One-loop renormalization of the four-dimensional theory for quantum dilaton gravity

We study one loop renormalization in the most general metric-dilaton theory with second derivative terms only. The classical action includes three arbitrary functions of the dilaton. The general theory can be divided into two classes. Models of one are equivalent to gravity conformally coupled to a scalar field and also to general relativity with a cosmological term. The models of the second class have one extra degree of freedom which corresponds to the dilaton. We calculate the one loop divergences for the models of the second class and find that the theory is not renormalizable off the mass shell. At the same time the arbitrary functions of the dilaton in the starting action can be fine-tuned in such a way that all the higher derivative counterterms disappear on shell. The only structures in both the classical action and counterterms, which survive on shell, are the potential (cosmological) ones. They can be removed by renormalization of the dilaton field that acquires the nontrivial anomalous dimension, which leads to the effective running of the cosmological constant. Another application of our calculations is the following. For some special choice of the arbitrary functions our dilaton model is equivalent to general relativity with an additional ${\mathit{R}}^2$ term. Such an equivalence holds at the quantum level if we do not introduce the external source for the dilaton field. Thus our calculations in a general dilaton model in original variables include quantum Λ+αR+β${\mathit{R}}^2$ theory as the particular case.