  We study the one loop renormalization in the most general metric-dilaton
theory with the second derivative terms only. 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 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.
