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 $\phi$ in theories in which scale symmetry is broken only spontaneously by the dilaton ($\sigma$). Its vev $\langle\sigma\rangle$ generates the DR subtraction scale ($\mu\sim\langle\sigma\rangle$), 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 $\langle\sigma\rangle\gg \langle\phi\rangle$, where such hierarchy is arranged by {\it one} initial, classical tuning. These operators emerge at the quantum level from evanescent interactions ($\propto\epsilon$) between $\sigma$ and $\phi$ that vanish in $d=4$ but are demanded by classical scale invariance in $d=4-2\epsilon$. 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.