Frozen up dilaton and the GUT/Planck mass ratio

Abstract

By treating modulus and phase on equal footing, as prescribed by Dirac, local scale invariance can consistently accompany any Brans-Dicke ω-theory. We show that in the presence of a soft scale symmetry breaking term, the classical solution, if it exists, cannot be anything else but general relativistic. The dilaton modulus gets frozen up by the Weyl-Proca vector field, thereby constituting a gravitational quasi-Higgs mechanism. Assigning all grand unified scalars as dilatons, they enjoy Weyl universality, and upon symmetry breaking, the Planck (mass) 2 becomes the sum of all their individual (VEV) 2 s. The emerging GUT/Planck (mass) 2 ratio is thus ∼ ωg 2 GU T /4π. Critical local Weyl invarianceThe Brans-Dicke theory [1] is described by the action(1)

Weyl gauge symmetry and its spontaneous breaking in the standard model and inflation

“…This is similar to the case with one scalar field in eq. (25). Thus, in the Einstein frame we have a massive vector boson and one (real) scalar field left (θ), while in Jordan frame we had two (real) scalar fields and a massless ω µ , so the number of degrees of freedom is again conserved.…”

“…Weyl geometry is a scalar-vector-tensor theory of gravity and thus provides a generaliza-tion (to classes of equivalence) of Brans-Dicke-Jordan scalar-tensor theory [13] and of other conformal invariant models [18]. It was used for model building [19,20] with renewed recent interest in [22][23][24][25][26][27][28][29] and applications to inflation, see e.g. [30][31][32][36][37][38][39][40][41].…”

“…Also note that the gauge kinetic term L g of ω µ , see eq. (10), is invariant under (22), (25). The scalar potential becomes a cosmological constant, V 0 = 3λ φ 4 /(2ξ 2 ), in Einstein frame.…”

“…The Weyl gauge symmetry is the natural extension for conformal invariant models e.g. [13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32]; the conformal transformation of the metric is extended by the associated gauge transformation of a Weyl gauge field (ω µ ) which is of geometric origin. Section 3 discusses how ω µ undergoes a Stueckelberg mechanism and becomes massive by "eating" the wouldbe-Goldstone field (dilaton ρ); here, the dilaton is the radial direction in the field space of scalar fields (φ j ) of different non-minimal couplings ξ j to R. The Weyl gauge symmetry is then spontaneously broken and there are no negative kinetic terms in the theory.…”

“…In the RG case, imposing the Weyl symmetry to avoid generating 1 The mechanism of the Weyl gauge symmetry breaking used here differs from that in [23] where a complex scalar is considered rather the (real) dilaton, and a Coleman-Weinberg mechanism (unitary gauge) is used (instead of a Stueckelberg mechanism), which breaks explicitly the Weyl symmetry by UV regularization. An explicit (classical) breaking was also considered in [25] for one scalar field case.…”