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 $\varphi$ in theories in which scale symmetry is broken only spontaneously by the dilaton ($\sigma$). Its vev $⟨\sigma ⟩$ generates the DR subtraction scale ($\mu \sim ⟨\sigma ⟩$), 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: ${\varphi }^6/{\sigma }^2$, ${\varphi }^8/{\sigma }^4$, etc, which generate an infinite series of higher dimensional polynomial operators upon expansion about $⟨\sigma ⟩\gg ⟨\varphi ⟩$, 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 $\varphi$ 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.

20 pages, LaTeX

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