Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Matter wave functions and Yukawa couplings in F-theory Grand Unification

A. Font, L. E. Ibanez

TL;DR

This work develops a local, bottom-up description of matter wavefunctions and Yukawa couplings in F-theory GUTs by solving the 8D equations on 7-branes and expanding the zero-mode profiles in magnetic fluxes and their derivatives. Yukawa couplings arise from overlapping three localized wavefunctions and are computed analytically as triple overlaps, yielding explicit expressions up to second order in the flux-derivative expansion and revealing a natural hierarchy controlled by U(1) charges and normalization. The results show that flux distortions on bulk and matter curves generate nontrivial Yukawa textures, with Y_{33} typically dominant and lighter-generation entries suppressed by flux parameters, while higher-derivative flux terms can be important for D-quarks and CKM structure. The framework provides a concrete mechanism to connect local geometric data to SM flavor hierarchies and CKM mixing, and suggests extensions to include closed-string fluxes, warping, and SUSY-breaking effects in F-theory unification.

Abstract

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.

Matter wave functions and Yukawa couplings in F-theory Grand Unification

TL;DR

This work develops a local, bottom-up description of matter wavefunctions and Yukawa couplings in F-theory GUTs by solving the 8D equations on 7-branes and expanding the zero-mode profiles in magnetic fluxes and their derivatives. Yukawa couplings arise from overlapping three localized wavefunctions and are computed analytically as triple overlaps, yielding explicit expressions up to second order in the flux-derivative expansion and revealing a natural hierarchy controlled by U(1) charges and normalization. The results show that flux distortions on bulk and matter curves generate nontrivial Yukawa textures, with Y_{33} typically dominant and lighter-generation entries suppressed by flux parameters, while higher-derivative flux terms can be important for D-quarks and CKM structure. The framework provides a concrete mechanism to connect local geometric data to SM flavor hierarchies and CKM mixing, and suggests extensions to include closed-string fluxes, warping, and SUSY-breaking effects in F-theory unification.

Abstract

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.

Paper Structure

This paper contains 23 sections, 131 equations, 4 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Review of F-theory unification
  3. Bulk fields
  4. Fields at intersections
  5. Zero modes at intersecting matter curves
  6. Zero modes in the absence of fluxes
  7. Zero modes with fluxes
  8. Zero modes with constant flux
  9. Zero modes with variable fluxes
  10. Evaluating the fluxes
  11. Yukawa couplings
  12. Computing Yukawa couplings
  13. The case of a constant $\chi_c$ wave function
  14. Yukawa matrices
  15. Fermion Yukawa couplings in F-theory GUT's
  16. ...and 8 more sections

Figures (4)

  • Figure 1: Intersecting matter curves
  • Figure 2: Triple intersection of three matter curves.
  • Figure 3: Intersecting matter curves and Yukawa couplings.
  • Figure 4: Matter curves intersecting to provide $U$-quark Yukawas.