A Composite Three-Body-Holonomy Toy Framework for Dynamical Electroweak-Scale Selection

Abstract

We propose a phenomenological toy framework promoted to a minimal three-state internal sector in $(3+1)$ dimensions and realized by composite strong dynamics. It is in a genuine three-body holonomy which originates in a relativistic $(1+1)$-dimensional Dirac system. The hypothesis is that the number three is not imposed first as a gauge symmetry, but is inherited from the primitive closed cyclic structure of the three-body holonomy. This threefold structure subsequently reappears as a three-state internal space, the three-site loop, a three-flavor hidden sector, and a cubic determinant interaction. We introduce an asymptotically free hidden $SU(3)_H$ gauge theory with three Standard-Model-singlet Dirac flavors. Strong hidden dynamics generates a composite condensate, while the inherited holonomy selects its cyclic internal orientation. The anomalous cubic determinant interaction produces a holonomy-dependent potential, and dimensional transmutation generates the characteristic composite scale. The radial composite excitation along the cyclic direction, denoted by $S_3$, couples to the Standard-Model Higgs doublet through a holonomy-dependent CP-even portal. The resulting condensate shifts the Higgs quadratic term and can trigger electroweak symmetry breaking. In the classically scale-invariant reference limit, the electroweak scale is induced from the dynamically generated hidden-sector scale through dimensionless portal couplings.