Renormalization group study of the higher derivative conformal scalar model

The second alternative conformal limit of the recently proposed general higher derivative dilaton quantum theory in curved spacetime is explored. In this version of the theory the dilaton is transformed, along with the metric, to provide the conformal invariance of the classical action. We find the corresponding quantum theory to be renormalizable at one loop, and the renormalization constants for the dimensionless parameters are explicitly shown to be universal for an arbitrary parametrization of the quantum field. The renormalization group equations indicate an asymptotic freedom in the IR limit. In this respect the theory is similar to the well-known model based on the anomaly-induced effective action. Introduction The gravitational effective action, generated by the trace anomaly of the conformal invariant matter in curved spacetime, is of considerable interest due to numerous physical applications (see [1] for the review) and also because of the hope to use such an action as an insight for the theory of quantum gravity [2, 3, 4, 5, 6]. The first solution for the effective action has been obtained by Reigert [7] and by Fradkin and Tseytlin [8] (see also [9, 10, 11]). The equation solved in [7, 8] links the effective action with the trace anomaly of the Energy-Momentum tensor − 2 √−g gμν δΓ δgμν = T μ μ . (1) The above equation remains unaltered if one changes the solution according to Γ → Γ+ Sc, where Sc(gμν) is an arbitrary conformal invariant functional. Therefore this equation doesn’t define Γ(gμν) completely, and in fact the most complicated part of the effective action remains hidden in Sc(gμν). As a result the nonconformal part of the effective action is ambiguous, and the particular version derived in [7, 8] fails to pass the test based on the calculation of the three point functions for the gravitational field [4, 12, 13]. Consequently, it is important to find a solution for (1) which assumes such a test, but this problem has not been solved yet [6]. Another discrepancy arises if one compares the solution of [7, 8] with the result of direct calculations of the effective action performed in [14]. The effective action for the conformal scalar field [14] is a nonlocal functional – as well as the solution of [7, 8] – but the nonlocalities are related with the Green functions of the second order conformal operator ∆2 = 2 − 1 6 R, (2) Electronic address: [email protected] Electronic address: [email protected]