Cosmological constant and higher dimensional dilatation symmetry

We discuss a possible solution to the cosmological constant problem based on the hypothesis of a fixed point for higher-dimensional quantum gravity coupled to a scalar. At the fixed point, which is reached only for an infinite value of the scalar field, dilatation symmetry becomes exact. For this limit we concentrate on the absence of a scalar potential since a polynomial potential is not consistent with dilatation symmetry in higher dimensions. We find generic solutions of the higher-dimensional field equations for which the effective four-dimensional cosmological constant vanishes independently of the parameters of the higher-dimensional effective action. Under rather general circumstances these are the only quasistatic stable extrema of the effective action which lead to a finite four-dimensional Planck mass. We discuss the associated higher-dimensional self-tuning mechanism for the cosmological constant. If cosmological runaway solutions approach the fixed point as time goes to infinity, the effective dark energy vanishes asymptotically. In the present cosmological epoch the fixed point is not yet reached completely, resulting in a tiny amount of dark energy, comparable to dark matter. We discuss explicitly higher-dimensional geometries which realize such asymptotic solutions for time going to infinity.