Abstract
We show that classically scale invariant gravity coupled to a single scalar field can undergo dimensional transmutation and generate an effective Einstein-Hilbert action for gravity, coupled to a massive dilaton. The same theory has an ultraviolet fixed point for coupling constant ratios such that all couplings are asymptotically free. However the catchment basin of this fixed point does not include regions of coupling constant parameter space compatible with locally stable dimensional transmutation. In a companion paper we will explore whether this more desirable outcome does obtain in more complicated theories with non-Abelian gauge interactions.
Highlights
In addition we seek a theory such that all dimensional couplings are asymptotically free (AF), with the region of DT within the basin of attraction of an ultra-violet stable fixed point (UVFP) for ratios of couplings
We show that classically scale invariant gravity coupled to a single scalar field can undergo dimensional transmutation and generate an effective Einstein-Hilbert action for gravity, coupled to a massive dilaton
NV are the numbers of scalar, fermion, and vector fields respectively. (Note that NF = 2N 1, the number of fermions as defined in ref. [1] and 2 earlier works.) CG, TF and TS are the usual quadratic Casimirs for the pure gauge theory and fermion and scalar representations, with the coefficients of TF and TS in eq (2.1b) reflecting our choices of two component fermions and real scalars. It is worth noting at this point that whereas g is AF for bg > 0 whether it is positive or negative, for a to be AF
Summary
One attractive property of pure renormalizable gravity is that it is asymptotically free (AF) [7, 8], and this property can be extended to include a matter sector with an asymptotically free gauge theory or even a non-gauge theory. [1] and 2 earlier works.) CG, TF and TS are the usual quadratic Casimirs for the pure gauge theory and fermion and scalar representations, with the coefficients of TF and TS in eq (2.1b) reflecting our choices of two component fermions and real scalars. It is worth noting at this point that whereas g is AF for bg > 0 whether it is positive or negative (the sign of the gauge coupling is not a physical observable), for a to be AF we must have a > 0; a < 0 corresponds to an unphysical phase with a Landau pole in the UV.
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