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Summary
One of the key features of quantum field theory is that a QFT can not predict a certain scale. However, to describe reality it is necessary to have observables and this requires the existence of physical scales. Once a scale is introduced in the action and the model parameters are specified at that scale, dimensional transmutation will let the parameters evolve when the scale is altered. This is the power of QFT and has been experimentally verified with high accuracy. It is this feature what makes it a quantum theory distinguishing it from a classical field theory.
The only scale we can postulate based on fundamental physics considerations is the Planck scale. This scale arises when gravity in four dimensions is not considered a QFT but treated as a classical background. By comparing the Compton wavelength and the Schwarzschild radius of an imaginary object one arrives at the scale where those become of same order. It is clear that no QFT known to us holds at the Planck scale, since there is no theory of quantum gravity available. This means, however, that there are no $M_{pl}^2$ contributions to physical masses one could expect from a QFT with degrees of freedom with masses of that scale.
The other crucial observation is that the standard model of particle physics (SM) is nearly conformal without any scale and only the mass terms of the scalar fields responsible for the breaking of $SU(2)_L \times U(1)$ symmetry violate conformal symmetry explicitly. If this scale is protected by a symmetry at tree level and generated entirely radiativly no hierarchy problem arises. However, since as stated above the QFT has no power to predict a scale, we have to choose the explicit scale where the classical Action is conformal. It appears natural to us to choose this to be the Planck scale. Thus gravity would be the reason for this fundamental scale. Note that it is crucial in this context that gravity is non renormalizable, otherwise it would be also impossible to predict an absolute scale as in a usual QFT. Note that the gauge hierarchy problem is one of the most severe naturalness problems of our time and has been addressed in many ways as introducing supersymmetry and other BSM extensions.
Starting at the Planck scale with a classically conformal invariant theory one can ask the question what the values of the masses will be at a lower scale. Dimension full mass terms can be generated, since the conformal symmetry is anomalous.
The particle content of the QFT in question determines the running of the masses and predicts their low scale values. This is known as the Coleman-Weinberg mechanism (CW). Despite its beauty it fails to predict the correct phenomenology in the SM. However, we know with certainty that the SM is incomplete, since neutrinos remain massless in contrast to observations. Furthermore, if one wishes to explain the astrophysical light to mass ratios by a new particle, there is no dark matter (DM) candidate in the SM.
In this work we focus on the neutrino mass aspect. The basic idea is that introducing additional bosonic degrees of freedom makes the running of the couplings such that spontaneous symmetry breaking is possible. Thus, the electroweak scale appears naturally given the particle content of the model directly from the Planck scale and hence from the gravity embedding. It is crucial to this mechanism that there is no intermediate scale involved, where additional fields could get a vev and thus contribute strongly to the Higgs mass, creating the necessity to invoke a fine tuning to generate this scale hierarchy.
We systematically study models that induce neutrino masses in conformally invariant theories and provide the correct Higgs mass phenomenology.
