Physical Vacuum – Part I Spectral vacuum mechanism (SVM) Lepton Mass Hierarchies

Abstract

We propose a structural vacuum model (SVM) in which the observed hierarchy of charged lepton masses arises as a spectral property of localized fluctuation modes around a vacuum configuration. The model is formulated in terms of a coupled second-variation (Hessian) operator acting on three fluctuation channels in a one-dimensional domain. Particle masses are identified with the square roots of positive eigenvalues of this operator, up to a single absolute calibration fixed by the electron mass.

At the purely spectral level, prior to any mass calibration, the eigenvalue structure of the Hessian operator naturally produces characteristic hierarchical ratios of order μ/e ≈ 206 and τ/e ≈ 3.5×10³, reflecting the ordering and localization of distinct fluctuation modes. Fixing the overall scale by the electron mass, the model reproduces the muon and tau masses without Yukawa couplings or multiparameter fitting, achieving relative accuracies at the level of 10⁻⁵–10⁻⁷.

The primary goal of this paper is not to postulate a microscopic Lagrangian, but to demonstrate that a minimal and internally consistent vacuum structure can simultaneously satisfy the conditions of stability, localization, and strong spectral uncoupling. The analysis focuses on structural necessity: identifying which features of the Hessian operator are unavoidable if such mass hierarchies are to exist at all.

Keywords: structural vacuum model, lepton mass hierarchy, spectral mechanism, vacuum fluctuations, hessian operator, localized modes, mass generation, absence of yukawa couplings, multi-scale vacuum structure, spectral stability, minimality principle, robustness analysis, electron muon tau hierarchy, variational vacuum approach, eigenvalue spectrum