Phase Encoding of Genuine Three-Body Interactions in a Relativistic Dirac System in $1+1$ Dimensions

Abstract

We show how genuine three-body phase information can enter the invariant mass of a relativistic three-particle Dirac system in $(1+1)$ dimensions. As a solvable reference system, we consider the Sakamoto--Munakata--Ino model with pairwise contact interactions $g_{ij}(1-α_iα_j)δ(x_i-x_j)$. These singular interactions can be transferred into sector-dependent phases and matching conditions by a discontinuous unitary transformation. Although the explicit contact terms are thereby removed, the nonzero constituent-mass operator is rotated and retains nontrivial spectral information. We introduce a genuine three-body holonomy generated by $Q_3=α_1α_2α_3$. The kinetic and pair-interaction parts commute with $Q_3$, while the constituent-mass operator anticommutes with it. Consequently, the massless system separates into the $Q_3=\pm1$ sectors, which acquire opposite holonomy phases $e^{\pm iθ_3}$, whereas nonzero constituent masses mix the two sectors. This phase-sector-mixing mechanism makes the relative three-body phase dynamically accessible to the bound-state spectrum and establishes an operator-level mechanism through which the three-body holonomy generates a $θ_3$ dependence of the physical three-body invariant mass. We further emphasize that the topological three-body holonomy is not automatically equivalent to a bare triple-contact potential; such an equivalence requires a regulated self-adjoint realization and a compatible interaction-dependent boost satisfying the Poincaré algebra. The resulting framework therefore connects genuine three-body phase information to the mass spectrum of a relativistic composite system while clearly separating the controlled holonomy construction from the unresolved short-distance triple-contact realization.