Toy models for hierarchy studies

Abstract

We provide a simple computation in order to estimate the probability of a given hierarchy between two scales. In particular, we work in a model provided with a gauge symmetry, with two scalar doublets. We start from a scale-invariant classical Lagrangian, but by taking into account the Coleman–Weinberg mechanism, we obtain masses for the gauge bosons and the scalars. This approach typically provides a light (L) and a heavy (H) sector related to the two different vacuum expectation values of the two scalars. We compute the size of the hypervolume of the parameter space of the model associated with an interval of mass ratios between these two sectors. We define the probability as proportional to this size and conclude that probabilities of very large hierarchies are not negligible in the type of models studied in this work.

Gravitational Coleman-Weinberg mechanism

“…with α ≡ αδ , and the relation with reduced Planck mass given by MPl = ξ φ 2[14,27]. Since we have slightly and explicitly broken the U(1) global symmetry, the angular mode can be understood as a pseudo-Goldstone boson (even in the case with θ = 0).…”