Geometric encoding of renormalization group $β-$functions in an emergent holographic dual description
First, we reformulate RG transformations in a recursive way with introduction of an order-parameter field. As a result, we manifest the RG flow of an effective field theory through the emergence of an extra dimensional space, where both RG $β-$functions of coupling functions and RG flow equation of the order-parameter field appear in the resulting effective action explicitly through the extra dimensional space. Second, we consider an effective dual holographic description derived recently, where the classical gravity theory of the large $N$ limit takes into account quantum corrections in the all-loop order. This non-perturbative nature turns out to originate from an intertwined renormalization structure between both RG flow equations of coupling functions and order-parameter fields in the emergent extra-dimensional space, where the IR boundary condition of the order-parameter field gives rise to a mean-field equation with fully renormalized interaction coefficients. Third, comparing the RG-reformulated effective field theory with this effective dual holographic description, we obtain term-by-term matching conditions in the level of an effective action. As a result, we express all RG coefficients of fields and interaction vertices in terms of the metric tensor of the dual holographic theory. Through this metric reformulation for the RG analysis, we propose a prescription on how to find RG $β-$functions of interaction coefficients in a non-perturbative way beyond the perturbative RG analysis. In particular, we claim that the present prescription of the RG flow generalizes the holographic RG flow of the holographic duality conjecture towards the absence of conformal symmetry, where the emergent holographic dual effective field theory has been derived from the first principle, thus being applicable even away from quantum criticality.