Scalar Field Theories with Polynomial Shift Symmetries

TL;DR

The paper addresses naturalness in Lifshitz-type nonrelativistic QFTs by studying scalar fields with polynomial shift symmetries, aiming to classify the lowest-dimension, symmetry-preserving interactions that deform Gaussian fixed points into self-interacting theories. It introduces a novel graph-theoretic framework to map P-invariants (invariants under degree-$P$ polynomial shifts) to diagrams, reproducing the known Galileon N-point invariants as equal-weight sums of labeled trees and generalizing to higher-degree shifts. Key contributions include: (i) a complete graphical method for constructing minimal invariants and proving their uniqueness for several $(P,N)$, (ii) the discovery of Medusa- and spider-based structures that organize total-derivative relations, and (iii) a superposition principle that builds higher-$P$ invariants from lower-degree data, with a conjecture that all P-invariants arise from exact invariants plus minimal loopless 1-invariants. The work illuminates how polynomial shift symmetries can protect multicritical dispersion relations, enable cascading fixed points, and refine nonrelativistic Goldstone theory, with potential implications for nonrelativistic quantum gravity and beyond.

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.

Figures (5)

• Figure 1: The most relevant 2-invariant for $N = 4$ from superposition of graphs.
• Figure 2: Superposition of graphs in \ref{['eq: n=4 star and path']} on trees in Figure \ref{['fig: spanning trees']}.
• Figure 3: Examples for the graphical representation of algebraic expressions.
• Figure 4: Two different linear combinations of $\star$-graphs result in an identical $\times$-relation for $P = 2$. In particular, the $\times$-graph with a coefficient 2 is associated with all three $\star$-graphs in the figure.
• Figure 5: All 1-invariant terms up to total derivatives, with $N =3$, $\Delta=4$ and $P =1$.

Theorems & Definitions (96)

• Definition 1
• Definition 2
• Definition 3: Variation Map
• Definition 4
• Definition 5: Derivative Map
• Definition 6: Relations
• Definition 7: $P$-Invariant
• Definition 8: Exact $P$-Invariant
• Definition 9: Associations
• Proposition 1