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

We study the one loop renormalization in the most general metric-dilaton theory with the second derivative terms only. Classical action includes three arbitrary functions of dilaton. The general theory can be divided into two classes, models of one are equivalent to conformally coupled with gravity scalar field and also to general relativity with cosmological term. The models of second class have one extra degree of freedom which corresponds to dilaton. We calculate the one loop divergences for the models of second class and find that the theory is not renormalizable off mass shell. At the same time the arbitrary functions of dilaton in the starting action can be fine-tuned in such a manner that all the higher derivative counterterms disappear on shell. The only structures in both classical action and counterterms, which survive on shell, are the potential (cosmological) ones. They can be removed by renormalization of the dilaton field which acquire the nontrivial anomalous dimension, that leads to the effective running of the cosmological constant. For some of the renormalizable solutions of the theory the observable low energy value of the cosmological constant is small as compared with the Newtonian constant. We also discuss another application of our result. In particular, our calculations in a general dilaton model in original variables give the possibility to estimate quantum effects in Λ + αR+ βR theory. On leave from Tomsk Pedagogical Institute, 634041 Tomsk, Russia. E-mail: [email protected] 2 E-mail: [email protected]