Hierarchy in double SU(2) models

TL;DR

This paper introduces a simple gauge theory model with multiple scalar doublets to study the probability distribution of hierarchies between two mass scales, revealing that large hierarchies are unlikely but possible.

Contribution

It provides a novel probabilistic framework for analyzing hierarchies in a double SU(2) gauge model with scalar doublets, incorporating various parameter scenarios and interactions.

Findings

Large hierarchies are logarithmically rare but feasible.

The model's probability distribution depends on parameter choices and interactions.

Including interactions affects the likelihood of hierarchy sizes.

Abstract

In this work, we provide a simple model that studies the probability to obtain a given hierarchy between two scales. In particular, we work in a theory with an $SU(2)_L\times SU(2)_H\times U(1)_X$ gauge symmetry and two scalar doublets. By the Coleman-Weinberg mechanism, the gauge bosons and scalars obtain different masses, corresponding to the light and heavy sectors. We analyze the mass ratio of these sectors in order to discuss the hierarchy between them, and we define a probability associated to this hierarchy. We analyze different cases in which one of the sectors is fixed or both of them have free parameters, and also study the effect of including an interaction between them. We conclude that the probability of obtaining very large hierarchies is (logarithmically) small but not negligible.