Abstract
We discuss two separate realizations of the diffeomorphism group for metric gravity, which give rise to theories that are classically equivalent, but quantum mechanically distinct. We renormalize them in d=2+epsilon dimensions, developing a new procedure for dimensional continuation of metric theories and highlighting connections with the constructions that previously appeared in the literature. Our hope is to frame candidates for ultraviolet completions of quantum gravity in d>2 and give some perturbative mean to assess their existence in d=4, but also to speculate on some potential obstructions in the continuation of such candidates to finite values of epsilon . Our results suggest the presence of a conformal window in d which seems to extend to values higher than four.
Highlights
A different perspective on the problem is achieved by noticing that Newton’s constant, G, is not dimensionful for every spacetime dimension d, hinting that the theory could be perturbatively renormalizable for some value of d
We have explored the renormalization of metric gravity with an Einstein-Hilbert-type action in d = 2+ dimensions with the intent of framing the discussion on the four-dimensional asymptotic safety conjecture from a different angle
Quantum gravity in d = 2 is asymptotically free, a simple dimensional argument shows that the beta function of Newton’s constant has a critical point in d = 2 +, which could represent a consistent ultraviolet completion of the theory and could circumvent the intrinsic limitations of the effective field theory approach caused by perturbative nonrenormalizability in d = 4
Summary
What we imply in this last step is that only after having regulated and renormalized the model, and obtained a beta function such as (12), we continue to d = 2 + > 2 dimensions by noticing that the coupling G must have negative mass dimension This step introduces the dimensionless coupling through the replacement G → Gμ− , where μ is the RG scale, effectively measuring the coupling constant in units of μ. The final result in an -expansion series that has d-dependent coefficients With these steps in mind, it should be clear that one can investigate the two dimensional limit by taking d = 2 + and → 0, but can estimate the four dimensional limit by taking d = 4 and extrapolating to → 2.
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