In the presence of spontaneous symmetry breaking, the alignment of the vacuumwith respect to the gauge group is often controlled by quadratically divergentoperators in the low energy non-linear sigma model. In principle the magnitudesand signs of these operators can be changed by making different assumptionsabout the ultraviolet physics, but in practice all known ways of regulatingthese theories preserve the naive vacuum alignment. We show that by"integrating in" different sets of heavy spin-one fields, it is possible to UVextend certain non-linear sigma models into two distinct UV insensitivetheories. These UV extensions have identical low energy degrees of freedom butdifferent radiative potentials, making it possible to engineer two differentvacuum alignments for the original non-linear sigma model. Our constructionemploys "non-square" theory spaces which generically violate the common lorethat the preferred vacuum alignment preserves the maximal gauge symmetry. By UVextending the SO(9)/(SO(4) X SO(5)) little Higgs model, we find a radiativepotential that deviates from the naive expectation but does not stabilize thecorrect vacuum for proper electroweak symmetry breaking.
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