Flavor structure in F-theory compactifications

Abstract

F-theory is one of frameworks in string theory where supersymmetric grand unification is accommodated, and all the Yukawa couplings and Majorana masses of righthanded 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 lowenergy 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 × 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.

Compact T-branes

“…integrates to zero, i.e. 10) JHEP09 (2017)132 which is nothing but the condition for vanishing D-term potential in the four-dimensional low-energy effective theory. Practically, equation (2.9) can only be solved analytically in few situations, because in general ϕ will depend non-linearly on g. Nevertheless this equation is always of elliptic type [13] and, as such, on a compact manifold it admits a unique smooth solution if the input function ϕ is smooth and provided that (2.10) is satisfied [38].…”

Phenomenological implications of a minimal F-theory GUT with discrete symmetry

“…This symmetry will be of the Z N type, which may serve as a matter parity, preventing unwanted dimension four proton decay operators. In this subsection, we give a brief introduction to this geometric symmetry, for more details see appendix D, as has been discussed in [65]. Given that, up to a phase, the spectral cover equation is invariant under:…”

“…We will present such a mechanism (firstly proposed in [65] and implemented on specific GUT constructions in [78]) in what follows.…”

“…A convenient simplification is based on the spectral cover description where the Higgs is replaced by its eigenvalues and the bundle by the corresponding eigenvectors. Since our primary interest is the reduction of E 8 to SU(5) × SU(5) ⊥ we focus in SU(5) group where the spectral surface is described by the equation: For a Z 2 symmetry discussed in [65] one requires a Z 2 background configuration, with a Z 2 action so that the mapping is: To see if this is a symmetry of the local geometry for a given divisor, we take local coordinates for the three trivialization patches. These can be defined as (t 1 , u 1 ) = (T /S, U/S),…”

MSSM from F-theory SU(5) with Klein monodromy

“…A significant part of the present work will be devoted to the viability of the corresponding two kinds of effective models. Another ingredient related to the predictability of the model, is the implementation of R-parity conservation, or equivalently a Z 2 Matter Parity, which can be realised with the introduction of new geometric symmetries [11] respected from the spectral cover. In view of these interesting features, we also investigate in more detail the superpotential, computing higher non-renormalisable corrections, analysing the D and F-flatness conditions and so on.…”