F-theory and Neutrinos: Kaluza-Klein Dilution of Flavor Hierarchy

TL;DR

The paper develops minimal F-theory GUT realizations of both Majorana and Dirac neutrinos in which neutrino masses arise from integrating out KK modes, yielding $m_ u \sim M_{\text{weak}}^2 / M_{\text{UV}}$ with $M_{\text{UV}}$ near the GUT scale. A key insight is that non-holomorphic KK wavefunctions dilute the flavor hierarchy, producing a normal hierarchy with $m_3:m_2:m_1 \sim 1:\alpha_{\text{GUT}}^{1/2}:\alpha_{\text{GUT}}$ and a PMNS matrix with large $\theta_{12},\theta_{23}$ and a relatively large $\theta_{13}$ around the Cabibbo angle, especially when neutrino and charged-lepton sectors unify at an $E_8$ point. The authors explore geometric realizations including monodromies, ${\mathbb{Z}}_2$ and more general finite-group quotients, and discuss how $U(1)_{PQ}$ and matter parity interplay with the neutrino sector. They also analyze Dirac scenarios with higher dimensional operators yielding similar Yukawa structures and quantify how KK modes adjust normalization via Green's functions. Overall, the work demonstrates that neutrino flavor can be naturally embedded in a highly constrained F-theory GUT framework with testable implications for mixing, mass scales, and neutrinoless double beta decay.

Abstract

We study minimal implementations of Majorana and Dirac neutrino scenarios in F-theory GUT models. In both cases the mass scale of the neutrinos m_nu ~ (M_weak)^2/M_UV arises from integrating out Kaluza-Klein modes, where M_UV is close to the GUT scale. The participation of non-holomorphic Kaluza-Klein mode wave functions dilutes the mass hierarchy in comparison to the quark and charged lepton sectors, in agreement with experimentally measured mass splittings. The neutrinos are predicted to exhibit a "normal" mass hierarchy, with masses m_3,m_2,m_1 ~ .05*(1,(alpha_GUT)^(1/2),alpha_GUT) eV. When the interactions of the neutrino and charged lepton sectors geometrically unify, the neutrino mixing matrix exhibits a mild hierarchical structure such that the mixing angles theta_23 and theta_12 are large and comparable, while theta_13 is expected to be smaller and close to the Cabibbo angle: theta_13 ~ theta_C ~ (alpha_GUT)^(1/2) ~ 0.2. This suggests that theta_13 should be near the current experimental upper bound.

Figures (10)

• Figure 1: Quiver diagram of the field theory associated to the Kaluza-Klein seesaw in the covering theory (left) and the quotient theory (right).
• Figure 2: Depiction of a minimal implementation of the Kaluza-Klein seesaw in which the fields of the covering theory are identified in the quotient theory. This geometrical action also identifies the two interaction points of the covering theory.
• Figure 3: Depiction of the $SU(7)$ toy model described in subsection \ref{['TOY']}. In the covering theory (left) $\widetilde{H}_{u}$ and $\widetilde{L}^{\prime}$ localize on the same curve, and the same is true for $\widetilde{H}_{u}^{\prime}$ and $\widetilde{L}$. As a consequence, in the quotient theory (right) $H_{u}$ and $L$ localize on the same matter curve.
• Figure 4: Depiction of the matter curves in the Kaluza-Klein seesaw associated with an $E_{8}$ intersection point. As opposed to the matter curve configuration of figure \ref{['su7maj']}, here $H_{u}$ and $L^{\prime}$ localize on different curves in the covering theory (left). In the quotient theory (right), $H_{u}$ and $L$ localize on two distinct matter curves.
• Figure 5: Depiction of a minimal F-theory GUT with a Majorana neutrino sector. In this case, the Higgs up curve forms a triple intersection with the lepton doublet curve and the right-handed neutrino curve. Integrating out the massive right-handed neutrino states generates the quartic operator $(H_{u}L)^{2}/\Lambda_{\text{UV}}$ in the low energy effective theory.