We study the bounded from below (BFB) conditions on a class of three Higgs doublet models (3HDM) constrained by the symmetry groups $U(1)\times U(1)$, $U(1)\times {\mathbb{Z}}_2$, and ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$. These constraints must be implemented on both the neutral (BFB-n) and charged (BFB-c) directions. The exact necessary and sufficient BFB conditions are unknown in the ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$ case. We develop a general strategy using lower bounds to find sufficient conditions for BFB-n and BFB-c and apply it to these symmetries. In addition, we investigate the concern that the use of safe sufficient conditions can ignore valid points which would yield distinct physical consequences. This is done by performing a full phenomenological simulation of the $U(1)\times U(1)$ and $U(1)\times {\mathbb{Z}}_2$ models, where exact necessary and sufficient BFB conditions are possible. We look specifically at the points allowed by exact solutions but precluded by safe lower bounds. We found no evidence of remarkable new effects, partly reassuring the use of the lower bounds we propose here, for those potentials where no exact necessary and sufficient BFB conditions are known.

• Received 19 August 2022
• Accepted 14 November 2022

DOI:https://doi.org/10.1103/PhysRevD.106.115010

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Published by the American Physical Society