Proton Decay, Yukawa Couplings and Underlying Gauge Symmetry in String Theory

TL;DR

This work argues that Yukawa couplings in string theory should originate from a single large gauge symmetry G that contains the SU(5) GUT, while prohibiting dimension-4 proton-decay operators by separating the origins of ar{f 5} and ar{H}(ar{f 5}) in the coset g/h. A Lie-algebraic bottom-up analysis shows E7 as the minimal viable symmetry, with E8 as an alternative, and this structure is realized concretely in Heterotic, M-theory, and F-theory vacua through reducible rank-5 vector bundles and local ALE-fibration models. In each framework, all Yukawa couplings descend from G Yang–Mills interactions, and proton decay operators are suppressed or forbidden by the residual symmetries and bundle data, with non-perturbative effects potentially contributing only at suppressed levels. The paper further develops intrinsic F-theory language for matter curves and chirality, connecting geometric data to the low-energy spectrum and Yukawa textures, and highlighting how different symmetry-breaking patterns yield testable constraints on the allowed EFT operators and neutrino sector. Overall, it links high-energy gauge structure to concrete low-energy operator content across string theory corners, offering a robust, model-agnostic pathway to realistic Yukawas with stable protons.

Abstract

In string theory, massless particles often originate from a symmetry breaking of a large gauge symmetry G to its subgroup H. The absence of dimension-4 proton decay in supersymmetric theories suggests that (\bar{D},L) are different from \bar{H}(\bar{\bf 5}) in their origins. In this article, we consider a possibility that they come from different irreducible components in $\mathfrak{g}/\mathfrak{h}$. Requiring that all the Yukawa coupling constants of quarks and leptons be generated from the super Yang--Mills interactions of G, we found in the context of Georgi--Glashow H=SU(5) unification that the minimal choice of G is E_7 and E_8 is the only alternative. This idea is systematically implemented in Heterotic String, M theory and F theory, confirming the absence of dimension 4 proton decay operators. Not only H=SU(5) but also G constrain operators of effective field theories, providing non-trivial information.

Figures (5)

• Figure 1: The (extended) Dynkin diagram of $E_6$, which not only describes the Lie algebra of $E_6$ symmetry but also the intersection form of $E_6$-type singularity. Each node is assigned a simple root $\alpha_i$ and a 2-cycle $C_i$. The $\mathfrak{su}(2)_2$ and $\mathfrak{su}(5)_{\rm GUT}$ subalgebra of $\mathfrak{su}(2)_2+\mathfrak{u}(1)+\mathfrak{su}(5)_{\rm GUT} \subset \mathfrak{e}_6$ in section \ref{['sec:BottomUp']} are generated by $\alpha_1$ and $\alpha_{3,4,5,6}$, respectively. The highest root $\theta$ is not linearly independent from other six simple roots: $\theta=\alpha_1+2\alpha_2+3\alpha_3+2\alpha_4+\alpha_5+2\alpha_6$.
• Figure 2: The (extended) Dynkin diagram of $E_7$. The $\mathfrak{su}(2)_2$ is generated by $\alpha_1$, and $\mathfrak{su}(6)_1$ by $\alpha_{3,4,5,6,7}$ ($\mathfrak{su}(5)_{\rm GUT}$ without $\alpha_7$). The highest root $\theta$ satisfies $-\theta + 2\alpha_1+3\alpha_2+4\alpha_3+3\alpha_4+2\alpha_5+\alpha_6 +2\alpha_7=0$.
• Figure 3: The extended Dynkin diagram of $E_8$ (I). The subalgebra $\mathfrak{su}(2)_2+\mathfrak{su}(2)_1+\mathfrak{su}(6)_1$ is generated by the simple roots $\alpha_1$, $-\theta$, and $\alpha_{3,4,5,6,7,8}$, respectively.
• Figure 4: The extended Dynkin diagram of $E_8$ (II), and its maximal subalgebra $\mathfrak{so}(10)+\mathfrak{su}(4)$. $\alpha_+ \equiv L_0-(L_3+L_4+L_5)$.
• Figure 5: Extended Dynkin diagram of $E_8$ (III), and its maximal subalgebra $\mathfrak{su}(3)_2 + \mathfrak{su}(2)_2+\mathfrak{su}(6)_2$. $\tilde{\alpha} \equiv 2L_0 - (L_1 +\cdots + L_6)$.