Renormalization group and logarithmic corrections to scaling relations in conformal sector of 4D gravity

We study the effective theory of the conformal factor near its infrared stable fixed point. The renormalization group equations for the effective coupling constants are found and their solutions near the critical point are obtained, providing the logarithmic corrections to the scaling relations. Some cosmological applications of the running of coupling constants are briefly discussed. Laboratoire Propre du CNRS UPR A.0014 It is quite an old idea to work with some effective theory for quantum gravity (QG) in the absence of a consistent theory. The traditional candidate for such an effective theory is usually Einstein gravity which is known to be non-renormalizable [1]. Despite this fact, already in this theory the loop corrections can be calculated and may become quite relevant for a variety of phenomena [2, 3] (for a general introduction to perturbative QG, see for example [4]). Recently, an interesting effective model aiming to describe QG in the far infrared, has been introduced [5]. This model is based on the effective theory for the conformal factor, and some of its properties have been further studied in refs.[6]-[9]. It was found that it possesses a non-trivial infrared (IR) stable fixed point which could become physically relevant at cosmological distance scales. Investigation of this critical point led to the derivation of exact scaling relations and, in particular, of the anomalous dimension of the conformal factor in analogy with 2D non-critical string theory. Furthermore, this model provides a dynamical framework for a solution of the cosmological constant problem [5]. The purpose of this letter is to study the effective theory of the conformal factor near its fixed point. We show that making the one-loop renormalization in a way which preserves global conformal invariance of counterterms, one can rigorously construct renormalization group (RG) equations for coupling constants. The solution of these equations near the critical point gives the logarithmic corrections to scaling relations. Consider the flat space theory described by the Lagrangian L = b1 (2σ) + b2 (∂μσ)2 2σ + b3 [ (∂μσ) 2 ]2 + b4(σ) (∂μσ) 2 + b5(σ) (1) where σ has classically canonical dimension zero, b1, b2, b3 are dimensionless coupling constants and b4, b5 are for the moment arbitrary dimensionful field-dependent generalized couplings. The one-loop divergences of this theory were computed in ref.[9] and for some choices of b4, b5 it is multiplicatively renormalizable in the usual sense. These choices correspond to the effective theory of the conformal factor [5] which we review below. We start for simplicity with N conformally invariant scalars in curved spacetime. The conformal anomaly in this case is [10] T μ μ = b (