Three-Body Holonomy and SU(3)→ SU(2)× U(1) Symmetry Breaking
Abstract
We show that a three-body holonomy can select the conventional SU(3)→ SU(2)× 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]=M0+2Δcos ((θ3+2πk)/3), k=0,1,2. At θ3=0 the spectrum splits as 3→2+1, and the induced mass operator is proportional to the Cartan generator T8. Its commutant is therefore SU(2)× 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.
Related
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- A Composite Three-Body-Holonomy Toy Framework for Dynamical Electroweak-Scale Selection
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BibTeX
@misc{yoshikawa2026threebody,
title = {{Three-Body Holonomy and $SU(3)\to SU(2)\times U(1)$ Symmetry Breaking}},
author = {Tadashi Yoshikawa},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.00235},
eprint = {2610.00235},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.00235},
}
RIS
TY - UNPB
TI - Three-Body Holonomy and SU(3)→ SU(2)× U(1) Symmetry Breaking
AU - Tadashi Yoshikawa
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.00235
UR - https://arxiv.org/abs/2610.00235
AB - We show that a three-body holonomy can select the conventional SU(3)→ SU(2)× 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+2Δcos ((θ_3+2πk)/3), k=0,1,2. At θ_3=0 the spectrum splits as 3→2+1, and the induced mass operator is proportional to the Cartan generator T_8. Its commutant is therefore SU(2)× 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.
ER -