Fermion masses and flavor mixings and strong CP problem

For all the success of the Standard Model (SM), it is on the verge of being surpassed. In this regard we argue, by showing a minimal flavor-structured model based on the non-Abelian discrete $S{L}_2\left(F_3\right)$ symmetry, that $U\left(1\right)$ mixed-gravitational anomaly cancellation could be of central importance in constraining the fermion contents of a new chiral gauge theory. Such anomaly-free condition together with the SM flavor structure demands a condition $k_1{X}_1/2=k_2{X}_2$ with $X_i$ being a charge of $U{\left(1\right)}_{X_i}$ and $k_i$ being an integer, both of which are flavor dependent. We show that axionic domain-wall condition $N_{\mathrm{DW}}$ with the anomaly free-condition depends on both $U{\left(1\right)}_X$ charged quark and lepton flavors; the seesaw scale congruent to the scale of Peccei–Quinn symmetry breakdown can be constrained through constraints coming from astrophysics and particle physics. Then the model extended by $S{L}_2\left(F_3\right)\times U{\left(1\right)}_X$ symmetry can well be flavor-structured in a unique way that $N_{\mathrm{DW}}=1$ with the $U{\left(1\right)}_X$ mixed-gravitational anomaly-free condition demands additional Majorana fermion and the flavor puzzles of SM are well delineated by new expansion parameters expressed in terms of $U{\left(1\right)}_X$ charges and $U{\left(1\right)}_X$-${\left[SU{\left(3\right)}_C\right]}^2$ anomaly coefficients. And the model provides remarkable results on neutrino (hierarchical mass spectra and unmeasurable neutrinoless-double-beta decay rate together with the predictions on atmospheric mixing angle and leptonic Dirac CP phase favored by the recent long-baseline neutrino experiments), QCD axion, and flavored-axion.