Three-Body Holonomy and $SU(3)\to SU(2)\times U(1)$ Symmetry Breaking

Abstract

We show that a three-body holonomy can select the conventional $SU(3)\to SU(2)\times U(1)$ breaking direction. Three particles are special in the present construction: after removal of the center-of-mass coordinate they are the minimal one-dimensional few-body system with a two-dimensional relative space, allowing a closed cycle around the triple-coincidence point once pairwise contacts are encoded by matching conditions. A cyclic three-state mass operator gives \[ M^{[k]}=M_0+2Δ\cos\!\left(\frac{θ_3+2πk}{3}\right),\qquad k=0,1,2. \] At $θ_3=0$ the spectrum splits as $3\to2+1$, and the induced mass operator is proportional to the Cartan generator $T_8$. Its commutant is therefore $SU(2)\times U(1)$. In a trinification-like embedding, distinct left- and right-sector holonomies leave the hypercharge combination which reproduces the Standard-Model hypercharges for the usual trinification matter assignment. The construction is a toy mechanism for selecting the breaking direction; additional dynamics are required for rank reduction and vacuum selection.