Microscopic derivation of the effective mass parameters

Derive the effective parameters $M_0$ and $\Delta$ from a self-adjoint microscopic three-body condition or a specific triple-coincidence operator in the one-dimensional relativistic three-body Dirac model.

Background

The paper introduces a Hermitian three-channel Peierls mass operator whose loop phase is the genuine three-body holonomy θ3\theta_3. This effective operator produces the three mass branches, but its common mass parameter M0M_0 and link amplitude Δ\Delta are not derived from the underlying relativistic three-body Hamiltonian.

The unresolved problem is to construct a self-adjoint microscopic triple-coincidence boundary condition or genuine three-body interaction and establish how it determines the effective parameters M0M_0 and Δ\Delta. Such a derivation would connect the phenomenological three-state mass matrix to the underlying configuration-space dynamics.

References

It does not yet constitute a microscopic derivation of the magnitude $\Delta$ from a specific triple-coincidence operator; deriving $M_0$ and $\Delta$ from a self-adjoint microscopic three-body condition remains a central open problem.

— Three-Body Holonomy as a Toy-Model Mechanism for Family Triplication and Mass Hierarchy in (1+1) Dimensions  (2609.24022 - Yoshikawa, 21 Sep 2026) in Section “Effective mass matrix and Peierls-type realization,” subsection “Peierls implementation of the loop phase”