Exact quantum conformal symmetry, its spontaneous breakdown, and gravitational Weyl anomaly

The classical Lagrangian of the Standard Model enjoys the symmetry of the full conformal group if the mass of the Higgs boson is put to zero. This is a hint that conformal symmetry may play a fundamental role in the ultimate theory describing nature. The origin of scales, such as the Higgs vacuum expectation value (VEV), may result from the spontaneous breakdown of the conformal symmetry by the dilaton field. In this work, we study whether this classical setup can be implemented in quantum theory and be phenomeno-logically viable by presenting an explicit construction where the exact conformal symmetry can be preserved and is anomaly-free while being spontaneously broken. Not only the Higgs mass but also the genuine quantum scales such as the QCD confinement radius are generated by the dilaton VEV. We also discuss the extension of these ideas to the theories with dynamical gravity and show that the only finite subgroup of the local Weyl transformations which is anomaly-free corresponds to the global scale symmetry. This means that the conformal invariance of the flat space theory is explicitly broken down to the scale symmetry by gravitational effects related to the Weyl anomaly.