The shape of quark flavors

We construct the Yukawa couplings of the quark sector as overlap integrals of Gaussian wave functions in extra spatial dimensions. Assuming that the wave function of each chiral fermion is localised at a point in the extra dimensions, while that of the Higgs doublet is flat, the Yukawa matrix elements are determined by the relative positions of the three left-handed quark doublets, $q_1$, $q_2$, $q_3$, and the six right-handed quark singlets, $u_1$, $u_2$, $u_3$, $d_1$, $d_2$, $d_3$. We employ numerical optimisation techniques from machine learning to identify field configurations that satisfy all current experimental constraints on the quark Yukawa sector. In this framework, the large hierarchy of quark masses arises naturally, since the overlap integrals of Gaussian wave functions are exponentially sensitive to ${\mathcal O}(1)$ separations in the extra dimensions.