Toy models for hierarchy studies

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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.
© The Autor(s) 2022 We thank F. J. Llanes-Estrada and V. Sanz for useful comments. This work was supported by Grants FPU16/06960 (MECD), the MICINN (Spain) projects PID2019-107394GB-I00/AEI/10.13039/501100011033 (AEI/FEDER, UE) and PID2019108655GB-I00/AEI/10.13039/501100011033 (AEI), the EU STRONG2020 project [grant no.824093], and STMS Grant from COST Action CA16108. JARC acknowledges support by Institut Pascal at Universite Paris-Saclay during the Paris-Saclay Astroparticle Symposium 2021, with the support of the P2IO Laboratory of Excellence (program "Investissements d-avenir" ANR-11-IDEX-0003-01 Paris-Saclay and ANR-10-LABX-0038), the P2I axis of the Graduate School Physics of Universite Paris-Saclay, as well as IJCLab, CEA, IPhT, APPEC, the IN2P3 master project UCMN and EuCAPT. This research was supported by the Munich Institute for Astro-and Particle Physics (MIAPP) which is funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany's Excellence Strategy-EXC-2094-390783311.
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