A relaxed version of the Koide formula from flavor nonets in the scalar potential model or the Yukawaon model.

We present a relaxed version of the Koide formula from a scalar potential model or a Yukawaon model, based on scalar fields set up in a nonet representation of the SU(3) flavor symmetry of the Standard Model. The Koide's character, which involves the Standard Model fermion mass ratios, is derived from the vacuum expectation value of the nonet field in either model. The scalar potential in the scalar potential model or the superpotential in the Yukawaon model is constructed with all terms invariant under symmetries. The resulting Koide's character, which is modified by two effective parameters, can fit the experimental data of masses of charged leptons, up quarks and down quarks. It offers a natural interpretation of the Standard Model fermion mass spectra.