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 $phi$ in theories in which scale symmetry is broken only spontaneously by the dilaton ($sigma$). Its vev $langlesigmarangle$ generates the DR subtraction scale ($musimlanglesigmarangle$), which avoids the explicit scale symmetry breaking by traditional regularizations (where $mu$=fixed scale). The two-loop potential contains effective operators of non-polynomial nature as well as new corrections, beyond those obtained with explicit breaking ($mu$=fixed scale). These operators have the form: $phi^6/sigma^2$, $phi^8/sigma^4$, etc, which generate an infinite series of higher dimensional polynomial operators upon expansion about $langlesigmaranglegg langlephirangle$, where such hierarchy is arranged by {it one} initial, classical tuning. These operators emerge at the quantum level from evanescent interactions ($proptoepsilon$) between $sigma$ and $phi$ that vanish in $d=4$ but are demanded by classical scale invariance in $d=4-2epsilon$. 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 $mu=$fixed scale. Therefore the running of the couplings enables one to distinguish between spontaneous and explicit scale symmetry breaking.