Yukawa hierarchies at the point of $E_8$ in F-theory

TL;DR

This study shows that Yukawa couplings in local SU(5) F-theory GUTs with an $E_8$ enhancement can be arranged to reproduce realistic fermion masses and mixings. By analyzing several $E_8$-based T-brane configurations, the authors identify a two-by-two plus one (2+2+1) model that yields a holomorphic Yukawa hierarchy of the form $(\mathcal{O}(1),\mathcal{O}(\epsilon),\mathcal{O}(\epsilon^2))$ which, together with non-perturbative effects and flux-induced wavefunction normalizations, produces physical Yukawas consistent with data for $\epsilon\sim 10^{-4}$ and large $\tan\beta$. The approach ties the CKM structure to the separation of two Yukawa points within S$_{GUT}$ and demonstrates that realistic charged fermion masses can be obtained in at least one $E_8$ model, with implications for constraints on local F-theory GUT constructions. The work also highlights the importance of the local geometry around the $E_8$ point and motivates embedding these local results into global string compactifications to connect flavour physics with moduli and fluxes."

Abstract

We analyse the structure of Yukawa couplings in local SU(5) F-theory models with $E_8$ enhancement. In this setting the $E_8$ symmetry is broken down to SU(5) by a 7-brane configuration described by T-branes, all the Yukawa couplings are generated in the vicinity of a point and only one family of quarks and leptons is massive at tree-level. The other two families obtain their masses when non-perturbative effects are taken into account, being hierarchically lighter than the third family. However, and contrary to previous results, we find that this hierarchy of fermion masses is not always appropriate to reproduce measured data. We find instead that different T-brane configurations breaking $E_8$ to SU(5) give rise to distinct hierarchical patterns for the holomorphic Yukawa couplings. Only some of these patterns allow to fit the observed fermion masses with reasonable local model parameter values, adding further constraints to the construction of F-theory GUTs. We consider an $E_8$ model where such appropriate hierarchy is realised and compute its physical Yukawas, showing that realistic charged fermions masses can indeed be obtained in this case.

Figures (4)

• Figure 1: Ratio of ratios (\ref{['ratiosa']}) where the horizontal axis is $x =-M_1/\tilde{N}_Y$ and the vertical is $y=(M_1+2M_2)/\tilde{N}_Y$. The right figure displays the value for $|d| = 0.1$ and the left one for $|d| = 0.02$. In the latter, the red dot corresponds to the values chosen in (\ref{['values2']}).
• Figure 2: Ratio (\ref{['ratiosb']}) with $x =-M_1/\tilde{N}_Y$, $y=(M_1+2M_2)/\tilde{N}_Y$. The red dot represents (\ref{['values2']}).
• Figure 3: Ratio $m_s/m_b$ that fixes the order of magnitude of $|\tilde{{\epsilon}} \tilde{N}_Y/2\rho_\mu|$.
• Figure 4: Ratio of $\tau$ and $b$ mass. The red dot corresponds to the values chosen in (\ref{['values2']}).