  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 $\beta-$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 $\beta-$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.
