Fitting fermion masses and mixings in F-theory GUTs

TL;DR

This work demonstrates that realistic fermion masses and CKM mixing can be obtained in ultra-local $SU(5)$ F-theory GUTs with $E_7$ enhancement by combining a non-commuting (T-brane) Higgs background with non-perturbative superpotential corrections. The authors derive holomorphic Yukawas via a residue formula, then incorporate local normalization to produce physical Yukawas, finding a hierarchical pattern controlled by the small parameter $oldsymbol{ ilde extε}$ and the Yukawa-point separation $oldsymbol{ extκ}$. They identify two viable matter-curve embeddings (Model A and Model B) and perform a detailed fit to MSSM fermion masses and CKM angles, showing that Model A can realize the observed spectrum over a broad parameter range, while Model B cannot due to chirality constraints. The results suggest a universal flavour structure in local F-theory GUTs that also applies to local $E_8$ enhancements, with implications for global model-building and the neutrino sector, albeit requiring further work on global embeddings and neutrino couplings.

Abstract

We analyse the structure of Yukawa couplings in local SU(5) F-theory models with $E_7$ enhancement. These models are the minimal setting in which the whole flavour structure for the MSSM charged fermions is encoded in a small region of the entire compactification space. In this setup the $E_7$ symmetry is broken down to SU(5) by means of a 7-brane T-brane background, and further to the MSSM gauge group by means of a hypercharge flux that also implements doublet-triplet splitting. At tree-level only one family of quarks and charged leptons is massive, while the other two obtain hierarchically smaller masses when stringy non-perturbative effects are taken into account. We find that there is a unique $E_7$ model with such hierarchical flavour structure. The relative simplicity of the model allows to perform the computation of Yukawa couplings for a region of its parameter space wider than previous attempts, obtaining realistic fermion masses and mixings for large parameter regions. Our results are also valid for local models with $E_8$ enhancement, pointing towards a universal structure to describe realistic fermion masses within this framework.

Figures (7)

• Figure 1: On the left the region in the $x-y$ plane for the ratio of masses \ref{['eq:ratio1']} compatible with the realistic value in (\ref{['rvalues']}). On the right the region in the $x-y$ plane for the ratio of masses \ref{['eq:ratio2']} compatible with (\ref{['rvalues']}), for different values of $\hat{d}$.
• Figure 2: Region in the plane $x-\tilde{{\epsilon}} \tilde{N}_Y/\mu_1^2$ for the ratio \ref{['eq:ratio3']} to be compatible with the range of values in table \ref{['tab:massas']}.
• Figure 3: Value of the ratio of ratios $(m_\mu/m_\tau)/(m_s/m_b)$ in the model B in the $\tilde{N}_Y - M_t$ plane, where are taking $M_t =- M_1 - \tilde{N}_Y > 0$ and $\tilde{N}_Y >0$ as dictated by eq.(\ref{['condBmodel']}).
• Figure 4: Region in the $x-\tilde{N}_Y$ plane with a ratio $Y_\tau/Y_b$ compatible with table \ref{['tab:massas']}.
• Figure 5: Regions in the $m$-$\tilde{N}_Y$ plane where all constraints are fulfilled for different values of $c$.