  We study the local structure of zero mode wave functions of chiral matter
fields in F-theory unification. We solve the differential equations for the
zero modes derived from local Higgsing in the 8-dimensional parent action of
F-theory 7-branes. The solutions are found as expansions both in powers and
derivatives of the magnetic fluxes. Yukawa couplings are given by an overlap
integral of the three wave functions involved in the interaction and can be
calculated analytically. We provide explicit expressions for these Yukawas to
second order both in the flux and derivative expansions and discuss the effect
of higher order terms. We explicitly describe the dependence of the couplings
on the U(1) charges of the relevant fields, appropriately taking into account
their normalization. A hierarchical Yukawa structure is naturally obtained. The
application of our results to the understanding of the observed hierarchies of
quarks and leptons is discussed.
