Topological Origin of Interaction Hierarchy in Elastic Vacuum: An Effective Field Theory Approach

This paper proposes a solution to the hierarchy problem of fundamental interactions by treating the vacuum as a system belonging to the universality class of elastic Cosserat media. Instead of fine-tuning elastic constants, we postulate that the vacuum microstructure is invariant under volume-preserving diffeomorphisms. This symmetry renders longitudinal modes non-physical (gauge) degrees of freedom, leaving only transverse excitations corresponding to photons. We rigorously define the effective vacuum thickness h not as a geometric dimension, but as a constitutive parameter derived from the ratio of bending to membrane stiffness (h ≡ √(D/Y)), identifying it with the UV-cutoff scale ~ ℓP. Within the framework of Effective Field Theory (EFT), we demonstrate that the low-energy action splits into a membrane term (~ h, charge) and a bending term (~ h³, mass). The resulting geometric suppression factor ξ ~ (h/L)², where L is the Compton wavelength, naturally reproduces the observed coupling constant ratio G/ke ~ 10⁻⁴⁰, establishing the hierarchy as a direct consequence of the scale separation between the Planck length and the Compton wavelength.