Preons, Braid Topology, and Representations of Fundamental Particles

TL;DR

The paper develops a representation-theoretic framework for preon ideas, focusing on Bilson-Thompson's helon and Lambek's 4-vector approaches. It shows that helon braids realize on-shell Lorentz spinor states with crossings encoding chirality and twists encoding charge, and maps these to the weight lattice of $SU(3)_c times U(1)_{em}$, while introducing a 5-vector representation from $D_2 ⊕ A_2 ⊕ A_1$ that unifies chirality with interactions. It demonstrates color-matched braid multiplication yields consistent composites and that CPT invariance selects the allowed helon braids, linking topological operations to Pin(1,3). It clarifies that the two constructions capture only the SU(3)_c × U(1)_{em} sector and outlines a path toward extending to the full electroweak group via higher-dimensional weight lattices, highlighting the potential of a diagrammatic, particle-centric view of fundamental interactions.

Abstract

In particle phenomenology, preon models study compositional rules of standard model interactions. In spite of empirical success, mathematical underpinnings of preon models in terms of group representation theory have not been fully worked out. Here, we address this issue while clarifying the relation between different preon models. In particular, we focus on two prominent models: Bilson-Thompson's helon model, and Lambek's 4-vector model. We determine the mapping between helon model particle states and representation theory of Lie algebras. Braided ribbon diagrams of the former represent on-shell states of spinors of the Lorentz group. Braids correspond to chirality, and twists, to charges. We note that this model captures only the $SU(3)_c\times U(1)_{em}$ sector of the standard model. We then map the twists of helon diagrams to the weight polytope of $SU(3)_c \times U(1)_{em}$. The braid structure maps to chiral states of fermions. We also show that Lambek's 4-vector can be recovered from helon diagrams. Alongside, we introduce a new 5-vector representation derived from the weight lattice. This representation contains both, the correct interactions found in 4-vectors and the inclusion of chirality found in helons. Additionally, we demonstrate topological analogues of CPT transformations in helon diagrams. Interestingly, the braid diagrams of the helon model are the only ones that are self-consistent with CPT invariance. In contrast to field-theoretic approaches, the compositional character of preon models offers an analogous particle-centric perspective on fundamental interactions.

Figures (14)

• Figure 1: Fermion assignments for a revised helon model are shown above. The up-quark states are updated to ensure the correct interactions.
• Figure 2: The three color states of the left-chiral down and up quarks are shown above.
• Figure 3: The braids claimed to be associated with the electroweak sector below the electroweak scale. Note that the Bilson-Thompson model does not have any data to associate the weak force with left-chiral fermions.
• Figure 4: Braid diagrams for gluons corresponding to the six non-Cartan generators of $SU(3)$.
• Figure 5: Candidates for quark composite braids are shown above. If color-matched braid multiplication was not used, then the proton braids would not always have one twist per ribbon and the neutron braids sometimes could contain twisted ribbons.