One of the primary challenges in Superfluid Vacuum Theory (SVT) is the description of emergent spacetime at the smallest scales, where the continuous medium approximation breaks down. In this paper, we discuss a model of a fractal hierarchy of the vacuum, the so-called "embedded vacuum, " where each lower scale is represented by a denser Bose-Einstein condensate (BEC) with a smaller coherence (healing) length. We show that the scaling parameters (q_ρ and q_σ) are uniquely related to the fractal dimension D₅, ₍ of the vacuum structure, taking into account the topological entanglement of the vortex network in 3D space, which obeys a renormalization group (RG) flow. A multilayer effective action is introduced along with an appropriate localization projector and homogenization procedure. We demonstrate how this hierarchy provides a natural regularization of ultraviolet (UV) singularities in quantum field theory without postulating an infinite cutoff.
Vakhtang Mchedlishvili (Sat,) studied this question.