Toy models for hierarchy studies

TL;DR

This paper introduces a simple model to estimate the likelihood of hierarchy between two scales, showing that large hierarchies are not unlikely in such gauge-scalar models with scale invariance and Coleman-Weinberg mechanism.

Contribution

It provides a quantitative estimate of hierarchy probabilities in a gauge-scalar model with scale invariance and Coleman-Weinberg symmetry breaking.

Findings

Large hierarchies have non-negligible probabilities in the studied models.

The model relates mass ratios to the parameter space volume.

Hierarchy likelihood depends on the parameter space structure.

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 hyper-volume 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 letter.