We consider the general problem of the quantum effects due to a scalar field coupled to a classical metric. Normally one expects a quartically divergent vacuum energy leading to a highly curved spacetime, incompatible with observations, unless one accepts an enormous fine tuning. This is the most striking form of the cosmological constant problem. It has been suggested by Becker and Reuter about a year ago that if one uses a dimensionless momentum cutoff and uses the equation of motion of the metric, things look very different.
I will review these arguments and further show that in this reatment the mass and quartic coupling only receive finite corrections.