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 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 (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 σ ≫ φ , where such hierarchy is arranged by one initial, classical tuning. These operators emerge at the quantum level from evanescent interactions (∝ ǫ) between σ and φ that vanish in d = 4 but are demanded by classical scale invariance in d = 4 − 2ǫ. 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. *

Scale symmetry without the anomaly

“…This was explicitly shown in [16] for the two-loop effective potential in the model of two scalars that we have considered here.…”

Classical Weyl transverse gravity

“…3, the meaning of "spontaneously broken" should be understood in the sense that the spurion field ϕ is assumed to be divided into two terms, ϕ = ϕ + σ where ϕ is the vacuum expectation value and σ is the Goldstone boson restoring conformal symmetry, respectively. Then the key technical idea in [9][10][11][12][13][14][15][16][17][18] is that the vacuum expectation value ϕ plays a role as the renormalization scale instead of the conventional fixed renormalization scale, by which the Weyl-invariant effective action can be obtained. Our conjecture that the fake Weyl symmetry has no anomaly is interpreted as a supplementary statement from the symmetry side, which supports this technical idea.…”

“…(20), or equivalently Eq. (18). The equivalence between the conformally invariant scalar-tensor gravity and the WTDiff gravity via the gauge-fixing procedure demands that the equations of motion in the WTDiff gravity should be obtained from Eq.…”

“…In a pioneering work by Englert et al [9], it has been clarified that the conformally invariant scalar-tensor gravity coupled to various matter fields is free of Weyl anomaly when the Weyl symmetry is spontaneously broken. This fact has been investigated and certified by subsequent papers [10][11][12][13][14][15][16][17][18]. Related to this work, in this article, we wish to put forward a new conjecture that the Weyl symmetry is not violated by radiative corrections if it is a fake Weyl symmetry and it is spontaneously broken.…”

“…Note that σ = 0 corresponds to the "unitary gauge" leading to general relativity or the WTDiff gravity depending on the choice of ϕ . The key idea is that we can use the vacuum expectation value of the spurion field, ϕ , as the renormalization scale instead of the conventional fixed renormalization scale μ [9][10][11][12][13][14][15][16][17][18]. With this idea, we have a conformally invariant effective potential without trace anomaly and the coupling constants still run with the momentum scale [15].…”