Natural stepwise breaking of gauge and discrete symmetries

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Abstract
We ask if it is possible for spontaneous symmetry breakdown to occur stepwise. The presence of massless non-Goldstone bosons in the tree approximation is necessary both for the continuous case (gauge hierarchies involving "pseudophotons") and for the discrete case, but it is not sufficient. Our findings on additional constraints are presented in the form of theorems. The first (concerning pseudophotons) depends only on the validity of perturbative expansions and on the Goldstone theorem. The other two require that all massless non-Goldstone bosons in the tree approximation arise from a vacuum symmetry.