In general 2HDM potentials with two local minima, all scalar masses remain bounded under perturbativity constraints on the quartic couplings.
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In real 3HDMs with spontaneous CP violation and discrete symmetry on quartics, perturbativity bounds keep new scalars near the electroweak scale despite large quadratic terms.
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On vacua and bounded masses in the general 2HDM
In general 2HDM potentials with two local minima, all scalar masses remain bounded under perturbativity constraints on the quartic couplings.
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Light states in real 3HDMs with spontaneous CP violation and softly broken symmetries
In real 3HDMs with spontaneous CP violation and discrete symmetry on quartics, perturbativity bounds keep new scalars near the electroweak scale despite large quadratic terms.