Effective Action, Conformal Anomaly and the Issue of Quadratic Divergences

TL;DR

This paper investigates the role of conformal symmetry in quantum field theory, showing it can address quadratic divergences and potentially replace supersymmetry in stabilizing the weak scale, under specific conditions.

Contribution

It demonstrates that exact conformal invariance can eliminate quadratic divergences in massless  theory and proposes conformal symmetry as an alternative to supersymmetry for hierarchy stabilization.

Findings

Quadratic divergences do not arise with exact conformal invariance.

Conformal symmetry can replace supersymmetry if no intermediate scales exist.

Running couplings must be free of Landau poles and instabilities.

Abstract

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.