A Generalized Gell-Mann-Nishijima Relation for Quark and Hadron Modelling

The electric charge of quarks is conventionally related to weak isospin and hypercharge through the Gell-Mann-Nishijima (GNN) relation. While this relation successfully reproduces the observed quark charges, the electroweak quantum numbers are introduced as representation labels of the gauge symmetry with no deeper physical motivation. In this work a generalized formulation is proposed in which the baryon number remains universal for all quarks and the distinction between leftand right-handed quarks is carried by a single chirality-dependent topological parameter identified as a vorticity quantum number Ω. Left-handed quarks carry zero vorticity; their doublet structure emerges from the interplay of intrinsic charge and universal baryon number alone. Right-handed quarks, being weak-isospin singlets, carry a flavour-dependent non-zero vorticity determined by the singlet condition. The proposed relation reproduces the observed electric charges and weak-isospin assignments of the first-generation quarks while maintaining a common baryonic contribution for every quark. The formulation is extended to light hadrons, demonstrating its applicability beyond the quark sector. Being independent of any particular microscopic model, it may serve as a generic parametrization of quark and hadron quantum numbers, with Ω providing a geometric and topological interpretation of chirality.