Systematic Operator Construction for Non-relativistic Effective Field Theories: Hilbert Series versus Young Tensor

TL;DR

The paper develops a unified, systematic approach to constructing non-relativistic EFT operator bases by combining a Hilbert-series counting framework (with the Lifshitz scaling to organize NR derivatives) and an extended Young tensor method using SU(2) SSYTs to build explicit, nonredundant operator sets. It resolves spin-related redundancies by employing the Spin(3) double cover, avoiding explicit spin operators, and demonstrates the equivalence between the Hilbert-series counts and the operator bases generated via the SSYT/Young tableau method across HPET, HQET, pionless EFT, and DM–nucleon interactions. The authors produce complete operator bases up to mass dimension nine for HPET and HQET, NN contact interactions up to $ ext{O}(Q^4)$ and 3N up to $ ext{O}(Q^2)$ in pionless EFT, and spin-1/2 DM–nucleon operators up to $ ext{O}(v^4)$, with explicit CP/T classification. This framework enables automated, redundancy-free operator enumeration, precise matching to relativistic theories, and scalable extension to other NR EFTs, offering a robust tool for EFT analyses and model-building in low-energy hadronic and dark-mell contexts.

Abstract

This work establishes a systematic framework for operator construction in the non-relativistic effective field theory, incorporating both the three dimensional Euclidean symmetry and the internal symmetries. By employing double cover of the rotation group, we extend the Hilbert series to the non-relativistic systems, and eliminates redundancies introduced by the spin operator. We also generalize the Young tensor method to the non-relativistic cases through the $SU(2)$ semi-standard Young tableaux, which allows for the construction of operator bases with repeated fields at any given mass dimension. Utilizing the Young tensor technique and Hibert series as cross-check, we obtain the complete operator bases for the following cases: heavy particle (and also heavy quark) effective theory operators up to mass dimension 9; pion-less effective theory operators, including nucleon-nucleon contact interactions up to $\mathcal{O}(Q^4)$ and three-nucleon interactions at $\mathcal{O}(Q^2)$; and finally the spin-1/2 dark matter-nucleon operators up to $\mathcal{O}(v^4)$.

Figures (4)

• Figure 1: This figure depicts the relationships among the Poincaré group, the Galilean group, and the group $\mathbb{R}^1\times \text{E}(3)$. The Galilean group is obtained from the Poincaré group via an Inönü-Wigner contraction, while the subgroup $\mathbb{R}^1\times \text{E}(3)$ arises from the Poincaré group through velocity projection. By imposing the corresponding boost invariance on the rotation group, the full Poincaré or Galilean symmetry can be restored.
• Figure 2: Schematic illustration of the manifold structures of $SU(2)$ and $SO(3)$ groups using their two-dimensional analogues. The $SO(3)$ group manifold is the real projective space $\mathbb{RP}^3$, obtained from the $S^3$ manifold of $SU(2)$ by identifying antipodal points. Although the actual manifolds are three-dimensional, this two-dimensional representation using $S^2$ and $\mathbb{RP}^2$ symbolically demonstrates the antipodal identification process: the blue region represents the resulting projective space after identifying opposite points on the sphere. Points $A$ and $B$ (antipodal on the sphere) are identified and projected to $A'$ in the projective region; similarly, $C$ and $C'$ are identified as $C'$ in the projective region.
• Figure 3: The DM-nucleon Contact Interaction.
• Figure 4: Schematic flowchart of the procedure to derive the non-relativistic operator bases in this work. The Euclidean group is the common subgroup of both the Poincaré group and the Galilean group. The Lifshitz algebra is used to obtain the NR Hilbert series, while the semi-standard Young tableaux of the $SU(2)$ group are utilized to explicitly construct the NR operators.