Flavor structure from misalignment of inner products in noncommutative geometry

In this letter, we consider an idea that induces flavor structure from inner products in noncommutative geometry. Assuming proper components of vectors $v_{(L,R)i}$ in enlarged representation space for fermions, we can induce the waterfall texture for Yukawa matrices retaining gauge interactions are universal. The hierarchy of the Yukawa interactions is a consequence of “misalignment” between the vectors $v_{Li}$ and $v_{Rj}$.