Effective Action, Conformal Anomaly and the Issue of Quadratic Divergences

TL;DR

Meissner and Nicolai investigate whether exact conformal invariance can stabilize the weak scale without intermediate scales by analyzing the one-loop effective action of massless $φ^4$ theory. They compute the lowest order non-local contributions to the effective action and derive the conformal anomaly from the non-local action, showing $T^μ{}_μ$ is a local expression proportional to $φ_c^4$ (in flat space) and related to the β-function via $T^μ{}_μ = \frac{1}{4} β(λ) (φ_c^2)^2$, with a generalization to $O(N)$ giving $(N+8)λ^2/(32π^2) (φ_c^2)^2$. They argue that requiring exact conformal invariance of counterterms eliminates quadratic divergences, drawing an analogy to SUSY, and that this mechanism can operate in a scenario with no intermediate scales up to the Planck scale $M_P$ and with running couplings free of Landau poles, thereby addressing naturalness without low-energy supersymmetry. The results point to conformal symmetry, possibly coupled to quantum gravity effects, as a plausible guide for ultraviolet completion and Planck-scale physics shaping low-energy mass scales.

Abstract

For massless $φ^4$ 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.