Breaking of Off-Shell Degeneracy in Rank-2 Separable Three-Body Models: Gauge Invariance, Spectral Restructuring, and Ultraviolet Stability (Paper LXVIII)

Abstract

We investigate three-body bound states in rank-2 separable Yamaguchi potentials with inter-channel mixing and establish three main results that clarify the nature of off-shell degeneracy in few-body systems. Building on our previous analysis of rank-1 separable models (Paper LXVII, DOI: 10.5281/zenodo.21824655), where off-shell degeneracy was observed as an apparent physical invariance, we demonstrate that this effect originates from a gauge-like redundancy of the separable representation rather than a fundamental symmetry of the three-body system. First, the triton binding energy B₃ is invariant under the mixing parameter λ₁₂ (variation < 0.002 MeV across λ₁₂ ∈ [0, 100]), confirming unitary equivalence of different rank-2 basis representations and establishing λ₁₂ as a gauge degree of freedom. This result provides a concrete realization of off-shell equivalence in few-body systems and sharpens its interpretation within the separable framework developed in earlier works on spectral structure and two-body input (Papers LI–LII, DOIs: 10.5281/zenodo.21678524, 10.5281/zenodo.21696330). Second, the spectator-channel strength λ₂₂ acts as a genuine physical parameter, controlling the emergence of additional bound states: a second bound state appears at B₃ ≈ 16 MeV for λ₂₂ ≳ 10⁵. This explicitly demonstrates that the degeneracy identified in rank-1 models is lifted once non-redundant dynamical degrees of freedom are introduced, consistent with previous observations of emergent structure and correlations in the three-body sector (Papers LIII and LVII, DOIs: 10.5281/zenodo.21701118, 10.5281/zenodo.21724064). Third, systematic numerical cutoff analysis (Λ_UV ∈ [88, 1407] MeV) reveals complete cutoff independence of the spectrum (scaling exponent α = 0.000 ± 0.001), demonstrating that the rank-2 separable structure acts as an intrinsic ultraviolet regulator without the need for additional three-body counterterms. This result is fully consistent with the previously established universality of the spectral scale and renormalization-group closure at the observable level (Papers LIX and LXIII, DOIs: 10.5281/zenodo.21762011, 10.5281/zenodo.21792923). We further show that β-scaling in Yamaguchi models does not constitute a clean zero-range test due to the intrinsic coupling between interaction range and strength, and therefore base our UV-stability conclusions on explicit cutoff variation rather than form-factor scaling. Taken together, our results establish that the off-shell degeneracy observed in rank-1 separable models is not a universal physical property but a manifestation of gauge redundancy in the separable basis. This redundancy is systematically broken when additional physical scales are introduced, highlighting the importance of distinguishing between gauge degrees of freedom and genuine physical parameters in nuclear few-body systems, and providing a direct bridge between separable potential methods and the broader HDG spectral framework.