  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.
