The Minimal Flavor Structure of Quarks and Leptons

A flavor structure with minimal parameters is proposed to address the fermion mass hierarchy and flavor mixing for quarks and leptons. Yukawa interaction is reconstructed in a new basis to show a homological flat structure for up-type quarks, down-type quarks, charged leptons and Dirac neutrinos. A $SO(2)_{LR}^f$ flavor symmetry is found from the hierarchy masses of quarks and leptons, which dominated CKM mixing for quarks and PMNS for leptons. Since the minimal flavor structure successfully addresses CKM and PMNS even in the mass hierarchy limit, mass hierarchy and flavor mixing are two independent questions. As a prediction, a sum rule on the mixing angles and CP violation phase is suggested, which explains the smallness of $s_{13}$ as a natural result of the mass hierarchy. Generalizing the flat structure to quarks and leptons, a unified Yukawa interaction is achieved for all fermions with only a single coupling.