Hilbert Series of Pseudoscalar Mesons: Operators and Sum Rules
We construct the complete non-redundant basis of operators built from pseudoscalar mesons using the Hilbert series method, subject to the global flavor symmetry $SU(3)_L \times SU(3)_R \times SU(2)_{L_H} \times SU(2)_{R_H}$, together with the underlying shift symmetry. The SM weak interactions are incorporated systematically by promoting the CKM matrix elements to spurion fields transforming under the global symmetry. The resulting operator basis provides a model-independent parametrization of two- and three-body weak decay amplitudes of heavy mesons, with the corresponding Wilson coefficients serving as reduced hadronic amplitudes. We demonstrate that all amplitude sum rules follow directly from the structure of the operator basis and propose an unfolding procedure that connects the effective operators to their possible ultraviolet origins. Our framework reproduces the known flavor relations while predicting additional identities arising from the symmetry-constrained operator basis.