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

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 ϕ 6 / σ 2 , ϕ 8 / σ 4 , 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.