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First, we reformulate RG transformations in a recursive way with introductionof an order-parameter field. As a result, we manifest the RG flow of aneffective field theory through the emergence of an extra dimensional space,where both RG $\beta-$functions of coupling functions and RG flow equation ofthe order-parameter field appear in the resulting effective action explicitlythrough the extra dimensional space. Second, we consider an effective dualholographic description derived recently, where the classical gravity theory ofthe large $N$ limit takes into account quantum corrections in the all-looporder. This non-perturbative nature turns out to originate from an intertwinedrenormalization structure between both RG flow equations of coupling functionsand order-parameter fields in the emergent extra-dimensional space, where theIR boundary condition of the order-parameter field gives rise to a mean-fieldequation with fully renormalized interaction coefficients. Third, comparing theRG-reformulated effective field theory with this effective dual holographicdescription, we obtain term-by-term matching conditions in the level of aneffective action. As a result, we express all RG coefficients of fields andinteraction vertices in terms of the metric tensor of the dual holographictheory. Through this metric reformulation for the RG analysis, we propose aprescription on how to find RG $\beta-$functions of interaction coefficients ina non-perturbative way beyond the perturbative RG analysis. In particular, weclaim that the present prescription of the RG flow generalizes the holographicRG flow of the holographic duality conjecture towards the absence of conformalsymmetry, where the emergent holographic dual effective field theory has beenderived from the first principle, thus being applicable even away from quantumcriticality.

Geometric encoding of renormalization group $\beta-$functions in an emergent holographic dual description