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

TL;DR

This paper develops a systematic method combining Hilbert series and Young tensor techniques to construct operator bases in non-relativistic effective field theories, accounting for symmetries and redundancies.

Contribution

It introduces a unified framework using Hilbert series and Young tensors for operator construction in non-relativistic EFTs, extending existing methods to include spin and internal symmetries.

Findings

Complete operator bases up to dimension 9 for heavy particles and quarks.

Operator bases for pion-less EFT including nucleon interactions up to old4.

Constructed spin-1/2 dark matter-nucleon operators up to old4.

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 $\mathcal{O}(v^4)$.