Nonrelativistic scalar field theories can exhibit a natural cascading hierarchy of scales, protected by a hierarchy of polynomial shift symmetries. Using a simple model, we argue that a high-energy cross-over to such nonrelativistic behavior naturally leads to light scalars, and thus represents a useful ingredient for technically natural resolutions of scalar mass hierarchies, perhaps even the Higgs mass hierarchy puzzle.

We investigate spontaneous global symmetry breaking in the absence of Lorentz invariance and study technical naturalness of Nambu-Goldstone modes for which the dispersion relation exhibits a hierarchy of multicritical phenomena with Lifshitz scaling and dynamical exponents z>1. For example, we find Nambu-Goldstone modes with a technically natural quadratic dispersion relation which do not break time reversal symmetry and are associated with a single broken symmetry generator, not a pair. The mechanism is protected by an enhanced "polynomial shift" symmetry in the free-field limit. © 2013 American Physical Society.

Without Lorentz invariance, spontaneous global symmetry breaking can lead to multicritical Nambu-Goldstone modes with a higher-order low-energy dispersion ω∼k^{n} (n=2,3,…), whose naturalness is protected by polynomial shift symmetries. Here, we investigate the role of infrared divergences and the nonrelativistic generalization of the Coleman-Hohenberg-Mermin-Wagner (CHMW) theorem. We find novel cascading phenomena with large hierarchies between the scales at which the value of n changes, leading to an evasion of the "no-go" consequences of the relativistic CHMW theorem.

We continue our study of naturalness in nonrelativistic QFTs of the Lifshitz type, focusing on scalar fields that can play the role of Nambu–Goldstone (NG) modes associated with spontaneous symmetry breaking. Such systems allow for an extension of the constant shift symmetry to a shift by a polynomial of degree P in spatial coordinates. These “polynomial shift symmetries” in turn protect the technical naturalness of modes with a higher-order dispersion relation, and lead to a refinement of the proposed classification of infrared Gaussian fixed points available to describe NG modes in nonrelativistic theories. Generic interactions in such theories break the polynomial shift symmetry explicitly to the constant shift. It is thus natural to ask: Given a Gaussian fixed point with polynomial shift symmetry of degree P, what are the lowest-dimension operators that preserve this symmetry, and deform the theory into a self-interacting scalar field theory with the shift symmetry of degree P? To answer this (essentially cohomological) question, we develop a new graph-theoretical technique, and use it to prove several classification theorems. First, in the special case of P = 1 (essentially equivalent to Galileons), we reproduce the known Galileon N-point invariants, and find their novel interpretation in terms of graph theory, as an equal-weight sum over all labeled trees with N vertices. Then we extend the classification to P > 1 and find a whole host of new invariants, including those that represent the most relevant (or least irrelevant) deformations of the corresponding Gaussian fixed points, and we study their uniqueness.