Three-Body Holonomy as a Toy-Model Mechanism for Family Triplication and Mass Hierarchy in (1+1) Dimensions
Abstract
We investigate the phenomenological consequences of a genuine three-body holonomy in a relativistic Dirac system in one spatial dimension. After removing the center-of-mass coordinate, the triple-coincidence point punctures the two-dimensional relative configuration space and permits a nontrivial U(1) winding phase, whereas the pairwise Sakamoto--Munakata--Ino contact interactions give trivial net matching around this point. At fixed intrinsic parity, the six particle-ordering sectors reduce to a three-dimensional cyclic space. A Hermitian C3-invariant effective mass operator admits a Peierls-type realization in which the gauge-invariant phase around the three links equals the three-body holonomy θ3. Its eigenvalues are M[k]=M0+2Δcos ((θ3+2πk)/3), k=0,1,2. The holonomy lifts the conjugate-channel degeneracy and can generate a parametrically light branch through cancellation between the common mass and the holonomy-induced shift. We further allow cyclic-symmetry breaking and consider a general Hermitian three-state mass matrix. Exact elimination of two heavy states by the Schur complement yields a low-energy correction containing the rephasing-invariant loop product Re(t12t23t31)∝cosθ3. Thus, even when only one branch is kinematically accessible, its effective mass can retain finite memory of the complete three-state loop. The complementary invariant Im(t12t23t31)∝sinθ3 is phase sensitive but does not alone imply CP violation. The construction provides a low-dimensional phenomenological proof of concept for family-like triplication, mass hierarchy, and infrared memory, rather than a microscopic theory of Standard Model fermion generations.
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BibTeX
@misc{yoshikawa2026threebody,
title = {{Three-Body Holonomy as a Toy-Model Mechanism for Family Triplication and Mass Hierarchy in (1+1) Dimensions}},
author = {Tadashi YOSHIKAWA},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.24022},
eprint = {2609.24022},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.24022},
}
RIS
TY - UNPB
TI - Three-Body Holonomy as a Toy-Model Mechanism for Family Triplication and Mass Hierarchy in (1+1) Dimensions
AU - Tadashi YOSHIKAWA
PY - 2026
DA - 2026/09/23
PB - arXiv
DO - 10.48550/arxiv.2609.24022
UR - https://arxiv.org/abs/2609.24022
AB - We investigate the phenomenological consequences of a genuine three-body holonomy in a relativistic Dirac system in one spatial dimension. After removing the center-of-mass coordinate, the triple-coincidence point punctures the two-dimensional relative configuration space and permits a nontrivial U(1) winding phase, whereas the pairwise Sakamoto--Munakata--Ino contact interactions give trivial net matching around this point. At fixed intrinsic parity, the six particle-ordering sectors reduce to a three-dimensional cyclic space. A Hermitian C_3-invariant effective mass operator admits a Peierls-type realization in which the gauge-invariant phase around the three links equals the three-body holonomy θ_3. Its eigenvalues are M^[k]=M_0+2Δcos ((θ_3+2πk)/3), k=0,1,2. The holonomy lifts the conjugate-channel degeneracy and can generate a parametrically light branch through cancellation between the common mass and the holonomy-induced shift. We further allow cyclic-symmetry breaking and consider a general Hermitian three-state mass matrix. Exact elimination of two heavy states by the Schur complement yields a low-energy correction containing the rephasing-invariant loop product Re(t_12t_23t_31)∝cosθ_3. Thus, even when only one branch is kinematically accessible, its effective mass can retain finite memory of the complete three-state loop. The complementary invariant Im(t_12t_23t_31)∝sinθ_3 is phase sensitive but does not alone imply CP violation. The construction provides a low-dimensional phenomenological proof of concept for family-like triplication, mass hierarchy, and infrared memory, rather than a microscopic theory of Standard Model fermion generations.
ER -