Flavor structure in F-theory compactifications

F-theory is one of frameworks in string theory where supersymmetric grand unification is accommodated, and all the Yukawa couplings and Majorana masses of right-handed neutrinos are generated. Yukawa couplings of charged fermions are generated at codimension-3 singularities, and a contribution from a given singularity point is known to be approximately rank 1. Thus, the approximate rank of Yukawa matrices in low-energy effective theory of generic F-theory compactifications are minimum of either the number of generations N-gen = 3 or the number of singularity points of certain types. If there is a geometry with only one E-6 type point and one D-6 type point over the entire 7-brane for SU(5) gauge fields, F-theory compactified on such a geometry would reproduce approximately rank-1 Yukawa matrices in the real world. We found, however, that there is no such geometry. Thus, it is a problem how to generate hierarchical Yukawa eigenvalues in F-theory compactifications. A solution in the literature so far is to take an appropriate factorization limit. In this article, we propose an alternative solution to the hierarchical structure problem (which requires to tune some parameters) by studying how zero mode wavefunctions depend on complex structure moduli. In this solution, the N-gen x N-gen CKM matrix is predicted to have only N-gen entries of order unity without an extra tuning of parameters, and the lepton flavor anarchy is predicted for the lepton mixing matrix. The hierarchy among the Yukawa eigenvalues of the down-type and charged lepton sector is predicted to be smaller than that of the up-type sector, and the Majorana masses of left-handed neutrinos generated through the see-saw mechanism have small hierarchy. All of these predictions agree with what we observe in the real world. We also obtained a precise description of zero mode wavefunctions near the E-6 type singularity points, where the up-type Yukawa couplings are generated.