨ σ⟩ ≫ ⟨ ϕ⟩ , where such hierarchy is arranged by one initial, classical tuning. These operators emerge at the quantum level from evanescent interactions (∝ ϵ) between σ and ϕ that vanish in d= 4 but are required by classical scale invariance in d= 4 - 2 ϵ. The Callan–Symanzik equation of the two-loop potential is respected and the two-loop beta functions of the couplings differ from those of the same theory regularized with μ= fixed scale. Therefore the running of the couplings enables one to distinguish between spontaneous and explicit scale symmetry breaking.