Generalized Schwinger Mass Formula

We generalize Schwinger’s original mass formula to the case of an additional isosinglet which mixes with the nonet mesons, by considering the corresponding 3×3 mass matrix in the most general case. We then make further generalization to either (i) an arbitrary number of additional isosinglets mixing with nonet mesons, or (ii) an arbitrary number of mesons with common J mixing with an additional isosinglet. In the former case, we present an explicit relation, while in the latter case, we show by numerical example that the new mass formula is only weakly affected by the inclusion of additional mesons, and hence holds with good accuracy for each of the 3× 3 mass sub-matrices.