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 (whereμ= 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/σ 2, φ8/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 (∝ e) between σ and φ that vanish in d = 4 but are required by classical scale invariance in d = 4 − 2e. 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.