Schwinger’s Nonet Mass Formula Revisited

We prove the validity of Schwinger’s quartic nonet mass formula for a general nonet structure with the isoscalar singlet mass being shifted from its “ideal” value. We apply this formula to the problem of the correct qq̄ assignment for the scalar meson nonet. The results favor scalar isoscalar singlet and isoscalar octet masses in the vicinity of 1 GeV and 1.45 GeV, respectively. We explain the failure of Schwinger’s formula for the pseudoscalar meson nonet, and suggest a new version of this formula, modified by the inclusion of the pseudoscalar decay constants, which holds with improved accuracy for this nonet. We also rederive the new Gell-Mann–Okubo mass formula 3M2(nn̄, I = 1) + 2M2(ss̄) = 4M2(sn̄) +M2(nn̄, I = 0) (n = u, d), suggested in our previous publications.