Quark and Lepton Masses and Mixing Angles from Superstring Constructions

We show that the observed structure of quark and lepton masses and mixing angles can arise entirely geometrically from superstring constructions, at the renormalizable level. The model we consider is a $Z_3$ orbifold compactification of heterotic string with two Wilson lines, where three families of particles of $SU(3)_c\times SU(2)_L\times U(1)_Y$, including Higgses, are automatically present. In orbifold models, Yukawa couplings can be calculated explicitly, and it is known that they get exponential supression factors depending on the distance between the fixed points to which the fields are attached. We find that in the $Z_3$ case, the quark and charged-lepton mass hierarchies can easily be obtained for reasonable values of the three moduli determining the radii of the compactified space, $T_i\sim 1$. For the neutrinos, due to the smallness of their Dirac masses, the required scale for the see-saw mechanism to give the correct masses is found to be within reach of the electroweak scale. Finally, we find that one of the small number of possibilities for quark and lepton mass matrices yields consistent results for the mixing angles and the weak CP violation phase. Although our scheme relies on the mixing between fields due to Fayet-Iliopoulos breaking, it is considerably more predictive than alternative models of flavour.