On holographic Wilsonian renormalization group of massive scalar theory with its self-interactions in AdS

Holographic model of massive scalar field with its self-interaction $\lambda\phi^n$ in AdS space is able to give a logarithmic scale dependence to marginal multi trace deformation couplings on its dual conformal field theory, where $\lambda$ is the self-interaction coupling of the scalar field, $\phi$ and $n$ is an integral number. In arXiv:1501.06664, the authors realize this feature by looking at bulk scalar solutions near AdS boundary imposing a specific boundary condition between the coefficients of non-normalizable and normalizable modes of the scalar field excitations. We study the same holographic model to see scale dependence of marginal deformations on the dual conformal field theory by employing completely different method: {\it holographic Wilsonian renormalization group}. We solve Hamilton-Jacobi equation derived from the holographic model of massive scalar with $\lambda\phi^n$ interaction and obtain the solution of marginal multi trace deformations up to the leading order in $\lambda$. It turns out that the solution of marginal multi trace deformation also presents logarithmic behavior in energy scale near UV region.