Effective Action, Conformal Anomaly and the Issue of Quadratic Divergences

For massless $\phi^4$ theory, we explicitly compute the lowest order non-local contributions to the one-loop effective action required for the determination of the trace anomaly. Imposing exact conformal invariance of the local part of the effective action, we argue that the issue of quadratic divergences does not arise in a theory where exact conformal symmetry is only broken by quantum effects. Conformal symmetry can thus replace low energy supersymmetry as a possible guide towards stabilizing the weak scale and solving the hierarchy problem, if (i) there are no intermediate scales between the weak scale and the Planck scale, and (ii) the running couplings exhibit neither Landau poles nor instabilities over this whole range of energies.