Systematic operator construction for non-relativistic effective field theories: Hilbert series versus Young tensor

A bstract 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 O(Q⁴) O Q 4 and three-nucleon interactions at O(Q²) O Q 2 ; and finally the spin-1/2 dark matter-nucleon couplings up to O(v⁴) O v 4 .