Symmetric Dual Breather Modes as a Dynamic Origin of Tiny Particle Masses

Spontaneous symmetry breaking (SSB) is conventionally understood through static vacuum selection in low-energy approximations. However, the nonperturbative dynamical pathways connecting symmetric configurations to asymmetric states remain an open question. In this paper, we propose an operator-theoretic framework that characterizes $Z_2$ symmetry breaking in scalar field theory through spectral branching. Specifically, we identify an effective invariant manifold of the nonlinear Klein-Gordon equation termed the “symmetric dual breather mode,” whose phase-space geometry is traced using a $t$-dependent finite-dimensional representation. Our analysis demonstrates that static SSB represents merely a small-amplitude limit ($A\ll {\phi }_{v1}$). As the amplitude increases into the heteroclinic regime ($A>{\phi }_{v1}$), the field dynamically overflows the central barrier, yielding a vanishing time-averaged order parameter ($⟨\phi ⟩=0$) and driving a geometric curvature cancellation ($⟨V^{\mathrm{\prime }\mathrm{\prime }}(\phi ){⟩}_t\to 0$). Governed by a universal scaling function $\mathfrak{F}(y)$, the logarithmic period divergence near the heteroclinic boundary naturally generates sub-eV effective mass scales without fine-tuning bare parameters. Furthermore, this dynamic barrier overflow naturally favors Majorana-like self-conjugate states over Dirac-type states. Additionally, nonlinear phase interference along the heteroclinic channel induces asymmetric phase-space contraction, decoupling the right-handed mode and dynamically giving rise to the left-handed Weyl equation. These results bridge nonperturbative wave dynamics, topological mass suppression, chiral selection, and the dynamic origin of active neutrino masses.