Scalar Field Theories with Polynomial Shift Symmetries

Abstract

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.

Extended Galilean symmetries of non-relativistic strings

“…40) 41) and yield the primary first class constraints [38] 43) with exactly the same equal-τ Poisson brackets (3.7) of the non-vibrating case. The canonical action is …”

“…An infinite set of non-central extensions in the algebra of conserved charges is also obtained. If should be emphasized that among the symmetries there is an infinite set of polynomial shift symmetries [41,42]. The presence of polynomial shift symmetries has been noticed in Galileon theories [43].…”

Cascading Multicriticality in Nonrelativistic Spontaneous Symmetry Breaking

“…[7,8], we showed that this type A-B dichotomy is further refined into two discrete families, labeled by a positive integer n: type A n NG modes are described by a single scalar with dispersion ω ∼ k n (and dynamical critical exponent z ¼ n), while type B 2n modes are described by a canonical pair and exhibit the dispersion relation ω ∼ k 2n (and dynamical exponent z ¼ 2n). These two families are technically natural, and therefore stable under renormalization in the presence of interactions [7].…”

“…Much progress in SSB has also been achieved in the nonrelativistic cases, where the reduced spacetime symmetries allow a much richer behavior, still very much the subject of active research (see, e.g., Refs. [2][3][4][5][6][7][8] and the references therein). Important novelties emerge already in the simplest case of theories in the flat nonrelativistic spacetime R Dþ1 [covered with Cartesian coordinates ðt; xÞ, x ≡ ðx i ; i ¼ 1; …; DÞ] and with the Lifshitz symmetries of spatial rotations and spacetime translations.…”

“…The lowest, least irrelevant interaction terms invariant under the polynomial shift were systematically discussed in Ref. [8] (see also Ref. [9]).…”