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^{†}We dedicate this work to the memory of Peter Hasenfratz.

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We study the proposal that a (

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The anti-de Sitter/conformal field theory (AdS/CFT) correspondence [

In a previous paper [^{1}

There are two key aspects in this proposal. First of all, the (

In this paper, we further investigate this proposal, explicitly considering a more general model, the O(

This paper is organized as follows. In

In this section, we briefly summarize the proposal of Ref. [

In this paper we consider the large

The

In Ref. [

The above

Even though the composite operator

Moreover, thanks to the large

In this paper, we consider the

This model describes the free massive scalar at

The partition function is evaluated as [

The large

The two-point function is then evaluated as

In the NLSM limit

In this paper, we consider the flow equation, given by

In the large

In the case of the interacting flow with

For the flow with

The two-point function of the flow field with

If we take the NLSM limit for the flow equation (

At

As mentioned before, to construct an RG transformation using the flow equation, which is merely a kind of smearing procedure for UV fluctuations, we introduce a normalization condition for the flow field.

There is no unique way to define the normalized flow field. In the standard block spin transformation, for example, the normalization factor for the field is so chosen so that some fixed point can appear for the defined RG transformation. In this paper, we propose the following normalization for the flow field:

For the interacting flow (

For the free flow (

We thus conclude that the normalized flow field defined by Eq. (

As proposed in

Using Eq. (

In the IR limit

In the opposite limit

In this section, we consider the VEV of the Einstein tensor

If the metric has the simple form

From Eq. (

Two scalar functions

For

Using the above results, we obtain

We next consider the behavior of

At

Similarly, we obtain

If we define the radius of the AdS space

We show this behavior more explicitly at

In this paper, we apply the method in Ref. [

The induced metric, and thus the geometry, from the flow field with the NLSM normalization, depends only on the renormalized mass

Since the information about the renormalized coupling constant appears at the next-to-leading order (NLO) in the large

S. A. thanks Dr H. Suzuki for useful discussions. He is supported in part by a Grant-in-Aid of

Open Access funding: SCOAP^{3}.

In this appendix, we show that an extra divergence appears in the flow field of the ^{2}

For simplicity, we consider the model at

We now consider the flow equation, given by

Setting

The two-point function for the flow field is given by

We finally obtain

We now consider this divergence from the result in the large

The incomplete gamma function of the second kind,

Other useful properties of

Using the above formulae, we obtain the asymptotic behavior of

^{1} There are also studies using a different method on the relation between O(

^{2} H. Suzuki, private communication.