Reverse ) Engineering Vacuum Alignment

In the presence of spontaneous symmetry breaking, the alignment of the vacuum with respect to the gauge group is often controlled by quadratically divergent operators in the low energy non-linear sigma model. In principle the magnitudes and signs of these operators can be changed by making different assumptions about the ultraviolet physics, but in practice all known ways of regulating these theories preserve the näıve vacuum alignment. We show that by “integrating in” different sets of heavy spin-one fields, it is possible to UV extend certain non-linear sigma models into two distinct UV insensitive theories. These UV extensions have identical low energy degrees of freedom but different radiative potentials, making it possible to engineer two different vacuum alignments for the original non-linear sigma model. Our construction employs “non-square” theory spaces which generically violate the common lore that the preferred vacuum alignment preserves the maximal gauge symmetry. By UV extending the SO(9)/(SO(4) × SO(5)) little Higgs model, we find a radiative potential that deviates from the näıve expectation but does not stabilize the correct vacuum for proper electroweak symmetry breaking. 1. Motivation Because a non-linear sigma model (NLΣM) describes degrees of freedom whose properties are defined entirely by spontaneous symmetry breaking, NLΣMs are ideal for understanding the low energy physics of QCD and technicolor, where strong dynamics obstructs calculability and only the symmetries are known a priori. Given the variety of possible symmetry breaking patterns, NLΣMs have also proven useful for building models with naturally light bosonic states. From the earliest theories of composite Higgs bosons [1, 2] to contemporary developments in little Higgs [3, 4, 5, 6, 7, 8, 9, 10], T -parity [11, 12, 13], holographic Higgs [14, 15], and twin Higgs theories [16, 17, 18], NLΣMs have enriched our understanding of electroweak physics beyond the standard model. However, the problem of vacuum alignment can hinder the construction of realistic models with the Higgs as a pseudo-Goldstone boson. Consider a theory with a global symmetry G spontaneously broken to a subgroup H ⊂ G. If G is an exact symmetry, then the orientation of H in G is arbitrary, and the Goldstone bosons in G/H parameterize a space of equivalent vacua. If a subgroup F ⊂ G is weakly gauged, then gauge interactions specify a second orientation in G which lifts the vacuum degeneracy and chooses a preferred alignment of H relative to F . This vacuum alignment is crucial to phenomenology because it sets the particle content and charges of the low energy theory [19]. In both the SU(6)/Sp(6) [6] and SO(9)/(SO(4) × SO(5)) [9] little Higgs theories, a näıve calculation indicates that the stable vacuum does not allow for electroweak symmetry breaking. Thus, it would be useful to have a method to engineer any desired vacuum alignment for a G/H NLΣM with F ⊂ G gauged. Of course, small tree level G-violating interactions can force such an alignment by hand, but we are looking for a method to fix the vacuum alignment from calculable one-loop corrections alone. In particular, when a NLΣM suffers from one-loop quadratic divergences, one cannot trust the näıve stability of a given vacuum alignment because it is controlled by UV sensitive operators with incalculable coefficients. Several remedies have been proposed to render such operators calculable, all of which more or less involve the addition of new spin-one modes. For example, hidden local symmetry [20, 21] can be used to introduce new gauge bosons which cut off quadratically divergent loop integrals. Another approach is to embed the theory in a higher dimensional spacetime, where nonlocality in the extra dimension softens UV sensitive operators [14]. But in both cases, the calculable vacuum alignment is the same as the vacuum alignment obtained from the näıve sign of quadratically divergent operators [21]. In QCD, the π+ is heavier than the π0, just as one would “predict” from the quadratically divergent photon loop, and regulating QCD with a ρ meson in the vector limit [22] confirms that prediction [23]. There are examples like Casimir and thermal effects where the näıve signs of quadratically divergent operators are just plain incorrect. But in the context of vacuum alignment, we know of no examples (until now) where different vacuum alignments can be chosen by making different assumptions about ultraviolet physics. In this paper, we present a novel method for UV extending low energy NLΣMs when the global symmetry breaking pattern is U(N) → U(M)× U(N −M). (1.1)