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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 $\langleσ\rangle$ generates the DR subtraction scale ($μ\sim\langleσ\rangle$), 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 $\langleσ\rangle\gg \langleϕ\rangle$, where such hierarchy is arranged by {\it one} initial, classical tuning. These operators emerge at the quantum level from evanescent interactions ($\proptoε$) between $σ$ and $ϕ$ that vanish in $d=4$ but are demanded by classical scale invariance in $d=4-2ε$. 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.